# Boiling in Nanopores through Localized Joule Heating:
Transition between Nucleate and Film Boiling

Soumyadeep Paul  Department of Mechanical Engineering, The University of
Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan  Wei-Lun Hsu  Department
of Mechanical Engineering, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku,
Tokyo 113-8656, Japan  Yusuke Ito  Department of Mechanical Engineering, The
University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan  Hirofumi
Daiguji  Corresponding author<EMAIL_ADDRESS>Department of
Mechanical Engineering, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku,
Tokyo 113-8656, Japan

###### Abstract

The transition from nucleate to film boiling on micro/nano textured surfaces
is of crucial importance in a number of practical applications, where it needs
to be avoided to enable safe and efficient heat transfer. Previous studies
have focused on the transition process at the macroscale, where heat transfer
and bubble generation are activated on an array of micro/nanostructures. In
the present study, we narrow down our investigation scale to a single
nanopore, where, through localized Joule heating within the pore volume,
single-bubble nucleation and transition are examined at nanosecond resolution
using resistive pulse sensing and acoustic sensing. Akin to macroscale
boiling, where heterogeneous bubbles can nucleate and coalesce into a film, in
the case of nanopores also, patches of heterogeneous bubbles nucleating on the
cylindrical pore surface can form a torus-shaped vapor film blanketing the
entire pore surface. In contrast to conventional pool boiling, nanopore
boiling involves a reverse transition mechanism, where, with increased heat
generation, film boiling reverts to nucleate boiling. With increasing bias
voltage across the nanopore, the Joule heat production increases within the
pore, leading to destabilization and collapse of the torus-shaped vapor film.

Keywords

Nanopore, Ionic Joule heating, Bubble nucleation, Boiling transitions,
Resistive pulse sensing, Acoustic sensing, Torus bubble

††preprint: APS/123-QED

##  I INTRODUCTION

Despite a century of research on vapor bubble dynamics and its ramifications
for boiling phenomena, a unified understanding of bubble nucleation at the
nanoscale and growth/transition to macrobubbles has yet to be established.
Little attention has been paid to establishing fundamental connections between
boiling characteristics at the nanoscale and macroscale, which may differ
significantly owing to confinement effects [1, 2]. Closing this fundamental
gap in knowledge is of paramount importance for fully leveraging the benefits
of phase change heat transfer in high-heat-flux applications, including, among
others, electronic cooling [3], inkjet printing [4], and spray quenching [5].
Starting with the work of Nukiyama [6], who heated a liquid by running
electricity through a metal wire, early research in this field was devoted to
establishing a boiling curve consisting of five regions of pool boiling [7]:
natural convection, isolated nucleate boiling followed by slug nucleate
boiling, transition boiling terminating at the Leidenfrost point, and
ultimately film boiling. Research has been focused on fundamental
understanding of pool boiling transitions with the aim of achieving practical
goals in engineering [3, 8, 9], such as enhancement of the heat transfer
coefficient [10] and prevention of early critical heat flux [3]. The latter is
the limiting heat flux beyond which the heat transfer to the liquid starts to
decrease owing to lack of liquid contact with the heated surface because of
horizontal coalescence of nucleating bubbles.

From a fundamental viewpoint, to understand the initial stage of boiling, it
is imperative to investigate the dynamics of single vapor nanobubbles. This is
especially challenging for two reasons: (i) heating must be focused such that
single-bubble nucleation can be achieved; (ii) the nanoscale–nanosecond
dynamics of the nanobubble post nucleation must be captured experimentally.
Because of these inherent difficulties, at the nanoscale, engineering research
on boiling involving heat transfer enhancement [11] and physics research on
single-vapor bubble dynamics involving equilibrium and stability
characteristics [12] have mostly proceeded in parallel but separately. On the
other hand, at the macroscale, engineering studies of boiling and of single-
vapor-bubble dynamics have been performed simultaneously. In this regard, we
would highlight the work of Dhir and co-workers [13] in which microgravity was
used to confine a macrobubble on a heater surface, thereby allowing a
prolonged investigation of spherical thermal bubble dynamics.

At the outset, we formulate three critical questions that need to be addressed
to bridge the gap between engineering boiling research and physics research on
single-vapor bubble dynamics at the nanoscale: (i) How do bubble growth and
transition dynamics at the nanoscale differ from those at the macroscale? (ii)
What is the connection between the governing dynamics at the two scales? (iii)
How can vapor bubble seeds improve the engineering limits of boiling heat
transfer. Both from the perspective of thermodynamics [1] and that of fluid
mechanics [2], the governing dynamics at the 100 nm scale are expected be
different owing to confinement effects, leading to different bifurcation
mechanisms of the boiling structure [14, 15, 16]. Owing to the lack of
sufficient experimental studies and theoretical understanding of nanoscale
boiling, we first need to address the first question and clarify boiling at
the single-nanobubble limit, before the scale gap between nanoscale and
macroscale boiling can be bridged.

Figure 1:  (a) 3D schematic view and (b) cross-sectional schematic view of the
acoustic and resistive pulse sensing experimental setup. The insets in (a)
show a schematic of the suspended membrane nanopore chip (a cylindrical hole
of diameter $D_{\mathrm{p}}$ on a silicon nitride thin layer of thickness $L$)
and an SEM image of the $D_{\mathrm{p}}=460$ nm nanopore. (c) Nanopore current
(blue) and hydrophone pressure (red) traces during nanopore boiling during a
$1468\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$–7 V voltage pulse
starting from $t=$ $0\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$. Before
$46\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$, when there is only
superheating and no boiling, both the current and pressure baselines are
steady. This is followed by stochastic nucleate boiling, when two processes of
bubble generation are predominant: (i) homogeneous bubble nucleation at the
pore center and (ii) patch heterogeneous bubble nucleation on the pore
surface. Here, discrete homogeneous and heterogeneous bubbles grow and shrink,
nonperiodic current blockage signals are seen in the current trace, and high-
amplitude nonperiodic pressure waves are observed. When nucleate boiling
transitions to film boiling after
$\sim$$16\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$, a torus-shaped vapor
bubble blankets the nanopore wall surface (iii). This bubble then performs
pinned volumetric oscillations in thermal resonance with the Joule heat
generation inside the pore liquid. Owing to the stability of the torus bubble
of volume comparable to the pore volume $V_{\mathrm{p}}$, the baseline current
steadily decreases by $\sim 42\%$ from $I_{\mathrm{b}}$ to $I_{\mathrm{b,a}}$.
The large shift in current baseline ($\Delta I_{\mathrm{b}}$) for a pore with
a high $D_{\mathrm{p}}/L$ ratio ($=4.6$) can only be explained by the
existence of a torus bubble that has a volume $V_{\mathrm{b}}$ scaling
proportionally with the pore diameter. Owing to the high sensitivity of the
nanopore current to the insulating bubble, oscillations of the bubble volume
about its mean value are represented by near-sinusoidal current oscillations
about the reduced baseline $I_{\mathrm{b,a}}$. Consequently, the hydrophone
also picks up high-frequency and periodic pressure waves generated as a result
of bubble volume oscillations. (d) 3D spectrogram analysis of the nanopore
current and hydrophone pressure. During nucleate boiling with stochastic
waiting times, the power spectrum is distributed across a wide frequency band,
and hence no clear frequency ridges are seen in the spectrograms. After the
film boiling transition at
$\sim$$62\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$, when the torus
bubble oscillates at a characteristic frequency, two ridges are seen at $f=6$
MHz and $12$ MHz, signifying the first and second harmonics, respectively.

To activate single-bubble nucleation, we utilize Joule heating, which directly
converts electric energy to thermal energy in confined liquid volumes,
circumventing interfacial heat transfer. Nagashima _et al._ [17] filled a
concentrated electrolyte solution in a nanopore connected with two large
solution reservoirs. An electric potential bias was imposed across a membrane
of thickness 71 nm via electrodes inserted in each side of the reservoir. The
electric field was focused around the nano-aperture on the thin membrane,
generating intense Joule heating in the nanospace. Homogeneous thermal bubble
generation in the liquid phase was successfully detected via ionic current
measurements in the gigahertz bandwidth of resistive pulses, based on the
volume exclusion effect of nonconductive bubbles. This method of bubble
generation and detection overcomes both the scale and time limitations of
traditional boiling studies using heater surfaces and high-speed cameras,
thereby serving as an ideal platform for tracking single nanobubbles in the
initial stage of boiling. Using the same platform, Paul _et al._ [18]
demonstrated that solely homogeneous nucleation occurs only in tiny apertures
at the nanoscale. As the pore diameter expands, heterogeneous bubbles
originating from the inner walls of nanopores become dominant. This transition
was attributed both to geometric confinement effects such as contact line
pinning and to a thermal confinement effect, namely, a large temperature
gradient within the nanopore.

In the present paper, we measure both ionic current variations and stress
waves created by a nanopore bubble [Figs. 1(a) and 1(b)], focusing
specifically on the transition between nucleate and film boiling. By
intensifying the heating rate through increasing the bias voltages across the
nanopore, we activate the transition from nucleate boiling to heterogeneous
film boiling on the rim of the nanopore. Owing to the cylindrical shape of the
pore, the film bubble resembles a torus in shape, as shown in Fig. 1(c)
[Appendix A]. A similar observation was also observed in Medvedev _et al._
[19], albeit at macroscale. Nanobubbles on flat surfaces have already been
shown to exhibit remarkable stability owing to the contact line pinning effect
[20, 21], and here we demonstrate that nano-torus thermal bubbles also exhibit
stability owing to similar pining effects within nanopores. Unlike nucleate
bubbles, which undergo rapid growth and collapse cycles separated by
significant reheating periods, the film bubble is stable and relatively long-
lived, undergoing pinned volumetric oscillations, which are captured by our
sensors in the form of pore current and hydrophone pressure oscillations. When
the bubble is moderately stable, the oscillations are weakly nonlinear [22,
23] leading to frequency dispersion, and only the fundamental and second
harmonics are observed beyond the noise level of our sensors. The 3D
spectrogram analysis of the torus bubble presented in Fig. 1(d) reveals two
ridges at 6 MHz and 12 MHz which correspond to these two harmonics. These
frequencies are one order higher than those recorded for oscillatory boiling
on microheaters [24], probably due to the larger bubble size compared to the
nanopore bubble. Intriguingly, our voltammetric studies reveal that for larger
pore diameters, with increasing voltage, the torus bubble gradually loses its
stability, ultimately leading to transition from film to nucleate boiling.

In this paper, we showcase the similarities in bubble dynamics and
thermodynamics between macroscale pool boiling and nanopore boiling, while
also highlighting the differences in the thermofluidic mechanisms driving the
transitions. The remainder of the paper is organized as follows. In Sec. II, a
brief overview of our experimental system is presented. Section III.1 explains
the nucleate-to-film boiling transition using current and hydrophone pressure
spectrograms for a 420 nm pore, and Sec. III.2 elucidates the film-to nucleate
boiling transition in a 460 nm pore. Section IV.1 discusses the effect on
Joule heat generation during the different nanopore boiling regimes for the
two pore sizes. In macroscale film boiling, the vapor film limits heat
transfer from the heated surface, whereas in nanopore boiling, the torus vapor
film limits Joule heat production through the volume exclusion effect. In Sec.
IV.2, a theoretical model is developed to capture the equilibrium size and
temperature of the torus vapor film at different applied voltages. In Sec.
IV.3, the stability of the torus vapor film is analyzed by perturbation
theory, and the reverse transition phenomenon (film-to-nucleate boiling) is
explained based on this analysis. Conclusions are presented in Sec. V/

##  II EXPERIMENTAL METHODS

Circular nanopores as shown in the scanning electron microscope (SEM) image in
Fig. 1(a) and in Fig. S1 of the Supplemental Material [25] were made on
silicon nitride chips (Model No. 4088SN-BA) purchased from Alliance Biosystems
Inc., each comprising a 100-nm-thick silicon nitride (Si3N4) membrane
deposited on a 200-µm-thick silicon substrate with an approximately square
$$50\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro
m}$\times$50\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro m}$$ opening at the
center. The nanopores were etched at the center of the free-standing part of
the membrane using a focused $\text{Ga}^{+}$ ion beam (SMI2050MS2, SII
Nanotechnology). Post fabrication, the nanopore chip was assembled between two
fluid tanks and wetted with ethanol before flushing with a 3M aqueous solution
of NaCl prepared by diluting 5M NaCl solution (Sigma-Aldrich) with deionized
(DI) water. Voltage pulses were applied through Ag/AgCl electrodes using a
pulse generator (Tektronix AFG3152C), and the ionic current flowing through
the nanopore was registered on a oscilloscope (Tektronix MSO56) by measuring
the voltage across a shunt resistor through an active power rail probe at 20
MHz terminal bandwidth (Tektronix TPR4000). In addition, a passive probe set
at 250 MHz terminal bandwidth was used to measure high-frequency current
oscillations across a second shunt resistor in series. The active probe, which
had a high signal-to-noise ratio but also high capacitance, was used to track
the baseline and frequency shifts of the nanopore current, while the passive
probe measurements taken at high bandwidth were used to measure the amplitude
of current oscillations. A needle hydrophone (sensor diameter 4 mm, Precision
Acoustics NH4000) encapsulated in a hollow quartz cell was also placed
vertically above the chip surface to collect the stress waves from the
nanopore bubbles at 20 MHz terminal bandwidth. The stress waves collected by
the piezoelectric sensor were converted into electrical signals by a pre-
amplifier and DC coupler, which were registered in the oscilloscope
concurrently with the current signals. To shield the piezoelectric element of
the hydrophone from the ionic current, the hydrophone was encapsulated in a
custom-made glass shell filled with DI water. A 20-µm-thick silicone film
(Wacker Asahikasei Silicone) [Fig. 1(b)] separated the DI water from the salt
solution, allowing acoustic signals to pass. Hydrophobic tape (3M Microfluidic
Diagnostic Tape 9965) was used to seal the junction of the silicone film and
quartz cell to prevent any electrolyte leakage. As a result of hydrophone
encapsulation, the clearance distance between the piezoelectric sensor and the
nanopore ($s=s_{1}+s_{2}$) had two components, $s_{1}$ and $s_{2}$ [Fig.
1(b)]. The distance between sensor and silicone film, $s_{1}$, was measured by
an optical microscope, while $s_{2}$ was measured first using a contact-type
distance sensor and this measurement was validated based on the delay time
between the current dip and the hydrophone peak signal for homogeneous bubble
nucleation (see Sec. S2 in the Supplemental Material [25]). To measure the
low-intensity stress waves emanating from torus bubble oscillations at an
acceptable signal-to-noise ratio, the net clearance needed to be as small as
possible. For the 460 nm pore experiments detailed in Figs. 1 and 3, $s=500\pm
20$ µm. Owing to the large difference between sensor diameter and clearance
distance, the spatial averaging effect [26] distorted the actual stress wave
amplitude. Nonetheless, a qualitative analysis of pressure amplitudes for
varying voltages and varying bubble frequencies still provides insight into
the bubble dynamics.

The acoustic signals registered in the oscilloscope as voltage signals
$V_{\mathrm{p}}(t)$ were then converted into pressure waveforms using the
following equation [27]:

$p(t)=F^{-1}\left\\{\frac{F\\{V_{\mathrm{p}}(t)\\}}{M(f)}\right\\},$ (1)

where $F$ and the $F^{-1}$ operators denote the Fourier and inverse Fourier
transforms, respectively, and $M(f)$ is the frequency response of the
hydrophone sensitivity as per plane wave calibration measurements performed by
the manufacturer, Precision Acoustics. The uncertainty in $M(f)$ was in the
range of 19–22%. The imaginary part of the acoustic pressure $p(t)$ obtained
after inverse Fourier transform is neglected.

##  III Results

Single-bubble boiling is activated by localized Joule heating inside a
submicrometer pore on a 100-nm-thick suspended silicon nitride membrane [Fig.
1(a)]. The nanopore is submerged inside a 3M NaCl electrolyte solution, which
leads to ionic current flow once bias voltages are applied across it through
Ag/AgCl electrodes. When homo- or heterogeneous bubbles are nucleated beyond
their respective superheating limits, the ionic current is altered, and these
changes are measured by a high-bandwidth oscilloscope. In addition, an
encapsulated piezoelectric hydrophone on top of the nanopore chip [Fig. 1(b)]
absorbs the stress waves generated by bubble motion and converts them into an
electrical signal to be recorded in the oscilloscope simultaneously. By
studying the amplitude and frequency variations of current and hydrophone
pressure, we can track the boiling transition within the pore. The current
signals are most sensitive to bubble dynamics within the pore, where minor
changes in bubble volume modulate the pore cross-section significantly.
Meanwhile, the hydrophone sensor catches any stress waves in the liquid
generated by bubble expansion or shrinkage, irrespective of the bubble
position relative to the pore.

###  III.1 Nucleate-to-film boiling transition

Figure 2 shows the boiling structure of a $D_{\mathrm{p}}=420$ nm pore. The
current–time traces in (a) and (c) show baseline current changes and
spectrograms (short time averaged Fast Fourier Transform) in (b) and (d) show
frequency changes, both providing phenomenological evidence of nucleate to
film boiling transition.

Figure 2:  Boiling transition with increasing bias voltage for a
$D_{\mathrm{p}}=420$ nm pore. (a) and (b) Nucleate homogeneous boiling at 6.3
V. (c) and (d) Transition to stable film boiling at 6.4 V. (a) and (c) are
plots of the transient shift in baseline current during the voltage pulse,
while (b) and (d) are spectrograms of the current corresponding to the bubble
oscillation frequency.

When a voltage pulse of 6–8 V is triggered, ionic current flowing through the
nanopore liberates Joule heat [28, 29], which transiently increases the
nanopore temperature at a rate of $\sim\\!10^{7}$ K/s from ambient conditions
of 298 K [30]. As the temperature rises, the ionic conductivity increases,
allowing more current flow and Joule heat generation in a feedback loop.
Unlike pool boiling, where the substrate temperature can be controlled
directly, in this case modulating the bias voltage only allows us to control
the heat generation rate, which can reach $\sim\\!10^{16}$
$\text{\text{~{}}}\mathrm{W}\mathrm{/}\mathrm{m}^{3}$ [30, 31]. Owing to the
rate of high heating within the confined space, a temperature gradient of the
order of 1 K/nm develops within the pore, which allows bubbles of different
volumes and temperature to exist in thermal equilibrium within the pore
liquid.

The current response to each pulse comprises an initial heating zone (A)
followed by a nucleate boiling zone (B), ultimately leading to torus film
boiling (C). At 6.3 V, except for an outlier (see Fig. S17 in the Supplemental
Material [25]), only zones A and B are present during boiling [Figs. 2(a) and
2(b)]. However, with a 6.4 V voltage pulse, all three zones can be clearly
identified, as shown in Fig. 2(c) and 2(d). This scheme is seen for multiple
voltage pulses of the same magnitude. Thus, there is an increasing probability
of film boiling transition as the voltage is increased from 6.3 V to 6.4 V.

In zone A, the Joule heat generation within the pore is balanced by heat
dissipation by the silicon nitride membrane and surrounding electrolyte. No
bubble-induced current blockage signals are seen, and the spectrograms also
show no effect [Figs. 2(a) and 2(b)]. Zone A terminates with nucleation of a
homogeneous bubble at the pore center when the local temperature reaches
$T_{\mathrm{c}}\sim 590\pm 10$ K. This temperature estimate was obtained
through numerical simulations, by fitting the experimental nanopore current
[the red trace in Fig. 2(a)] as described in Appendix B. Before this bubble
nucleation, although the pore surface temperature matches the patch
heterogeneous nucleation temperature of $472\pm 35$ K [32], these bubbles are
suppressed. Our previous paper [18] on this topic showed that during Joule
heating, unstable vapor clusters can form homogeneously at the pore center
with temperature $T_{\mathrm{c}}$ and heterogeneously on the pore surface with
temperature $T_{\mathrm{w}}$. Depending on the value of the cross-pore
temperature difference $\Delta T_{\mathrm{p}}=T_{\mathrm{c}}-T_{\mathrm{w}}$,
a cluster ripening competition is established between these two cluster groups
at the two nucleation sites. When $\Delta T_{\mathrm{p}}$ is higher,
homogeneous cluster growth requires less free energy to grow, and hence the
heterogeneous clusters are suppressed.

Figure 2(a) shows only the nucleate boiling regime (B) involving quasiperiodic
homogeneous bubble formation and ejection at 6.3 V for a 420 nm pore. It
should be noted that unlike patch bubbles, which grow and collapse in a pinned
state [20, 18, 33] on the pore surface, the homogeneous bubbles are ejected
from the pore by the electric field force acting on its negatively charged
surface (the typical surface charge is
$-23\text{\text{~{}}}\mathrm{m}\mathrm{C}\mathrm{/}\mathrm{m}^{2}$ [34]). The
homogeneous bubble is retained near the pore access region, where it rebounds
spherically and volumetrically, emitting high-amplitude stress waves.
Additionally, owing to the limited influence of bubble volume fluctuations on
the ion current in the pore access region, low-amplitude current waviness is
seen in the reheating current trace [Fig. 1(c)] after the first growth cycle
post nucleation [23]. This phenomenon is illustrated in Fig. S2 of the
Supplemental Material [25]. This is markedly different from pool boiling,
where the departure of the bubble from the nucleation site is mainly buoyancy-
driven.

Figure 3:  Boiling transition with increasing bias voltage for a
$D_{\mathrm{p}}=460$ nm pore. (a) Transient current, (b) current spectrogram,
and (c) hydrophone pressure spectrogram during stable oscillatory film boiling
at 6.6 V. (d) Transient current, (e) current spectrogram, and (f) hydrophone
pressure spectrogram during transition boiling at 7 V. (g) Transient current
of the whole voltage pulse and (h) a zoomed view during nucleate boiling at 8
V. (i) Current spectrogram at 8 V. Owing to stochastic nucleation, no clear
band is seen, but high oscillation power can be seen in the 0–5 MHz range.

The transition from nucleate to film boiling is seen in Fig. 2(c). Following
zone B, zone C-i starts, during which the baseline current decreases but does
not stabilize. In addition, the spectrograms in Fig. 2(d) show unstable
frequency bands in this zone. Compared with zone B, the frequency bands are
discrete but unsteady, indicating that a single unstable torus bubble is
oscillating within the pore. In this regime, the torus bubble is not in
thermal equilibrium with the Joule heating, which leads to rapid variations in
the mean bubble size, in addition to highly nonlinear volumetric pinned
oscillations about the mean size. Eventually, the torus bubble reaches the
equilibrium size corresponding to the applied voltage, and the current
oscillations stabilize about a stable mean value. This marks the beginning of
the stable oscillatory torus boiling regime (C-ii). In this regime, the
baseline current stabilizes, and the current and pressure spectrograms show
steady and narrow frequency bands. This regime continues until the end of the
voltage pulse. For the 420 nm pore [Fig. 2(d)], there is a stable oscillation
frequency of 8.9 MHz, whereas for the 460 nm pore [Fig. 3(b)], there is a
pseudostable oscillation frequency of 4.3 MHz. Even in this regime, low-power
frequency bands at secondary harmonics are also seen, indicating that the
oscillations are weakly nonlinear [22]. For the 420 nm pore, we also performed
a boiling structure analysis at 6.5 V and 6.75 V. We found that the stable
oscillation zone (C-ii) essentially disappeared and was replaced by low-
frequency ($\sim$5 MHz) nonlinear oscillations indicative of an unstable torus
bubble (C-i). This gradual loss of torus stability with voltage is observed
more prominently for the 460 nm pore and is discussed in Sec. III.2.

###  III.2 Film-to-nucleate boiling transition

As the torus bubble forms on the pore circumference [Fig. 1(c)], its volume
scales linearly with the pore diameter, $V_{\mathrm{b}}\propto
D_{\mathrm{p}}$, while the cylindrical pore volume where the Joule heat is
generated scales quadratically, $V_{\mathrm{p}}\propto D_{\mathrm{p}}^{2}$.
Thus, for torus bubbles forming on the same pore length $L$ [Fig. 1(c)], with
increasing $D_{\mathrm{p}}$, more Joule heat will be liberated within the pore
volume than can be blocked by the bubble through the volume exclusion effect.
Hence, according to our hypothesis, the torus bubble will be forced to bulge
outward, triggering instability and eventually a reverse transition. Through
our experiments, we have found this transition to manifest in the pore
diameter range of 400–500 nm. We therefore chose the 460 nm pore results to
showcase the boiling structure during this process.

First, compared with the 420 nm pore at 6.3 V, which has a
$700\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$ duration of homogeneous
nucleate boiling, the 460 nm pore at 6.6 V has a much shorter nucleate boiling
zone of less than $100\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$ [zone B
in Fig. 3(a)]. Additionally, for the 460 nm pore, a much higher rate of patch
heterogeneous nucleation is observed, owing to the lower cross-pore
temperature difference $\Delta T_{\mathrm{p}}$ corresponding to a given wall
temperature $T_{\mathrm{w}}$. With increasing pore diameter, the specific
Joule heat density within the pore volume decreases as the pore volume offers
less electrical resistance than the access region [35]. This decreases the
temperature gradient from the pore center to the pore walls [18], resulting in
a lower $\Delta T_{\mathrm{p}}$.

Owing to uncertainty in the heterogeneous nucleation temperature [32], the
waiting times between bubble nucleations are stochastic, leading to
nonperiodic bubble signals in the nucleate boiling regime, as shown in Fig.
1(c). Consequently, in zone B, a wide or nonspecific frequency band (0–5 MHz)
is noticeable in the current and pressure spectrograms [Fig. 3(b)]. Each
blockage signal in this zone indicates a separate bubble event comprising
bubble nucleation, inertial and evaporation-induced growth to several
micrometers, and eventual collapse back to the liquid phase owing to the lack
of sufficient sensible heat supply. The baseline current is recovered after
each bubble collapse. Owing to the abundant heterogeneous nucleation inside
the 460 nm pore, the probability of eventual coalescence of patch nuclei
toward formation of a film-like torus bubble on the pore surface is increased
compared with the 420 nm pore. As a result, zone B is short-lived, and is
followed by zone C-i at 6.6 V.

The film-to-nucleate boiling transition is activated at higher bias voltages
and is best captured at 7 V. As the voltage is increased from 6.6 V to 7 V, a
steady and pseudostable oscillating torus bubble is never seen, as is evident
from the continuously varying baseline current [Fig. 3(d)]. Also, the
spectrograms reveal intermittent narrow frequency bands (C-ii) separated by
unstable torus oscillations (C-i) and nucleate boiling (B) [Fig. 3(e)]. This
signifies that the torus bubble is only temporarily stable, and boiling
switches chaotically [16] between nucleate boiling and film boiling multiple
times. This effect is similar to the intermittent film boiling seen during
transition pool boiling before the Leidenfrost point. From the first work by
Nukiyama onward, much attention has been paid to intermittent film boiling,
but an overall model has yet to be established [11]. Actually, for many years,
the sudden dip in the boiling curve from the critical heat flux (CHF) until
the Leidenfrost point was generally represented by a dashed and broken line
rather than a well-characterized continuous curve [7]. In this paper, we show
that by precisely controlling the bias voltage and capturing fast current
transitions with a high-bandwidth oscilloscope, this region can now be
characterized in minute detail.

Figure 4:  (a) Schematic showing the torus bubble within nanopore for varying
pore diameters [340 nm, 420 nm, 460 nm]. (b) Current baseline dip ratio
($\Delta I_{\text{b}}/I_{\text{b}}$) and (c) first harmonic frequency of
stable or pseudo-stable torus bubble oscillations as seen in experiments for
the three pore diameters. (d) Volume exclusion ratio
$f_{\text{v}}=V_{\text{b}}/V_{\text{p,c}}$ variation with contact angle for
the three pore diameters. $V_{\text{b}}$ is the volume of the torus bubble and
$V_{\text{p,c}}$ is the cylindrical pore volume as shown in Fig. 6a.

A further increase in the bias voltage to 8 V reveals the strange phenomenon
of nanopore boiling reverting to completely nucleate boiling. Figures 3(g) and
3(h) reveal a recovery in the baseline current between discrete blockage
signals, and the spectrogram in Fig. 3(i) reveals wideband oscillation power
at 0–5 MHz range akin to the nucleate boiling regime (A) in Figs. 3(b) and
3(e). The reverse transition (film-to-nucleate) with increasing bias voltage
is because of a higher Joule heat generation rate, which destabilizes the
torus bubble, causing it to collapse. It should be noted that thermal/vapor
bubbles are always in unstable equilibrium [12], and oscillations are stable
only for small-amplitude driving forces [36].

As the total Joule heat generation within the pore also scales with the pore
diameter, it can be expected that at a given voltage, the torus bubble will be
more stable for smaller pore diameters. This can be observed by comparing the
spectrograms for the 460 nm pore at 7 V [Fig. 3(e)] with those for the 340 nm
pore at 7 V [Fig. S14(c) [25]]. While the 460 nm pore exhibits intermittent
boiling at this voltage, the 340 nm pore exhibits stable film boiling at
$\sim$15 MHz for more than 2 ms. Also, we can compare Fig. S11(b) [25], which
shows the spectrogram for an expanded 250 nm pore under 7 V. Steady frequency
bands in the 10 MHz range are observed. Because the experiments were repeated
multiple times, the pore expanded from its initial size of 250 nm to the
300–400 nm range owing to erosion caused by nanobubble vibrations.

In the Supplemental Material [25], we examine the long-term stability of the
torus bubble inside the 460 nm pore at 6.3 V and 6.5 V (see Figs. S15 and S16,
respectively). We find that the 6.3 V torus bubble underwent highly nonlinear
oscillations, with many subharmonic and superharmonic frequency bands. In the
case of 6.5 V, the torus bubble oscillated weakly nonlinearly for
$\sim$$400\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$, before returning to
nonspecific frequency bands indicative of its instability. Although we do not
exactly understand the nonlinear behavior of these bubbles, we can still
safely infer that for the 460 nm pore, the torus bubble is either pseudostable
or unstable. Comparing the baseline current dip ratio ($\Delta
I_{\text{b}}/I_{\text{b}}$) during stable film boiling (Fig. 4b), we find that
the 340 nm pore has a larger ratio than the 420 nm pore, while the 460 nm
pores have nearly the same ratio as the 420 nm pore. This can be explained by
comparing the volume exclusion ratios, $f_{\text{v}}$ of the torus bubble
inside the three pore diameters. As the dip in baseline current is
proportional to the volume exclusion effect posed by the torus bubble, we can
expect that $f_{\text{v}}\propto\Delta I_{\text{b}}/I_{\text{b}}$. As shown in
Fig. 4d, $f_{\text{v}}(340\text{nm})>f_{\text{v}}(420\text{nm})\approx
f_{\text{v}}(460\text{nm})$. This is responsible for the current dip ratio
trend shown in Fig. 4b. Also, the fundamental frequency of the bubble is
observed to increase with decreasing pore diameter (Fig. 4c). As the
oscillating bubble takes a torus shape, its volume ($V_{\mathrm{b}}$) will be
directly proportional to the pore diameter which will cause the frequency, $f$
to vary inversely with diameter as frequency scales inversely to volume
according to [37]. In Sec. IV.2, we discuss the underlying mechanism of torus
bubble instability under a confined heating and pinning effect using a
theoretical model. The effects of pore diameter and bias voltage on stability
are also elucidated.

##  IV Discussion

###  IV.1 Heat generation during nanopore boiling

Table 1: Variation of Joule heat generation during boiling regimes. |  | Zone A | Zone B | Zone C-i | Zone C-ii
---|---|---|---|---|---
$D_{\mathrm{p}}$ (nm) |  $V_{\mathrm{app}}$ (V) |  $t_{\mathrm{z}}$ ($\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$) |  $H$ ($\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro W}$) |  $t_{\mathrm{z}}$ ($\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$) |  $H$ ($\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro W}$) |  $t_{\mathrm{z}}$ ($\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$) |  $H$ ($\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro W}$) |  $t_{\mathrm{z}}$ ($\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$) |  $H$ ($\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro W}$)
420 | 6.3 | $261$ | $758$ | $239$ | $710$ | $0$ | $0$ | $0$ | $0$
| 6.4 | $205$ | $772$ | $116$ | $669$ | $65$ | $484$ | $114$ | $517$
460 | 6.6 | $263$ | $1187$ | $50$ | $900$ | $627$ | $725$ | $525$ | $777$
| 7.0 | $44$ | $1389$ | $614$ | $984$ | $510$ | $822$ | $297$ | $789$

In the previous section, nucleate boiling, transition boiling and film boiling
were categorized and analyzed minutely within each voltage pulse by
deciphering changes in baseline current and oscillation frequency. As the
bubble volume, pore volume, and heating volume are all comparable, the
presence of a bubble inside the pore severely restricts Joule heat generation.
In this section, we report the effect of the nanopore boiling regime on the
heat generation rate. This effect is similar to pool boiling, where the
transition to film boiling reduces the heat flux at the solid surface by
cutting off liquid contact from the solid wall. However, the critical
advantage of studying nanopore boiling is that it allows us to focus on a
single nucleation site, which is not possible in the case of pool boiling,
where heat transfer properties are spatially averaged over multiple nucleation
spots. Another advantage of nanopore boiling is that it allows us to
investigate the transient effect of boiling transitions, thus highlighting the
nonequilibrium states lying between nucleate boiling and stable film boiling.
This property is specifically useful in understanding intermittent film
boiling during transition boiling, an area of profound scientific interest
since the work of Nukiyama.

Table 1 shows the average Joule heating rate $H=V_{\mathrm{app}}I$ in
different boiling zones for a single voltage pulse of duration
$t_{\mathrm{p}}$. $t_{\mathrm{z}}$ denotes the total duration of each zone
during the pulse. For the 420 nm and 460 nm pores, $t_{\mathrm{p}}$ is taken
as $500\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$ and
$1468\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$, respectively. The 420 nm
and 460 nm pore results were captured using the passive and active probes,
respectively.

Some clear trends can be seen in Table 1. For example, the duration of the
initial heating zone (A) becomes shorter with higher voltage, signifying a
more rapid temperature rise to nucleation conditions. Compared with the regime
in zone A, $H$ decreases during the nucleate boiling (B) and film boiling (C)
regimes, with the latter exhibiting a greater decrease. For the 420 nm pore,
during nucleate boiling (B), $H$ is higher at 6.3 V than at 6.4 V. This is
because the periodicity of bubble nucleation increases, i.e., the waiting
times decrease, with increasing bias voltage, leading to more ionic current
blockages. In the case of the 460 nm pore, we find that the duration of the
nucleate boiling zone (B) is significantly longer at 7 V than at 6.6 V,
because nucleate boiling regimes (B) appear repeatedly, suggesting the start
of transition boiling. For the same reason, the duration of stable film
boiling (C-ii) becomes shorter with increasing voltage. $H$ increases by 17%
in zone A, but by only 1.5% in zone C-ii. This is due to the volume exclusion
effect imposed by the torus bubble.

###  IV.2 Nano-torus bubble at quasi-equilibrium

Figure 5: Schematic explanation of the heat transfer mechanism through the
torus bubble sandwiched between the Joule heated liquid and the silicon
nitride membrane which acts like a heat sink. The ion flux $\mathbf{J}$
through the unrestricted pore volume generates Joule heat $H$, part of which
is transferred through the torus bubble to the silicon nitride walls.
Continuum heat transport takes place in the liquid and solid while ballistic
heat transport is takes place through the bubble where the mean free path of
vapor molecules is in the same order as the bubble height, $h$. Figure 6:  (a)
Steady-state Joule heat density and (b) contour plot of the steady-state
temperature distribution for a 420 nm pore under 6.3 V with a $\theta=69$°
torus bubble sitting on top of $L=100$ nm thick silicon nitride membrane
walls. The temperature distribution along the bubble midline $r_{\mathrm{v}}$
is shown on the left of (b). (c) Variation of bubble temperature with contact
angle under thermal steady-state conditions (lines with markers). The solid
blue line shows the variation of bubble temperature under mechanical (Laplace)
equilibrium conditions. Solution of the thermal and mechanical equilibrium
conditions give the overall equilibrium bubble contact angle
$\theta_{\mathrm{e}}$. (d) Variation of the net Joule heat generation
$H_{\mathrm{n}}$, the Joule heat generation $H_{\mathrm{c}}$ inside
$V_{\mathrm{p,c}}$, and the heat transfer coefficients of the membrane
$h_{\mathrm{tm}}$ when a torus bubble with equilibrium contact angle
$\theta_{\mathrm{e}}$ occupies the pore volume.

In Sec. III.1, we showed that in zone C-ii, steady-state current oscillations
are seen with respect to a stable yet reduced baseline. The reduction in
baseline current was attributed to the formation of a pseudostable torus
bubble with oscillation amplitude proportional to the volumetric oscillation
amplitude of the bubble. In this subsection, we model the mean torus bubble
size inside the 420 nm pore by matching the experimental steady-state current
in zone C-ii with steady-state Joule heating simulations. As the bubble exists
in a pinned state, decreasing the contact angle causes an increase in bubble
volume.

When a torus bubble is introduced om the cylindrical silicon nitride pore
surface as shown in Fig. 5 and Fig. 6a, the Joule heat density in the liquid
increases, while the total Joule heat production decreases, because of the
volume exclusion effect. The torus bubble shape is calculated [Appendix A]
assuming that it has a constant mean surface curvature (CMC) and it remains
pinned on the two pore edges as shown in Fig. 5. The vapor molecules inside
the torus bubble transports heat collides between the liquid/vapor and
solid/vapor interfaces (axisymmetric view in Fig. 5), transporting heat
ballistically with no or minor inter-molecular collisions. The ballistic heat
flux for a given temperature drop is much more powerful and faster than
continuum heat transport in the liquid [Eqs. (12) in Appendix B]. Hence, the
torus bubble offers no thermal resistance to heat transfer between the
liquid/vapor and solid/vapor interfaces. We solve the energy conservation
equations [Eqs. (7) in Appendix B] in the electrolyte and the silicon nitride
membrane, taking into account the Joule heat source term
$H=\mathbf{J}\cdot\mathbf{E}\approx\sigma|\mathbf{E}|^{2}$, where $\sigma$,
$\mathbf{J}$ and $\mathbf{E}$ are the electrical conductivity, ion flux
density and electric field, respectively. When the heat generation inside the
pore and the heat diffusion through the membrane are balanced [Eqs. (11) and
Fig. S7(b) [25]], the steady-state nanopore and bubble temperatures are
obtained [Fig. 6(b)]. In this configuration, the bubble has a vapor
temperature $T_{\mathrm{v}}=$450.2 K. The net Joule heat production,
$H_{\mathrm{n}}$ is the sum total of the heat generation in the cylindrical
pore region around the bubble [$H_{\mathrm{c}}$ inside $V_{\mathrm{p,c}}$,
shown in Fig. 6(a)] and the heat generation in the access region away from the
bubble ($H_{\mathrm{a}}$), i.e.,
$H_{\mathrm{n}}=H_{\mathrm{c}}+H_{\mathrm{a}}$. For
$\theta_{\mathrm{e}}=69\degree$, as shown in Fig. 6(b), $H_{\mathrm{n}}=499$
$\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro W}$, $H_{\mathrm{c}}=214$
$\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro W}$, and $H_{\mathrm{a}}=283$
$\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro W}$. $H_{\mathrm{c}}$ is balanced
by the diffusive heat flux through the liquid surrounding the pore
($H_{\mathrm{ta}}=52$ $\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro W}$) and
the diffusive heat flux through the silicon nitride membrane
($H_{\mathrm{tm}}=162$ $\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro W}$). It
should be noted that the majority of the Joule heat
($\upsilon=H_{\mathrm{tm}}/H_{\mathrm{c}}=0.76$ or 76%) produced within the
cylindrical pore region is consumed by the bubble and transferred to the
silicon nitride surface on top of which it sits. Also,
$H_{\mathrm{tm}}=h_{\mathrm{tm}}2\pi R_{\mathrm{p}}L(T_{\mathrm{v}}-T_{0})$,
which can be solved to obtain the heat transfer coefficient of the membrane at
$V_{\mathrm{app}}=6.3$ V, namely, $h_{\mathrm{tm}}=8.06$ MW m-2 K-1 [Fig.
6(d)].

Figure 6(c) shows the variation of torus bubble temperature with contact angle
according to mechanical equilibrium (blue line) and thermal steady-state
conditions (lines with markers) for different voltages. The vapor temperature
for mechanical equilibrium has been obtained according to Eq. (13) in Appendix
C, assuming saturated vapor pressure inside the bubble, which is balanced by
the Laplace pressure. In the thermal steady state, there is no heat
accumulation in the bubble, and also there exists no temperature drop across
the interface to cause a net evaporation flux [Eqs. (11)]. We can see that for
a given voltage, there is only one solution for the contact angle at which
both mechanical equilibrium and steady-state conditions are simultaneously
satisfied, namely, $\theta_{\mathrm{e}}=69\degree$ when $V_{\mathrm{app}}=6.3$
V. This steady-state size and temperature of the bubble can also be termed a
quasi-equilibrium state [38]. It should be noted here that the growth–collapse
cycle of nucleate bubble (either homogeneous or heterogeneous) is a highly
transient and out-of equilibrium process [30], and the torus film bubble can
reach a quasi-equilibrium state that is experimentally observed to be stable
for longer than $100\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$. This is
due to the high volume of the torus bubble, which restricts Joule heat
production within the pore and arrests the transient liquid temperature rise,
enabling a thermal steady state to exist. It is interesting to note that at
$\theta_{\mathrm{e}}=69\degree$, the nanopore current from the simulations
[Fig. S6(a) [25]] and the mean oscillation current observed experimentally
[$I_{\mathrm{b,a,e,Q}}$ in Fig. S17(c) [25]] are in agreement with each other.
This justifies the attribution of the baseline current dip to the amount of
volume occlusion in the pore volume provided by the torus bubble.

It should also be noted that with increasing voltage, the equilibrium
temperature of the bubble $T_{\mathrm{v,e}}$ remains almost constant, while
the pinned bubble grows (i.e., $\theta_{\mathrm{e}}$ decreases) significantly.
Because of the negative curvature of the pore surface and the pinning effect,
the curvature of the torus bubble is less sensitive to decreasing contact
angle [pinned expansion, see Fig. S3(c) [25]], as a result of which the vapor
temperature as given by the Laplace equation remains quasi-uniform. Also, the
heat transfer capacity of the silicon nitride membrane is limited, which
forces the bubble to grow to the limit $H_{\mathrm{c}}$ such that a thermal
steady state can be established. This can be explicitly observed in Fig. 6(d),
where for $\theta=\theta_{\mathrm{e}}$, the net Joule heat generation
$H_{\mathrm{n}}$ decreases while $H_{\mathrm{c}}$ increases slightly with
increasing bias voltage. Qualitatively, this finding supports our experimental
results presented in Table 1, where increasing the voltage from 6.6 V to 7 V
caused $H$ to rise by only 1.5% in zone C-ii. This analysis shows that,
similar to vapor films in pool boiling, which impede the solid-to-liquid heat
flux, nanopore torus bubbles also exert a limiting effect, but on Joule heat
generation in the bulk liquid, through volume exclusion. As Joule heating is
restricted by the torus bubble, the pore center temperature is unable to rise
to 600 K, thereby preventing homogeneous nucleation.

###  IV.3 Stability of Nano-torus bubble

Figure 7:  Variation of torus bubble stability parameters with oscillation
frequency, $f$ under different bias voltages $V_{\mathrm{app}}$: (a) curvature
coefficient $\eta_{2}$; (b) volume expansion coefficient $\psi$; (c) phase
difference $\delta$ between bubble temperature and volume; (d) mechanical
stiffness $\kappa$.

Using nanopore Joule heating, we reduced the boiling scale to the 100 nm
range, thus entering the domain of single-bubble dynamics. At this scale,
buoyancy become insignificant, while surface tension dominates, as
characterized by the very low Bond number
$Bo=\rho_{\mathrm{w}}gR^{2}/\gamma=\mathcal{O}(10^{-9})$. Also, hydrodynamic
instabilities can be ruled out at this scale, since the critical wavelengths
for surface waves according to Rayleigh–Taylor ($\lambda_{\mathrm{c}}=14$ cm)
and Kelvin–Helmholtz ($\lambda_{\mathrm{c}}=2.3$ mm) instabilities [39]are
four orders of magnitude larger than the dimensions of the bubble. For
Kelvin–Helmholtz instability, we have assumed zero vapor velocity
($u_{\mathrm{v}}=0$) because of the pinning effect, while we have assumed the
velocity of the liquid surrounding the bubble to be equal to the typical
electro-osmotic flow velocity inside the nanopore, $u_{\mathrm{l}}=5$ m/s
[40].

We judge the stability of thermal bubble equilibrium from first principles, by
perturbing the bubble slightly and checking whether there is a positive
restoring force [36, 41]. Hence, when a thermal bubble initially in quasi-
equilibrium is radially perturbed ($\Delta R$), the restoring force $F$ can be
written as

$F=S_{\mathrm{b}}\Delta(P_{\mathrm{v}}-P_{\mathrm{w}}-K\gamma)=-\kappa\Delta
R.$ (2)

Here, $K$ and $S_{\mathrm{b}}$ are the curvature and liquid/vapor interfacial
surface area of the bubble, respectively (Fig. S3 [25]). $\kappa$ is the
mechanical stiffness, which must satisfy $\kappa>0$ for stable equilibrium.
$P_{\mathrm{v}}$ and $\gamma$ are the vapor pressure and surface tension,
respectively, of water at saturation temperature. $P_{\mathrm{w}}$ is the
atmospheric pressure. It should be noted that submicrometer-scale thermal
spherical bubbles are regarded to be in unstable equilibrium [12]. For a
spherical vapor bubble with radius $R_{\mathrm{e}}$ and temperature
$T_{\mathrm{v}}$ in mechanical and thermal equilibrium in an infinite
reservoir, the linear oscillation model, Eq. (2), can be written as [41]

$\rho_{\mathrm{w}}R_{\mathrm{e}}{\omega}^{2}=\frac{h_{\mathrm{fg}}\rho_{\mathrm{v}}}{\rho_{\mathrm{w}}c_{\mathrm{w}}T_{\mathrm{v}}}\sqrt{\frac{\omega}{2D_{\mathrm{th}}}}-\frac{2\gamma}{{R_{\mathrm{e}}}^{2}},$
(3)

with $\kappa=\mathcal{M}{\omega}^{2}$, where
$\mathcal{M}=4\pi\rho_{\mathrm{w}}R_{\mathrm{e}}^{3}$ is the reduced mass of
the spherical bubble [42]. There exists a real solution for $\omega$ only when
$R_{\mathrm{e}}\geq$14\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro m}$$. This
is due to the inverse relation between curvature and bubble radius, $K=2/R$,
which causes the surface tension force to dominate for smaller bubbles,
leading to $\kappa<0$ and preventing stable equilibrium. This analysis,
however, begs the question: how does a $\mathcal{O}(100~{}\mathrm{nm})$ torus
bubble attain stability within the nanopore, making stable film boiling
possible in the first place.

Next, we judge the thermal stability of the torus bubble equilibrium. Assuming
that thermal equilibrium is maintained at the liquid–vapor and solid–vapor
interfaces, ($T_{\mathrm{v}}=T_{\mathrm{w}}=T_{\mathrm{m}}$) and that the
vapor temperature varies along the saturation line ($\Delta
P_{\mathrm{v}}/\Delta
T_{\mathrm{v}}=h_{\mathrm{fg}}\rho_{\mathrm{v}}/T_{\mathrm{v}}$) [36], Eq. (2)
can also be written as

$\displaystyle\kappa$
$\displaystyle=-\frac{S_{\mathrm{b}}^{2}}{\psi}\\!\left(\frac{h_{\mathrm{fg}}\rho_{\mathrm{v}}}{T_{\mathrm{v}}}\cos\delta-\psi\gamma\frac{dK}{dV_{\mathrm{b}}}\right)$
$\displaystyle={S_{\mathrm{b}}^{2}}\\!\left(\eta_{2}-\frac{h_{\mathrm{fg}}\rho_{\mathrm{v}}}{T_{\mathrm{v}}}\frac{\cos\delta}{\psi}\right)=\mathcal{M(\theta_{\text{e}})}{\omega}^{2}.$
(4)

$\mathcal{M(\theta_{\text{e}})}$ is the reduced mass of the torus bubble
having quasi-equilibrium contact angle, $\theta_{\text{e}}$. A detailed
derivation is provided in Appendix D [Eqs. (14)–(23)]. $h_{\mathrm{fg}}$ and
$\rho_{\mathrm{v}}$ are the latent heat and vapor density, respectively,
assuming saturation conditions. For linear oscillations of the bubble volume
$V_{\mathrm{b}}$ and temperature $T_{\mathrm{v}}$ given by
$V_{\mathrm{b}}=V_{\mathrm{b,e}}+V_{\mathrm{b,a}}\exp(j\omega t)$ and
$T_{\mathrm{v}}=T_{\mathrm{v,e}}+T_{\mathrm{v,a}}\exp[j(\omega t+\delta)]$,
with $j=\sqrt{-1}$, $\psi=V_{\mathrm{b,a}}/T_{\mathrm{v,a}}$ denotes the
volume expansion coefficient (i.e., the ratio of volume and temperature
amplitudes) and $\delta$ ($-90\degree\leq\delta\leq 90\degree$) denotes the
phase difference between them. $\eta_{2}$ is the curvature coefficient
capturing the variation of bubble curvature with contact angle for a given
pore size as shown in Fig. S4 [25].

Thus, for stable equilibrium, the mechanical stiffness of the bubble must
satisfy $\kappa>0$, which necessitates that there be a positive restoring
force responsible for returning the bubble back to equilibrium conditions
after it has been perturbed. As the bubble expands, the Joule heat density
inside the pore increases [$d\hat{h}/d\theta<0$ in Fig. S6(c) [25]]. This rise
in heat generation is balanced by the internal energy rise of the system, the
latent heat of evaporation into the bubble, the $P$–$V$ work done by the
bubble, and the rise in heat flux from the bubble to the silicon nitride
membrane [Eq. (16) in Appendix D]. Thus, by perturbing the energy balance of
the bubble, $\psi$ and $\delta$ are obtained. Excess heat generation leads to
$\psi>0$. On the other hand, if the bubble expansion consumes more heat than
is being generated, then $\psi<0$.

Figure 7 summarizes the four stability parameters $\eta_{2}$, $\psi$,
$\delta$, and $\kappa$ for the equilibrium torus bubble inside the 420 nm
pore. Owing to the pinning effect, the curvature perturbation $\eta_{2}\geq 0$
for $\theta_{\mathrm{e}}\geq$ 90°, which causes $\kappa$ to increase [Eq.
(4)], providing a positive restoring force following volume perturbation. In
this $\theta$ range, at the high-frequency limit where $\psi<0$, temperature
perturbations will also tend to increase $\kappa$, thus lending an extra
stabilization effect. Owing to the dual effect of curvature stability
($\eta_{2}>0$) and thermal stability ($\psi<0$), nano-torus film bubbles
inside a nanopore can exist in stable equilibrium. This behavior is, however,
not possible for spherical bubbles at the same scale.

However, when $\psi>0$ [Fig. 7(b)] and $|\delta|<90\degree$ in the low-
frequency range [Fig. 7(c)], the thermal perturbation

$-\frac{h_{\mathrm{fg}}\rho_{\mathrm{v}}}{T_{\mathrm{v}}}\frac{\cos\delta}{\psi}$

becomes negative and can overpower the curvature stability, triggering
destabilization. This can be seen for the $\kappa$ trace [Fig. 7(d)] at 6.0 V,
which enters the stable equilibrium zone ($\kappa>0$) only beyond a critical
frequency (3 MHz). In the stable equilibrium zone, the solution for $\kappa$
according to Eq. (4) (solid line) and $\kappa=\mathcal{M}{\omega}^{2}$ [dashed
line in Fig. 7(d)] gives a resonant frequency $f=20.8$ MHz for the 93° torus
bubble at 6.0 V and $f=8.8$ MHz for the 90° torus bubble at 6.025 V. Here,
$\omega=2\pi f$ is the angular frequency and $\mathcal{M}=2.23\times 10^{-17}$
kg is the reduced mass for the 93° torus bubble inside the 420 nm pore. The
method for calculating the reduced mass numerically is described in Appendix E
[Eq. (26)]. By comparison, in experiments, we observed a stable oscillation
frequency of 8.9 MHz for the 420 nm pore at 6.3 V. Compared with the 93°
torus, the curvature coefficient $\eta_{2}$ is smaller for the 90° torus,
leading to a smaller stiffness and resonant frequency. Again, for
$\theta<90\degree$, $\eta_{2}$ decreases, leading to bubble instability.
Hence, we can say that $f=8.8$ MHz is the minimum frequency for stable torus
oscillations. It should also be noted that at $\theta_{\mathrm{e}}=90\degree$,
the curves of $\kappa=\mathcal{M}{\omega}^{2}$ and Eq. (4) are tangent to each
other. Also, at this limit where the modeled and experimental resonant
frequencies match, the error of $\sim$0.3 V is acceptable given the
approximations in our model. We should also cite the analytical model by
Dockar _et al._ [37], which was formulated for adiabatic bubble oscillations
on a flat surface. According to that model, the natural frequency of a 90°
patch heterogeneous bubble pinned on a 100 nm flat surface is in the region of
450 MHz. The one order smaller frequency for the torus bubble observed here is
due to (i) a larger volume compared with a patch bubble, which leads to a
larger reduced mass $\mathcal{M}$, and (ii) lower stiffness and smaller volume
expansion coefficient magnitude $|\psi|$ owing to Joule heat density
amplification with bubble enlargement.

However, with increasing bias voltage, the equilibrium bubble volume increases
(i.e., $\theta_{\mathrm{e}}$ decreases), which causes the bubble to enter the
$\eta_{2}<0$ zone, where the surface tension force tends to reduce $\kappa$.
This can be observed in Fig. 7(a), where for 6.0 V, $\eta_{2}>0$, but for 6.3
V and 6.75 V, $\eta_{2}<0$. In this range, even for $\psi<0$ and
$\delta\xrightarrow{}0\degree$, a positive $\kappa$ is not obtained, and thus
the torus bubble equilibrium is unstable. This trend is supported by our
experiments, where with increasing voltage, the stable torus bubble zone
(C-ii) decreases in extent and eventually disappears, paving the way toward
transition back to nucleate boiling.

This theory also explains the higher stability of torus bubble films with
decreasing pore size. As shown in Fig. 7(a), for the 340 nm pore, a bias
voltage of 6.4 V results in an equilibrium torus bubble contact angle in the
$\theta_{\mathrm{e}}>90\degree$ range, where curvature perturbation enhances
bubble stability ($\eta_{2}>0$). As the heat transfer capacity of the silicon
nitride membrane ($h_{\text{tm}}$) is limited, the bubble volume would
increase to limit $\text{H}_{\text{c}}$ rise with increasing voltage, so that
thermal steady-state is achieved. Therefore, smaller pore diameters, which
have a higher bubble volume exclusion ratio [$f_{\mathrm{v}}$ in Fig. 4d] for
the same $\theta$, allow thermal steady-state to be established at
$\theta_{\mathrm{e}}\xrightarrow[]{}90\degree$ or
$\theta_{\mathrm{e}}>90\degree$. This indicates that stable film boiling will
persist to higher voltages as the pore diameter is decreased. Our experiments
also support this hypothesis, where stable film boiling exists for 6.7 V and 7
V for the 340 nm pore, whereas for the 420 nm pore, loss of stable film
boiling is triggered at 6.5 V.

It should be noted that this model only checks stability in the quasi-
equilibrium state of the bubble. To avoid the zone of curvature instability
($\theta_{\mathrm{e}}<90\degree$), the mean bubble size may exist in an out-
of-equilibrium state, with $\theta_{\mathrm{e}}\approx 90\degree$, where it
can be quasi-stable with nonlinear and large-amplitude oscillations as seen in
experimental spectrograms. The linear model described here is not able to
capture the complex out-of-equilibrium behavior of the torus bubble, and a
complete multiphase computational fluid dynamics (CFD) treatment may be
necessary.

##  V Conclusions

In summary, we have presented a consolidated picture of boiling inside a
cylindrical nanopore, which differs significantly from traditional pool
boiling owing to the confined nature of Joule heat generation and bubble
dynamics. Measuring the ionic current flow reduction with bubble formation
within the nanopore volume and stress waves generated by bubble motion enable
investigation of vapor nanobubbles at nanosecond resolution. Two modes of
boiling are detected: (i) nucleate boiling inside the nanopore, involving
discrete homogeneous and patch heterogeneous bubble nucleation and (ii)
oscillatory film boiling, involving a torus bubble that restricts Joule heat
generation, thus preventing nucleate boiling at the same time. Interestingly,
with increasing bias voltage, a reverse transition from film to nucleate
boiling is observed. The dynamics of the torus bubble are explained with the
help of theoretical models. Owing to the pinning effect, the torus bubble,
albeit at the nanoscale, is able to attain a stable equilibrium in the low-
bias-voltage range. However, with increasing voltage, the labile vapor bubble
expands in a pinned state to limit Joule heat generation and establish quasi-
thermal equilibrium. However, the mechanical stiffness of the bubble decreases
with increasing volume, and beyond a critical voltage, the stiffness becomes
negative, making the equilibrium unstable and ultimately destabilizing the
torus bubble.

###### Acknowledgements.

This work was supported by the Japan Society for the Promotion of Science
(JSPS) KAKENHI Grant Nos. 20H02081 and 20J22422. Part of this work was
conducted at the Advanced Characterization Nanotechnology Platform of the
University of Tokyo, which was supported by the Nanotechnology Platform of the
Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan,
Grant No. JPMXP09A21UT0089.

##  Appendix A Torus bubble shape calculations

Assuming that the torus bubble has uniform mean curvature $K$, this can be
expressed as [43]

$K=\frac{b^{\prime}\sin\nu+b\cos\nu}{(a+b\cos\nu)\sqrt{b^{2}+b^{\prime
2}}}-\frac{bb^{\prime\prime}-2b^{\prime 2}-b^{2}}{(b^{2}+b^{\prime
2})\sqrt{b^{2}+b^{\prime 2}}},$ (5)

where $b$ is the radial distance from the first principal center of curvature,
which varies with $\nu$. Thus, $b^{\prime}=db/d\nu$. From symmetry, we know
that $b^{\prime}=0$ when $\nu=180\degree$. Let the value of $b$ at
$\nu=180\degree$ be $b_{\pi}$. Now, for a given value of $K$, we can solve for
$b^{\prime\prime}$, which we can use to incrementally obtain $b$ and
$b^{\prime}$ for decreasing values of $\nu$ from $\nu=180\degree$ to
$\nu=0\degree$, thus tracing the constant-mean-curvature (CMC) torus bubble
interface. Mathematically,

$\displaystyle b^{\prime}(\nu+\Delta\nu)$
$\displaystyle=b^{\prime}(\nu)+b^{\prime\prime}(\nu)\Delta\nu,$ (6)
$\displaystyle b(\nu+\Delta\nu)$
$\displaystyle=b(\nu)+b^{\prime}(\nu+\Delta\nu)\Delta\nu,$

where $\Delta\nu=-0.0018\degree$. We find that only when
$K=(a-2b_{\pi})/[b_{\pi}(a-b_{\pi})]$ is the torus bubble surface free of
perturbations and the surface area is minimized as shown. Now, for different
values of $[a,b_{\pi}]$, $[K,l,\theta]$ are calculated. Here, $l$ is the
bubble length on the pore surface and $\theta$ is the contact angle, as shown
in Fig. S3(a) [25]. The values of $[a,b_{\pi}]$ that satisfy the pinning
condition $l=L$ are selected, and geometric properties such as the bubble
volume $V_{\mathrm{b}}$ and surface area $S_{\mathrm{b}}$ are obtained by
numerical integration over the tracer points ($R,Z$) used to construct the
surface [Figs. S3(e) and S3(f) [25]].

##  Appendix B Joule heating model

To obtain the temperature distribution responsible for the bubble behavior,
numerical simulations are employed to solve the energy-conservation equations
in the liquid and the silicon nitride membrane:

$\displaystyle\begin{aligned} \rho_{\mathrm{w}}c_{\mathrm{p,w}}\frac{\partial
T}{\partial t}&=\frac{1}{r}\frac{\partial}{\partial
r}\\!\left(k_{\mathrm{w}}r\frac{\partial T}{\partial
r}\right)+\frac{\partial}{\partial z}\\!\left(k_{\mathrm{w}}\frac{\partial
T}{\partial z}\right)+\sigma{|\mathbf{E}|}^{2},\\\\[12.0pt]
\rho_{\mathrm{m}}c_{\mathrm{m}}\frac{\partial T}{\partial
t}&=\frac{1}{r}\frac{\partial}{\partial
r}\\!\left(k_{\mathrm{m}}r\frac{\partial T}{\partial
r}\right)+\frac{\partial}{\partial z}\\!\left(k_{\mathrm{m}}\frac{\partial
T}{\partial z}\right).\end{aligned}$ (7)

Here, $\rho_{\mathrm{w}}$, $c_{\mathrm{p,w}}$, and $k_{\mathrm{w}}$ are the
temperature-dependent water density, specific heat, and thermal conductivity,
respectively [44, 45], and $t$ is time. $\rho_{\mathrm{m}}$, $c_{\mathrm{m}}$,
and $k_{\mathrm{m}}$ are the density, specific heat and thermal conductivity
of the silicon nitride membrane. $\sigma$ is the electrical conductivity of
the electrolyte, which is captured using an empirical relation first
established by Levine _et al._ [44],

$\sigma=mT-b-\frac{(T-T_{0})^{\alpha}}{\beta},$ (8)

where $m=0.391$ S m-1 K-1, $b=0.391$ S/m, $\alpha=2.65$, $\beta=5.6\times
10^{4}$, and $T_{0}=293.15$ K. Here, $\alpha$ is the only fitting parameter
that is varied to fit the pre-boiling baseline current $I_{\mathrm{b}}$ of the
nanopore. The electric field inside the liquid is obtained by combining the
ion flux balance ($\nabla\cdot\mathbf{J}=d\rho_{\mathrm{e}}/dt$) and Poisson’s
equation ($\nabla\cdot\epsilon\mathbf{E}=\rho_{\mathrm{e}}/\varepsilon_{0}$):

$\nabla\cdot(\sigma\nabla U)=\frac{d}{dt}[\varepsilon_{0}\nabla\cdot(\epsilon
U)],$ (9)

where $\mathbf{J}=\sigma\mathbf{E}$ is the ionic flux, and $U$,
$\rho_{\mathrm{e}}$, $\varepsilon_{0}$, and $\epsilon$ are the electric
potential, induced charge, dielectric permittivity of free space, and
temperature-dependent dielectric constant of water, respectively [18, 44]. The
boundary conditions for $T$ and $J$ applied on the simulation boundaries are
shown in Figs. S7(a) and S7(b) [25].

In an axisymmetric reference frame with origin at the center of the nanopore,
$r$ represents the radial coordinate and $z$ the axial coordinate. Equations
(7) are solved on a finite-volume mesh with appropriate boundary conditions
and numerical discretizations [46] (Sec. S3 of the Supplemental Material
[25]). As the nanopore temperature increases, so too does the electrolyte
conductivity, allowing greater current flow for the same voltage:

$\displaystyle I_{\mathrm{b}}$
$\displaystyle=2\pi\int_{0}^{R_{\mathrm{p}}}{\sigma\mathbf{E}}\cdot\mathbf{\hat{z}}r\,dr,$
(10) $\displaystyle I_{\mathrm{b,a}}$
$\displaystyle=2\pi\int_{0}^{r_{\mathrm{v}}}{\sigma\mathbf{E}}\cdot\mathbf{\hat{z}}r\,dr,$

where $I_{\mathrm{b}}$ denotes the baseline current development before the
onset of bubble nucleation as shown by the red dashed line in Fig. S17(a)
[25]. As can be seen, the experimental and simulation current development
match reasonably well except for the initial
$20\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro}\sec$, when membrane capacitive
charging affects the nanopore current [18]. At the point of first homogeneous
bubble nucleation, the temperature at the pore center reaches 584 K, which is
the range of theoretical homogeneous nucleation temperatures [47].

Next, we apply the same model to a nanopore-bubble system where a torus bubble
blankets the cylindrical pore surface, as shown in Fig. 6(b). We assume that
the liquid and silicon nitride are in thermal equilibrium with the bubble
temperature, i.e.,

$\begin{gathered}T_{\mathrm{v}}=T\|_{S_{\mathrm{b}}}=T\|_{S_{\mathrm{p}}},\\\\[6.0pt]
\int_{S_{\mathrm{b}}}{k_{\mathrm{w}}\nabla
T\cdot\mathbf{\hat{n}}\,dS}=\int_{S_{\mathrm{p}}}{k_{\mathrm{m}}\nabla
T\cdot\mathbf{\hat{n}}\,dS},\end{gathered}$ (11)

where $\mathbf{\hat{n}}$ is the normal vector to the surface concerned,
$S_{\mathrm{b}}$ denotes the liquid–vapor interface and $S_{\mathrm{p}}=2\pi
R_{\mathrm{p}}L$ the vapor-solid interface. Equations (11) signify that the
vapor inside the bubble has a uniform temperature from the liquid/vapor
interface to the vapor/solid interface and that the heat flux from the liquid
to the bubble is balanced by the heat flux from the bubble to the silicon
nitride wall on top of which it sits.

The above assumptions are valid when the vapor inside the bubble is assumed to
be a Knudsen gas (i.e., the Knudsen number $\mathrm{Kn}>1$) [38, 48].
According to Craig [49], bubbles smaller than 100 nm typically satisfy the
Knudsen gas definition. The Knudsen number is given by
$\mathrm{Kn}=\lambda/h$, where
$\lambda={k_{\mathrm{B}}T_{\mathrm{v}}}/{\sqrt{2}\sigma_{\mathrm{A}}P_{\mathrm{v}}}$
is the mean free path of vapor molecules and $h$ is the distance between the
liquid/vapor and vapor/solid interfaces as shown in Fig. 5.
$\sigma_{\mathrm{A}}$ is the molecular cross-sectional area and
$k_{\mathrm{B}}$ is Boltzmann’s constant. Taking the diameter of a water
molecule to be $d_{\mathrm{w}}=2.7$ Å and
$\sigma_{\mathrm{A}}=\pi{d_{\mathrm{w}}}^{2}$, $\lambda=20.6$ nm. For the
bubble shown in Fig. 5, $h\sim R_{\mathrm{p}}-r_{\mathrm{v}}=69$ nm, which
leads to a Knudsen number $\mathrm{Kn}=0.299$, which falls in the transition
regime ($0.1<\mathrm{Kn}<10$). On the other hand, the collision length of
liquid water molecules in the pore is 0.13 nm, which is much smaller than the
pore diameter (340–460 nm). In this situation, the heat transfer in the pore
liquid is governed by continuum transport. On the other hand, the heat flux
through the bubble can be assumed to be governed by the ballistic heat flux
according to kinetic theory ($q_{\mathrm{b}}$), which is given by [38]

$q_{\mathrm{b}}=\alpha_{\mathrm{c}}\rho_{\mathrm{w,n}}\sqrt{\frac{2k_{\mathrm{B}}^{3}}{m}}\,\big{(}T_{\mathrm{lv}}^{3/2}-T_{\mathrm{sv}}^{3/2}\big{)}.$
(12)

Here, $\alpha_{\mathrm{c}}$ is the accommodation coefficient, which can be
assumed to be 1 [50]. $m$ is the mass of one water molecule.
$\rho_{\mathrm{w,n}}$ is the liquid molecule number density on the
liquid/vapor interface. $T_{\mathrm{lv}}$ and $T_{\mathrm{sv}}$ are the
temperatures on the liquid/vapor and solid/vapor interfaces. For a Knudsen gas
at $T_{\mathrm{lv}}=450$ K and $T_{\mathrm{sv}}=445$ K,
$q_{\mathrm{b}}=$1982.4\text{\text{~{}}}\mathrm{M}\mathrm{W}\mathrm{/}\mathrm{m}^{2}$$.
On the other hand, the heat flux on the solid/vapor interface,
$k_{\mathrm{m}}\nabla T$, in Fig. 6(b) comes out to be
$1227\text{\text{~{}}}\mathrm{M}\mathrm{W}\mathrm{/}\mathrm{m}^{2}$. These
calculations show that a mere 5 K temperature drop between the two interfaces
can account for the huge heat flux through the bubble. Physically, this high
heat transfer rate is possible when vapor molecules evaporating from the
liquid/vapor interface at high temperature travel to and collide with the
solid/vapor interface at a lower temperature without interacting with other
vapor molecules inside the bubble, thereby transporting heat without creating
a temperature gradient. Also, according to kinetic theory, the root-mean-
square velocity of vapor molecules within the bubble,
$u_{\mathrm{v}}=\sqrt{3k_{\mathrm{B}}T_{\mathrm{v}}/m}=788$
$\text{\text{~{}}}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ when the vapor
temperature $T_{\mathrm{v}}=450$ K. This gives the ballistic heat transport
time scale as $\sim\\!h/u_{\mathrm{v}}=\mathcal{O}(0.1)$ ns. In our
experiments, we find the torus film bubble to oscillate at time periods in the
range of $\tau_{\omega}\sim\mathcal{O}(100)$ ns. Thus we can safely say that
thermal equilibrium is established across the bubble interfaces during
oscillations. On the other hand, the time scale of thermal relaxation in the
water in the pore volume can be obtained as
$\tau_{\mathrm{w}}\sim\alpha_{\mathrm{th}}/{D_{\mathrm{p}}}^{2}=\mathcal{O}(100)$
ns, where
$\alpha_{\mathrm{th}}=k_{\mathrm{w}}/(\rho_{\mathrm{w}}c_{\mathrm{p,w}})$ is
the thermal diffusivity of water. This is in the same order as
$\tau_{\omega}$, while the lifetime of the stable torus bubble has been
observed for more than $100\text{\text{~{}}}\mathrm{\SIUnitSymbolMicro s}$.
This bodes well with our theory that the nanopore temperature distribution
approaches steady state for the mean torus bubble size.

However, in a real situation, the heat transport through the bubble will be a
combination of both diffusive and ballistic transport, since the Knudsen
number of the bubble is close to the $\mathrm{Kn}=1$ limit. In our model, for
the sake of simplicity, we have neglected continuum diffusive transport
through the bubble, assuming that the heat flux in the liquid surrounding the
bubble can be balanced by the ballistic heat flux without incurring a huge
temperature drop within the bubble. Thus, the vapor temperature within the
bubble remains uniform and the heat flux is balanced, ensuring thermal quasi-
equilibrium of the bubble. It should be noted that when this assumption is
adopted, the steady-state liquid temperature distribution within the pore is
such that the corresponding steady-state current $I_{\mathrm{b,a}}$ [Fig.
S6(b) [25]] obtained according to the second of Eqs. (10) in Appendix B
matches the experimental mean current for a steady-state torus bubble [Fig.
S17(c) [25]], thus justifying our model.

##  Appendix C Bubble mechanical equilibrium

For net equilibrium, the torus bubble should be in mechanical equilibrium in
addition to thermal steady-state, which can be quantified by the Young–Laplace
equation

$P_{\mathrm{v}}=P_{\mathrm{w}}+\gamma K,$ (13)

where $K$ is the curvature of the torus bubble surface and $\gamma$ is the
temperature-dependent surface tension of water. Assuming chemical equilibrium
between liquid and vapor, the vapor pressure of the bubble,
${P_{\mathrm{v}}}$, is related to the saturation vapor pressure
$P_{\mathrm{sat}}$ and the vapor temperature ${T_{\mathrm{v}}}$ according to
$P_{\mathrm{v}}=P_{\mathrm{sat}}\exp[(P_{\mathrm{w}}-P_{\mathrm{sat}})M_{\mathrm{w}}/(N_{\mathrm{A}}\rho_{\mathrm{w}}k_{\mathrm{B}}T_{\mathrm{v}})]$,
where $P_{\mathrm{sat}}$ is evaluated at $T_{\mathrm{v}}^{*}$, assuming
saturation conditions. Here, $M_{\mathrm{w}}$, $N_{\mathrm{A}}$, and
$k_{\mathrm{B}}$ are the molecular weight of water, Avogadro’s number, and
Boltzmann’s constant, respectively.

##  Appendix D Linear model for bubble oscillations

To model the thermal oscillation frequency of the torus bubble, we adopt and
extend the linear and approximate model described in Hao and Prosperetti [36]
and originally proposed by Alekseev [51]. We assume that the bubble contains
saturated vapor and is in thermal equilibrium with the surrounding liquid and
the silicon nitride membrane surface on which it sits:
$T_{\mathrm{v}}=T_{\mathrm{w}}=T_{\mathrm{m}}$. Also, for small-amplitude
oscillations in contact angle, the vapor pressure inside the bubble is assumed
to vary along the saturation line. So, for a change in bubble volume of
$\Delta V_{\mathrm{b}}$ corresponding to a contact angle change of
$\Delta\theta$, the expansion of bubble mass is given by

$\Delta m_{\mathrm{b}}=\rho_{\mathrm{v}}\Delta
V_{\mathrm{b}}+V_{\mathrm{b}}\Delta\rho_{\mathrm{v}}=\rho_{\mathrm{v}}\frac{dV_{\mathrm{b}}}{d\theta}\Delta\theta+V_{\mathrm{b}}\frac{d\rho_{\mathrm{v}}}{dT_{\mathrm{v}}}\Delta
T_{\mathrm{v}},$ (14)

where $\rho_{\mathrm{v}}$ is the saturation vapor density. When there is no
external heating, the increase in bubble mass is caused by the evaporation of
liquid on the interface, the latent heat of which causes a reduction in
interface liquid temperature by $\Delta T_{\mathrm{v}}$. Therefore,

$-S_{\mathrm{b}}\sqrt{\frac{2D_{\mathrm{th}}}{\omega}}\,\rho_{\mathrm{w}}c_{\mathrm{w}}\Delta
T_{\mathrm{w}}=h_{\mathrm{fg}}\Delta m_{\mathrm{b}},$ (15)

where $S_{\mathrm{b}}$ is the liquid/vapor surface area of the torus bubble.
Here, $h_{\mathrm{fg}}$ and
$D=k_{\mathrm{w}}/(\rho_{\mathrm{w}}c_{\mathrm{w}})$ are the latent heat and
thermal diffusion coefficient, respectively. $k_{\mathrm{w}}$,
$\rho_{\mathrm{w}}$, and $c_{\mathrm{w}}$ are the thermal conductivity,
density, and specific heat capacity of water as functions of the liquid
temperature. For low-amplitude oscillations at low frequencies, it is safe to
assume that the thermal equilibrium at the bubble/liquid and bubble/membrane
interfaces is not lost: $\Delta T_{\mathrm{v}}=\Delta T_{\mathrm{w}}=\Delta
T_{\mathrm{m}}$. The term $\sqrt{2D_{\mathrm{th}}/\omega}$ [12] represents the
thermal relaxation length over which the temperature decrease $\Delta
T_{\mathrm{w}}$ is observed. Here, $\omega$ is the angular frequency of bubble
oscillation. Now, in the presence of Joule heating, this energy balance
equation needs to be modified to account for the additional Joule heat
generation due to bubble expansion, $\Delta V_{\mathrm{b}}$, which tends to
increase the temperature in the thermal relaxation length. Thus,

$\displaystyle\frac{d\hat{h}_{\mathrm{c}}}{dV_{\mathrm{b}}}S_{\mathrm{b}}$
$\displaystyle\sqrt{\frac{2D_{\mathrm{th}}}{\omega}}\Delta
V_{\mathrm{b}}-h_{\mathrm{tm}}S_{\mathrm{p}}\Delta T_{\mathrm{v}}$
$\displaystyle={}$
$\displaystyle(h_{\mathrm{fg}}\rho_{\mathrm{v}}+P_{\mathrm{v}})\frac{\Delta
V_{\mathrm{b}}}{\Delta t}$
$\displaystyle+\left(S_{\mathrm{b}}\sqrt{\frac{2D_{\mathrm{th}}}{\omega}}\rho_{\mathrm{w}}c_{\mathrm{w}}+S_{\mathrm{p}}\sqrt{\frac{2D_{\mathrm{m}}}{\omega}}\rho_{\mathrm{m}}c_{\mathrm{m}}\right.$
$\displaystyle\hskip
18.0pt\left.\vphantom{\sqrt{\frac{2D_{\mathrm{th}}}{\omega}}}+h_{\mathrm{fg}}V_{\mathrm{b}}\frac{d\rho_{\mathrm{v}}}{dT_{\mathrm{v}}}\right)\\!\frac{\Delta
T_{\mathrm{v}}}{\Delta t},$ (16)

where $S_{\mathrm{p}}=2\pi R_{\mathrm{p}}L$ is the cylindrical nanopore
surface area of the silicon nitride membrane and
$\hat{h}_{\mathrm{c}}=H_{\mathrm{c}}/V_{\mathrm{p,c}}$ is the average Joule
heat density over the cylindrical pore volume $V_{\mathrm{p,c}}$.
$H_{\mathrm{c}}$ is calculated by simulating the steady-state Joule heat and
temperature distributions in the presence of the torus bubble following the
model described in Eqs. (7)–(11). $V_{\mathrm{p,c}}$ is the cylindrical pore
volume, which extends for $1.66L$ across the pore midline ($z=0$), enclosing
the torus volume $V_{\mathrm{b}}$ [Figs. 6(a) and 6(b)].
$D_{\mathrm{m}}=k_{\mathrm{m}}/(\rho_{\mathrm{m}}c_{\mathrm{m}})$ is the
thermal diffusivity of the silicon nitride membrane. As discussed in Appendix
B, in the equilibrium bubble position, the net Joule heat production rate in
$V_{\mathrm{p,c}}$ is balanced by the heat dissipation through the surrounding
liquid and the bubble, thereby rendering a steady-state temperature
distribution.

Using Joule heating simulations, we have calculated the increase in Joule heat
density $\hat{h}_{\mathrm{c}}$ for different contact angles
$\theta_{\mathrm{e}}$ of the torus bubble during pinned expansion [Fig. S6(c)
[25]] and have found that for large $\theta_{\mathrm{e}}$, when the bubble
expands to constrict the pore volume, the Joule heat density increases in the
cylindrical pore region: ${d\hat{h}_{\mathrm{c}}}/{dV_{\mathrm{b}}}>0$. This
increase in heat generation is in turn balanced by a change in heat transfer
through the membrane ($h_{\mathrm{tm}}S_{\mathrm{p}}\Delta T_{\mathrm{v}}$),
latent heat consumption by the bubble ($h_{\mathrm{fg}}\rho_{\mathrm{v}}\Delta
V_{\mathrm{b}}/\Delta t$), $P$–$V$ work done by the bubble, and sensible heat
rise in the thermal relaxation length in the liquid and the silicon nitride.
Here, the internal energy rise of the bubble is neglected because of its
comparatively lower heat capacity than the liquid layers surrounding it. Thus,
in short, Eq. (16) accounts for the energy balance of the bubble and the
thermal boundary layers on its interfaces.

As the Eq. (16) contains time-derivative terms, the solution for
$T_{\mathrm{v}}$ oscillations will not be in the same phase as that for
$V_{\mathrm{b}}$ oscillations. We can write

$\displaystyle\theta$
$\displaystyle=\theta_{\mathrm{e}}-\theta_{\mathrm{a}}\exp(j\omega t),$ (17)
$\displaystyle V_{\mathrm{b}}$
$\displaystyle=V_{\mathrm{b,e}}+V_{\mathrm{b,a}}\exp(j\omega t),$
$\displaystyle T_{\mathrm{v}}$
$\displaystyle=T_{\mathrm{v,e}}+T_{\mathrm{v,a}}\exp[j(\omega t+\delta)],$

where $j=\sqrt{-1}$. Here, the oscillation amplitudes of contact angle, bubble
volume, and bubble temperature are taken to be positive by convention, i.e.,
$\theta_{\mathrm{a}}>0$, $V_{\mathrm{b,a}}>0$, and $T_{\mathrm{v,a}}>0$.
$\delta$ is the phase difference between volume expansion and vapor bubble
temperature rise. Substituting Eq. (17) into Eq. (16), we obtain

$V_{\mathrm{b,a}}(B-jA)=T_{\mathrm{v,a}}\exp(j\delta)(D+jC),$ (18)

where

$\displaystyle
B=\frac{d\hat{h}_{\mathrm{c}}}{dV_{\mathrm{b}}}S_{\mathrm{b}}\sqrt{\frac{2D_{\mathrm{th}}}{\omega}},\qquad
A=\omega(h_{\mathrm{fg}}\rho_{\mathrm{v}}+P_{\mathrm{v}}),$ $\displaystyle
D=h_{\mathrm{tm}}S_{\mathrm{p}},\qquad
C=\omega\\!\left(S_{\mathrm{b}}\sqrt{\frac{2D_{\mathrm{th}}}{\omega}}\,\rho_{\mathrm{w}}c_{\mathrm{w}}+h_{\mathrm{fg}}V_{\mathrm{b}}\\!\frac{d\rho_{\mathrm{v}}}{dT_{\mathrm{v}}}\right).$

Equating the real parts and imaginary parts of Eq. (18), we arrive at

$\displaystyle\tan\delta$ $\displaystyle=\frac{AD+BC}{AC-BD},$ (19)
$\displaystyle\cos\delta$ $\displaystyle=\psi\\!\left(\frac{BD-
AC}{D^{2}+C^{2}}\right).$

Because $A>0$, $B>0$, $C>0$, and $D>0$, for small positive values of $\delta$,
$\tan\delta>0$ will be satisfied only when $AC>BD$. Here
$\psi={V_{\mathrm{b,a}}}/{T_{\mathrm{v,a}}}$. Now, the ratio of bubble volume
amplitude and bubble temperature amplitude can be solved for from Eq. (19) as

$\psi=-\operatorname{sign}(\delta)\left[\frac{D^{2}+C^{2}}{\sqrt{(AC-
BD)^{2}+(AD+BC)^{2}}}\right].$ (20)

It is interesting to note that when this ratio is negative, with increasing
bubble volume, the vapor temperature will fall, and thus the vapor pressure
inside the bubble is also expected to decrease along the saturation line
$\Delta P_{\mathrm{v}}=(dP/dT)_{\mathrm{sat}}\Delta
T_{\mathrm{v}}={h_{\mathrm{fg}}\rho_{\mathrm{v}}}/{T_{\mathrm{v}}}\Delta
T_{\mathrm{v}}$. Hence, a resisting force opposing bubble expansion, $F$, is
developed:

$F=S_{\mathrm{b}}\Delta(P_{\mathrm{v}}-P_{\mathrm{w}}-K\gamma)=-\kappa\Delta
R=-\kappa\frac{\Delta V_{\mathrm{b}}}{S_{\mathrm{b}}},$ (21)

i.e.,

$S_{\mathrm{b}}\\!\left(\frac{h_{\mathrm{fg}}\rho_{\mathrm{v}}}{T_{\mathrm{v}}}\Delta
T_{\mathrm{v}}-\gamma\frac{dK}{dV_{\mathrm{b}}}\Delta
V_{\mathrm{b}}-K\frac{d\gamma}{dT_{\mathrm{v}}}\Delta
T_{\mathrm{v}}\right)=-\kappa\frac{\Delta V_{\mathrm{b}}}{S_{\mathrm{b}}},$
(22)

where $\Delta R=\Delta V_{\mathrm{b}}/S_{\mathrm{b}}$ is the expansion in
bubble size. Now, substituting $\Delta
V_{\mathrm{b}}=V_{\mathrm{b,a}}\exp(j\omega t)$, $\Delta
T_{\mathrm{v}}=T_{\mathrm{v,a}}\exp[j(\omega t+\delta)]$, and
$\psi=V_{\mathrm{b,a}}/T_{\mathrm{v,a}}$ into Eq. (22) and solving for the
real part, we obtain

$\displaystyle\kappa=$
$\displaystyle-{S_{\text{b}}}^{2}\left[\frac{h_{\text{fg}}\rho_{\text{v}}}{\psi
T_{\text{v}}}\cos(\delta)-\gamma\frac{dK}{dV_{\text{b}}}-K\frac{d\gamma}{dT_{\text{v}}}\frac{\cos(\delta)}{\psi}\right]$
$\displaystyle\kappa=$
$\displaystyle{{S_{\text{b}}}^{2}}\left[\eta_{2}-\frac{h_{\text{fg}}\rho_{\text{v}}}{T_{\text{v}}}\frac{\cos(\delta)}{{\psi}}+\eta_{1}\frac{\cos(\delta)}{{\psi}}\right]$
(23)

The imaginary part of Eq. (22) would correspond to the damping term associated
with temperature being out of phase with volume perturbation. Here, $\kappa$
is the mechanical stiffness of the torus bubble and $\gamma$ is the
temperature-dependent surface tension of water. $\eta_{1}$ and $\eta_{2}$ are
the curvature coefficients depending on the bubble curvature, which in turn
depend on contact angle and pore size as shown in Fig. S4. It should be noted
that $\psi\eta_{2}$ is one order of magnitude higher than $\eta_{1}$, so
variations in $\eta_{1}$ can be neglected for the sake of this discussion. It
should be noted that we ignore the recoil force applied on the bubble
interface due to evaporation/condensation. The recoil pressure, which is given
by
${|\dot{m}|}^{2}(1/\rho_{\mathrm{v}}-1/\rho_{\mathrm{w}})\approx\rho_{\mathrm{v}}\omega^{2}\Delta
R^{2}$, is proportional to the square of the bubble radius perturbation and
thus can be neglected for small-amplitude oscillations.

##  Appendix E Reduced mass of pinned torus bubble

Under the constraint of pinning, as a torus bubble with contact angle $\theta$
oscillates with a contact angle velocity $\dot{\theta}$, the liquid
surrounding the bubble also develops a velocity according to the continuity
equation. Let the kinetic energy of the liquid be given by

$E_{\mathrm{k}}=\tfrac{1}{2}\mathcal{M}(\theta)\dot{R}^{2},$ (24)

where $\mathcal{M}(\theta)$ is the reduced mass of the bubble [12, 42] and
$\dot{R}$ is the velocity of the bubble interface, which can be approximated
by

$\dot{R}=\frac{1}{S_{\mathrm{b}}}\frac{dV_{\mathrm{b}}}{d\theta}\dot{\theta}.$

For a spherical bubble of radius $R$, $\mathcal{M}$ can be determined
analytically to be $\mathcal{M}=4\pi\rho_{\mathrm{w}}R^{3}$. However, for a
torus bubble, we need to calculate $\mathcal{M}$ numerically, by implementing
potential flow theory [52] on a finite element method (FEM) grid in MATLAB
[53].

First, at a given $\theta$, ${dV_{\mathrm{b}}}/{d\theta}$ is calculated from
the slope of $V_{\mathrm{b}}$ versus $\theta$ in Fig. S3(f) [25]. The
continuity equation is invoked in the liquid surrounding the bubble in an
axisymmetric coordinate system:

${\nabla}^{2}\phi=0,$ (25)

where $\phi$ is the velocity potential. For a contact angle velocity
$\dot{\theta}$, we calculate the radial ($\Delta R/\Delta t$) and axial
($\Delta Z/\Delta t$) velocities of the tracer points on the bubble surface
[Fig. S8(b) [25]]. These velocities are interpolated on the mesh points of the
bubble surface, which act as boundary conditions. A far-field boundary
condition $\phi=\phi_{0}$ is applied on the liquid boundary far from the
bubble. On the remaining boundary surfaces, zero-flux boundary conditions are
applied: $d\phi/dn=0$, where $n$ is the normal direction to the surface. The
solution for the velocity potential is used to obtain the radial liquid
velocity $u_{r}=\partial\phi/\partial r$ and axial liquid velocity
$u_{z}=\partial\phi/\partial z$, which are integrated over the mesh triangles
to obtain the total liquid kinetic energy

$E_{\mathrm{k}}=\int_{V}\rho_{\mathrm{w}}\tfrac{1}{2}(u_{r}^{2}+u_{z}^{2})\,dV.$
(26)

Here, the density of the liquid is taken as $\rho_{\mathrm{w}}=1000$
$\text{\text{~{}}}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$. From Eqs.
(24) and (26), $\mathcal{M}(\theta)$ is obtained.

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