3. Electrical forces in the double layer: Electroosmotic and AC Electroosmotic Pumps

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1 3. Electrical forces in the double layer: Electroosmotic and AC Electroosmotic Pumps At the interface between solid walls and electrolytes, double layers are formed [1] due to differences in electrochemical potentials of both phases. The solid surface becomes charged and counterions coming from the bulk liquid shield this surface charge. At equilibrium, the electrostatic attraction between the charged surface and the counterions is balanced by thermal agitation. The liquid is electrically neutral except for a charged layer close the charged wall. The characteristic thickness of this layer is the Debye length λ D given by Here n i is the number density of species far from the wall. The Debye length for a z-z electrolyte can be written as λ av D = εd / σ where D av is the average diffusion coefficient. Therefore, the Debye length can be seen as the distance an ion travels by diffusion during the charge relaxation time. The application of an electric field along the surface produces a Coulomb force on the diffuse charge and the liquid is set into motion. Because the thickness of the Debye layer is very small (1 100 nm in water) the Coulomb force is considered as a superficial force. There are two main ways of generation of charge at a solid surface: 1) a chemical mechanism: such as ionization of bound surface groups; 2) an electrostatic mechanism: a solid metal surface gets charged when subjected to a potential difference with respect to bulk electrolyte. In both cases, a double layer is formed in order to screen the wall surface charge. In the diffuse layer, Coulomb attraction to the wall is balanced by thermal diffusion. We can identify two kinds of pumps based on these charging mechanisms: Electroosmosis and AC Electroosmosis. 3.1 Electroosmotic Pump The Electroosmotic (EO) pump makes use of the electric double layer that spontaneously develops between liquid in contact with solid. The chemical state of the solid surface is commonly altered because of the presence of the electrolyte. For example, in the case of silica-based ceramics, like glass, with SiOH groups at the

2 surface, a fraction of the Si OH bonds change into Si O releasing H+ when immersed in water. Application of an electric field along the surface causes the motion of the mobile ions of the double layer, dragging the fluid with them. This is called Electroosmosis. V E(x) diffuse charge fixed charge (in glass) λ D λ D velocity profile zero back pressure velocity profile back pressure EO phenomena have been known for 200 years (F.F. Reuss discovered electroosmosis in 1809). Flows generated by EO pumping are used in a range of applications, including soil remediation or contaminant removal from groundwater, and have been used in chemical and biological analysis since a long time. Pretorius et al. (1974) [2] proposed electroosmosis for high-speed chromatography. It has been proposed to use Electroosmosis in miniaturized systems for chemistry and life sciences. Examples are: flow injection analysis (Dasgupta and Liu, 1994) [3], on-chip electrophoretic separation (Manz et al, 1994) [4], and on-chip liquid chromatography (Jaconson et al, 1994) [5]. Yao and Santiago (2003) [6] presented a detailed description of the history and development of high pressure EO pumps Pump Principle The most basic EO pump is simply a capillary or micro-pipe with electrodes submerged within end-channel reservoirs. Let us consider a cylindrical capillary of radius a [7], with the wall of the capillary at potential ζ, the zeta potential. Zeta potential is a key parameter in the theory of electrokinetics. Classical theory describes the electrical double layer as divided into the Stern layer and Gouy-Chapman diffuse layer [1]. The Stern or compact layer is formed of ions (and/or solvent molecules) absorbed onto the wall, while the ions of the Gouy-Chapman layer are diffuse. The plane of separation between these two layers is called the shear plane, and the potential at this plane is the zeta potential. Let us consider a 1-1 electrolyte inside a cylindrical capillary. The bulk ionic concentration is n. The ions near the wall follow the Boltzmann distribution.

The Poisson-Boltzmann equation in this cylindrical geometry is In the Debye-Hückel limit ζ << k B T/e, we get with I 0 the zero order modified Bessel function

We now can integrate the velocity equation (at zero back pressure): The solution that holds u(a)=0 (non slip condition) is 1 0.8 0.6 0.

8 1 In the case that the radius is much greater than the Debye length, a>>λ D, the velocity varies from zero at the wall to the asymptotic value u s = -εζe z /η.

3 The Poisson-Boltzmann equation in this cylindrical geometry is In the Debye-Hückel limit ζ << k B T/e, we get with I 0 the zero order modified Bessel function of first kind and For a/λ D >>1 charge and potential are only different from zero near the wall. We now can integrate the velocity equation (at zero back pressure): The solution that holds u(a)=0 (non slip condition) is In the case that the radius is much greater than the Debye length, a>>λ D, the velocity varies from zero at the wall to the asymptotic value u s = -εζe z /η. Furthermore, the bulk of the liquid in the capillary moves with velocity u s, known as the electroosmotic slip velocity. For this case of a>> λ D, the slip velocity expression is valid in the general case where z is not restricted to be smaller than k B T/e. Integrating the velocity, we get with and I 1 the modified Bessel of first kind of order one.

2 Mechanical Characteristics We can write maximum flow-rate and maximum pressure.

approximation of small zeta potential, the conductivity is written with the bulk ionic concentration

includes threshold voltages for Faradaic reactions and junction voltages at inlet and outlet of

4 The flow-rate as a function of a* gets to be very small as a* goes to zero Mechanical Characteristics We can write maximum flow-rate and maximum pressure. The power consumption comes from IV so we obtain firs the total current conduction + advection In the approximation of small zeta potential, the conductivity is written with the bulk ionic concentration The energy efficiency is (neglecting charge advection) where V app the applied potential that also includes threshold voltages for Faradaic reactions and junction voltages at inlet and outlet of capillary. Taking into account the relation between conductivity and Debye length, we can write the efficiency as We can see that maximum efficiency is obtained for a*~1, or a~λ D.

3.1.3 Practical Case: Porous Structures. In order to increase the generated pressure, the diameter of the pipe can be reduced. This, however, reduces the flow-rate.

5 3.1.3 Practical Case: Porous Structures. In order to increase the generated pressure, the diameter of the pipe can be reduced. This, however, reduces the flow-rate. A way of increasing the pressure, while not reducing much the flow-rate, is to use a bundle of parallel micropipes with small diameter. In practice, high-pressure EO micropumps employ porous structures so that each pore acts as a tortuous capillary. The porous structure can be visualized as a bundle of tortuous micropipes. L L e = n. capillaries 2a L e α L The total flow-rate is given by the product of the number of capillaries times the flowrate of one capillary. Let us define A as the cross-sectional area of the porous structure and ψ as its porosity (void volume divided by total volume of porous material). The ψ A area occupied by the pores is given by ψ A. The number of capillaries is π a 2 cosα, where π a 2 is the cross-sectional area of one tortuous capillary and cosα=l/l e. The flow-rate generated by one capillary is due to E e =V eff /L e : = flow-rate per capillary The total flow-rate is then The generated pressure is that of one micropipe The energy efficiency of a porous structure will be given approximately by that of a capillary since in the model, both Q and I are proportional to the number of capillaries in parallel and Δp and V are the same for all the capillaries. (Δp Q)/(IV) Some examples Yao et al (2003) [8] working with borate buffer at 100 V (total volume of micropump 1300 mm 3 ) obtained:

6 - Maximum pressure around Pa - Maximum flow-rate around 550 mm 3 /s - Energy efficiency around for a =0.5 μm and for a =1.2 μm Debye length around 13 nm << a Wang et al (2006) [9] working with water at 6 kv (0.5 mm 3 micropump) obtained: - Maximum pressure around Pa. - Maximum flow-rate around 0.05 mm 3 /s - Energy efficiency around with a=2 μm and λ D =0.1 μm They also worked with acetonitrile with eff=0.022, the main difference is due to conductivity: σ(water)~10-4 S/m and σ(aceton.)~10-6 S/m. The higher efficiency of acetonitrile is mainly due to its lower conductivity Working Liquids. Electrokinetic phenomena tend to disappear at high ionic concentrations. When the electrolyte conductivity is high, the Debye length becomes very small and comparable or smaller than the thickness of the Stern layer. Since the total double-layer voltage is shared between the compact and the diffuse layers, the reduction of the diffuse part leads to a reduction of the potential across it, i.e., a reduction of the zeta potential ζ. Applying a simple model of two capacitors in series (the compact-layer capacitor in series with the diffuse-layer capacitor) leads to the conclusion that ζ decreases with the squared-root of electrolyte concentration n. For values of σ ~1 S/m, both capacitances are of the same order, (the Stern layer capacitance is of the order of 0.3 Fm 2 ). The prediction that ζ ~ 1/(n ) 0.5 fails when the ph of the solution controls the surface charge [6]. However, ζ is still a decreasing function of ion concentration. A lower bound for the conductivity of the liquid comes from a comparison between the electric field inside the double layer and the applied electric field. In the EO theory, it was supposed that the applied electric field does not perturb the original distribution of ions in the double layer. We expect that if the applied electric field is of the order of the DL electric field, the distributions of ions in the capillary would be much distorted. Therefore, it is not expected that electroosmosis can work properly at very low conductivity. For an applied field E z ~ 10 4 V/m, ζ V, this condition implies σ>10-7 S/m for water (or σ>10-8 S/m for nonpolar liquid with ε r = 2). When the charge on the surface is governed by the process of deprotonation, as in the case of silica surfaces, the ph of the liquid is another parameter to be taken into consideration. The zeta potential is a strong function of ph, typically saturating at high and low ph. According to Yao et al. [8], low buffer concentrations (below 0.1 mm) are impractical because pump operation generates prohibitively large fluctuations in ph. Liquids that have been pumped using electroosmosis have conductivities that range from 10-6 to 1 S/m according to Chen and Santiago [10].

7 3.1.6 Problems Electrolysis generates gas bubbles. Redox reactions at the electrodes will eventually change the ph. Traditionally some of these problems are avoided by separating the electrodes from the EO pumping channels with an ion exchange system (Dasgupta and Liu, 1994 [3]). Electrodes are placed in reservoirs separated by the channels, exchange of ions is allowed but not of bulk flow. The reservoirs are filled with buffers to reduce the changes in ph. Joule heating can also be problematic. The temperature rise can be excessive for conductivities around 1 S/m estimation for a capillary These pumps usually require high voltage (traditionally in the kilovolt range). This can be a drawback, especially for portable chips. A low voltage EO pump that produces high pressure (around 25 kpa at 10 V) has been realized by using a cascade pump design [11] Applications Saline solutions with σ 1 S/m can be pumped; therefore, possible applications range from biomedical to chemical analysis, such as, the previously mentioned, on-chip liquid chromatography and on-chip electrophoresis separation. The energy efficiency can be very low for conductivities around 0.1 S/m, nevertheless there are many applications where this is not a limiting issue. Potential applications include the replacement of high-pressure pumps in micro-total-analysissystems: for drug delivery, sample analysis, separation, and mixing processes. Another application is the use of closed-loop electroosmotic microchannels as cooling systems for microelectronics. Arrays of micropumps, temperature sensors, and fluid flow networks could be used to redistribute heat from hot spots that arise during microchip operation.

3.2 AC Electroosmotic Pump An alternating-current/induced-charge electroosmotic pump makes use of the Coulomb force on the induced-charge in the double layer [12, 13, 14].

8 3.2 AC Electroosmotic Pump An alternating-current/induced-charge electroosmotic pump makes use of the Coulomb force on the induced-charge in the double layer [12, 13, 14]. Typically, the normal component of the electric field charges the double layer at the electrode/electrolyte interface, while the tangential electric field component produces a force on the induced charge in the diffuse layer that results in fluid motion. Unidirectional fluid flow is typically obtained in two ways: arrays of asymmetric pairs of electrodes subjected to a single AC signal [15] or arrays of symmetric electrodes subjected to travelling-wave signals [16]. The asymmetric array breaks the left-right symmetry of the flow rolls that form on top of the electrodes and, eventually, unidirectional flow is generated. In the travelling-wave case, the potential wave induces a charge wave in the double layer. Because of the finite time to charge the DL, the charge wave lags behind the potential wave and a longitudinal electrical force appears pulling the fluid in the direction of the travelling wave Pump Principle Let us analyse the case of travelling-wave electroosmosis. We use the thin double-layer and Debye-Hückel (low voltage) approximations. We assume that the electrodes are perfectly polarisable, i.e. there are no Faradaic currents from the electrodes to the liquid. We approximate the applied voltage at the electrodes as a pure single mode travelling i( t kx) wave of the formv( x, t) = Re[ V0e ω ]. The frequency ω is much smaller than the charge relaxation frequency σ/ε, so that the DL is in quasi-equilibrium. Under these conditions the DL behaves in a capacitive manner and the bulk in a resistive manner. The DL is characterised by its surface capacitance C DL and the bulk by its conductivity σ. The electric current in the bulk follows Ohm s law j = σe and, from the continuity equation of current ( σ E ) = 0, the electric potential holds Laplace s equation 2 ( ) Φ= 0. The electric potential in the bulk is of the form Φ ( xt, ) = Re[ Ae i ωt kx kz ]. The boundary condition at the electrodes comes from the charge conservation equation: the current arriving at the double layer charges the double-layer capacitor: The potential solution in the bulk is

We can now obtain the slip velocity given by the time-averaged of the Helmholtz-

where ζ is the potential drop across the diffuse part of the double layer.

C d C DL where C s and C d are, respectively, the capacitances of the Stern (or compact)

velocity Ω=1 maximum velocity Nondimens.

9 We can now obtain the slip velocity given by the time-averaged of the Helmholtz- Smoluchowski expression [17]. where ζ is the potential drop across the diffuse part of the double layer. In our model, and Λ is the ratio between diffuse to total potential drop in DL given by C s C d C DL where C s and C d are, respectively, the capacitances of the Stern (or compact) layer and diffuse layer. The slip velocity is with Nondimens. velocity Ω=1 maximum velocity Nondimens. frequency For constant applied voltage, the slip velocity is maximum at a frequency ω=σk/c DL Mechanical Characteristics L (w >> h>>k -1 ) w h Let us consider a channel like the one in the figure. The maximum flow-rate that the travelling-wave array generates is

The maximum pressure is The power consumption is given by The energy efficiency is then electric Reynolds For a typical velocity of 1 mm/s, h~10-5 m, the energy efficiency is eff~λ10-7 /σ (σ in SI

10 The maximum pressure is The power consumption is given by The energy efficiency is then electric Reynolds For a typical velocity of 1 mm/s, h~10-5 m, the energy efficiency is eff~λ10-7 /σ (σ in SI units). These pumps can be very inefficient. Ideally, we can obtain In order to increase the generated pressure, we can increase L and reduce k -1 and h. Reducing h leads to a reduction of the flow-rate. If kh < 1, but λ D <<h, the slip velocity expression is the same but Ω is now Experimental Characteristics Quadratic dependence on voltage is only observed at very low voltages. The slip velocity tends to a lower increment with voltage. This may be explained because of, at least, two mechanisms acting at the same time: (a) the potential ζ can not grow without limit because the charge in the diffuse double layer has an upper limit given by the size of the ions. Therefore, ζ saturates and the velocity becomes proportional to V; (b) if the potential is beyond the threshold voltage for the appearance of Faradaic currents, the double layer starts to leak charge. Experimentally, saturation of the velocity with voltage is observed. At still higher voltages, the flow changes direction, and generation of bubbles start to appear easily [18, 19]. It seems that experimentally the AC EO mechanism provides a maximum slip velocity around 1 mm/s.

11 3.2.4 Working Liquids As for DC Electroosmosis, we expect that there is a range of conductivities for this kind of actuation. The upper bound seems to be more restrictive than for DC EO, σ < 0.1 S/m from experiments by Studer et al, 2002 [18]. If we set C d ~ C s as a limit, this leads to ε/λ D ~ 0.2 F/m 2 that gives a conductivity σ~0.11 S/m. We expect that there should be a lower bound similar to DC EO. The applied electric field should not be greater than the electric field inside the DL: In this case, this is similar to say that λ D should be smaller than typical distance k Applications Water saline solutions with σ 0.1 S m 1 can be pumped, therefore, possible applications range from biomedical to chemical analysis. Typically, the generated pressure is small. However, these kinds of pumps have many potential applications for manipulating particles and fluids in closed microdevices, which do not require high pressure. For example, it may have applications in circular chromatography [20]. References [1] Hunter R.J., Zeta Potential in Colloid Science. Academic Press, San Diego (1981). [2] Pretorius V., Hopkins B.J. and Schieke J.D., Electro-osmosis: A new concept for high-speed liquid chromatography, J. Chrom., 99, (1974) [3] Dasgupta P.K. and Liu S., Electroosmosis: A reliable fluid propulsion system for flow injection analysis, Anal. Chem., 66, (1994) [4] Manz A., Effenhauser C.S., Burggraf N., Harrison D.J., Seiler K. and Fluri K., Electroosmotic pumping and electrophoretic separations for miniaturized chemical analysis systems, J. Micromech. Microeng., 4, (1994) [5] Jacobson S.C., Hergenroder R., Koutny L.B. and Ramsey J.M., Open-channel electrochromatography on a microchip, Anal. Chem., 66, (1994) [6] Yao S. and Santiago J.G., Porous glass electroosmotic pumps: Theory, J. Colloid Interface Sci., 268, (2003) [7] Rice C.L. and Whitehead R., Electrokinetic flow in a narrow cylindrical capillary, J. Phys. Chem., 69, (1965) [8] Yao S., Hertzog D.E., Zeng S., Mikkelsen J.C. and Santiago J.G., Porous glass electroosmotic pumps: Design and experiments, J. Colloid Interface Sci., 268, (2003) [9] Wang P., Chen Z. and Chang H.H., A new electro-osmotic pump based on silica monoliths, Sensors Actuators B, 113, (2006) [10] Chen C.H. and Santiago J.G., A planar electroosmotic micropump, J. Microelectromech. Syst., 11, (2002)

12 [11] Takamura Y., Onoda H., Inokuchi H., Adachi S., Oki A. and Horiike A., Lowvoltage electroosmosis pump for stand-alone microfluidics devices, Electrophoresis, 24, (2003) [12] Ramos A., Morgan H., Green N.G. and Castellanos A., AC electric-field induced fluid flow in microelectrodes, J. Colloid Interface Sci., 217, (1999) [13] Ajdari A., Pumping liquids using asymmetric electrode arrays, Phys. Rev. E, 61, R45 R48 (2000) [14] Green N.G., Ramos A., González A., Morgan H. and Castellanos A., Fluid flow induced by non-uniform AC electric fields in electrolytes on microelectrodes I: Experimental measurements. Phys. Rev. E, 61, , (2000) [15] Brown A.B.D., Smith C.G. and Rennie A.R., Pumping of water with AC electric fields applied to asymmetric pairs of microelectrodes, Phys. Rev. E, 63, (2000) [16] Cahill B.P., Heyderman L.J., Gobrecht J. and Stemmer A., Electro-osmotic streaming on application of traveling-wave electric fields, Phys. Rev. E, 70, (2004) [17] González A., Ramos A., Green N.G., Castellanos A. and Morgan H., Fluid flow induced by non-uniform AC electric fields in electrolytes on microelectrodes II: A linear double-layer analysis. Phys. Rev. E, 61, , (2000) [18] Studer V., Pepin A., Chen Y. and Ajdari A. An integrated AC electrokinetic pump in a microfluidic loop for fast and tunable flow control. Analyst, 129, (2004) [19] García-Sánchez P., Ramos A., Green N.G. and Morgan H., Experiments on AC electrokinetic pumping of liquids using arrays of microelectrodes, IEEETrans. Dielectr. Electr. Insul., 13, (2006) [20] Debesset S., Hayden C.J., Dalton C., Eijkel J.C.T. and Manz A., An AC electroosmotic micropump for circular chromatographic applications, Lab Chip 4, (2004)

AC Electro-osmotic Flow

AC Electro-osmotic Flow Synonyms Induced-charge electro-osmosis, AC pumping of liquids, traveling-wave electro-osmosis. Definition AC electro-osmosis (ACEO) is a nonlinear electrokinetic phenomenon of