# Mechanically activated ionic transport across 2nm carbon nanotubes

Alice Marcotte, Timothée Mouterde, Antoine Niguès, Alessandro Siria⋆, Lydéric
Bocquet⋆ Laboratoire de Physique de l’Ecole normale Supérieure, ENS,
Université PSL, CNRS,
Sorbonne Université, Université de Paris, 75005 Paris, France
<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Mechanically activated ionic transport across single digit carbon nanotubes

Alice Marcotte, Timothée Mouterde, Antoine Niguès, Alessandro Siria⋆, Lydéric
Bocquet⋆ Laboratoire de Physique de l’Ecole normale Supérieure, ENS,
Université PSL, CNRS,
Sorbonne Université, Université de Paris, 75005 Paris, France
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Fluid and ionic transport at nanoscale recently highlighted a wealth of exotic
behaviours Bocquet2010 ; SDN2019 ; Bocquet2020 ; Majumder2005 ; Noy2006 ;
Noy2016 ; Secchi2016 ; Noy2017 ; Siria2013 ; Siria2017 ; Radenovic2016 ;
Radenovic2016_2 ; Keyser2017 ; Karnik2013 . However, the artificial
nanofluidic devices Esfandiar2017 ; Radha2016 ; Radha2018 ; Mouterde2019 are
still far from the advanced functionalities existing in biological systems,
such as electrically and mechanically activated transport Anishkin2014 ;
Cox2019 . Here we focus on the ionic transport through 2 nm-radius individual
multiwalled carbon nanotubes (CNT), under the combination of mechanical and
electrical forcings. Our findings evidence mechanically activated ionic
transport under the form of an ionic conductance which depends quadratically
on the applied pressure. Our theoretical study relates this behaviour with the
complex interplay between electrical and mechanical drivings, and shows that
the superlubricity of CNT Lindsay2011 ; Noy2017 ; Noy2016 ; Noy2006 ;
Majumder2005 ; Secchi2016 is a prerequisite to attain mechanically activated
transport. The pressure sensitivity shares similarities with the response of
biological mechanosensitive ion channels Anishkin2014 ; Cox2019 observed here
for the first time in an artificial system. This paves the way to build new
active nanofluidic functionalities inspired by the complex biological
machinery.

Nanofluidics, the field studying transport of fluids at the nanoscales, has
now reached a stage of maturity. The manifold systems which have been
developped over the recent years, from nanopores, nanotubes, to angström scale
channels, together with the new instrumentation developped, now allow
investigating fluid transport at the frontier of hydrodynamics. This points to
key knowledge gaps in the so-called single digit nanopores with size below ten
nanometers SDN2019 .

Still, such artificial systems remain far from the impressive complexity of
the biological ionic machinery, which exhibits a plethora of advanced
transport functionalities. In particular, the recent identification of
mechanotransduction channels has highlighted complex response involving
excitatory mechanically activated ionic current under mechanical pressure or
stretching Anishkin2014 ; Cox2019 . Reproducing such advanced response in
artificial systems was up to now unreachable.

In the quest to reproduce such advanced response, we investigate here the non-
linear transport in individual ’single-digit’ CNT with radius $\approx$ 2 nm
and explore how mechanical pressure does affect the ionic transport in these
channels. CNT are unique materials in that they highlight ultra-low water
friction on their surface Secchi2016 and electro-dynamic couplings are
accordingly expected to be exalted. However, probing transport in individual
CNT with nanometric radius under coupled voltage and pressure drops has been
so far a considerable challenge, notably in terms of fabrication and
sensitivity, and most experimental studies in such tiny NT have been
restricted up to now to electric forcings Lindsay2010 ; Lindsay2011 ;
Strano2013 ; Jourdain2017 . To overcome this bottleneck, we build on the
transmembrane nanotube methodology, initially developed for the measurement of
the ionic transport with larger 10 - 50 nm radii nanotubes, Siria2013 , but
here applied to 2 nm nanotubes. The experimental set-up is shown in Fig. 1b:
an individual arc-discharge CNT (SIGMA ALDRICH, Product number: 406074)
connects two macroscopic fluid reservoirs through a 50 nm thick silicon-
nitride (SiN) membrane (NORCADA). We investigated CNTs with inner radii of
$R\approx 2.0\pm 0.6$ nm and lengths of $L\approx$ 1 $\mu$m. The length of the
CNTs is measured during the fabrication procedure under Scanning Electron
Microscope (SEM) imaging (see Fig. 1a) and the inner radii distribution for
the CNT has been obtained by Transmission Electron Microscopy (TEM)
observations and is displayed on Fig. 1b. Transmembrane individual CNT (t-CNT)
devices are realized by in situ SEM nanomanipulation, using the method
previously described in Siria2013 (see Methods for more details). The membrane
is then sealed between two reservoirs, which are filled with a potassium
chloride (KCl) solution at various concentrations (denoted as $c$) and pHs.
One Ag/AgCl electrode is immersed in each reservoir, and connected to a patch-
clamp amplifier (Axopatch 200B) which allows for ionic current measurements
with a current resolution of 0.1 pA. Both reservoirs can be connected to a
pressure controler, enabling us to impose a pressure drop ($\pm$ 0 - 2 bar) in
both directions across the membrane. This setup allows measuring the ionic
current driven by a pressure drop $\Delta P$ and the combined effect of
additional forcing $\Delta V$ applied along the t-CNT. In this study, three
independent t-CNT devices with similar geometries were studied, and gave the
same response as described below. Their IV characteristics at c = 1 M and pH
5.5 are given by Supplementary Figure 1.

In a typical experiment, we measured the pressure-driven component of the
ionic current $I$, referred to as the streaming current and defined here as
$I_{\rm str}=I(\Delta P,\Delta V)-I(\Delta P=0,\Delta V)$, see Methods. This
current results from the transport of (excess) ions carried by a pressure-
induced flow (inset Fig. 1c) and probes the interplay between ion and mass
transport. A pressure drop between 50 mbar and 1.5 bar is applied in both
directions across the t-CNT. We do not observe a temporal evolution of the
system and the conductance is the same before and after pressure measurements
are performed. Let us focus first on the pressure-response of the system in
the absence of voltage bias ($\Delta V=0$). As shown on Figure 1c and on
Supplementary Figure 2 for $c$ = 1 M and $\Delta V=0$, once the pressure drop
is applied, the streaming current overshoots and then quickly relaxes to a
stationary value.

We report in the Figure 1d, the stationary value reached by $I_{str}$ as a
function of the applied pressure $\Delta P$. As usually expected for such
measurements without applied voltage, the streaming current varies linearly
with the applied pressure, $I_{\rm str}\propto\Delta P$, see Fig. 1d, and the
slope increases with pH. The sign of the current is furthermore consistent
with an excess transport of positive charges flushed by water flow. This
response is in agreement with results obtained for CNT with larger diameters
SecchiPRL and was explained in terms of hydroxide adsorption on the CNT walls.
Another contribution can originate from the differential mobility of the ions
in confinement, as discussed recently Mouterde2019 .

In the present study, we rather focus on the pressure response in the presence
of a constant bias voltage $\Delta V$, typically $\Delta V\in[-100$ mV$;100$
mV$]$. Figure 2a shows the time-dependent response of the streaming current
while applying pressure drops of $\pm$200 mbar and $\pm$400 mbar, when a bias
voltage of 25 mV and of -25 mV is imposed along the CNT. We measure that for a
given bias voltage, the streaming current is of the same sign independently of
the sign of the pressure drop. This is best illustrated in Fig. 2b, which
shows that the streaming is now a non-linear function of the pressure drop.
This response is evidenced for all investigated systems, salt concentrations
and pHs, see Supplementary Figures 3, 4 and 5. The experimental data is very
well described by a second order polynomial function of the pressure drop:

$I_{\rm{str}}(\Delta P)=\mu_{EO}\Delta P+\beta(\Delta V)\times\Delta P^{2},$
(1)

where the first term is the (linear) electro-osmotic term with mobility
$\mu_{EO}$ and the second quadratic term is the dominant contribution to the
current, Fig. 2b. Furthermore, the coefficient $\beta(\Delta V)$ is shown to
depend linearly on the gating voltage, $\Delta V$, as indicated on the inset
of Fig. 2c.

Gathering the streaming and bare conductance contributions to the ionic
current, one finds that the total current can be rewritten as

$I_{\Delta P}(\Delta V)=\mu_{EO}\times\Delta P+G(\Delta P)\times\Delta V$ (2)

where the conductance is now a quadratic function of the applied pressure,
$G(\Delta P)=G_{0}+G_{2}\times\Delta P^{2}$. This result highlights that
narrow carbon nanotubes display mechano-sensitive ionic conduction, with a
conductance which is tuned by the mechanical pressure. Figure 2c represents
the conductance $G(\Delta P)$ as a function of the pressure drop $\Delta P$,
here for a solution of salt concentration of 1 M and pH 5.5. The collapse of
the $G$ curves for various $\Delta V$ shows that the pressure-dependent
conductance is the physically relevant quantity here. This behaviour was
measured for all investigated pHs and salt concentrations. Supplementary
Figure 6 shows that $G_{2}$ increases with salt concentration (for both pH 5.5
and pH 10.5) with only a weak dependence on pH. This suggests that this result
does not stem from a surface charge effect contrary to the case of the – much
smaller – streaming currents without applied bias displayed in the Figure 1c.
In the latter case, the pH-dependent surface charge of the CNT is expected to
be the main contribution to the bare streaming current. Furthermore, both the
effects of the stretching of the supporting membrane, and the change of radius
of the CNT can be discarded as the origin of this phenomenon, since both are
negligible due to the high stiffness of the materials (in contrast to
explanations for the mechano-sensitive response of biological channels
Anishkin2014 ; Cox2019 ). Interestingly the observed response in CNT is also
very different from that observed recently in 2D Angström slits with carbon
confinement Mouterde2019 , where only a linear dependence of the ionic current
as a function of pressure drop could be observed whatever the applied voltage.
Here the response reveals a robust, quadratic, pressure-dependence of the
conductance. Interestingly, such pressure-sensitive behaviour shares
similarities with the mechanosensitive response of biological channels
Anishkin2014 ; Cox2019 . In the latter systems, a pressure drop converts into
a ionic current, showing a non-linear dependence on the applied pressure.

In order to rationalize this behaviour, we have developed a theoretical model
capturing the main ingredients of ionic transport in narrow carbon nanotubes.
A key feature is the low friction expected for water on the CNT surface, which
results in fast flows inside the CNT under imposed pressure drop. We
anticipate that this is at the root of the strong interplay between the ionic
transport and mechanical forcings.

The theoretical description proceeds with a modified Poisson-Nernst-Planck-
Stokes (PNPS) description for the ion transport inside the CNT, building on
the framework introduced in Mouterde2019 ; the details of the derivation are
provided in the Supplementary Note 1. In short, we consider the 1-dimensional
transport of ions inside the CNT, under the combined effects of the local
electric field and (fast) flow of water. Ion mobilities are not necessarily
equal and we account for the friction of ions on both water and surface walls.
Last but not least, water slippage on the CNT surface is assumed. The PNPS
framework writes the conservation equations for each specie and water along
the channel.

Under limited approximations, this extended PNPS model can be solved
analytically in 1D and one obtains an analytical prediction for the ionic
current under the combined effects of pressure and voltage drops. Remarkably
the theory predicts that the conductance is a function of a Péclet number
${\rm Pe}$, defined as

${\rm Pe}=\frac{Q_{w}}{D}\times\frac{L}{\pi R_{0}^{2}}$ (3)

where $Q_{w}$ is the water flow rate and $D$ a typical ion diffusion
coefficient. Note that the definition of the Péclet number involves the length
$L$ (and not the radius $R$) of the channel. To linear order in $\Delta V$ as
in the experiments, but keeping the full dependence on the Péclet number, one
predicts

$I=\mu_{EO}\times\Delta P+G({\rm Pe})\times\Delta V+\dots$ (4)

where $\mu_{EO}$ is an electro-osmotic mobility, which is a function of the
various kinetic parameters – see Supplementary Information for details –, and
$G({\rm Pe})$ is a Péclet-dependent conductance. The theory predicts a
transparent expression for the conductance as:

$G({\rm Pe})=G_{0}\times{{{\rm Pe}\over 2}\over\tanh{{\rm Pe}\over 2}}$ (5)

which can be expanded as $G({\rm Pe})=G_{0}\times(1+{{\rm Pe}^{2}\over
12}+\dots)$. One expects the pressure-driven contribution to the flow to be
the main contribution to the Péclet number, so that ${\rm Pe}={Q_{w}L/D\pi
R_{0}^{2}}={K_{\rm app}\over D}\times\Delta P$, with $K_{\rm app}$ the
apparent permeance of the CNT. The ionic conductance $G({\rm Pe})$ is now a
function of the pressure drop, varying quadratically for low $\Delta P$. This
is in full agreement with experimental measurements, see Fig.2 and Eq. (2).
Physically, this square-like dependence of the conductance on the Péclet
number results from the balance between electric and mechanical forces. This
leads to an overall increase of the ionic concentration in the nanotube for
both $\Delta P>0$ and $\Delta P<0$ – as can be verified from the direct
calculation of the ionic density profiles in the channel, see Supplementary
Note 1. –, hence an increase in the conductance of the nanotube which is
symmetric for positive and negative pressure drops. A key remark is that the
Péclet number (hence flow) cannot be neglected here to describe ionic
transport in tiny CNTs, in strong contrast to usual channels: typical values
for the Péclet number may be inferred from the current-pressure
characteristics such as in Fig. 2b, yielding ${\rm Pe}\sim 1-10$ depending on
the conditions and nanotube size. In contrast, a Poiseuille flow prediction
with no-slip boundary condition on the surface would yield a Péclet number in
the range ${\rm Pe}\sim 10^{-2}$ (considering a 2 nm-radius channel under a
maximum pressure drop of 0.5 bar as applied here). But this estimate for ${\rm
Pe}$ is amplified by a factor $1+8b/R$ when surface slippage, with slip length
$b$, occurs on the CNT surface. Hence, large slippage, i.e. low surface
friction on the CNT surface, leads to a strong sensitivity of the conductance
on the pressure drop, as highlighted in Eq.(5). This makes the slippery carbon
nanotube unique to obtain the mechanosensitive conductance.

In order to further assess these predictions, we solved numerically the full
PNPS ionic transport model using a finite-element approach (COMSOL) (see
Supplementary Note 2). Overall, these numerical calculations fully confirm the
results of the analytical model described above. A linear dependence of the
streaming current on the pressure drop is recovered when no bias voltage is
applied, and the magnitude of the streaming current is as expected much larger
in the low friction regime than in the high friction regime, see Supplementary
Figure 7.

Now, applying simultaneously voltage and pressure drops, the numerical
calculations highlight a quadratic response of the conductance as a function
of the pressure drop, Fig. 3, and furthermore show that this response is
intimately connected to a low surface friction (i.e. strong slippage); see the
high versus low friction response in Supplementary Figure 8, and the $G_{2}$
dependency on the slip length $b$ on Supplementary Figure 9. A linear response
in $\Delta P$ is recovered in the high friction regime, and is reminiscent of
the results reported in Mouterde2019 in Angström slits (slippage was shown to
be larger in nanotubes than on flat graphite Secchi2016 ). But a quadratic
response in $\Delta P$ is obtained for the low friction surface as observed
here for the CNT.

Overall we unveil that the conductance of the CNT is quadratically modulated
by the applied pressure. It is interesting to remark that this result is
reminiscent of the response of mechanosensitive biological channels. However,
in the case of mechano-sensitive biological channels the mechanism at stake
rather involves the sensing of changes in membrane tension coupled to
conformational changes in the channel protein structures. As discussed above,
such mechanism does not occur for the transmembrane CNT considered here. Our
results rather point to the unique properties of the ’single digit’ CNTs to
obtain such pressure-sensitive response, whereby a pressure-dependent
accumulation of ionic species occurs in the CNT under a voltage bias. The low
friction of water on the carbon surface is accordingly shown to be a
prerequisite to the quadratic pressure dependence of the ionic response in the
CNT. Single digit CNTs are thus able to respond to very local pressure signal,
with nanometric spatial resolution.

Hence, this artificial system reproduces a pressure-sensitive response. Since
the device properties are robust and well calibrated, it provides a unique
platform to study such advanced response in an artificial system. In addition,
the possibility of inserting CNTs into lipid membranes, as developed in Refs
Noy2017 ; Wu2013 , provides promising applications to conceive new types of
ultra-local pressure sensors using e.g. assemblies of vesicles as electric
compartments with pressure-modulated potential. This would constitute a
promising building-block to fabricate sensitive soft materials, enabling one
to reproduce artificial sensing devices for (light) touch or proprioception.
Such hybrid devices offer manifold perspectives in the development of
iontronic systems for which the response intensity can be tuned. Finally, our
discovery should benefit to the flourishing field of bottom-up artificial
cells fabrication for synthetic ion channels design, which mechanosensitive
response can be rationally engineered, via our theoretical model, by tuning
the nanochannel nature, length or size.

Acknowledgements
A.S. acknowledges funding from the EU H2020 Framework Programme/ERC Starting
Grant agreement number 637748-NanoSOFT. L.B. acknowledges funding from the EU
H2020 Framework Programme/ERC Advanced Grant agreement number 785911-Shadoks
and ANR project Neptune. L.B. and A.S. acknowledge support from the Horizon
2020 program through Grant No. 766972- FET-OPEN-NANOPHLOW.

Author contributions
L.B. and A.S. designed and directed the project. A.M., A.S., and A.N.
fabricated the devices. A.M. performed the measurements with inputs from the
other authors. All authors analyzed the data and contributed to discussions.
L.B. performed the theoretical analysis and A.M. the numerical modelling.
A.M., A.S and L.B. wrote the manuscript with inputs from T.M.

Data availability
The data that support the plots within this paper and other findings of this
study are provided with this paper as source data and in Supplementary
Information.

Competing interests
The authors declare no competing interests.

## Figures

Figure 1:  Experimental setup and pressure-driven current without bias voltage
($\Delta V=0$). a, Top: CNT glued at the apex of an electro-chemically etched
tungsten tip (SEM imaging). Bottom: CNT introduced into the hole drilled in a
Si/SiN membrane (SEM imaging) b, Left: schematic of the nanofluidic cell for
pressure and voltage-driven current measurements through a single CNT. Right,
top: inner radii distribution for a sample consisting in 57 CNTs observed
under TEM imaging. Right, bottom: a CNT observed under TEM imaging. c, Ionic
current as a function of time, at a salt concentration of 1 M and pH 10.8
without bias voltage ($\Delta V=0$). A sliding average of the current over
four data points is performed. Current overshoots once the pressure is
applied, and then saturates to a steady-state value. Inset: Sketch for the
pressure-driven current through a CNT. d, Streaming current as a function of
the pressure drop $\Delta P$ at different pHs without applied voltage ($\Delta
V=0$). For each pH, the dashed line represents the best linear fit. The error
bars in d represent the standard error over five experimental replicates.

Figure 2:  Pressure-driven current while applying a bias (experiments): a,
Streaming current as a function of time, at $c$ = 1 M and at pH 10.5 while
applying a bias voltage $\Delta V$ = 25 mV (red curve) and $\Delta V$ = -25 mV
(blue curve). b, Streaming current as a function of the pressure drop $\Delta
P$ while applying a bias voltage $\Delta V\in\left[-100\text{ mV},100\text{
mV}\right]$, at $c$ = 1 M and at pH = 5.5 (left) and pH = 10.5 (right). For
each applied bias, the dashed line represents the best parabolic fit. c,
Mechano-sensitive conductance $G$ as a function of the pressure drop $\Delta
P$ for different bias voltages $\Delta V\in\left[-25\text{ mV},25\text{
mV}\right]$, at $c$ = 1 M and pH = 5.5. The dashed line represents the best
parabolic fit. Inset: Second-order coefficient from the best parabolic fit in
b, as a function of the bias $\Delta V$, for $c$ = 1 M and pH = 5.5. The
dashed line represents the best linear fit. Error bars represent: b, standard
error over 25 replicates; c, inset: uncertainty in the fit value obtained in
b.

Figure 3:  Theoretical analysis of the electro-hydrodynamic coupling. a,
Streaming current as a function of the pressure drop $\Delta P$ with different
bias voltages $\Delta V\in\left[-50\text{ mV},50\text{ mV}\right]$. For each
bias, the dashed line represents the best parabolic fit (numerical
simulations). b, Mechano-sensitive conductance $\Delta G=G(\Delta P)-G_{0}$ as
a function of the pressure drop $\Delta P$, for different bias voltages
$\Delta V\in\left[-50\text{ mV},50\text{ mV}\right]$. The dashed line
represents the best parabolic fit. Inset: second-order coefficient from the
best parabolic fit in a, as a function of the bias $\Delta V$. The dashed line
represents the best linear fit.

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## Methods

Device fabrication. Our devices were fabricated following previously reported
procedures Siria2013 ; SecchiPRL . Briefly, a single multiwall carbon nanotube
(SIGMA ALDRICH, Product number: 406074) glued at the apex of an
electrochemically etched tungsten tip is inserted into a 200 nm hole drilled
in a 50 nm thick silicon nitride (SiN) membrane hanging on a centimetric
silicium (Si) frame of 500 $\mu$m in thickness (NORCADA). The insertion is
performed in the chamber of a Scanning Electron Microscope using a built-in
nano manipulator consisting in three axis piezo-inertial step motors of
nanometric precision controlled by an outside joystick. The sealing of the
tube in the membrane nanohole is ensured by localized cracked naphthalene
deposition. Eventually, the sealed nanotube is cut free from the tungsten tip
by highly energetic electron beam exposure in water vapor. The centimetric
Si/SiN chip is then squeezed between two macroscopic reservoirs filled with a
potassium chloride (KCl) solution of various concentrations. All steps of this
procedure follow those described in detail in the Supplemental Materials of
Siria2013 .

Streaming current measurements. Ag/AgCl electrodes are immersed into the
reservoirs and connected to an external patch-clamp amplifier (Axopatch 200B,
Molecular Devices) which allows for electrical forcing as well as for
electrical measurements, with a resolution in the tenths of picoamperes. The
ionic current is measured versus time, while steps of pressure ranging from 50
mbar to 1.5 bar are applied via a pressure pump (AF1, Elveflow). Each step
typically lasts 15 s, and the delay between two successive steps is 15 s. The
streaming current is defined as the difference between the plateau value
reached by the ionic current while the pressure is applied, and its value when
the pressure is released. For each drop of pressure, this measurement is
performed between 2 and 25 times. Supplementary Figure 2 gives an example of
an ionic current measurement as a function of time, when no bias voltage is
applied.

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