VI. Electrokinetics. Lecture 31: Electrokinetic Energy Conversion

Transcription

1 VI. Electrokinetics Lecture 31: Electrokinetic Energy Conversion MIT Student 1 Princiles 1.1 General Theory We have the following equation for the linear electrokinetic resonse of a nanochannel: ( ) ( )( ) Q K P K EO P = I K EO K E V The basic idea 1 is to aly P and to try to harvest the streaming current I or streaming voltage V Oen-circuit Potential (Streaming Voltage) I = K EO P + K E V KEO I = 0 V O = P Where K EO P is the streaming current and K E V is the Ohmic current. 1 J.F. Osterle, A unified treatment of the thermodynamics of steady state energy conversion, Al. Sci. Res. 12 (1964), J.F. Osterle, Electro-kinetic energy conversion, J. Al. Mech. 31 (1964), The same idea was also discovered by Quincke in the 1800s. K E 1

2 Lecture 31: Electrokinetic energy conversion P = P in P out V O = V in V out Then KEO 2 K P K E. Q = K P P + K EO V K 2 EO = KP 1 P K P K E = (1 α)k P P Where α = Pressure-driven flow is reduced by electro-osmotic back flow. Net flow is 0 when α = 1, but this is not ossible Second Law of Thermodynamics Work done on system to drive motion er time (ower inut): P = Q P + I V > 0 ( ) P = ( P V)K > 0 V P must be ositive since irreversible work roduces heat in the system. The conductance tensor K must be ositive definite, since this inequality holds for any P, V. So det(k) > 0 and thus α < 1. So, we can t aly ressure and get a flow in the reverse direction! (Q = (1 α)k P P). 2

3 1.1.3 Streaming Current Harvesting Consider harvesting the streaming current via two electrodes connected by a total load resistance R L (= R external + R internal from interfaces). Then V = IR L. Also: I = K EO P + K E V = K EO K E R L I ( ) K EO = P 1 + K E R L Let θ = K E R L = R L R E = external load internal resistance. So: K EO P I = 1 + θ So, alied ressure leads to a current source via streaming current, which flows through internal and external resistors in arallel. We have the equivalent circuit: Q = K P P + K EO V K 2 = K P P EOR L V (1 1 + θ αθ ) = K P P ( 1 + θ ) 1 + θ αθ = K P P 1 + θ 3

4 Efficiency P out I V I 2 R L ε EK = = = P in Q P Q P ( ) K EO P 2 1+θ R L αθ = = = ε K P P 2 1+θ αθ (1 + θ)(1 + θ αθ) 1+θ Maximum at load resistance given by: α ε max = 4 for α 1 α + 2( 1 α + 1 α) EK Also: 1 θ max = 1 α Since α < 1 and θ max > 1, R L > R E = 1 K E for otimal energy effi ciency. The same results aly to the efficiency of electro-osmotic uming through an external fluidic resistance R P = 1 K P. 2 Porous Media Porous or comosite materials are useful to scale u microfluidic henomena to macroscoic volumes and useful flow rates. We will discuss general theories of transort in orous media later in the class. For now, we just need some basic concets to facilitate our discussion of electrokinetic energy conversion, using orous media or microchannels. First, define the orosity ɛ as the ratio of the volume of ores to the total volume of the system: Pore Volume ɛ = Total Volume 4

5 Figure 1: Sketch of an examle ore. Similarly we can define a measure of the ore wall area density as: Pore Surface Area a = Total Volume Now consider how to define a mean ore size, h. It is clear that ɛ h = h has the correct dimensions, lets consider the case of cylindrical ores to confirm that it is a reasonable aroximation. For cylindrical ores: ɛ = πr 2 L a = 2πr L r h = 2 so we conclude that h is a good measure of the mean ore size. The last arameter we will define is the tortuosity τ, which measures how much longer is the mean distance L between two oints traveling through the ores comared to the straight line distance L between them: L τ = L 3 Linear Electrokinetics: Scaling Analysis First we will consider a simle scaling analysis of the electrokinetic behavior of a ore as sketched in Figure 1. As discussed in the revious lecture we can relate the flow rate Q, and current I, to the alied ressure and voltage differences P, V through the conductance tensor K by: ( ) K K K = eo (1) K sc 5 K e

6 Additionally, from Onsager s symmetry rincile we know that K must be a symmetric matrix so K sc = K eo. In order to erform our scaling analysis we will develo aroximate exressions for each of the comonents of K. 3.1 Hydraulic Conductance An estimate of the hydraulic conductance K is given by: K h 2 L k where k is given by: h 2 k η lugging this in gives the scaling of K as: 3.2 Electrical Conductance In general a scaling estimate of K e is given by: h 4 K (2) ηl h 2 K e k e (3) L where k e is the electrical conductance of the material in the ore. We will consider two limits of k e corresonding to the thin double layer, small surface charge regime and the thick double layer, large surface charge regime Thin Double Layers (ɛ = λ d << h 1) In this limit the conductance of the bulk electrolyte dominates the conductance of the double layer giving: 2 c De k k bulk 0 Dɛ e e (4) k b T λ 2 d 6

7 3.2.2 Thick Double Layers In the limit of thick double layers and large surface charge the ore 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 ore wall we have: De qs ke (5) k b T h Now lugging in Equations 4 and 5 back in to Equation 3 we obtain the scaling of the electrical conductance in each limit: K e D ɛ h 2 L λ 2, thin double layers or low charge d (6) thick double layer, large charge De q s h, k b TL 3.3 Electroosmotic Conductance As for the electrical conductance we will estimate the electroosmotic conductance in the limits of thin and thick double layers. First, the electroosmotic conductance is given generally by: ɛ K eo dxdy(ζ ψ) (7) ηl Aty Thin Double Layers In the limit of thin double layers ψ = 0 over most of the area of the ore so the integral reduces to ζh 2 giving: ɛ K eo ζh 2 ηl Since we are more interested in the case of constant surface charge than constant zeta otential we use the relation to obtain: q s λ d ζ ɛ h 2 K eo q s λ d (8) ηl 7

8 3.3.2 Thick Double Layers For the case of thick double layers we estimate the order of magnitude of (ζ ψ) from the caacitance of the ore giving: lugging this in to Equation 7 yields: q s qsh (ζ ψ) ɛ 3.4 Energy Conversion Efficiency C ore q s h 3 K eo (9) ηl Now that we have scaling estimates for each comonent of K we will estimate the maximum efficiency that can be obtained from an electrokinetic device using the relation Keo 2 ɛ max (10) K K e derived last lecture for the limit of low efficiencies. By substituting the values of K eo, K, and K e from Equations 9, 8, 2, 6 we get the scaling estimate for ɛ max in the two limits of interest: q 2 sλ 4 d 2, thin double layers or low charge ɛ max ηɛdh (11) q s k b T ηd e h, thick double layers, large charge It is imortant to note here that ɛ 2 max scales as 1/h for the thin double layer case, and as h in the thick double layer case. Since the thick double layer case corresonds to small mean ore sizes this means that the efficiency increases with ore size in the small ore limit, but decreases with ore size for large ores. Therefore there must be a maximum efficiency that occurs for intermediate ore sizes on the order of λ d, this effect is deicted in Figure 2. Two recent aers by van der Heyden et al have solved the full non-linear set of equations to determine ɛ max and erformed exeriments to test the theory [1, 2]. The theoretical and exerimental efficiency curves are lotted in Figures 3(a), and 3(b) resectively. References [1] Frank H. J. van der Heyden, Douwe Jan Bonthuis, Derek Stein, Christine Meyer, and Cees Dekker. Electrokinetic energy conversion efficiency in nanofluidic channels. Nano Letters, 6(10): , October

9 Figure 2: Scaling of the maximum efficiency with ore size [2] Frank H. J. van der Heyden, Douwe Jan Bonthuis, Derek Stein, Christine Meyer, and Cees Dekker. Power generation by Pressure-Driven transort of ions in nanofluidic channels. Nano Letters, 7(4): , Aril

10 h σ (nm x C/m 2 ) Li + 10 εmax(%) K + H + ε max (%) TAmA + η = η m Na + Li + K µ/µ k h/22 Image by MIT OenCourseWare. (a) Theoretical efficiency P (W) ε (%) R L (GΩ) (b) Exerimental efficiency Image by MIT OenCourseWare. Figure 3: Plots of the theoretical and exerimentally observed efficiencies of electrokinetic energy conversion from the work of van der Heyden et al [1, 2] 10

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