i i i iDc FIV. Transport Phenomena Lecture 35: Porous Electrodes (I. Supercapacitors) MIT Student (and MZB) 1. Effective Equations for Thin Double Layers For supercapacitor electrodes, convection is usually negligible, and we drop out convection terms here. Let’s focus on effective equations governing the transports and electrostatics in electrolyte. Figure 1. Flat Electrode Surface Species conservation equations, Nernst-Plank flux constitutive equations, and Poisson equation make up Poisson-Nernst-Plank (PNP) set of equations (bold fonts indicate that the variables are in vector quantity): 0iictF(1) (2) (3) 2iiiz ec     Electrostatic constrain and flux constrains on the surface specify boundary conditions. Electrostatic constrain can be interpreted differently, given different specified variables. When there is a specified amount of surface charge, we can have Gaussian law satisfying the electrostatic constrain:ˆ()sq   nˆiiRnFLecture 35 10.626 Electrochemical Energy Systems (2011) Bazant (4) On the other hand, when given a specified surface potential, we can use the following approximation, Equation (5), instead of Equation (4), satisfying the electrostatic constrain. This boundary condition could be more simplified assuming negligible capacity in Stern layer. ˆ(x = 0) (x = 0)(x = 0)ese        n(5) We can use either Equation (4) or (5) to satisfy the electrostatic constrain, depending on specified variables at the surface. In addition, flux constrains of species specify the rest of necessary boundary conditions: (6) We now apply the above set of equations (PNP) as well as boundary conditions to the porous electrode with double layer thickness far thinner than the pore length scale. Figure 2. Thin Double Layer in a Pore When the pore length scale is far larger than the length scale of double layer, , we have separation of length scales. The mathematical structure of thin double layer problem is well understood from the perspective of singular perturbation analysis, in which each of two regions requires a different approximation. Two different approximations are constrained by matched asymptotic expansions. We first assign notations for different variables. From now on throughout this lecture, we use the following notations. i in Double Layer : Concentration of Species i in Bulk Electrolyte : Concentration of Specieslimccˆi limixxˆ 0Si i i S iD   FLecture 35 10.626 Electrochemical Energy Systems (2011) Bazant i in Double Layer : Chemical Potential of Species i in Bulk Electrolyte : Chemical Potential of Species Figure 3. Variables in Two Different Regions In quasi-neutral bulk electrolyte, we can use the quasi-neutral approximation, and conclude with zero divergence of current density. (7) 0iiiz ec0iiize  jF(8) As stated above discussion, asymptotic expansions of approximated variables in two different regions should be matched with the corresponding pair respectively. In a pore, variables are well approximated by the quasi-equilibrium approximation, and chemical potentials in the two regions are approximately constant across the pore. (9) (10) Now, conditions constraining the variables in double layer can be found by defining surface variables as shown in Figure 3 (Stern layer capacity ignored):  0ˆˆlimxiic c dxi  (11) (12) Then the effective boundary condition using the variables above is (Chu and Bazant, 2007): ˆiiˆSSiS i i iRt    F n F(13)pDLhLecture 35 10.626 Electrochemical Energy Systems (2011) Bazant where : Excess Surface Concentration of Species i per Area i: Surface Flux of Species i: Dimensionless Chemical Potential of Species  Using Bulk Variables ( i: Net reaction rate of Species Total diffuse charge density can be calculated with excess surface concentrations. (14) iiiq z e2. Porous Electrodes Formal derivation by volume averaging (homogenization) goes from the thin double layer equations inside pores to macroscopic partial differential equations. An appropriate model would involve volume averages of variables over a volume element small compared to the overall dimensions (L), but large than the pore structure length scale (hp). Hence, to have this model valid, the following condition is necessary. (15) Figure 4. Length Scales in Porous Electrode Not always, but for supercapacitors, we do not deplete the salt ions in electrolyte. Thus it is a valid approximation to assume that concentrations ( and potentials ( and ) are varying slowly in the macroscopic viewpoint. This assumption justifies the volume-averaging or homogenization. In this macroscopic treatment, we do not consider the actual geometric details of the pores. Rather, we define macroscopic potential in electrolyte, potential in solid material, and ion concentrations to be continuous and well-defined functions of space coordinates. As a result, the porous electrode in this model is represented by the superposition of two continuous mediai p i p ic c a  i i i iDc  FSi i p i i i p i iD c D a     FLecture 35 10.626 Electrochemical Energy Systems (2011) Bazant without microstructure, one corresponding to the electrolyte solution and the other corresponding to the solid material matrix. In this model both media are defined in the whole domain. Therefore, in this macroscopic model, potential in electrolyte ( ) as well as potential in conducting solid material ( ) are defined in the whole domain of space, whereas they were only defined in each phase formerly. Volume-averaged concentrations in macroscopic viewpoint are different from and related to the concentrations in bulk electrolyte and in double layers as shown  in the following equation (in former lectures, was used for the concentrations in reservoirs or  in inlet flows of fuel. In this lecture, is used for volume-averaged concentrations in macroscopic viewpoint). The macroscopic ion concentrations are defined throughout the whole volume as well. (16) The first term on right hand side corresponds to the contributions from bulk electrolyte, and the rdsecond term is from excess concentrations in double layers. Newman’s book (3 Ed, 2004) does not take account of the second term. However,