Concentration Polarization

IV. Transport Phenomena Lecture 16: Concentration Polarization We have previously discussed open circuit voltage, which can be derived from the Nernst equation, and activation overpotentials, which can be de­ rived from the Butler-Volmer equation. This can accurately describe the behavior of electrochemical cells at low currents, but for sufciently large currents, the transport of reactants to the reaction sites begins to become limiting, and the concentration at the reaction sites will be significantly lower than the bulk concentration. As the current gets even larger, the re­ action will completely run out of reactants, and the voltage required to push the reaction will diverge as the current approaches a limiting current, Ilim . This behavior is commonly described by a concentration polarization. For this lecture, we will examine how the di↵usion of reactants can be modeled to describe this process. 1 Linear Di↵usion and Convection In a dilute (quasi-binary) solution, the transport of solutes can be described in terms of a flux density F~ , which in the absence of convection can be described be Fick’s Law: F~ = -DrC, where D is the di↵usivity of the particles, and C is their concentration. In the presence of fluid velocity ~u, the more general expression for Fick’s Law becomes: F~ = -DrC + ~uC. (1) Note that a more accurate model of gas transport in porous media is the Dusty Gas Model, as has been shown for the case below of a solid oxide fuel cell1 , but Fick’s Law is a reasonable simple model for most electrochemical systems. 1 Y. Fu et al., J. Electrochem. Soc. 162, F613 (2015). 1 Lecture 16: Concentration polarization 10.626 (2011) Bazant The di↵usivity D can be derived from a molecular random walk, which yields a result of the form x2 D= , (2) 2 t where x is the mean free path of the solute and t is the mean time between collisions of the solute. Keep in mind that this expression would need to be modified for concentrated solutions because it relies upon the fact that there are no interparticle interactions in order to obtain independent random walks. In order to apply Fick’s law, we need a final relationship between flux and concentration, which we obtain from conservation of mass. There are many good derivations of the conservation of mass relation available in textbooks, [?] but for our purposes, it is adequate to observe that for a point in space, the flux inwards plus any local particle generation must be equal to the flux outwards, or else there will a change in local concentration. For the case where there are no volumetric sources, this relationship is described by: @C + r · F~ = 0. @t (3) If we assume a constant di↵usivity and incompressible flow, so that r · (DrC) = Dr2 C and r · ~u = 0, we can combine equations 1 and 3 to obtain the linear convection-di↵usion equation: @C + ~u · rC = Dr2 C, @t (4) which is the primary tool for describing mass transport in many systems. 2 Limiting Current Density Concentration polarization can be defined as the loss of voltage due to concentration changes out of equilibrium, or while current is passing. There are three basic ways for this to occur. The first is that the local equilibrium voltage depends on the local concentration, as described by the Nernst Equation. The second is that the local exchange current, and therefore the activation overpotential, also depend on local concentration, as described by the Butler-Volmer equation. The third process, which is often described on its own as the concentration polarization, is that the conductivity of the electrolyte will also change as a function of local concentration, as described by the Nernst-Planck equation. The first two of these e↵ects are commonly referred to as “external” e↵ects because they are dependent upon the reactant 2 Lecture 16: Concentration polarization 10.626 (2011) Bazant Anode Electrolyte Cathode (porous) (solid) (porous) H2 gas O2 gas O2 H2 O la le lc Reactant Concentration Profile: c̄O2 c̄H2 cH2 c O2 0 Let cO2 represent the concentration of O2 in the cathode at the interface with 1: (Image taken from 2009 10.95 Scribe notes by Charles Moore.) theFigure electrolyte and c̄O represent the concentration of O2 being supplied to the Schematic of a solid2 oxide fuel cell. Oxygen is supplied to the cathode, free surface of the cathode. Similarly, define cH2 and c̄H2 respectively to be the where it is reduced to oxygen ions that migrate across the electrolyte to the concentration of H2 in the anode at the interface with the electrolyte and at the free anode, where they oxidize hydrogen that has adsorbed onto the anode. At surface. open circuitprofiles voltage; voltage the absence steady Also state let forVIO>denote 0, thethe concentration of the oxygen andin hydrogen in of any c̄Hshown voltage of the O2 = c̄O2 and 2 =as 2 ). The thediffusion cathode gradients and anode(celectrodes willcHbe inequilibrium the lower part of the cell is given by the Nernst equation: figure. ✓ ◆ ✓ ◆ cH2 kT kT cO2 ln while Veq = VO +outside + ln of the electrolyte, and product concentrations the third e↵ect 2e c̄O2 c̄H2 4e is an “internal” e↵ect because it stems from changes taking place inside the electrolyte. Assumptions inherent in this derivation are: constant H2 O 3• aExample: Solid-Oxide Fuel Cell • a constant O2 Consider the case of a solid oxide fuel cell. This system, with parameters as shown in figure 1, consists of two (since porousa gas electrodes (typically • neglect diffuse charge effect soliddi↵usion electrolyte has a constant backnickel anode and LSM cathode) on either side of a solid oxide electrolyte ground charge) (typically YSZ, yttrium stabilized zirconia) that can be treated as ideally In steady to state, reactants through the electrodes results in linear permeable O2-diffusion ions, andofimpermeable to everything else. In practice, these systems are comprised of three layers, generally concentration profiles. The diffusive fluxceramic is simply givenand by Fick’s firstoperate law. When multiplied by the electrons extracted per reactant and the surface area of the cathode, Ac we obtain the current: 3 2 Lecture 16: Concentration polarization 10.626 (2011) Bazant at temperatures between 600 and 1000 C. The half reactions are as follows: anode: cathode: net: H2 + O2 1 2 O2 ! H2 O + 2e ! O2 + 2e H2 + 21 O2 ! H2 O (5) For the simple case of steady state di↵usion in the porous electrodes where we neglect di↵usion, the transport equations simplifies to @ 2 c/@x 2 = 0, which has a linear solution. If we let the ’bulk’ concentrations be ci and the concentrations at the surface of the electrodes facing the electrolyte be ci , then the problem is fully defined, and it is trivial to specify the flux F = D ci /li . The current is connected to the particle flux by I = 2eAa FH2 = eAc FO2 , so the current can be written in terms of the flux across either the anode or the cathode: cH 2 c H 2 2eAa la cO2 cO2 DO2 lc 4eAc . anode reactant: I = DH2 cathode reactant: I = (6) Physically, concentrations must always be positive, so the particle flux through the porous electrode cannot exceed the value it achieves when the reactant concentration at the electrode surface reaches zero. This corresponds to anode and cathode limiting currents, which occur when one or the other reactant cannot di↵use fast enough to allow for any more current to pass. These can be written as c Ilim = a Ilim = 4eAc DO2 cO2 lc 2eAa DH2 cH2 la (7) We can now express c in terms of I and Ilim : ◆ ✓ ✓ ◆ c O2 cH2 c a I = Ilim 1 = Ilim 1 c O2 cH2 c O2 I cH 2 I =1 , =1 a Ilic m cH2 Ilim cO2 (8) We can also relate the current to the outward flux of the water vapor reaction product at the anode, anode product: I = DH2 O 4 cH2 O c H2 O la 2eAa Lecture 16: Concentration polarization 10.626 (2011) Bazant and thus relate the water concentration at the anode surface to the current c H2 O I =1+ w , cH 2 O Ilim (9) where 2eAa DH2 O cH2 O (10) la w , when the cell is run in reverse as which is the limiting current, I = Ilim an electrolyzer, splitting water electrochemically into hydrogen and oxygen gases, by applying a reverse voltage and putting energy into the system. Now that we have have expressions for c(I) at either electrode, we can plug them into our model: w = Ilim V (I) = cathode anode electrolyte (11) where cathode = anode = electrolyte c metal a metal c electrolyte a electrolyte = = c eq a eq c + ⌘act (I) a + ⌘act (I) = IRel . From the Nernst equation (assuming dilute gases), we have ◆ ✓ kT cO 2 c , ln p eq = Uc cO2 2e ✓ ◆ kT c H2 O a ln eq = Ua + 2e c H2 cO 2 (12) so if we let V0 correspond to the open circuit voltage at the ’bulk’ conditions cO2 and cH2 , and then plug our result from equation 8 into the Nernst equation, we get: ✓ ◆ kT c H2 O p ln Ua V 0 = Uc 2e c H 2 c O2 ◆ ◆ ✓ ✓ ✓ ◆ cO2 cH 2 cH 2 O kT kT kT Veq (I) = V0 + + ln ln ln 2e 2e cH 4e cO 2 c̄H2 O ◆ ✓ ◆ ✓ 2 kT I kT I + = V0 + ln 1 ln 1 a c 4e Ilim 2e Ilim ✓ ◆ kT I ln 1 + w (13) 2e Ilim i , the equilibrium voltage diverges because di↵usion cannot supAs I ! Ilim ply enough reactants to the reaction sites to maintain a steady state. Note 5 Lecture 16: Concentration polarization 10.626 (2011) Bazant c a . As noted that the voltage will diverge at the minimum of Ilim and Ilim w due to di↵usion limi­ above, the voltage also diverges to +1 as I ! -Ilim tation of water vapor during electrolysis in reverse operation. As we discussed earlier, variations in concentration also a↵ect reaction kinetics, as manifested in the Butler-Volmer equation. We will limit our analysis to symmetric (↵ = 1/2) reactions in a dilute gas for now, but one could obtain similar results for more complicated models. In our simple case, Butler-Volmer takes the form: ◆ ✓ I kT a ⌘act = sinh 1 e 2I a ✓ 0◆ I kT c ⌘act = sinh 1 e 2I0c where: v u r u 1 I aI cH2 c̄H2 O a a at lim I 0 = K0 = K0 cH 2 c H2 O 1 + I wI lim s r I c O2 I0c = K0c 4 = K0c 4 1 (14) c . Ilim c O2 All the parameters that are not a function of current are folded into the K0i constants so that we can focus on the limiting current behavior. When we plug these results back into equation 11, we obtain a full model for the e↵ect of concentration, which has a current versus potential relationship shown in i , we can take asymptotic figure 2. It is worth mentioning that for I ! Ilim expansions of the activation overpotentials to obtain simpler expressions. [Note: For large z, sinh-1 (z) ⇠ ln(2z)] ! ✓ ◆ I kT kT I p ⇠ ln 1 ln ⌘a ⇠ 2e Iliam e K0a 1 I/Iliam ! ✓ ◆ I kT kT I p ⌘c ⇠ ln ⇠ ln 1 . (15) c c 4e Ilim e K0c 1 I/Ilim The last comment to make about this result is that although concentration polarization is often only significant for relatively large currents, in many real world systems it can be a major concern in system design. If we look at power rather than voltage, as shown in the lower part of figure 2, we obtain a result with a sharp peak power near the limiting current followed by a steep drop in output as the system approaches super galvanic operation 6 Lecture 16: Concentration polarization 10.626 (2011) Bazant 1 Without Concentration Polarization With Concentration Polarization 0.9 0.8 Voltage (V) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 2 Current Density (A/cm ) 0.8 1 0.45 Without Concentration Polarization With Concentration Polarization 0.4 Power Density (W/cm2) 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 2 Current Density (A/cm ) 0.8 1 Figure 2: Voltage and power versus current under simplified fuel cell condi­ c a = 1 A/cm2 , K c = K a = 10- 5 A/cm2 ) = Ilim tions. (Rel = 0.1 ⌦ · cm2 , Ilim 0 0 When concentration polarization e↵ects are included, voltage diverges as current approaches the limiting current. Peak power occurs quite near to limiting current density, so concentration polarization becomes the main limitation on maximum power density. 7 Lecture 16: Concentration polarization 10.626 (2011) Bazant (V < 0). For many mobile systems, power density is often of greater impor­ tance than efciency, so the hard limit on power density that concentration polarization imposes can be a significant design limitation. For further reading, see O’Hare et al., Fuel Cell Fundamentals, Ch. 5.2. 8