Counterion flow through a deformable and charged nanochannel

A theoretical model is developed that describes nonlinear coupling between wall deformation and water and ion flows in a charged, deformable nanochannel whose viscoelasticity is governed by the Kelvin-Voigt model. Using continuum mean-field theories for mass and momentum conservation of the solid-liquid coupled system, a set of one-dimensional nonlinear partial differential equations is derived to capture the dynamics of wall deformations. For elastic but non-viscous walls undergoing small deformation, the problem simplifies to one of advection-diffusion type which is analytically solvable at first-order perturbation. Results unveil a rich coupling between the elasticity and charge distribution of the channel walls, which vanishes in the limit of weakly charged channels. This coupling significantly alters the quantitative response of the walls' relaxation dynamics and the channel's electrokinetic transport, thereby having important consequences for the description and understanding of electrokinetic flow through charged, viscoelastic media.

Figure 1

Schematic illustration of the examined configuration: (a) equilibrium nanchannel in 2D cross section and (b) 3D deformed nanochannel.

Figure 2

The deformation profile for a nanochannel after application of a pressure step for a small deformation ($\mathrm{\Delta }{\tilde{a}}_i=0.1$). Solid lines represent analytical solution in the limiting case of non-viscous flow, given by Eq. (38); symbols correspond to the numerical solution of the fully coupled model. The reference case corresponds to $\tilde{a}=1$. Parameters used are: ${\mathcal{K}}_Y=0.5,{\mathcal{K}}_{\text{osm}}=0.3,\kappa =2$, and $\alpha =0.5$.

Figure 3

Variation of the Peclet number with the channel surface charge density for different values of (a) elasticity constant $Y$ and (b) initial channel radius $R_0$. Other parameters used are: $L_0=50\mu \mathrm{m}$, $\mathrm{\Delta }{V}_0=0.1$ V, $R_0=2$ nm, $Y=0.05$ GPa, and $\alpha =0.5$.

Figure 4

(a) Variation of the relaxation time of the walls with the position along the channel. (b) Variation of the averaged relaxation time with the charge density at different elasticity constants. Among the fixed parameters are: $L_0=50\mu \mathrm{m}$, $\mathrm{\Delta }{V}_0=0.1$ V, $R_0=2$ nm, $Y=0.1$ GPa, and $\alpha =0.5$.

Figure 5

Profiles of total proton flux along the nanochannel at different times. The reference case corresponds to ${\overline{J}}_z/{\overline{J}}_{\text{ref}}=1$. Reference parameters are: $L_0=50\mu \mathrm{m}$, $\mathrm{\Delta }{V}_{\text{ref}}=0.1$ V, $R_{\text{ref}}=2$ nm, $Y_{\text{ref}}=0.1$ GPa, ${\sigma }_{\text{ref}}=-0.5$ C/${\mathrm{m}}^2$, $\mathrm{\Delta }{\tilde{p}}_i=0.05$, and ${\alpha }_{\text{ref}}=0.5$.

Figure 6

Variation of the average proton flux at steady state with surface charge for different (a) elasticity constants $Y$ and (b) channel charge parameters $\alpha$. Reference parameters are: $L_0=50\mu \mathrm{m}$, $\mathrm{\Delta }{V}_{\text{ref}}=0.1$ V, $R_{\text{ref}}=2$ nm, $Y_{\text{ref}}=0.1$ GPa, ${\sigma }_{\text{ref}}=-0.2$ C/${\mathrm{m}}^2$, $\mathrm{\Delta }{\tilde{p}}_i=0.05$, and ${\alpha }_{\text{ref}}=0.5$.

Figure 7

Variation of the average convective, diffusive and migration proton fluxes against surface charge density. Reference parameters are: $L_0=50\mu \mathrm{m}$, $\mathrm{\Delta }{V}_{\text{ref}}=0.1$ V, $R_{\text{ref}}=2$ nm, $Y_{\text{ref}}=0.1$ GPa, ${\sigma }_{\text{ref}}=-0.2$ C/${\mathrm{m}}^2$, $\mathrm{\Delta }{\tilde{p}}_i=0.05$, and ${\alpha }_{\text{ref}}=0.5$.

Figure 8

Variation of the average proton flux at steady state with surface charge for different (a) voltage drops and (b) channel radii. Reference parameters are: $L_0=50\mu \mathrm{m}$, $\mathrm{\Delta }{V}_{\text{ref}}=0.1$ V, $R_{\text{ref}}=2$ nm, $Y_{\text{ref}}=0.1$ GPa, ${\sigma }_{\text{ref}}=-0.2$ C/${\mathrm{m}}^2$, $\mathrm{\Delta }{\tilde{p}}_i=0.05$, and ${\alpha }_{\text{ref}}=0.5$.

Figure 9

Variation of the average water flux at steady state with surface charge for different elasticity constants. Reference parameters are: $L_0=50\mu \mathrm{m}$, $\mathrm{\Delta }{V}_{\text{ref}}=0.1$ V, $R_{\text{ref}}=2$ nm, $Y_{\text{ref}}=0.1$ GPa, ${\sigma }_{\text{ref}}=-0.2$ C/${\mathrm{m}}^2$, $\mathrm{\Delta }{\tilde{p}}_i=0.05$, and ${\alpha }_{\text{ref}}=0.5$.