Ionic heat dissipation in solid-state pores.

Energy dissipation in solid-state nanopores is an important issue for their use as a sensor for detecting and analyzing individual objects in electrolyte solution by ionic current measurements. Here, we report on evaluations of heating via diffusive ion transport in the nanoscale conduits using thermocouple-embedded SiNx pores. We found a linear rise in the nanopore temperature with the input electrical power suggestive of steady-state ionic heat dissipation in the confined nanospace. Meanwhile, the heating efficiency was elucidated to become higher in a smaller pore due to a rapid decrease in the through-water thermal conduction for cooling the fluidic channel. The scaling law suggested nonnegligible influence of the heating to raise the temperature of single-nanometer two-dimensional nanopores by a few kelvins under the standard cross-membrane voltage and ionic strength conditions. The present findings may be useful in advancing our understanding of ion and mass transport phenomena in nanopores.

INTRODUCTION

When electrical charges pass through a medium under a potential gradient, their kinetic energy is dispersed as heat upon inelastic interactions. This phenomenon is known as Joule heating, ubiquitous in any conductors larger than a characteristic mean free path. Electric field–driven ion transport through solid-state nanopores (–) is also predicted to involve heat dissipation as cations and anions move in crowds of water molecules. The fact that the ionic conductance is well described by a Maxwell model (–) clearly suggests the diffusive nature of the ion motions and concomitant Joule heat generation even in single-nanometer-scale nanopores. On the other hand, it remains almost unexplored how the ionic heating actually occurs because of the technical difficulty to probe local heat at the nanoscale conduits (), despite the tremendous efforts devoted to single-particle and single-molecule analyses via resistive pulse measurements (–), along with the expected impact on translocation dynamics (, ) as well as the ion transport properties (, ). For example, high voltage applied to a solid-state nanopore was reported to induce characteristic ionic current fluctuations indicative of nanobubble nucleation (, ). Furthermore, optical measurements using a fluorescent probe observed temperature increase in and outside a conical micropore in a polymeric membrane (). Although these studies inferred substantial increase in the local temperature by energy dissipation (–), direct assessments of self-heating are yet to be performed. Here, we addressed this issue by embedding a nanothermometer in the vicinity of a nanopore to simultaneously record the temperature change upon voltage sweeps across the membrane.

RESULTS

The device consists of a 40-nm-thick SiN thin film suspended on a silicon wafer (Fig. 1, A to C; see also fig. S1 for details about the fabrication procedure). On the membrane, we formed a 100-nm-sized point contact made of Au and Pt nanowires that served as a thermometer. At the side of the thermal probe, a pore of diameter dpore was sculpted. The whole structure was coated with a SiO2 layer for preventing current leakage. The thermocouple was calibrated in individual experiments by measuring the change in the thermovoltage ΔVth between the Au and Pt nanowires upon external heating by a resistive heater as described elsewhere (fig. S2) (, ).

Fig. 1.

Local heating in a solid-state nanopore.

(A) A schematic model depicting simultaneous measurements of ionic current Iion and local temperature at a nanopore. Diffusive ion transport in salt solution under the applied cross-membrane voltage Vb induces energy dissipation to local heating at the nanopore, whose effect was evaluated by recording the thermovoltage change Vth at the thermocouple embedded in the vicinity of the channel. (B and C) Optical image (B) and false-colored scanning electron micrograph (C) of a 300-nm-sized thermometer-embedded nanopore. Square region in (B) is the SiO2/SiN membrane. The thermometer consisted of a 100-nm-sized point contact made of Pt and Au nanowires lithographed at the side of the nanopore (C).

The ionic current through the thermometer-embedded pore Iion was recorded together with ΔVth under varying cross-membrane voltage Vb to probe local heat generated via energy dissipation (Fig. 1A). The Iion-Vb characteristics of a 300-nm-sized nanopore demonstrated ohmic behavior in phosphate-buffered saline (PBS) containing 1.37 M NaCl (red curve in Fig. 2A, inset). At the same time, ΔVth exhibited nonlinear rise under the positive and negative cross-membrane voltage (Fig. 2A). The corresponding temperature at the Au/Pt point contact Tt (fig. S2) suggested prominent heating of the nanopore by more than 30 K from the ambient at 1 V. Meanwhile, whereas the Tt response became weaker in diluter electrolyte solution of lower ionic strength (blue plots in Fig. 2A), we found that the results at different salt concentrations share the same linear relationship between Tt and the input power P = IionVb (Fig. 2B). This can be interpreted as signifying a steady-state change in the nanopore temperature under the balance between the local heating via the electric field–driven diffusive ion transport and the associated heat transport (, ) [see also fig. S3 for the heating properties in a nanopore under a salt gradient, where we observed a highly asymmetric Tt-Vb behavior reflecting the electroosmotically driven diode-like Iion-Vb characteristics ()].

Fig. 2.

Simultaneous measurements of ionic current and local temperature at a nanopore.

(A) Changes in the thermovoltage ΔVth and the corresponding local temperature Tth at the thermocouple against the voltage Vb applied across the 40-nm-thick SiO2/SiN membrane holding a nanopore of 300 nm diameter. Red and blue plots are the results obtained in phosphate-buffered saline (PBS) of ion concentration 1.37 M and 137 mM, respectively. Inset is the simultaneously recorded ionic current Iion versus Vb curves. (B) Plots of Tt of (A) as a function of the electric power P = IionVb. Color code is the same as that in (A). (C) Temperature response against larger input power in a 300-nm-sized nanopore. The curve became irretraceable when excessive power was loaded over 30 μW. In contrast, the Iion-Vb characteristics was observed to be stable as shown in the inset, indicating failure of the thermocouple under the dissipated heat at the nanopore.

It is interesting to see how high the temperature can become by the local heating. For instance, theoretical estimations predicted superheating of electrolyte solution in a nanopore to above the boiling point (, ). In the present study, on the other hand, whereas we observed linear Tt versus P characteristics in the repetitive Vb scans up to P = 18 μW in the nanopore of dpore = 300 nm (green plots in Fig. 2C), further enlargement of the electrical load eventually led to irretraceable plots to immeasurable thermovoltage, wherein the pore temperature was raised to above 440 K (orange plots in Fig. 2C). In contrast, the Iion-Vb curve remained stable even under the high-power condition, meaning no notable deformation of the robust SiN channel structure (). The instability of the ΔVth measurements at the high-Vb conditions thus indicated malfunction of the metallic nanothermometer by the ionic heating. Checking the endurance of the thermocouple, we observed severe damage on the Au nanowire when advertently heating it to 573 K while leaving the Pt component intact (fig. S4). The phenomenon detected at the high-P regime is hence attributed to the thermally induced breakdown of the nanoscale Au line () by the dissipated heat in the nanopore. The results unambiguously prove the significant role of energy dissipation capable to heat up the nanopore to above the boiling point of water (see also figs. S5 to S7 for the ionic current characteristics under Vb > 3 V) (, ).

The pronounced local heating anticipates marked change in the ion transport characteristics (, ). In this regard, the open pore conductance G was found to increase steadily with increasing Vb (Fig. 3A), ascribable to the influence of the energy dissipation. Specifically, G is given by the Maxwell model (–) as G = (Rpore + Racc)−1, where Rpore = 4ρLpore/πdpore2 and Racc = ρ/dpore are the resistance inside and outside the pore, respectively. Assuming the depth Lpore and the diameter dpore of the nanopore remain unaltered during the voltage scans, which is very likely to be the case since no hysteresis was found in the Iion versus Vb curves, contributions of the electrical heating were expected to enter through the conductivity σ = ρ−1 = nceμc + naeμa, where nc,a and μc,a are, respectively, the number and the mobility of cations and anions. Here, μc,a is further given as μc,a = Q/4πηrc,a in the framework of Stokes law (), predicting viscous drag force Fdrag = 4πηrc,avc,a acting on the cations and anions moving at the drift velocity vc,a in a liquid of viscosity η. Consequently, we arrive at a simple expression G ~ β/η, where β is the coefficient defined by the nanopore structure as well as the ion radii and concentrations tentatively taken as constants. Using semiempirical formula (), we deduced η as a function of Tt where we approximated the viscosity of the electrolyte buffer to be the same as that of water. It predicted less viscous fluid in the self-heated pore at the elevated Vb (Fig. 3B). The result demonstrated linear dependence of G on η−1 (Fig. 3C), consistent to the classical Walden rule (). While this manifests the notable effects of ionic heating on the open pore conductance, its actual influence is not so significant, increasing the conductance by only about 10% under 100 K rise in Tt, which is presumably due to the fact that whereas the conductance is dominated by the conductivity outside the pore as Racc >> Rpore for dpore = 300 nm and Lpore = 50 nm (), the temperature rise takes place mainly in the nanopore, thus leaving a large portion of the access resistance unchanged.

Fig. 3.

Ion transport in self-heated nanopores.

(A) The ionic current Iion (red) and the conductance G (blue) in a 300-nm-sized nanopore filled with electrolyte buffer containing 1.37 M NaCl under a voltage scan from 0 to 3 V. (B) Theoretically predicted water viscosity change η associated with the temperature increase at the nanopore. (C) Linear dependence of G on η−1 revealing pronounced effects of heat dissipation–mediated viscosity change on the ionic current characteristics. Red line is a linear fit defining a slope β. (D) Linear relation between β and the pore diameter dpore from 100 nm to 10 μm. Dashed line is a linear fitting to the plots.

To investigate whether this is a common feature in fluidic channels of different sizes, we extended the measurements to thermometer-embedded pores of various diameters ranging from 10 μm to 100 nm (see figs. S8 to S12). All of these conduits displayed linear η−1 versus G characteristics. Meanwhile, the slopes β were found to scale linearly with dpore (Fig. 3D). This signifies equal effectiveness of the local heating on the ionic conductance irrespective of the channel size, considering the fact that G ~ Racc−1 ~ dpore, and hence, β ~ dpore, for the low depth-to-diameter aspect ratio pores (–).

The local heat dissipation is also expected to yield large noise considering its outcome to induce thermal agitation of ions at the nanopores (). Fast Fourier transform analyses of Iion traces provided noise spectra at various Vb (Fig. 4A; see also fig. S13). Under low cross-membrane voltage, the power spectrum density Inoise showed a characteristic profile typical for solid-state nanopores composed of surface charge fluctuation–derived 1/f feature at the low-frequency regime below 1 kHz and capacitance-mediated noise at above 10 kHz (). Meanwhile, increase in Vb led to monotonic up-shift of the spectra (fig. S13). The trend can be seen more clearly in the plots of the noise intensity Ith at 2 kHz, demonstrating linear rise with the input electrical power (Fig. 4B), which suggests steady increase in the thermal noise in response to the increase in the nanopore temperature via the local heating. At higher temperatures, on the other hand, the noise intensity levels off due presumably to the fact that the thermal noise became more pronounced than the capacitance-derived high-frequency component ().

Fig. 4.

Cross-membrane voltage–dependent noise spectra.

(A) Overplots of noise spectra obtained at Vb = 0.3 V (green), 0.6 V (blue), and 0.9 V (red), which showed a monotonic upward shift with increasing Vb. Dashed line indicates f = 2 kHz. (B) Plots of the noise intensity Ith at f = 2 kHz as a function of the input electrical power. Inset shows a magnified view at a low-power regime. Dashed lines are linear fits at IionVb < 100 W.

Last, we inspect the mechanism underlying the energy dissipation by comparing the heating properties in pores of different diameter. Measuring the local temperature at the point contact of the nanothermocouple under the applied ramp cross-membrane voltage, Tt tended to increase linearly with P regardless of the size of the fluidic channels ranging from dpore = 10 μm to 100 nm (Fig. 5A). Meanwhile, as evident in the figure, it costed higher electrical power to heat up larger pores. To shed light on the scaling law, the efficiency of the ionic heating was evaluated by deducing C = P/Tt from the linear dependence of the nanopore temperature on the load power. Plotting as a function of the pore diameter, C was found to scale linearly with dpore (Fig. 5B). The trend can be qualitatively explained in a framework of Fourier’s law that renders the interplay of the local power dissipation P at the nanopore and concomitant heat flow via the effective thermal conductance K at the both sides of the channel as P = 2 K(Tt − T0), where T0 is the surrounding temperature. Assuming hemispheres of water with diameter dw were involved in the heat spread, K ~ dw and hence P/Tt = C ~ dw. Figure 5A can thus be interpreted as reflecting dw ~ dpore, suggestive of energy dissipation at more expansive area at the orifices of larger pores that consumes more power to heat up the channel.

Fig. 5.

Local heating in solid-state pores.

(A) Tt-P characteristics in pores of various sizes from 100 nm to 10 μm. Images are the false-colored scanning electron micrographs displaying the structure of the thermometer-embedded pores used. The local temperature tends to be raised under lower electric power for smaller channels, attributed mainly to the lower water heat capacity involved in the heating. (B) Plots of the power cost C to raise the local temperature by 1 K with respect to dpore. The gray curve is a linear fit with zero intercept indicating C increase with dpore. Insets are an equivalent circuit depicting steady-state heat flow under the temperature difference of Tt and the bath temperature T0 via the thermal resistance 1/K and a magnified view at small dpore. (C) Predicted change in the temperature ΔT at nanopores of diameter dpore = 10 nm in a 10-nm-thick membrane (blue) and a 1-nm-sized channel mimicking a graphene nanopore structure (red) under the applied voltage Vb in 1 M KCl. A case of 4 M KCl is also shown for the 1-nm nanopore (pink).

The above physical picture presumes that the thermal conduction through water plays a predominant role on the heat spread at the nanopores. Meanwhile, the electroosmosmotic flow in the voltage-biased nanopores anticipates heat transfer via convection. We therefore evaluated these two heat transport mechanisms by numerically solving a continuum heat transfer problem with a finite element method (figs. S14 to S16). The effectiveness of the convection to carry the ionic heat away from the nanopores was found to be little compared to the heat conduction counterpart.

The electrical heating mechanism was further verified by examining the nanopore measurements in liquid of different thermodynamic properties. As a model case, we used glycerol having specific heat capacity of cgly = 2390 J/kgK, which is about a factor of 2 smaller than that of water cw = 4182 J/kgK. The local temperature at a 3-μm-sized pore in 1: 1 mixture of glycerol and PBS showed linear increase with P, analogous to that in the electrolyte buffer (fig. S17). In contrast, the rate of Tt increase was found to be more rapid due to the lower heat capacity involved in the local heating. More quantitatively, the specific heat capacity of the glycerol/PBS cmix is given as cmix = (mwcw + mglycgly)/(mw + mgly), where mw and mgly are the mass of the water and the glycerol, respectively (assuming the density and specific capacity of the electrolyte buffer as those of water). With the density of 1.25 g/ml of glycerol, cmix is deduced as 3186 J/kgK. On the other hand, the heating efficiency C is deduced from the Tt-P plots (fig. S17C) as Cmix = 849 nW/K and CPBS = 1145 nW/K. Comparing the results, we found that the ratio between Cmix and CPBS (CPBS/Cmix = 1.35) is in fair agreement with the relative difference in the specific heat capacities (cw/cmix = 1.31). This manifests the important role of the liquid heat capacity at the pore orifice where the electrokinetic energy of ions is mostly dissipated, which, in turn, indicates a minor influence of solid-state compartments such as the SiN wall and the metallic thermometer due, in part, to the relatively small specific heat capacities on the order of 100 J/kgK. More specifically, the finite element analyses of the ionic heat dissipation elucidated rather uniform temperature distributions, with only 21% underestimation of the 300-nm nanopore temperature by Tt (fig. S18A). Meanwhile, we add to note that this error becomes slightly larger for smaller pores, 24% for the 100-nm-sized nanopore (fig. S18B), ascribed to the more pronounced influence of heat loss at the SiN for their deteriorated cooling capability via heat conduction through water. This may be a reason why the plots deviate more at the low dpore regime in Fig. 5B.

Extrapolating the dpore dependence (Fig. 5C), we estimated the impact of the local heating in smaller nanopores of diameter 10 and 1 nm with low–aspect ratio structure, the designs of which have been widely used for sensing proteins (–) and genomes (). In 1 M KCl, for instance, it predicts a notable increase in the pore temperature by 10 K above the ambient for a 10-nm-sized nanopore of 10 nm depth under Vb = 1.0 V, where we obtained the open pore conductance from the Maxwell model () with a bulk conductivity of the KCl solution (). The result is similar in the case of a 1-nm-diameter pore in a two-dimensional material of 0.6 nm effective thickness (). Increasing the salt concentration to 4 M KCl, on the other hand, the local heating becomes more pronounced, raising the temperature at the nanochannel by more than 57 K at 1 V. It would thus be of crucial importance to take the Joule heating into account in the nanopore sensing as these conditions are normally used for single-molecule analyses by ionic current.

DISCUSSION

We elucidated the pronounced effects of energy dissipation in solid-state nanopores causing significant local heating under typical experimental conditions. While it was demonstrated to affect the ionic conductance via the induced change in the viscosity, the electrically formed heat spot is also anticipated to alter the capture-to-translocation dynamics of analytes. For example, it foresees structural change of soft biomolecules such as proteins and amyloids (, ) that should be considered in an attempt to resolve the conformational changes by the blockade current (, ). Meanwhile, it would facilitate DNA to denature into single-stranded forms, an important aspect for implementing single-molecule sequencing with solid-state nanopores (, ). Moreover, the induced temperature gradient at the orifices may bring thermophoretic forces to draw analytes into pores that contributes to better throughput of resistive pulse sensing (, ). Future efforts should be devoted to verify the roles of self-heating on these unique capabilities of solid-state nanopores.

MATERIALS AND METHODS

Fabrication of thermometer-embedded nanopores

A 40-nm-thick SiN-coated, 4-inch silicon wafer was diced into 30 mm–by–30 mm pieces with a dicer. On the SiN surface, microelectrodes were formed by photolithography followed by radio-frequency magnetron sputtering (Samco) of a 20-nm-thick Ti/Au layer and subsequent liftoff by ultrasonication in N,N-dimethylformamide (DMF; Wako). A nanowire was then delineated by electron beam lithography (Elionix) using ZEP520A (Zeon). In the pattern, there was also a nanoscale cross drawn at a predefined position. After development, the entire surface was coated with a 20-nm-thick Pt layer by sputtering and was lifted off in DMF. The series of the electron beam lithography and metal deposition processes were then implemented to form a 20-nm-thick Au nanowire so as to create the nanothermocouple consisting of a 100-nm-sized Au/Pt point contact, wherein the nanoscale cross was used as a marker to align the tips of the two nanowires. After that, the Si beneath the thermometer was wet-etched in heated KOH aq. (Sigma-Aldrich) through a 1-mm-sized window made in the SiN layer on the back side of the substrate by reactive ion etching with a metal mask. As a result, we obtained a SiN membrane supporting the thermometer. A pore of various diameters ranging from 10 μm to 100 nm was then sculpted at the vicinity of the thermometer by electron beam lithography and reactive ion etching, again using the nanocross as a marker. Last, the entire surface was covered with a 40-nm-thick SiO2 from the top by chemical vapor deposition.

Thermometer calibration

The Pt/Au thermometer was calibrated by a procedure described elsewhere (, ). Briefly, the thermometer sample was sealed in a vacuum chamber and evacuated to a level below 10−5 torr. The point contact was then heated via electrical heating of a resistive heater to a set temperature. Confirming that the temperature became stable, the thermovoltage at the Pt/Au thermometer was recorded using a Keithley 2182A nanovoltmeter. The series of the processes were performed in a temperature range from 300 to 600 K. The temperature dependence of the thermovoltage was used to calculate the thermometer temperature rise during the cross-membrane voltage scans.

Nanopore sealing

Polyimide blocks having microchannels at one side of the surface were prepared by curing polydimethylsiloxane (PDMS; Sylgard184, Dow) precursor on a photo-patterned SU-8 mold on a Si wafer, followed by baking for polymerization and cutoff with a knife. In each PDMS block, we punched three through-holes. The PDMS and the thermometer-embedded nanopore chip were then treated with oxygen plasma for surface activation. After that, they were put together for eternal bonding of the PDMS on the nanopore substrate. The process was repeated again to adhere another PDMS block to the other side of the Si chip.

Ionic current and nanopore temperature measurements

PBS (Wako) was poured into the nanopore through the inlet and outlet holes punched in the PDMS. The ionic current Iion between a pair of Ag/AgCl rods placed in the remaining holes in the polymer blocks was then measured under the applied cross-membrane voltage Vb using a Keithley 6487 (Keithley) picoammeter. At the same time, the thermovoltage Vth of the Au/Pt thermometer was recorded through the microelectrodes using a Keithley 2182A nanovoltmeter. The measurement setup was General Purpose Interface Bus (GPIB)-controlled under a program coded in Visual Basic to record Vth upon Vb change by 1 mV.

Finite element simulations

The temperature distribution in a nanopore under applied cross-membrane voltage was described by a heat equation based on Joule heating source terms of J·E, where J and E are the current density and the applied electric field, respectively (). The steady-state heat equations for (Eq. 1) solids and (Eq. 2) fluids are respectively expressed asandHere, κ, T, ρd, Cp, and u are the thermal conductivity, temperature, density, heat capacity, and fluid velocity field, respectively. The current density in these equations was coupled to the continuity equation at a steady stateFrom Ohm’s law, J is proportional to E as J = σE, where σ is the conductivity. The temperature dependence of the bulk electrical conductivity for the electrolyte solution was evaluated from the relation of G(T) = (4ρLpore/πdpore2 + ρ/dpore)−1. The mobilities of anions (μa) and cations (μc) were obtained as μa,c = σ/(20eNAc0) where NA = 6.02 × 1023 mol−1 is the Avogadro constant. The number density distribution of anions (na) and cations (nc) for the nanopore was calculated by using the Nernst-Planck equationHere, kB, e, za,c, and φ are the Boltzmann constant (kB = 1.38 × 10−23 J/K), elementary charge (e = 1.60 × 10−19 C), charge number of ions (zc = 1 for Na+ and za = −1 for Cl−), and electrostatic potential. The distribution of φ was solved by using the Poisson-Boltzmann equationHere, ρQ and εw are the electric charge density and permittivity of water. The temperature dependence of εw = (249 − 0.790T + 7.30 × 10−4 T2)ε0 was assumed where ε0 is the electric constant of 8.85 × 10−12 F/m. Through the coupling between ρQ and E, the force ρQE generates the electroosmotic flow. The resulting heat convection was simulated by using the incompressible Navier-Stokes equation for hydrodynamic pressure p and fluid velocity field uEqs. 1 to 6 were simultaneously solved by using a software package for finite element methods of COMSOL Multiphysics 5.5 (COMSOL Inc., Stockholm, Sweden).