VI. Electrokinetics Lecture 31: Electrokinetic Energy Conversion MIT Student 1 Principles 1.1 General Theory We have the following equation for the linear electrokinetic response of a nanochan-nel: ! Q " KPKEO∆P = ! I KEO KE "!∆V "The basic idea1 is to apply ∆P and to try to “harvest” the streaming current I or streaming voltage ∆V. 1.1.1 Open-circuit Potential (Streaming Voltage) I = KEO∆P + KE∆V KI = 0 ⇒ VEO∆O = − ∆PKE Where KEO∆P is the streaming current and KE∆V is the Ohmic current. 1J.F. Osterle, A unified treatment of the thermodynamics of steady state energy conversion, Appl. Sci. Res. 12 (1964), pp. 425-434. J.F. Osterle, Electro-kinetic energy conversion, J. Appl. Mech. 31 (1964), pp. 161-164. The same idea was also discovered by Quincke in the 1800s. 1Lecture 31: Electrokinetic energy conversion 10.626 (2011) Bazant ∆P = Pin − Pout ∆VO = Vin − Vout Then Q = KP∆P + KEO∆V K2= KEO P 1 −  ∆P KPKE = (1 − α)KP∆P K2 Where α = EOKPKE . Pressure-driven flow is reduced by electro-osmotic back flow. Net flow is 0 when α = 1, but this is not possible. 1.1.2 Second Law of Thermodynamics Work done on system to drive motion per time (power input): P = Q∆P + I∆V > 0=! ∆P (∆P∆V)K > 0 ∆V "P must be positive since irreversible work produces heat in the system. The conductance tensor K must be positive definite, since this inequality holds for any ∆P, ∆V. So det(K) > 0 and thus α < 1. So, we can’t apply pressure and get a flow in the reverse direction! (Q = (1 − α) KP∆P). 2 Lecture 31: Electrokinetic energy conversion 10.626 (2011) Bazant 1.1.3 Streaming Current Harvesting Consider harvesting the streaming current via two electrodes connected by a total load resistance RL (= Rexternal + Rinternal from interfaces). Then ∆V = −IRL. Also: I = KEO∆P + KE∆V = KEO∆ − KERLI! " KEO = ∆P1 + KERL Let θ = KERL = RL external load = RE internal resistance . So: KEO∆PI = 1 + θ So, applied pressure leads to a current source via streaming current, which flows through internal and external resistors in parallel. We have the equivalent circuit: Q = KP∆P + KEO∆V K2 −EORL∆V = KP∆P 1 + θ αθ = KP∆P )1 − 1 + θ *! 1 + θ = KP∆P − αθ 1 + θ "3 Lecture 31: Electrokinetic energy conversion 10.626 (2011) Bazant Efficiency Pout I∆V I2RLεEK = = −= Pin Q∆P Q∆P+ K EO∆P,21+θ RL αθ = = = εKP∆P21+θ−αθ (1 + θ)(1 + θ − αθ)EK 1+θ Maximum at load resistance given by: α εmax = √∼ 4 for α α + 2( 1 − α + 1 − α) % 1Also: 1 θmax = √1 − α Since α < 1 and θmax > 1, RL > RE = 1 for optimal energy effi KE ciency.The same results apply to the efficiency of electro-osmotic pumping through an external fluidic resistance RP = 1 KP . 2 Porous Media Porous or composite materials are useful to “scale up” microfluidic phenomena to macroscopic volumes and useful flow rates. We will discuss general theories of transport in porous media later in the class. For now, we just need some basic con-cepts to facilitate our discussion of electrokinetic energy conversion, using porous media or microchannels. First, define the porosity $p as the ratio of the volume of pores to the total volume of the system: Pore Volume $p = Total Volume 4 Lecture 31: Electrokinetic energy conversion 10.626 (2011) Bazant Figure 1: Sketch of an example pore. Similarly we can define a measure of the pore wall area density as: Pore Surface Area ap = Total Volume Now consider how to define a mean pore size, hp. It is clear that $php = hp has the correct dimensions, lets consider the case of cylindrical pores to confirm that it is a reasonable approximation. For cylindrical pores: $p = πr2 pL ap = 2πrpL rph p = 2 so we conclude that hp is a good measure of the mean pore size. The last parameter we will define is the tortuosity τ, which measures how much longer is the mean distance Lp between two points traveling through the pores compared to the straight line distance L between them: Lpτ = L 3 Linear Electrokinetics: Scaling Analysis First we will consider a simple scaling analysis of the electrokinetic behavior of a pore as sketched in Figure 1. As discussed in the previous lecture we can relate the flow rate Q, and current I, to the applied pressure and voltage differences ∆P, ∆V through the conductance tensor K by: K = ! Kp KeoKscKe " (1) 5 Lecture 31: Electrokinetic energy conversion 10.626 (2011) Bazant Additionally, from Onsager’s symmetry principle we know that K must be a sym-metric matrix so Ksc = Keo. In order to perform our scaling analysis we will develop approximate expressions for each of the components of K. 3.1 Hydraulic Conductance An estimate of the hydraulic conductance Kp is given by: h2 pK p ! kLpp where kp is given by: h2 pk p ! η plugging this in gives the scaling of Kp as: h4 pKp ! (2) ηLp 3.2 Electrical Conductance In general a scaling estimate of Ke is given by: h2 pKe ! ke (3) Lpwhere ke is the electrical conductance of the material in the pore. We will consider two limits of ke corresponding to the thin double layer, small surface charge regime and the thick double layer, large surface charge regime. 3.2.1 Thin Double Layers ($ = λd<< hp 1) In this limit the conductance of the bulk electrolyte dominates the conductance of the double layer giving: 2ck Dekbulk0 D$ e ! e ! ! (4)kbT λ2 d 6Lecture 31: Electrokinetic energy conversion 10.626 (2011) Bazant 3.2.2 Thick Double Layers In the limit of thick double layers and large surface charge the pore is filled almost entirely with counter ions and they dominate the electrical conductance. Since we know that the counterions will balance out the surface charge on the pore wall we have: De qks | e! |(5)kbT hp Now plugging in Equations 4 and 5 back in to Equation 3 we obtain the scaling of the electrical conductance in each limit: $ h2Dp 2 , thin double Lp layers or low chargeKe ! λd (6) De|qs|hp , kbTLp thick double layer, large charge 3.3 Electroosmotic Conductance As for the electrical conductance we will estimate the electroosmotic conductance in the limits of thin