Non-Linear Oscillation in Ionic Current Due to Size Effect in Glass Nanopipette

We studied the size effect of the ionic current in glass pipette, and found an interesting 2.7 mHz oscillation at 50 nm. In this study, we would like to discuss the mechanism of the non-linear oscillation. Cation-rich layer with its Debye length \textit{$\lambda $} exists in nanopipette, and its conductivity \textit{$\sigma $}$_{d}$ is lower than that in the central bulk layer \textit{$\sigma $}$_{b}$ in this study. The pressure difference \textit{$\Delta $P} = \textit{$\Delta $cRT} where \textit{$\Delta $c} is the difference in concentrations between in and out of the pipette. Then, the ionic current $I$ can be estimated by using Hagen-Poiseuille equation; \[ I=\frac{\pi }{8\eta }\frac{\Delta cRT}{\ell }\left\{ {\sigma _d r^4+\left( {\sigma _b -\sigma _d } \right)\left( {\lambda -r} \right)^2\left( {r^2+2r\lambda -\lambda ^2} \right)} \right\}. \] ($r$: inner radius, \textit{$\ell $}: pipette length, \textit{$\eta $}: viscosity) The last term indicates the non-linear oscillation. Moreover, we roughly estimated \textit{$\lambda $} = 2.08$\times $(2$r)^{1/2}$. Then, the bulk layer appears appropriately when 2$r\sim $50 nm, which causes the effective ionic current oscillation.