1. Conduction. Percolation model (which assumes isotropic media) gives

Transcription

1 1. Conduction In the revious lecture, we considered conduction of electricity (or heat conduction or mass diffusion) in a comosite medium, where each comonent has a nonzero conductivity. In a orous medium, we focus on one ore hase and assign zero conductivity to the matrix hase. From the notation of the revious lecture, φ 1 = ε = orosity σ 1 = σ = ore hase conductivity σ = 0, for i > 1 i Note: the Hashin-Shtrikman and Wiener lower bounds are zero, since any volume fraction of nonconductive material can be distributed so as to comletely block conduction through the orous medium (i.e. if there is no ercolating ath of the conductive hase). The uer bounds are Wiener σ max = φ 1 σ 1 = ε σ anisotroic ores HS σ 1 2 φ 1 φ 2 ε 1 ε ε σ max = φ 1 σ 1 φ2 σ 1 + σ 1 = σ ε 2 ε = σ 2 ε isotroic ores Percolation model (which assumes isotroic media) gives σ erc ~ ε ε c t For ε just above the critical oint ε c, where the exonent t=2 is believed to have a universal value for any 3D model. A simle form which catures this effect is σ erc σ ε ε 1 εc 0 c 2, ε c ε,0 ε 1 ε c

2 since σ erc σ as ε 1. Note that this lies between the Hashin-Shtrikman bound and the lower bound. In electrochemical engineering, it is common to use the emirical Bruggeman formula σ B = ε 3/2 σ As shown in the figure below (where ε c is assumed to be 0.24), this lies above the Hashin- Shtrikman uer bound for large ε, so it is inconsistent to as for isotroic media. It also neglects ercolation transition at small ε. However, it is fairly close to the Hashin-Shtrikman uer bound, so it may be reasonable for a continuous orous hase with isolated solid matrix articles (as in the core-shell model for the HS uer bound model). σ σ Wiener Bound Bruggeman Bound Percolation Hashin-Shtrikman Bound ε c ε 2. Diffusion The effective conductivity determines the macroscoic current density J = σ φ. In the case of diffusion, the net flux is F = σ d c, where c is the concentration in the ores (which is constant in equilibrium, even if the orosity varies in sace). Macroscoic conservation of mass requires

3 c + F = 0 Where c = ε c is the volume-averaged concentration. c = σ d 2 c = D 2 c or c = D 2 c Where D = σ ε d is the effective diffusivity in the orous medium. Note: the Wiener uer bound imlies D D, since σ ε σ. Therefore, we can interret the reduction of D in the ores via an effective extension of the ath length for diffusion by a factor called tortuosity τ. L = τ L 2 = τ c = D c D = D τ2 So we can recover the same diffusion equation as in the free solution only with a stretched satial coordinate system. Thus, we arrive at the following interretation of the diffusive mean conductivity. σ d = D ε τ 2 Note: if σ d is a tensor, then tortuosity is also a tensor given by τ = D ε σ 1 d 1/2. For the models and bounds above, we have the following tortuosity (noting that τ = 1 when ε = 1): Wiener τ = 1 (lower bound, attained by aligned stries); HS τ = 2 ε (lower bound for isotroic ores); 1/4 τ B = ε (Bruggeman emirical formula);

4 τ erc ε 1 εc ε εc, ε c ε 1,0 ε ε c Note: tortuosity makes no sense when conductivity becomes significantly reduced by loss of ercolation, since it is not the longer ath length but rather the many dead ends and few ercolating aths that lower the conductivity. The figure below shows the tortuosity according to the above models (note, here ε c is assumed to be 0.24) τ Tortuosity according to Wiener Bound Tortuosity according to Hashin-Shtrikman Bound Tortuosity according to Bruggeman Bound Tortuosity according to Percolation ε c ε

5 3. Ion transort (neglect convection and electroosmotic flow) Surface charge qs on the ore wall Microscoic Nernst-Planck equation in the ores (NOT Lalace s equation as before): c i + Fi = 0 F i = D i c i μ i μ = μ /k T i i B μ = k T ln(γ c ) + z eφ i B i i i Poisson s equation gives: ε 2 φ = ρ = i z i ec i Boundary condition (no reaction, fixed surface charge): ε n φ = q s n F i = 0 Macroscoic PNP equation: c i + F i = 0 F i = D i c i μ i μ constant, across ores and small length scales, even though c and φ vary quickly. i

6 Define ρ = z i ec i (ε φ ) = ρ + ρ Where ρ s = a qs, a = i s ore surfcae area volume. In the limit of the double layer, c i constant in the ore bulk solution = ci/ε, and φ φ = constant outside double layer and ε 0: ρ + ρ s = 0 Thin double layer ρ s surface charge = volume This is just macroscoic neutrality condition.