HOMOGENIZATION OF ELECTROKINETIC FLOWS IN POROUS MEDIA: THE ROLE OF NON-IDEALITY.

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1 Ion transport in porous media 1 G. Allaire HOMOGENIZATION OF ELECTROKINETIC FLOWS IN POROUS MEDIA: THE ROLE OF NON-IDEALITY. Grégoire ALLAIRE, Robert BRIZZI, CMAP, Ecole Polytechnique Jean-François DUFRECHE, Université Montpellier 2 Andro MIKELIC, ICJ, Université Lyon 1 Andrey PIATNITSKI, Narvik University. 1. Introduction 2. Partial linearization of the model 3. Homogenization and macroscopic Onsager properties 4. Numerical results on the effective coefficients NM2PorousMedia, September 29 - October 2, 24, Dubrovnik

2 Ion transport in porous media 2 G. Allaire -I- INTRODUCTION We consider ion transport in a charged porous medium. Coupled model: Poisson equation for the electrostatic potential Ψ ǫ, Stokes equations for the fluid velocity and pressure u ǫ,p ǫ ), Nernst-Planck convection-diffusion) equations for the N species concentrations n ǫ j. Non-ideal model for the ion diffusion: activity coefficients given by the MSA model. Our goal is to homogenize this model and compare the effective properties in the ideal and MSA cases. Small pores: Debye length of the order of the pore size. We choose a scaling of the model for which all unknowns are varying at the pore scale similar to Looker and Carnie 2006).

3 Ion transport in porous media 3 G. Allaire Assumptions and notations Rigid solid part of the porous medium. Saturated incompressible single phase flow containing N dilute charged species with valence z j and same ion radius σ. For each species: diffusion coefficient D 0 j, Péclet number Pe j, diffusive flux j ǫ j. Surface charge Σ on the pore walls. Small hydrostatic force f and external potential Ψ ext,.

4 Ion transport in porous media 4 G. Allaire ε Small parameter ǫ = ratio between the period and a macroscopic lengthscale. Periodic porous medium Ω: fluid part Ω ǫ, solid part Ω\Ω ǫ. Ω

5 Ion transport in porous media 5 G. Allaire Adimensionalized equations Poisson + Stokes + Nernst-Planck): ǫ 2 Ψ ǫ = β z j n ǫ j in Ω ǫ, j=1 ǫ 2 u ǫ p ǫ = f + z j n ǫ j Ψ ǫ in Ω ǫ, j=1 divu ǫ = 0 in Ω ǫ, div j ǫi +Pe i n ǫiu ) ǫ = 0 in Ω ǫ, i = 1,...,N, j ǫ i = L ǫ ij Mj ǫ j=1 L ǫ ij = n ǫ i δ ij + k BT Di 0 and M ǫ j = ln Ω ij ) 1+R ij ) Boundary conditions on the pore walls: n ǫ jγ ǫ je z jψ ǫ), i,j = 1,...,N, ǫ Ψ ǫ ν = Σ, u ǫ = 0, j ǫ i ν = 0 on Ω ǫ, i = 1,...,N.

6 Ion transport in porous media 6 G. Allaire Non-ideality: MSA model γ ǫ j = activity coefficient Ideal case: γ ǫ j = 1, Ω ij = 0 and R ij = 0 L ǫ ij = nǫ i δ ij). Non-ideal case: γ ǫ j = γ HS exp{ L BΓ ǫ Γ c z 2 j 1+Γ ǫ Γ c σ) } and Γǫ ) 2 = k=1 n ǫ k z2 k 1+Γ c Γ ǫ σ) 2 where Γ ǫ is the screening parameter and γ HS is the hard-sphere term γ HS = exp{pξ)} with pξ) = ξ 8 9ξ +3ξ2 1 ξ) 3 and ξ = πn c 6 n ǫ kσ 3 k=1 Complicated formulas for Ω ij and R ij but the Onsager tensor L ǫ ij is symmetric).

7 Ion transport in porous media 7 G. Allaire External boundary conditions For simplicity we choose a cube domain Ω = 0,L) d with d = 2,3. On the outer boundary Ω ǫ Ω we can thus impose periodic boundary conditions. The fluid velocity and pressure u ǫ,p ǫ ) and the concentrations n ǫ j are L-periodic. Given an external potential Ψ ext, x), the total electrokinetic potential Ψ ǫ +Ψ ext, is L-periodic. The forcing is caused by Ψ ext,, the surface charge density Σ and the hydrodynamic force f.

8 Ion transport in porous media 8 G. Allaire Strategy for the homogenization process Find so-called equilibrium solutions in the absence of exterior forcing. In the ideal case it yields the non-linear) Poisson-Boltzmann equation. For small exterior forcing f and Ψ ext, but large surface charge Σ ), linearize the transport model. Homogenize the linear model on a non-linear electrostatic background.

9 Ion transport in porous media 9 G. Allaire Bibliography Reference book on the model: Karniadakis-Beskok-Aluru, Microflows and Nanoflows. Fundamentals and Simulation. Interdisciplinary Applied Mathematics, Vol. 29, Springer, New York, 2005). Many numerical works on the upscaling: Adler, Coelho, Marino, Shapiro, Smith... Linearization process: O Brien and White 1978). Formal two-scale asymptotic expansions: Auriault-Strzelecki 1981), Moyne-Murad 2002, 2003, 2006), Looker and Carnie 2006). Homogenization: Schmuck 20), Ray 21)... Our own work in the ideal case: Journal of Mathematical Physics, 51, ). MSA case: Physica D, 282, )

10 Ion transport in porous media 10 G. Allaire -II- PARTIAL LINEARIZATION OF THE MODEL Following the lead of O Brien and White 1978) we perform a partial) linearization. In the ideal case, this is the same as in Looker and Carnie 2006). We assume that the forcing terms Ψ ext, and f are small, but not the surface charge density Σ which can still be large. We denote by n 0,ǫ i,ψ 0,ǫ,u 0,ǫ,p 0,ǫ the equilibrium quantities for f = 0 and Ψ ext, = 0. At equilibrium we look for a solution with vanishing fluxes u 0,ǫ = 0 and j 0,ǫ i = 0 and an ǫ-periodic electrostatic potential Ψ 0,ǫ x) = Ψ 0 x ǫ )

11 Ion transport in porous media 11 G. Allaire Equilibrium solution Consequence of the zero ionic flux j 0,ǫ i = 0: Mj ǫ = 0 with Mj ǫ = ln n ǫ jγje ǫ z jψ ǫ) Thus n 0,ǫ j x) = n 0 j )γ 0 j ) exp{ z j Ψ 0,ǫ x) } γ 0,ǫ j x) where n 0 j ) are constants called infinite dilution concentrations) and γ0 j ) are the constant activity coefficients for zero potential.

12 Ion transport in porous media 12 G. Allaire Poisson-Boltzmann equation at equilibrium y Ψ 0 y) = β y Ψ 0 ν = Σ z j n 0 jy) in Y F, j=1 y Ψ 0 y) is 1 periodic, n 0 jy) = n 0 j )γ 0 j ) exp{ z j Ψ 0 y) } with the activity coefficient defined by γ 0 j y), γ 0 jy) = γ HS y)exp{ L BΓ 0 y)γ c z 2 j 1+Γ 0 y)γ c σ) } and Γ0 y)) 2 = on Y F \ Y, k=1 n 0 k y)z2 k 1+Γ c Γ 0 y)σ) 2, γ HS = exp{pξ)} with pξ) = ξ 8 9ξ +3ξ2 1 ξ) 3 and ξy) = πn c 6 n 0 ky)σ 3. k=1

13 Ion transport in porous media 13 G. Allaire Poisson-Boltzmann equation at equilibrium We impose the bulk electroneutrality condition, i.e., for Ψ 0 = 0, z j n 0 j ) = 0. j=1 Theorem. Assuming that the ion radius σ is not too small and that the characteristic concentration n c is not too large, there exists a solution Ψ 0 of the Poisson-Boltzmann equation. Remark. In the ideal case, γj 0 y) = 1, the Poisson-Boltzmann equation has always a unique solution since it corresponds to the minimization of a convex energy. The MSA model destroys this convexity property.

14 Ion transport in porous media 14 G. Allaire Linearization n ǫ ix) = n 0,ǫ i x)+δn ǫ ix), Ψ ǫ x) = Ψ 0,ǫ x)+δψ ǫ x), u ǫ x) = u 0,ǫ x)+δu ǫ x), p ǫ x) = p 0,ǫ x)+δp ǫ x), Trick O Brien and White): introduce the ionic potential Φ ǫ i defined by n ǫ ix)γ ǫ ix) = n 0 i )exp{ z i Ψ ǫ x)+φ ǫ ix)+ψ ext, x))}. In the ideal case, this trick is useful because it yields the following change of variables ) δn ǫ ix) = z i n 0,ǫ i x) δψ ǫ x)+φ ǫ ix)+ψ ext, x) However, in the non-ideal case the algebra is much more complex! In particular, each δn ǫ i involves all Φǫ k.

15 Ion transport in porous media 15 G. Allaire After linearization and some algebra!) we obtain the problem we want to homogenize: ǫ 2 u ǫ P ǫ = f j=1 z j n 0,ǫ j Φ ǫ j +Ψ ext, ) in Ω ǫ, divu ǫ = 0 in Ω ǫ, u ǫ = 0 on Ω ǫ \ Ω, N ) Kijz ǫ j Φ ǫ j +Ψ ext, )+Pe i u ǫ = 0 in Ω ǫ, i = 1,...,N, divn 0,ǫ i K ǫ ij = j=1 δ ij + k BT D 0 i Ω ij ) 1+R ij ), i,j = 1,...,N, Kijz ǫ j Φ ǫ j +Ψ ext, ) ν = 0 on Ω ǫ \ Ω, j=1 u ǫ, P ǫ, Φ ǫ j are L periodic.

16 Ion transport in porous media 16 G. Allaire In the previous equations, n 0,ǫ j and Kij ǫ are ǫ-periodic coefficients evaluated at equilibrium by solving the non-linear Poisson-Boltzmann equation). Ψ 0,ǫ x) = Ψ 0 x ǫ ), n0,ǫ j x) = n 0 j x ǫ ), Kǫ ijx) = K ij x ǫ ). The linearization is thus partial because Ψ 0 is solution of a highly) non-linear equation. Lemma. The linearized problem admits a unique solution. Remark. It is a crucial assumption that all ions have the same diameter.

17 Ion transport in porous media 17 G. Allaire -III- HOMOGENIZATION AND TWO-SCALE LIMIT We assume that the porous medium is periodic. Periodic unit cell Y = 0,1) n = Y F Σ 0 with fluid part Y F. Fast variable y = x ǫ. Two-scale asymptotic expansions: u ǫ x) = u 0 x,x/ǫ)+ǫu 1 x,x/ǫ)+..., P ǫ x) = p 0 x)+ǫp 1 x,x/ǫ)+..., Φ ǫ jx) = Φ 0 jx)+ǫφ 1 jx,x/ǫ)+...

18 Ion transport in porous media 18 G. Allaire Theorem. The solution satisfies u ǫ x) u 0 x, x ǫ ), Pǫ x) p 0 x)+ǫp 1 x, x ǫ ), Φǫ jx) Φ 0 jx)+ǫφ 1 jx, x ǫ ), where u 0,p 0,p 1,{Φ 0 j,φ1 j }) is the solution of the two-scale homogenized problem which admits a unique solution). Remark. The difficulty is to extract from the two-scale homogenized problem a macroscopic homogenized model and to study its Onsager properties. Remark. The oscillating) concentrations are recovered from the ionic potentials by n ǫ ix) n0 i )γ0 i ) γ 0 i x ǫ ) exp{ z i Ψ 0 x ǫ )+Φ0 ix)+ψ ext, x))}.

19 Ion transport in porous media 19 G. Allaire Two-scale homogenized problem y u 0 x,y)+ y p 1 x,y) = x p 0 x) f x) + z j n 0 jy) y Φ 1 jx,y)+ x Φ 0 jx)+e x)) in Ω Y F, j=1 ) div y u 0 x,y) = 0 in Ω Y F, div x u Y 0 dy = 0 in Ω, F N div y n 0 iy) j=1 N div x n Y 0 iy) F K ij z j y Φ 1 jx,y)+ x Φ 0 jx)+e x) ) ) +Pe i u 0 x,y) j=1 u 0 x,y) = 0 on Ω Y F, = 0 K ij z j y Φ 1 jx,y)+ x Φ 0 jx)+e x) ) ) +Pe i u 0 x,y) dy = 0 K ij z j y Φ 1 j + x Φ 0 j +E ) ν = 0 on Ω Y F. j=1 The macroscopic forcing terms are in red and blue.

20 Ion transport in porous media 20 G. Allaire Factorization of the two-scale functions We want to separate the slow x and fast y variables. Our approach is different from that of Looker and Carnie. We decompose d ) p u 0 x,y) = v 0,k 0 y) +fk x)+ x k p 1 x,y) = Φ 1 jx,y) = k=1 d ) p π 0,k 0 y) +fk x)+ x k k=1 d θ 0,k p 0 j y) +fk x k k=1 ) x)+ i=1 i=1 i=1 v i,k y) Ek + Φ0 i x k π i,k y) Ek + Φ0 i x k θ i,k j y) Ek + Φ0 i x k where v i,k,π i,k,θ i,k j ), for 0 i N, are solutions of cell problems. ) ) x) ) ) x) ) ) x)

21 Ion transport in porous media 21 G. Allaire Definition of effective or homogenized) quantities We define the following effective hydrodynamic velocity: ux) = 1 u 0 x,y)dy. Y F Y F We also introduce the effective electrochemical potential of the jth species ) µ j x) = z j Φ 0 jx)+ψ ext, x), and the effective ionic flux of the jth species j j x) = 1 Y F N n 0 jy) Y F l=1 z l K jl y Φ 1 Pe lx,y)+ x Φ 0 lx)+e x) ) ) +u 0 dy j We are now able to write the homogenized equations for the above effective fields.

22 Ion transport in porous media 22 G. Allaire Theorem homogenized equations) The macroscopic equations in Ω are u {j i } div x u = 0 and div x j i = 0, for i = 1,...,N, with J 1 K... z 1 = M p0 +f D 11 L 1 and M = z 1 { µ i } D N1 L N z 1 J N z N D 1N z N D NN z N Furthermore the tensor M is symmetric positive definite Onsager properties).

23 Ion transport in porous media 23 G. Allaire The matrices J i, K, D ji and L j are defined by their entries {J i } lk = 1 v i,k y) e l dy, {K} lk = 1 v 0,k y) e l dy, Y F Y F Y F Y F {D ji } lk = 1 Y F {L j } lk = 1 Y F n 0 jy) v i,k y)+ Y F m=1 n 0 jy) v 0,k y)+ Y F K jm z m Pe j δim e k + y θ i,k m y) )) e l dy, m=1 z ) m K jm y θm 0,k y) Pe j e l dy.

24 Ion transport in porous media 24 G. Allaire First cell problem: imposed pressure gradient y v 0,k y)+ y π 0,k y) = e k + j=1 j=1 z j n 0 jy) y θ 0,k j y) in Y F div y v 0,k y) = 0 in Y F, v 0,k y) = 0 on Y F, N ) div y n 0 iy) K ij y)z j y θ 0,k j y)+pe i v 0,k y) = 0 in Y F j=1 K ij y)z j y θ 0,k j y) ν = 0 on Y F.

25 Ion transport in porous media 25 G. Allaire Second cell problem: imposed electrostatic field y v l,k y)+ y π l,k y) = j=1 j=1 z j n 0 jy)δ lj e k + y θ l,k j y)) in Y F, div y v i,k y) = 0 in Y F, v i,k y) = 0 on Y F, N div y n 0 iy) K ij y)z j δlj e k + y θ l,k j y) ) ) +Pe i v l,k y) = 0 in Y F, j=1 K ij y)z j δlj e k + y θ l,k j y) ) ν = 0 on Y F.

26 Ion transport in porous media 26 G. Allaire -IV- NUMERICAL RESULTS All computations are done with FreeFem++. Aqueous solution of NaCl at 298 K. Cation Na + with diffusivity D 0 1 = 13.33e 10m 2 /s. Anion Cl with diffusivity D 0 2 = 20.32e 10m 2 /s. The concentrations of the species in the bulk are considered equal : n 0 1 ) = n 0 2 ) = 0.1mole/l. The dynamic viscosity η is equal to 0.89e 3 kg/m sec). The pore size is l = 5.e 8m = 50nm. The ion radius is σ = 2nm. The surface charge density is minus) Σ = 0.129C/m 2.

27 Ion transport in porous media 27 G. Allaire Ideal case: cation concentration left), anion concentration right)

28 Ion transport in porous media 28 G. Allaire Averaged ion concentration as a function of n 0 1 ) = n 0 2 )

29 Ion transport in porous media 29 G. Allaire Rescaled anion concentration N2mean/n 0 2 ) as a function of n 0 1 ) = n 0 2 )

30 Ion transport in porous media 30 G. Allaire Permeability tensor rescaled): variation of n 0 1 ) = n 0 2 )

31 Ion transport in porous media 31 G. Allaire Diffusion tensor for the cation: variation of n 0 1 ) = n 0 2 )

32 Ion transport in porous media 32 G. Allaire Coupling tensors L i : variation of n 0 1 ) = n 0 2 )

33 Ion transport in porous media 33 G. Allaire Permeability tensor : variation of the pore size nm)

34 Ion transport in porous media 34 G. Allaire Averaged cell concentration : variation of the pore size nm). Donnan effect

35 Ion transport in porous media 35 G. Allaire Variation of porosity: 0.19, 0.51 and 0.75

36 Ion transport in porous media 36 G. Allaire Variation of porosity: permeability

37 Ion transport in porous media 37 G. Allaire Diffusion versus porosity: cation left), anion right)

TRANSPORT IN POROUS MEDIA

1 TRANSPORT IN POROUS MEDIA G. ALLAIRE CMAP, Ecole Polytechnique 1. Introduction 2. Main result in an unbounded domain 3. Asymptotic expansions with drift 4. Two-scale convergence with drift 5. The case