Electrohydrodynamic Micropumps

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Electrohydrodynamic Micropumps Antonio Ramos Dept. Electrónica y Electromagnetismo Universidad de Sevilla Avda. Reina Mercedes s/n 41012 Sevilla. Spain 1.

1 Electrohydrodynamic Micropumps Antonio Ramos Dept. Electrónica y Electromagnetismo Universidad de Sevilla Avda. Reina Mercedes s/n Sevilla. Spain 1. Introduction Microfluidics deals with the pumping, control and manipulation of micro-litre to picolitre volumes of fluids. The typical length of the system ranges from 100 nm to 1mm. In our case, we are going to deal with liquids in microsystems: sub-continuum effects that are important for gases in microfluidics (or, in general, in nanofluidics) are not important here. Knudsen number (molecular mean free path length / typical length) is much smaller than one. Liquid manipulation can be divided loosely into: - digital microfluidics: liquid divided into droplets (e.g. Electrowetting); - continuous microfluidics: liquid transported inside conduits. In this work, we deal with continuous microfluidics. Possible applications of microfluidics: dispensing therapeutic agents into the body; cooling of microelectronic devices; handling of liquids in space exploration, where miniaturization is required; micropropulsion, as another potential application of microfluidics in space (electrospray propulsion for small satellites); handling of liquids for the Lab-on-a-chip technology, i.e. micro-devices for chemical or biological analysis of chip-format. Burns et al An integrated nanoliter DNA analysis device. Science 282: Microfluidic transport can sometimes be achieved by passive mechanisms (surface tension or gravity). Other applications employ macroscale pumps: syringe pumps,

2 pressure/vacuum chambers and valves. However, many applications would benefit from an on-chip active micropump. According to Laser and Santiago [1], there are two categories of micropumps: (a) displacement pumps, which exert pressure forces on the working fluid by one or more moving boundaries and (b) dynamic micropumps, which exert forces directly on the liquid, without moving parts. Piezoelectric and electro-wetting micropumps are examples of displacement pumps. Most of electrohydrodynamic (EHD) micropumps are dynamic pumps: they use electric forces directly on the liquid. An integrated pump design that has been studied intensively over the years is the piezo-actuated micropump. In this, a membrane is displaced to create a pulsating flow that is rectified using valves. However, the moving parts make the fabrication and operation delicate. In this context, the requirement of an integrated micropump with no moving parts can be fulfilled by using EHD micropumps. In all EHD dynamic micropumps, electrical current flows through the working liquid. Different pumping techniques appear depending on liquid electrical properties (mainly conductivity). EHD Micropumps may be classified into: - Pumps that exert electric forces in the liquid bulk: 1) Injection, 2) Conduction, 3) Induction, - Pumps that exert electric forces in the double layer: 4) Electroosmosis, 5) AC Electroosmosis 1.2 Equations in Electro-Quasi-Statics The equations that govern the electric field are those of Electro-Quasi-Statics. Magnetic effects can be neglected when the typical electric energy is much greater than the typical magnetic energy. 1 2 WM 2 μh = << W ε E E 2 The magnetic field generated by the electric currents in the liquid can be estimated from Ampère-Maxwell law: Either H~σ E l or H~ε ω E l : 2 2 WM μσ l H~σ E l leads to ~ 1 W ε E H~ε ω E l leads to W W M E l 2 2 ~ μεω 1 for (σ 1 S/m, l ~ 1 mm, ω 2π10 7 rad/s).

3 Electric fields in our system are governed by Quasi-Electro-Static Maxwell s equations: In effect, the electric field is almost irrotational. From Faraday s law: H~σ E l leads to H~ε ω E l leads to which are much smaller than unity for (σ 10 4 S/m, l ~1mm, ω 2π10 7 rad/s) 1.3 Basic Features of Conduction in Liquids Charge carriers in liquids are electrons, holes, and ions [2]. Measurements of electron and hole mobility require extremely pure dielectric liquids in order for the electron or hole to survive attachment or charge transfer. Except for liquid metals and/or ultrahigh purified noble liquids, there exist many situations where free electrons in liquids are quickly trapped by electronegative impurities or molecules of the liquid. In most cases, the charge carriers in liquids will be ions in dissolution, and this is the situation we are going to consider. The electrical current density is written as valence electro-migration diffusion convection proton charge with z i the valence, e proton charge, μ i mobility, D i diffusivity, n i number density, u fluid velocity. The first term is the electromigration: under the action of the electric field, ions can be considered to move at terminal velocity μ i E. A good model is to consider charged sphere in liquid affected by electric field. Stokes-Einstein μ= m 2 /sv for K + at T=293 K leads to R=0.1 nm Walden s rule

4 This model leads to Walden s rule: mobility μ times viscosity η is almost constant. Ion motion is also affected by thermal diffusion, which leads to the term Di ni. The ratio between diffusion and mobility is the Nernst-Einstein relation Nernst-Einstein = V at T=293 K for mono-valent ion In many situations common in microsystems, the diffusion current is negligible in front of the electro-migration current. diffusion volts migration ~ ΔΦ In many situations, the voltage drop across the typical distance is much greater than thermal voltage. However, diffusion current can be important near electrodes or walls, where double layers are formed. The ratio between convection and migration is of the order of convection migration electric Reynolds number This ratio is much smaller than unity for electrolytes. For insulating dielectric liquids the ratio can be of greater or smaller than one. Each ionic and neutral species satisfies a conservation equation [3] rate of production For neutral species μ=0. Some examples: For strong electrolytes, ions come from totally dissociated salt. There are no neutral species and no reactions in the bulk ( r i =0 per each ionic species). Electrochemical reactions take place only at the electrodes. For unipolar injection, there is only a single ionic species, and there is no generation of charges in the bulk r i =0. Ions are produced at the electrodes. For weak z-z electrolyte, there are three species: n +, n -, n 0 (positive, negative and neutral). Ions are produced by reactions at the electrodes and by chemical reactions in the bulk: where k d and k r are the dissociation and recombination rate constants.

5 Ions in dielectric liquids are originated from dissociation of impurities (or electrolytes) in the bulk, or of the liquid itself [4]. C+D (A + B - ) A + + B - (A + B - ) A + + B - ionophore electrolyte ionogen electrolyte Ionophore electrolytes are true electrolytes (sodium chloride is an example). Ionogen electrolytes or potential electrolytes react with the solvent to form ions. In both cases a good model is that of three species: equilibrium, n + =n - n 0 concentration of ionic pair. The conductivity of a dielectric liquid is highly dependent on its dielectric constant. Ions recombine if the energy of Coulomb attraction is greater than the thermal energy k B T at the point of closest approach of the ions. + - a + + a - The distance at which Coulomb attraction energy equals k B T is called Bjerrun length l B =e 2 /(4πεk B T). Because the probability to occur dissociation is proportional to exp[-l B /(a + +a - )], the conductivity is highly dependent on the dielectric constant ε r. ciclohexane chlorobenzene nitrobenzene water 2 ε r ~ l B (nm) ~ 280 ~ 110 ~ 16 In polar liquids (ε r >> 2), all impurities tend to be dissociated. In non-polar liquids (ε r ~ 2) there is equilibrium between dissociation and recombination. To establish a DC current in a liquid, electrons should be transferred from the metal electrodes to the molecules in the liquid, and viceversa. This can happen by electron emission from surface irregularities, from electrochemical redox reactions at the surface or by other means. From the mathematical point of view, the different mechanisms are described by similar parameters. V e- X X- Y + Y X X- Z X + + Y - Y + Y red-ox reactions or electron emission

6 A simple picture is to consider that neutral molecules can be charged at the electrodes: on the cathode X+e =X, on the anode X e =X+; and that ions can be discharged at the electrodes: on the cathode X + +e =X, on the anode X e =X. These reactions at electrodes, together with the homogeneous reactions of ion production in the bulk, form a simple dissociation-injection picture of conduction in liquids. The current-versus-voltage characteristic for a dielectric liquid typically has a form as shown in the figure [2]. Generally, different behaviours can be distinguished. electrode electrode X Y X - Y + neutralization of ions I I II III V Current -Voltage characteristic of a semi-insulating dielectric liquid electrode X + X ion injection At low voltages (region I), the current is caused by the dissociation of impurities in the liquid, and ohmic behaviour is observed. As the voltage increases (electric fields around 10 5 V/m), ions generated by the dissociation of impurities can not be created as fast as they are neutralized at the electrodes and the current density reaches saturation (region II). This region II is always present in gases but is ill-defined or may not be present in dielectric liquids. At higher voltages, a steep increase in the current is observed (region III). The beginning of region III can be due to enhancement of the dissociation process of impurities by the electric field strength. Nevertheless, at further increase of the electric field strength, ion injection from the electrodes becomes the dominating process. The current then tends to the I V 2 law of the space-charge-limited emission. Finally, at electric fields of sufficient high strength (of the order of 100 MV/m), breakdown may take place. The breakdown electric field is very dependent on the liquid, impurities, and geometry. In the case of AC currents, the charging of the double layer at the electrodes should also be considered. A simple model is to consider the electric double layer of the electrodes as a capacitor and a resistor in parallel and both connected to a resistive liquid bulk. The current through the capacitor represents the charging of the double layer, while the current through the resistor represents the Faradaic current. In the AC case, if the voltage drop across the double layer is not high, the electrochemical reactions at the electrodes can be very much reduced. In this case, the continuity of electric current can be guaranteed with no electrochemical reactions (no Faradaic current) by the charging of the double layer.

7 C DL V(t) ~ u s u a lly n e g lig ib le C b C DL R DL R b R DL double la ye r bulk double la y e r For AC currents, double layer charging m ust be considered 1.4 Mechanical Aspects of Micropumps Liquid motion is governed by Navier-Stokes for incompressible fluids. In microsystems, the Reynolds number is usually very small (inertial force / viscous force) R e = ρ m ul /η < 0.1 (u < 1000 μm/s, l < 100μm). Under steady-state conditions, we get to Stokes equations The equations relating fluid velocities to forces and pressure gradients are linear, and this linearity allows for the usual representation of microfluidic networks in terms of equivalent electrical circuits: resistors and generators. Flow-rate Q and pressure drop Δp are equivalent to electrical current intensity I and potential drop ΔV. Applied force is commonly concentrated in a region of the microfluidic system, this is the micropump region. micropump The relation between pressure drop and flowrate for a straight channel of circular crosssection is where D is the diameter and L the length of the channel.

8 Two important parameters of a pump are: Δp max maximum pressure and Q max maximum flow-rate that the pump can generate. The maximum pressure is the pressure difference required to stop the flow that the pump generates. The maximum flow rate is obtained when the pump actuates without an external load, that is, when the pressure difference between the inlet and outlet of the pump is zero. In many situations, there is a linear relation between flow-rate and back pressure under constant operating conditions. Power consumption and energy efficiency are important operational parameters of pumps. When comparing different pumps, we would prefer the one that consumes less power for obtaining a given flow rate or a given pressure. It is defined the energy efficiency as maximum delivered hydraulic power divided by consumed power. The last equality is obtained for linear relationship between Δp and Q. We will use this expression in order to estimate the efficiencies of micropumps, even in cases where the linear relationship does not hold. References [1] Laser D.J. and Santiago J.G., A review of micropumps, J. Micromech. Microeng., 14, R35 R64 (2004). [2] Schmidt W.F., Conduction mechanisms in liquids. In: Bartnikas R. (ed) Electrical Insulating Liquids. Engineering Dielectrics, Vol III. American Society for Testing and Materials, Philladelphia, PA, (1994) [3] Saville D.A., Electrohydrodynamics: The Taylor-Melcher leaky dielectric model. Annu. Rev. Fluid Mech, 29, (1997) [4] Castellanos A., Basic concepts and equations in Electrohydrodynamics. Part I. In: Castellanos A. (ed) Electrohydrodynamics. Springer-Verlag, New York, (1998)

Electrohydromechanical analysis based on conductivity gradient in microchannel

Vol 17 No 12, December 2008 c 2008 Chin. Phys. Soc. 1674-1056/2008/17(12)/4541-06 Chinese Physics B and IOP Publishing Ltd Electrohydromechanical analysis based on conductivity gradient in microchannel