Universal Scaling of Robust Thermal Hot Spot and Ionic Current Enhancement by Focused Ohmic Heating in a Conic Nanopore

A stable nanoscale thermal hot spot, with temperature approaching $100°\mathrm{C}$, is shown to be sustained by localized Ohmic heating of a focused electric field at the tip of a slender conic nanopore. The self-similar (length-independent) conic geometry allows us to match the singular heat source at the tip to the singular radial heat loss from the slender cone to obtain a self-similar steady temperature profile along the cone and the resulting ionic current conductance enhancement due to viscosity reduction. The universal scaling, which depends only on a single dimensionless parameter $Z$, collapses the measured conductance data and computed temperature profiles in ion-track conic nanopores and conic nanopipettes. The collapsed numerical data reveal universal values for the hot-spot location and temperature in an aqueous electrolyte.

Figure 1

(a) Current voltage characteristic curves for the polymer nanopore. The rectification ratio for all data points is below 1.05. (b) Schematic of temperature profile in the $z$ direction inside a cone with different but self-similar temperature profiles in the $r$ direction at two positions on the cone surface.

Figure 2

(a) Measured conductance of a single conic nanopore ion-track PET membrane, charge-free nanopipette, and patch pipette, normalized by the zero-voltage conductance of each experiment, as a function of voltage. A quadratic increase with respect to voltage is observed, in violation of Ohmic law with constant conductance. (b) Membrane, nanopipette, and patch pipette data collapsed by the scaling theory into a linear correlation with respect to a normalization factor that contains cone angle $\theta$ (2.6°, 2.5° and 4.5° degrees for polymer nanopore, nanopipettes, and patch pipette, respectively), $k=0.15$ and $0.6\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ for the PET membrane pore and the silica pipettes, coefficient of temperature-dependent electrical conductivity $m$, thermal conductivity outside the cone $k$, conductivity $\sigma$ at indicated ionic strength at room temperature, and voltage $V$. Insets show cavitation in both the nanopore membrane and patch pipette at the corresponding voltages. Cavitating bubbles are first observed within the patch pipette at some distance from the tip, as shown in the second inset, but will eventually grow sufficiently to exit the pipette, as is observed at the membrane surface.

Figure 3

Collapse of simulation result of highest temperature vs normalized conductance for the nanopipette and nanopore with voltages ranging from 2V to 25V. $R_t$ indicates tip radius, which varies by a factor of 20 from polymer nanopores to nanopipettes. $C$ indicates molarity of KCl. Contour plots of the temperature near the tip of the nanopipette and polymer nanopore at 24V are shown in the inset. Room temperature is $25°\mathrm{C}$.

Figure 4

Simulation results of the cutoff position for different voltages and cone geometry using a cutoff temperature rise of $15°\mathrm{C}$. The theory that does not account for varying conductivity inside the cone is only valid at low voltages when conductivity variation throughout the cone is small. Once conductivity variation is considered, the theory is valid at both high and low temperatures.