Numerical homogenization of electrokinetic equations in porous media using lattice-Boltzmann simulations

We report the calculation of all the transfer coefficients which couple the solvent and ionic fluxes through a charged pore under the effect of pressure, electrostatic potential, and concentration gradients. We use a combination of analytical calculations at the Poisson-Nernst-Planck and Navier-Stokes levels of description and mesoscopic lattice simulations based on kinetic theory. In the absence of added salt, i.e., when the only ions present in the fluid are the counterions compensating the charge of the surface, exact analytical expressions for the fluxes in cylindrical pores allow us to validate a new lattice-Boltzmann electrokinetics (LBE) scheme which accounts for the osmotic contribution to the transport of all species. The influence of simulation parameters on the numerical accuracy is thoroughly investigated. In the presence of an added salt, we assess the range of validity of approximate expressions of the fluxes computed from the linearized Poisson-Boltzmann equation by a systematic comparison with LBE simulations.

Figure 1

Simulation box for a cylindrical charged pore, using a volumic (a) or surface (b) charge distribution inside the solid. The black circles represent the fluid nodes, and others denote the solid ones. The color of the latter indicate their charge: uncharged (white), weak (pink), or strong (red). Periodic boundary conditions are used in all directions and computed properties may depend on the width $w$ of the solid phase.

Figure 2

Relative error on the counterion density obtained with the LBE simulations, compared to the analytical result [Eq. (14)] in a cylindrical channel of radius $R=5$ nm and surface charge density $\sigma =-0.08$ $e$ nm${}^{-2}$, in the absence of added salt. The simulation results depend on how the charge is distributed inside the solid nodes [in the whole volume (top) or only at the surface (bottom)] and on the size of the solid in which the channel is embedded [$w=\Delta x$ (left) and 50$\Delta x$ (right), see Fig. 1]. In order to use the same color code for all cases, in the top left part all errors larger than $\pm 10$$\%$ appear as red or blue (they can be as large as $\pm 60$$\%$).

Figure 3

Electro-osmotic permeability for cations $K_{+}^{\mathrm{eo}}$ in a cylindrical channel of radius $R=5$ nm and surface charge density, $\sigma =-0.08$ $e$nm${}^{-2}$, as a function of the salt concentration ${\rho }_{\mathrm{salt}}$. LBE results are given for different lattice spacings, $\Delta x$.

Figure 4

Electro-osmotic permeability for cations $K_{+}^{\mathrm{eo}}$ as a function of the surface charge density $\sigma$ in a cylindrical channel of radius $R=5$ nm, in the absence of added salt (${\rho }_{\text{salt}}=0$). LBE simulations (symbols) are compared to the analytical results (29) obtained with the nonlinear PB equation (dashed line) and with the linearized one (28) (solid line).

Figure 5

Relative error due to the linearization of the Poisson-Boltzmann equation on the electro-osmotic permeability for cations $K_{+}^{\mathrm{eo}}$, as a function of the surface charge density $\sigma$ and the cylinder radius $R$, in the absence of added salt (${\rho }_{\text{salt}}=0$). Note that the analytical solution is known for both the linearized [Eq. (28)] and the nonlinear [Eq. (29)] cases. The lines indicate the regions corresponding to given values of the reduced potential $\beta e\varphi$, which is assumed to be small in the linearization procedure. As expected, large errors on the transport coefficient $K_{+}^{\mathrm{eo}}$ are obtained when the $\beta e\varphi \gtrsim 1$.

Figure 6

Relative error obtained with the LBE simulations compared to the nonlinear solution [Eq. (29)] for the electro-osmotic permeability for cations $K_{+}^{\mathrm{eo}}$, as a function of the surface charge density $\sigma$ and the cylinder radius $R$, in the absence of added salt (${\rho }_{\text{salt}}=0$). The simulations are performed with the parameters indicated in the text, in particular a lattice spacing $\Delta x=0.208$ nm.

Figure 7

Hydrodynamic permeability $K_{\pm }^P$ (red upper and blue lower lines for cations and anions, respectively), for ions in a cylindrical channel of radius $R=5$ nm and surface charge density $\sigma =-0.08$ $e$nm${}^{-2}$, as a function of salt concentration ${\rho }_{\text{salt}}$. LBE simulations for cations ($•$) and anions ($\blacksquare$) are compared to the analytical results [Eq. (23)] obtained with the linearized PB equation (line).

Figure 8

Electro-osmotic permeability for the solvent $K_s^{\psi }$ as a function of salt concentration ${\rho }_{\text{salt}}$ in a cylindrical channel of radius $R=5$ nm. LBE simulations (symbols) are compared to the analytical results [Eq. (25)] obtained with the linearized PB equation (line).

Figure 9

Effect of a potential gradient on the cations ($•$) and anions ($\blacksquare$) in a cylindrical channel of radius $R=5$ nm and surface charge density $\sigma =-0.08$ $e$ nm${}^{-2}$, as a function of salt concentration ${\rho }_{\text{salt}}$. The total electrical permeability $K_{\pm }^{\psi }=K_{\pm }^{\mathrm{eo}}+K_{\pm }^{\mathrm{NE}}$ [(left) red upper and blue lower solid lines for cations and anions, respectively] as well as the Nernst-Einstein $K_{\pm }^{\mathrm{NE}}$ [(left) orange upper and green lower dashed lines for cations and anions, respectively] and the electro-osmotic contribution $K_{\pm }^{\mathrm{eo}}$ [(right) red upper and blue lower solid lines for cations and anions, respectively] are indicated. LBE simulations (symbols) are compared to the analytical results of Eqs. (27) and (28) obtained with the linearized PB equation (lines).

Figure 10

Relative error on the electro-osmotic permeability for cations $K_{+}^{\mathrm{eo}}$ due to the linearization of the PB equation, compared to the nonlinear one (as determined by LBE simulations), as a function of the surface charge density $\sigma$ and the salt concentration ${\rho }_{\text{salt}}$. The radius of the cylindrical channel is $R=5$ nm.

Figure 11

Effect of a concentration gradient on the solvent ($▴$) and ions ($•$, $\blacksquare$) in a cylindrical channel of radius $R=5$ nm and surface charge density $\sigma =-0.08$ $e$nm${}^{-2}$, as a function of salt concentration ${\rho }_{\text{salt}}$. For ions, the total coupling parameter $K_{\pm }^{\rho }=K_{\pm }^d+K_{\pm }^{\mathrm{co}}$ (red upper solid and blue upper dashed lines for cations and anions, respectively) and the Fickian contribution $K_{\pm }^d$ (orange lower solid and green lower dashed lines for cations and anions, respectively) are indicated. LBE simulations (symbols) are compared to the analytical results [Eqs. (30), (31), and (32)] obtained with the linearized PB equation (lines).