Superheating and Homogeneous Single Bubble Nucleation in a Solid-State Nanopore

Gaku Nagashima,1 Edlyn V. Levine,1 David P. Hoogerheide,2 Michael M. Burns,3 and Jene A. Golovchenko1,2,*1School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA3The Rowland Institute at Harvard, Cambridge, Massachusetts 02142, USA

(Received 13 March 2014; published 9 July 2014)

We demonstrate extreme superheating and single bubble nucleation in an electrolyte solution within ananopore in a thin silicon nitride membrane. The high temperatures are achieved by Joule heating from ahighly focused ionic current induced to flow through the pore by modest voltage biases. Conductance,nucleation, and bubble evolution are monitored electronically and optically. Temperatures near thethermodynamic limit of superheat are achieved just before bubble nucleation with the system atatmospheric pressure. Bubble nucleation is homogeneous and highly reproducible. This nanoporeapproach more generally suggests broad application to the excitation, detection, and characterizationof highly metastable states of matter.

DOI: 10.1103/PhysRevLett.113.024506 PACS numbers: 47.55.dp, 47.55.db, 85.35.-p

In the course of exploring the limits and consequencesof extreme current densities in solid state nanopores [1],we have discovered that matter can be brought to highlylocalized, excited metastable thermal states in the pore.Moreover, these states can be probed and studied in aremarkably detailed and simple way and potentially usedfor practical purposes in many disciplines. We demonstratethis capability by bringing an aqueous electrolyte solutionin a nanopore to extreme levels of superheat, culminating inhomogeneous nucleation and growth of the vapor phase.This is achieved under highly repeatable conditions ame-nable to modern multiphysics based modeling.The homogeneous nucleation and growth of vapor

bubbles in liquids is a phenomenon whose thermal, kinetic,and mechanical aspects have been experimentally exploredand theoretically modeled to varying degrees of sophisti-cation from the time of Gibbs to the present [2–4]. Inclassical nucleation theory, a superheated liquid must over-come a surface tension induced energy barrier to bubbleformation by means of localized density fluctuations. Thedegree of superheat and rates of bubble nucleation have beencharacterized by kinetic models appealing to microscopicmechanisms involved in nucleation [5–10]. The superheatlimit of liquids has been studied using a variety of exper-imental methods. Among the most successful are micro-capillary boiling [11,12], heating in a host liquid [13], andpulse heating of a filament [14,15]. Bubble nucleation anddynamics associated with the phenomena of cavitation [16],sonoluminescence [17,18], laser induced heating of nano-particles [19], and heterogeneous bubble formation inmacroscopic pores [20] have also been areas of intensestudy recently.Here, we present new methods for observing and

studying superheating and homogeneous single bubblenucleation in the extreme environment created in a single

nanopore. We begin by reporting electrical conductivitymeasurements on a nanopore immersed in an electrolyte;these indicate extreme Joule heating, followed by bubblenucleation, probed on nanosecond time scales. Highfrequency periodic bubble nucleation and growth phenom-ena are presented. Optical measurements are used todetermine the bubble nucleation site and to estimate theinitial bubble growth rate. Calculated results are thendiscussed, accounting for and expanding on the varietyof phenomena revealed in the experiments.A schematic of the experiment is shown in Fig. 1. A

single nanopore was fabricated with a focused ion beammachine in a free-standing silicon nitride membrane affixedto a silicon-dioxide–silicon frame. Silicon nitride waschosen because it is highly wettable and has a higherthermal conductivity than the electrolyte, both of which areimportant for extreme superheating, minimizing hetero-geneous nucleation (compare with Ref. [20]). It wasmounted in a fluidic cell in which the membrane separatedtwo fluid chambers connected electrically only through thepore. A 3M NaCl solution prepared in deionized, degassedwater was added to each chamber and contacted with

FIG. 1 (color). Cross sectional schematic of the experimentalsetup.

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Ag=AgCl electrodes. A pulse generator (HP 8110A),current sensing resistor, and high bandwidth (500 MHz)oscilloscope are connected to the fluidic cell with acompensation circuit to minimize the effect of capacitancebetween the two fluidic chambers.Figure 2(a) shows the time-dependent nanopore con-

ductance observed when 11 μs voltage pulses, rangingfrom 4 to 8.22 V, with 30 ns rise time were applied across a53.5 nm radius, 71 nm thick nanopore. The initial nanoporeconductance is 1.15 μS (aside from an initial capacitancespike due to imperfect compensation) and increases withtime and applied voltage to a value of 3.5 μS and a currentdensity of 3.3 × 109 A=m2. The rise is expected due to timedependent Joule heating of the electrolyte in and near thepore and the positive temperature dependence of electrolyteconductivities, which are strongly influenced by the

temperature dependence of the water viscosity [21]. Thenoise in the data belongs to the oscilloscope amplifiers.Figure 2(b) shows a bubble nucleation event at 10.4 μs

in the continuation of the 8.22 V conductance data. Itconsists of a rapid drop in conductance when the bubbleblocks the ionic conduction through the pore. After thebubble collapse, subsequent bubble events are seen to occurwith quasiregular periodicity. The duration of each bubbleevent is approximately 16 ns with 120 ns between events.The behavior is that of a relaxation oscillator whosetime constant is determined by thermal dynamics discussedbelow.Figure 3(a) shows a second experimental setup, using

a larger 1.9 μm radius, 2.5 μm thick pore, designed foroptically probing the onset and location of the bubblenucleation in the pore. (Larger pores exhibit similarquasiperiodic bubble nucleation to that observed in the53.5 nm radius pore albeit with lower frequencies atcomparable voltage bias). Optical transmission of a focused514 nm, 0.5 mW, cw laser through the pore is measuredduring the nucleation process. A 60× water immersionobjective lens brings the laser to a beam waist diameterof 350 nm. The transmitted optical beam was capturedand brought to a focus with f ¼ 0.62 optics onto a 1 nsresponse time silicon photodiode (Thorlabs DET10A). Thephotodiode current was monitored simultaneously withthe time-dependent electrical signal from the ionic currentpassing through the pore. For this experiment, the responsetime for the ionic current measurement was determinedby the capacitance of the pore membrane. The lateralx-y position of the beam waist could be accurately movedto different positions across the pore with beam steeringoptics.Figure 3(b) shows both the electrical conductance

and photodiode current falling rapidly at the onset of anucleation event, stimulated by the application of an 18-V22-μs pulse. At this voltage, the events consistently occur14 μs after the pulse is applied. For these data, the laserbeam waist was positioned near the pore center. We definean event onset time for both the optical and ionic currentsignal to be the intercept of the prebubble current level withthe extrapolated linear region of the current drop. Theseare labeled in the figure as ti and tp for the ionic andphotodiode current, respectively. Details of the opticalsignal beyond its initial rapid drop are not currently wellunderstood but are unimportant for our current purposes.When the laser beam waist is moved to the periphery of

the pore, there is a clear increase in the delay in tp withrespect to ti as seen in Fig. 3(c). This suggests that thebubble was formed at the center of the pore, and as a resultof its finite growth velocity there is a delay until it scattersthe incident laser beam at the pore periphery.Figure 3(d) confirms this view. It shows the offset Δt ¼

tp − ti as a function of the laser position across the pore intwo perpendicular directions. The points corresponding to

FIG. 2 (color). (a) The conductance of a 53.5 nm radius, 71 nmthick nanopore at 4, 5, 6, 7, and 8.22 volts. The data are filtered at13 MHz from 0 to 1 μs and 20 MHz from 1 to 10 μs by an eight-pole Bessel filter. Computed conductance curves for each voltageare shown along with the calculated maximum temperaturesachieved in the pore. (b) Continuation of the 8.22 V conductancedata after 10 μs. The data are filtered at 200 MHz; the faded linein the background is the unfiltered measured conductance data.

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the data in Figs. 3(b) and 3(c) are indicated. The symmetryof the data around the pore center confirms that the bubblenucleation events are homogeneous and occur at the centerof the pore. The bubble radius growth velocity obtainedfrom the slope of the data in the figure is 52.1� 1.6 m=sfor the y-axis scan and 49.2� 1.7 m=s for the x-axis scan.A straightforward interpretation of the data presented

above involves rapid Joule heating of the electrolyte inand near the nanopore that ultimately results in nucleationof a vapor bubble at the pore center. The vapor bubbleexpands, cutting off the Joule heating when it reaches thepore periphery. It continues to grow due to fluid inertia andthermal energy stored in the superheated liquid. The bubbleultimately reaches a maximum size and then collapses afterthe vapor pressure in the bubble decreases below ambientpressure in the liquid sufficiently to overcome the inertialforces of liquid expansion.We have calculated the coupled space- and time-depen-

dent ionic current density and temperature fields in and nearthe pore prior to bubble formation. The time-dependentconductance of the pore was then modeled for comparison

with the experimental results. The nonlinear, inhomo-geneous heat equation with a Joule heating source termwas solved for the nanopore geometry using the COMSOL

MultiPhysics program (COMSOL, Inc.) [22]. The temper-ature dependence of the heat capacity, thermal conductivity,and density were taken to be those of superheated water atatmospheric pressure, given by the IAPWS-95 formulation[23–25]. The temperature dependence of the electricalconductivity of 3M NaCl solution at atmospheric pressurewas chosen to fit the experimental results by extrapolatingmeasured high temperature data taken at pressures aboveatmospheric [21,26].The resulting computed pore conductance curves are

shown as the solid smooth curves in Fig. 2(a). Excellentagreement has been achieved with the experimental datawith reasonable temperature dependent properties of thesolution. The predicted temperature attained at the porecenter after 10 μs for each voltage pulse is indicated inFig. 2(a) for each conductance curve.A contour plot of the temperature field within the

nanopore at 10.4 μs is shown in the inset of Fig. 4 for

FIG. 3 (color). (a) Schematic of the optical experiment. (b),(c) Electrical conductance and photodiode current at the onset of anucleation event with the focused laser at (b) the center and (c) the periphery of the pore. (d) Δt ¼ tp − ti as a function of the laserposition for x- and y-axis scan. Each point contains ten measurements; the error bars show the standard deviations from the mean. Themeasurements from the data shown in Figs. 4(b) and 4(c) were used in the points labeled (b) and (c). Linear fitting is performed for thescans across each axis.

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8.22 V applied across the 53.5 nm radius pore. Thelocalization of the extreme superheating at the pore centeris evident. The maximum temperature there is 600 K, anddrops to 470 K at the pore wall. This maximum is about 5%greater than the experimentally measured limit of superheatof pure water [2]. The high concentration of NaCl insolution may contribute to this increase in a manner similarto boiling point elevation in electrolyte solutions [27]. Theregion of superheated liquid extends outward to 230 nmfrom the pore center. The thermally stored energy availablefor bubble formation and expansion in this superheatedliquid immediately before nucleation is approximately 5 pJ,assuming a spherical temperature distribution. The upperlimit for the radius of a spherical bubble corresponding tothis energy is 620 nm, accounting for latent heat ofvaporization and assuming no thermal losses due todiffusion. Similar results with regard to electrical conduct-ance, temperature profile, and temperature at nucleation areobtained for the 1.9 μm radius pore.The magnitude of the localized temperature maximum

calculated at the nanopore center dramatically increasesthe likelihood that a bubble will nucleate homogeneouslythere. This can be demonstrated using the nucleation rate,obtained from classical nucleation theory [4], with thesurface tension of the liquid vapor interface taken to bethat along the saturation curve [28]. In Fig. 4, we plot thisnucleation rate within the 53.5 nm radius pore as a functionof distance from the center. The rate is sharply peaked,supporting homogeneous single bubble nucleation at thepore center for this case.The behavior of the relaxation oscillator [Fig. 2(b)] can

be understood with a simple model of the heating andcooling dynamics. The initial 16 ns bubble event is taken tocutoff of the ionic current. During this event lifetime, the

maximum temperature in the pore drops by about 200 Kdue to thermal diffusion. The time needed to reheat the porecenter to 600 K once ionic current resumes is calculatedto be approximately 120 ns. This corresponds well to theexperimentally measured time to the second bubble of117 ns. The dynamics of bubble growth in an unbounded,uniform temperature, superheated liquid have been wellstudied [29–31]. Inertial effects govern early growth drivenby the high vapor pressure inside the bubble. We calculatean initial radial growth velocity of 126 m=s at 600 K, usingthe Rayleigh-Plesset theory [30], which applies to spheri-cally symmetric, free expansion in an unconfined liquid.The presence of the pore walls will decrease the calculatedgrowth speed. Nevertheless, the result corresponds reason-ably well to the measured velocity (50 m= sec) obtainedfrom the optical probing data.The inertial growth modeling does not include mass

transfer at the boundary or heat transport effects. At hightemperature, the surface tension is greatly diminished,decreasing its effect on early growth. The effects of heattransport are only manifest in later stages of bubble growth[29]. A comprehensive analysis of the bubble dynamics isas yet unavailable. The time evolution of heat transport andpressure in the bubble with a moving boundary are requiredto understand the dynamics of the bubble over its lifetime.The preceding experimental and theoretical discussion

provides strong evidence for extreme superheating, homo-geneous nucleation, and growth of vapor bubbles consistentwith the experimental conditions studied. Quantitativeunderstanding of all the details is a challenge, in partbecause of lack of certain knowledge of many physicalproperties of strong electrolytes in superheated metastablestates. The simple model we have used to explain the mostelementary aspects of the phenomena must also beextended to more accurately and completely describe theelectrical, optical, and fluidic phenomena reported here.This will require more comprehensive multiphysics mod-eling. Given the potential electrical, chemical, optical,fluidic, and acoustic phenomena that may be excited andobserved in the high field and extreme environment of asolid-state nanopore we believe that this new platform iscertainly worth further attention.

G. N. (experiments) and E. V. L. (theoretical modeling)each contributed equally to the research reported in thisLetter. We thank R. D. Aubut and A. T. Kuan for technicalassistance. This work was supported by the NationalInstitutes of Health Grant No. HG003703 to J. A. G. andD. Branton. E. V. L. received support from the NSF-GRFPand the NDSEG fellowship programs.

*golovchenko@physics.harvard.edu[1] J. Li, D. Stein, C. McMullan, D. Branton, M. J. Aziz, and

J. A. Golovchenko, Nature (London) 412, 166 (2001).[2] F. Caupin and E. Herbert, C. R. Phys. 7, 1000 (2006).

FIG. 4 (color). The nucleation rate as a function of distancefrom the center of the nanopore. This corresponds to thetemperature computed for the 53.5 nm radius nanopore of Fig. 2at 10.4 μs, just before bubble nucleation occurs. A contour plot ofthe temperature at this time is shown in the inset.

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[3] C. T. Avedisian, J. Phys. Chem. Ref. Data 14, 695 (1985).[4] M. Blander and J. L. Katz, AIChE J. 21, 833 (1975).[5] P. G. Debenedetti, Metastable Liquids: Concepts and Prin-

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[22] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.113.024506 for calcu-lation details.

[23] W. Wagner and A. Pruß, J. Phys. Chem. Ref. Data 31, 387(2002).

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Supplemental Material for “Superheating and Homogeneous Single Bubble Nucleation in a Solid-State

Nanopore”

Gaku Nagashima, Edlyn V. Levine, David P. Hoogerheide, Michael M. Burns, and Jene A. Golovchenko

This document provides supplemental information regarding the COMSOL calculation used to

determine the temperature of the electrolyte in the solid-state nanopore during the nanopore

heating experiments.

The calculation required solving the heat equation in the nanopore with inclusion of a Joule

heating source term, appropriate boundary conditions and temperature dependent material

properties of the electrolyte and membrane. The heat equation must be coupled with charge

continuity and Ohm’s law to determine the source term. This is done using the Joule Heating (jh)

and Electric Currents (ec) modules in COMSOL. The equations take the form · · 0 · ·

where is the temperature, is the electrical potential, is the electric field, and is the current

density. The membrane is assumed to be non-conducting and the applied voltage is specified on

the boundary far from the pore. The temperature condition imposed on the boundary was ,

where is the ambient temperature measured at the start of the experiment. The initial condition

was everywhere. The density, , heat capacity at constant pressure, , thermal

conductivity, , and electrical conductivity, , are all temperature dependent properties of the

electrolyte. Material data for superheated aqueous 3M sodium chloride solution at atmospheric

2

pressure is not available for the extremely high temperature regime that we reach experimentally.

Therefore, we used values for the density, heat capacity and thermal conductivity available from

the IAPWS-95 formulation for the equation of state of water [1-3].

Experimental data for the electrical conductivity at such high temperatures under atmospheric

pressure is not available either. We therefore fit the electrical conductivity to the experimentally

measured conductance curve using two free parameters. Our resulting conductivity curve is

compared to those reported by Bannard [4] in Figure 1. The high temperature behavior is

understood to arise from the temperature dependent behavior of the solvent, water, as described

in Quist and Marshal’s paper [5] on the subject.

Figure 1. The electrical conductivity of aqueous 3M NaCl solution as a function of temperature. The data points are those reported in Figure 7 of [4] and are shown here for reference. The conductivity curve we fit to our experimental results and used for temperature calculations is also shown.

[1] W. Wagner and A. Pruß, J. Phys. Chem. Ref. Data 31, 387 (2002).

3

[2] Revised Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of

Ordinary Water Substance for General and Scientific Use, (The International Association

for the Properties of Water and Steam, Doorwerth, The Netherlands, September, 2009,

http://www.iapws.org/relguide/IAPWS-95.html).

[3] Release on the IAPWS Formulation 2011 for the Thermal Conductivity of Ordinary

Water Substance (The International Association for the Properties of Water and Steam,

Plzen, Czech Republic, pp. 1–15, September, 2011,

http://www.iapws.org/relguide/ThCond.html).

[4] J. Bannard, J. Appl. Electrochem 5, 43 (1975).

[5] A. S. Quist and W. L. Marshall, J. Phys. Chem 72, 684 (1968).