Avalanche effects near nanojunctions

In this article, we perform a computational investigation of a nanopore connected to external fluidic reservoirs of asymmetric geometries. The asymmetry between the reservoirs is achieved by changing the cross-sectional areas, and the reservoirs are designated as the micropore reservoir and macropore reservoir. When an electric field is applied, which is directed from the macropore towards the micropore reservoir, we observe local nonequilibrium chaotic current oscillations. The current oscillations originate at the micropore-nanopore interface owing to the local cascade of ions; we refer to this phenomenon as the “avalanche effects.” We mathematically quantify chaos in terms of the maximum Lyapunov exponent. The maximum Lyapunov exponent exhibits a monotonic increase with the applied voltage and the macropore reservoir diameter. The temporal power spectra maps of the chaotic currents depict a low-frequency “$1/f$”-type dynamics for the voltage chaos and “$1/f^2$”-type dynamics for the macropore reservoir chaos. The results presented here offer avenues to manipulate ionic diodes and fluidic pumps.

Figure 1

(a) Schematic illustration of multiple nanopores interacting with asymmetric fluidic reservoirs (b) Simulation setup of a single micro-nano-macropore system. The electric field is applied from macropore to micropore reservoir. Ionic current is calculated at four different regions: at bulk micropore reservoir $(B_{\mathrm{micro}},x=1\mu \text{m}$ from the origin), nanopore junction $\left(NJ\right)$ at a distance of $0.5\mu \text{m}$ from the micropore-nanopore entrance, center of the nanopore $(NP,x=9\mu \text{m}$ from the origin), and bulk macropore reservoir $(B_{\mathrm{macro}},x=13.5\mu \text{m})$.

Figure 2

Variation of the output current (in magnitude) at the center of the nanopore with (a) applied voltage (current is averaged over $4\text{ms}$ time interval for each voltage); inset shows the variation of the standard deviation of the current over the time interval considered for each voltage, (b) time (data shown for the last $2\text{ms})$, and (c) its corresponding histogram for $5\text{V}$ (solid blue line) and $100\text{V}$ (solid red line), respectively. (d) Variation of the maximum Lyapunov exponent with applied voltage. The ratio of macropore to micropore reservoir diameter is 4 in (a) to (d).

Figure 3

Variation of the output current (in magnitude) with (a) macropore to micropore radius $\left(R=d_{ma}/d_{mi}\right)$, current averaged over $4\text{ms}$ time interval for each $R)$; inset shows the variation of the standard deviation of the current, over the time interval considered for each $R$, (b) time (data shown for the last 1 ms), and (c) its corresponding histogram for $R=2$ (solid blue line) and $R=20$ (solid red line), respectively. (d) Variation of the maximum Lyapunov exponent with $R$. The applied voltage is $5\text{V}$, and the micropore reservoir diameter is $0.05\mu \text{m}$ in (a) to (d).

Figure 4

Avalanche effect mechanism for (a) a symmetric reservoir $\left(R=1\right)$, ions are enriched near the drain micropore-nanopore interface and an ion depletion zone is developed near the source micropore-nanopore interface. (b) An asymmetric reservoir $\left(R>1\right)$, cluster of ions accumulate near the micropore-nanopore interface, an “avalanche zone” is created, and a relatively weak ion depletion zone is developed near the nanopore-macropore interface. Concentration distribution of sodium ions (counterions) along the micro-nano-macropore reservoir system for (c) $5\text{V}$ and $R=1$, inset reveals distribution of an ion enriched zone; (d) $5\text{V}$ and $R=20$, inset reveals dynamic clustering of ions near the micropore-nanopore interface; (e) $5\text{V}$ and $R=4$, inset reveals weak oscillation of ions near the enriched zone; and (f) $100\text{V}$ and $R=4$, inset reveals strong enrichment of ions and an inhomogeneous distribution of ions near the enriched micropore-nanopore interface.

Figure 5

Current-time dynamics at (a) bulk micropore $\left(x=1\mu \text{m}\right)$, (b) nanojunction $\left(x=6.5\mu \text{m}\right)$ (c) center of nanopore $\left(x=9\mu \text{m}\right)$, (d) bulk macropore $\left(x=13.5\mu \text{m}\right)$. The macropore reservoir diameter is $0.2\mu \text{m}$, micropore reservoir diameter is $0.05\mu \text{m}$. Ratio of macropore to micropore reservoir is 4. The applied voltage is $100\text{V}$.

Figure 6

Variation of potential (a) along the micro-nano-macropore system. The inset reveals the unstable potential distribution near the enriched micropore-nanopore interface (b) with time near the enriched micropore-nanopore interface. The macropore reservoir diameter is $1\mu \text{m}$, micropore reservoir diameter is $0.05\mu \text{m}$. Ratio of macropore to micropore reservoir is 20. The applied voltage is $5\text{V}$.

Figure 7

Power spectral density (PSD) analysis for (a) an applied voltage of $100\text{V}$, inset shows PSD for $70\text{V},R=4$ is used for both the applied voltages; (b) a macropore diameter of $1\mu \text{m}$, inset shows PSD for a macropore diameter of $0.5\mu \text{m}$. The applied voltage is $5\text{V}$.