Pressure-sensitive ion conduction in a conical channel: Optimal pressure and geometry

Abstract

Using both analytic and numerical analyses of the Poisson–Nernst–Planck equations, we theoretically investigate the electric conductivity of a conical channel which, in accordance with recent experiments, exhibits a strong non-linear pressure dependence. This mechanosensitive diodic behavior stems from the pressure-sensitive build-up or depletion of salt in the pore. From our analytic results, we find that the optimal geometry for this diodic behavior strongly depends on the flow rate with the ideal ratio of tip-to-base-radii being equal to 0.22 at zero-flow. With increased flow, this optimal ratio becomes smaller and, simultaneously, the diodic performance becomes weaker. Consequently an optimal diode is obtained at zero-flow, which is realized by applying a pressure drop that is proportional to the applied potential and to the inverse square of the tip radius, thereby countering electro-osmotic flow. When the applied pressure deviates from this ideal pressure drop the diodic performance falls sharply, explaining the dramatic mechanosensitivity observed in experiments.

Ion Current Rectification and Long-Range Interference in Conical Silicon Micropores

“…Currently in the literature there are two, complementary, theories for current rectification without EDL overlap for pores with large aspect ratio. The theory by Cengio and Poggioli describes ICR through the variation of the surface conductance over the pore length but neglects electro-osmotic flow, while the theory of ref (developed by some of the present authors) does account for this flow but fails at extremely low salt concentrations. Hence both theories are complementary rather than mutually exclusive: refs and are valid at all concentrations, while the theory of ref is valid at all flow rates.…”

“…The theory by Cengio and Poggioli describes ICR through the variation of the surface conductance over the pore length but neglects electro-osmotic flow, while the theory of ref (developed by some of the present authors) does account for this flow but fails at extremely low salt concentrations. Hence both theories are complementary rather than mutually exclusive: refs and are valid at all concentrations, while the theory of ref is valid at all flow rates. We will find that our experiments show characteristic flow-sensitive behavior, and therefore we build on the theory of ref .…”

“…In steady state this salt current must be laterally constant to prevent the buildup of salt through the pore, π R 2 ( x ) ∂ t ρ̅ s = −∂ x J = 0, where ··̅· = 2π(π R 2 ( x )) −1 ∫ 0 R(x) ··· r d r denotes the cross-sectional average of the salt concentration. The condition of a divergence-free flux (eq ), which is necessary for a steady-state solution, leads to a differential equation for the cross-sectionally averaged salt concentration for x ∈ {0, L } D ∂ x ( π R 2 ∂ x ρ̅ s + 2 π R σ e ∂ x ψ k B T ) − Q ∂ x ρ̅ normals = 0 with the electric field −∂ x ψ = (Δψ p / L ) R b R t / R 2 ( x ) and the electro-osmotic flow in a conical channel Q = −Δψ p (π R t R b / L )(εψ 0 /η) (as derived beneath eq 2 in ref ). In eq the first term represents diffusion of salt through the bulk of the pore, the second term conduction of salt through the EDL, and the third term advection of salt through the bulk of the pore.…”

“…For Δψ < 0 this source term is positive, and thus the salt concentration increases. Solving for the cross-sectional average concentration profile ρ̅ s ( x ) leads to a nontrivial solution ρ̅ normals ( x ) − 2 ρ normalb = normalΔ ρ Pe [ x L R t R false( x false) − exp true( x L R t 2 R normalb R ( x ) Pe true) − 1 exp true( R normalt R normalb Pe true) − 1 ] infix= normalΔ ρ 2 | Pe | ( R b R false( x false) true( 1 − x L ( 1 + R t R b ) …”

“…The ICR originates from an asymmetry in the ionic current along the length of the channel due to a varying relative contribution to the ionic current of the charge-selective electric double layer (EDL) that screens the charge on the channel walls. Typically, ICR is demonstrated in nanoscale conical channels, where EDL overlap occurs on the narrow end of the channel. , In general, the ICR mechanism for a geometrically asymmetric, or tapered, channel can be understood by considering that the relative contribution of the salt current through the EDL to the total current is smaller at the wide opening than at the narrow opening. , This results in an asymmetry of the transference (i.e., the partial current due to either ionic species). Considering a channel with a negative surface charge on its wall, resulting in an EDL with excess positive ionic charge, an electric field directed toward the narrow end leads to more ions leaving the small opening than entering the large opening (before steady state is reached), resulting in depletion of ions inside the channel and a suppressed conductance .…”