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# Ionic Coulomb blockade as a fractional Wien effect

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Recent advances in nanofluidics have allowed the exploration of ion transport down to molecular-scale confinement, yet artificial porins are still far from reaching the advanced functionalities of biological ion machinery. Achieving single ion transport that is tunable by an external gate—the ionic analogue of electronic Coulomb blockade—would open new avenues in this quest. However, an understanding of ionic Coulomb blockade beyond the electronic analogy is still lacking. Here, we show that the many-body dynamics of ions in a charged nanochannel result in quantized and strongly nonlinear ionic transport, in full agreement with molecular simulations. We find that ionic Coulomb blockade occurs when, upon sufficient confinement, oppositely charged ions form ‘Bjerrum pairs’, and the conduction proceeds through a mechanism reminiscent of Onsager’s Wien effect. Our findings open the way to novel nanofluidic functionalities, such as an ion pump based on ionic Coulomb blockade, inspired by its electronic counterpart.
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Articles
https://doi.org/10.1038/s41565-019-0425-y
Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité,
Paris, France. *e-mail: lyderic.bocquet@ens.fr
Ionic transport is key to numerous processes, from neurotransmis-
sion to ultrafiltration14. Over the past decade, it has been exten-
sively investigated in biological systems, evidencing advanced
functionalities such as high selectivity, ionic pumping and elec-
trical or mechanical gating57. However, it is only recently that
experimental progress in nanoscience has allowed the fabrication
of artificial pores with controlled material properties and channel-
or slit-like geometries. These new systems have reached unprec-
edented nanometre- and even angström-scale confinements811, yet
they are still far from exhibiting the same functions as biological
ionic machines.
At these scales, ion transport is usually described by the Poisson–
Nernst–Planck (PNP) framework2, which couples ion diffusive
dynamics to electric interactions. Although it may account for
nonlinear (for example, diode-type) effects2, the PNP framework is
intrinsically continuum and mean field. Building bio-inspired func-
tions, however, may require control of transport at the single ion
level, which is out of the reach of a mean-field description. Single
charge transport in fact echoes the canonical Coulomb blockade
(CB) phenomenon, which has been thoroughly explored in nano-
electronics. CB is typically observed in a single electron transistor:
under fixed bias, the current between source and drain peaks at
quantized values of the gating voltage12. The origins of this effect
stem from the many-body Coulomb interactions between electrons
and from the discreteness of the charge carriers13. Similar physi-
cal ingredients are at play in a nanoscale channel filled with a salt
solution (Fig. 1a): the ions also interact via Coulombic forces, and a
variable surface charge on the channel walls can play the role of the
gating voltage. One may therefore expect to observe an ‘ionic CB’,
namely peaks in the ionic conductance of a nanochannel at quan-
tized values of its surface charge. It is thus of interest that molecular
dynamics simulations1417 and experiments1820 have shown what
might be indirect signatures of ionic CB (although in the absence
of a gating voltage), and conductance gating by a surface charge
has been demonstrated in simulations of a biological ion channel
model21,22. These observations remain surprising, because ionic
systems in water at room temperature have specific features con-
trasting with electronic systems that may preclude the occurrence
of ionic CB. Beyond the absence of quantum effects, the fact that
ions are of both signs— while electrons are only negative—results
in Debye screening, which is expected to greatly weaken the many-
body interaction. It remains unclear under which conditions these
aspects may suppress ionic transport quantization.
Although pioneering analytical efforts have translated the results
established for electrons13 to the ionic case2224, a general theory
for ionic CB, incorporating the unique features of ionic systems in
contrast to their electronic counterparts, is still lacking. We develop
Model definition and numerical results
Our theory is based on a simple but general model of a nanochan-
nel which confines ions in one dimension (Fig. 1a). The channel
LR
, as opposed to nanopores which have
length L R. The nanochannel is filled with water, which under
confinement exhibits a priori an anisotropic dielectric permittivity
ϵ
(refs. 25,26), and it is embedded in a membrane with low permit-
tivity
ϵm
(whenever needed, we use
ϵ=2
m
). Under such conditions
(Fig. 1b), the electric field lines produced by an ion stay confined
inside the channel over a characteristic length ξ (ref. 27). This leads
to a stronger Coulomb interaction than in the bulk solution, which
is well described by the exponential potential
ξ
=ξ−∣ ∣∕
Vx kTxe() (1
)
x
B
T
This introduces a thermal length xT (ref. 28), which quantifies the
strength of the interaction. We detail in Supplementary Section 3
how the parameters ξ and xT are related to the channel geometry
and to the various dielectric constants. If the permittivity of con-
fined water is assumed to be the same as in the bulk, one has ξ 7R
and
=∕x R 2
T2B
, where
ℓ= .07nm
B
is the Bjerrum length in bulk
water. We shall use these relations in the following, keeping in mind
that taking into account the anisotropic permittivity would result in
a stronger interaction for a given confinement.
A charge is imposed on the confining surface and acts as a gate
on the system; here, we reduce the surface charge to a point-like
charge Q. The ions interact between themselves and with the sur-
face charge through the potential given in equation (1); depending
on conditions, an electric field E may be applied along the channel.
Ionic Coulomb blockade as a fractional Wien effect
Nikita Kavokine , Sophie Marbach , Alessandro Siria and Lydéric Bocquet *
Recent advances in nanofluidics have allowed the exploration of ion transport down to molecular-scale confinement, yet arti-
ficial porins are still far from reaching the advanced functionalities of biological ion machinery. Achieving single ion transport
that is tunable by an external gate—the ionic analogue of electronic Coulomb blockade—would open new avenues in this quest.
However, an understanding of ionic Coulomb blockade beyond the electronic analogy is still lacking. Here, we show that the
many-body dynamics of ions in a charged nanochannel result in quantized and strongly nonlinear ionic transport, in full agree-
ment with molecular simulations. We find that ionic Coulomb blockade occurs when, upon sufficient confinement, oppositely
charged ions form ‘Bjerrum pairs’, and the conduction proceeds through a mechanism reminiscent of Onsager’s Wien effect.
Our findings open the way to novel nanofluidic functionalities, such as an ion pump based on ionic Coulomb blockade, inspired
by its electronic counterpart.
NATURE NANOTECHNOLOGY | VOL 14 | JUNE 2019 | 573–578 | www.nature.com/naturenanotechnology 573
... Water notably becomes anisotropic, with a dielectric constant ε ⊥ ∼ 2 in the direction of confinement -while other directions remain unaffected, with ε ∼ 80. Overall, one should typically replace ε w by √ ε ⊥ ε in Eqs. (3) and (4) 31,39 . Hence, modifications to the dielectric properties of water at the nanoscale generally result in stronger electrostatic effects. ...
... In particular, this phenomenon replicates the physics of voltage-gated ion channels, and its modelization has opened the way for the design of solid-state channels with key properties. Examples include a single-ion pump (see Fig. 3A) 39 and ionic memristors (see Fig. 3B) 31 . On the experimental side, voltage-gating was observed in both short pores and 2D slit-like channels (see Fig. 3B and C) 32,40 . ...
... • Nanofluidic machines: In terms of circuitry, we believe that gated channels discussed in Section II are the building blocks which offer some of the most promising perspectives 32,39,57 . They potentially allow to control ion transport at the molecular level, through the use of 'switches' on the surface of the pores -playing a role similar to transistors in electronics. ...
Article
Nanofluidics, the field interested in flows at the smallest scales, has grown at a fast pace, reaching an ever finer control of fluidic and ionic transport at the molecular level. Still, artificial pores are far from reaching the wealth of functionalities of biological channels that regulate sensory detection, biological transport and neurostransmission - all while operating at energies comparable to thermal noise. Here, we argue that artificial ionic machines can be designed by harnessing the entire wealth of phenomena available at the nanoscales and exploiting techniques developped in various fields of physics. As they are generally based on solid-state nanopores, rather than soft membranes and proteins, they should in particular aim at taking advantage of their specific properties such as their electronic structure or their ability to interact with light. These observations call for the design of new ways of probing nanofluidic systems. Nanofluidics is now at the crossroads, there are new avenues to build complex ionic machines and this may allow to develop new functionalities inspired by Nature.
... The corresponding effective Coulomb interactions are stronger than in bulk water, their strength increasing with decreasing channel width 1 . In channels with diameters smaller than about 2 nm, these were predicted to cause strong ionic correlations, which in turn result in Bjerrum pairing and ion-exchange phase transitions [8][9][10]12 , leading to deviations from Ohm's law transport in the form of Wien effect conduction 13 and Coulomb-blockade-like phenomena 11,13 . The concept of reinforced Coulomb interactions due to a dielectric contrast was later extended to a twodimensional nano-slit geometry 14 , where it was found to produce even more striking non-linear ion transport: under the effect of an electric field, the ions may undergo a dynamical phase transition and assemble into dense clusters termed Bjerrum polyelectrolytes [14][15][16] . ...
... We find that, in all but the perfect metal cases, the measured current displays a non-linear dependence on the applied electric field and the channel conductance is hindered with respect to the Ohm's law prediction: this is a signature of the second Wien effect. The Wien effect has been historically known to govern the conduction in weak electrolytes; it has recently been predicted to occur as well in strong electrolytes when confined in nanoscale channels, because oppositely charged ions form Bjerrum pairs due to the reinforced Coulomb interactions in confinement 13,14 . In refs. ...
... In refs. 13,14 , the Wien effect was studied in the case of insulating channel walls. Here, we find that the Wien effect can persist even in the presence of electronic screening and the magnitude of the effect can be tuned by the channel wall's electronic properties. ...
... We first assume for simplicity that the channel is filled with water that has isotropic dielectric permittivity w = 80, and that it is embedded in an insulating medium with much lower permittivity m (for a lipid membrane 6 , m ∼ 2). The effective Coulomb interaction V (x) between two monovalent ions separated by a distance x on the channel axis can be computed exactly by solving Poisson's equation 7,11,12 . A simple approximate expression can be obtained for ...
... Here, we detail the analytical solution for the partition function of a 1D Coulomb gas-like system that was first introduced in ref. 12 . We set k B T = 1 until the end of Sec. ...
... We may check that the prefactor in Eq. (46) is the correct one by evaluating analytically the expression in Eq. (26) in the low concentration limit z T ≡ zx T 1. An analytical expansion of the function χ(k) in powers of z T was derived in ref. 12 ...
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Ion transport measurements are widely used as an indirect probe for various properties of confined electrolytes. It is generally assumed that the ion concentration in a nanoscale channel is equal to the ion concentration in the macroscopic reservoirs it connects to, with deviations arising only in the presence of surface charges on the channel walls. Here, we show that this assumption may break down even in a neutral channel, due to electrostatic correlations between the ions arising in the regime of interaction confinement, where Coulomb interactions are reinforced due to the presence of the channel walls. We focus on a one-dimensional channel geometry, where an exact evaluation of the electrolyte's partition function is possible with a transfer operator approach. Our exact solution reveals that in nanometre-scale channels, the ion concentration is generally lower than in the reservoirs, and depends continuously on the bulk salt concentration, in contrast to conventional mean-field theory that predicts an abrupt filling transition. We develop a modified mean-field theory taking into account the presence of ion pairs that agrees quantitatively with the exact solution and provides predictions for experimentally-relevant observables such as the ionic conductivity. Our results will guide the interpretation of nanoscale ion transport measurements.
... In particular, this phenomenon replicates the physics of voltage-gated ion channels, and its modelization has opened the way for the design of solid-state channels with key properties. Examples include a single-ion pump (see Fig. 3A) 32 and ionic memristors (see Fig. 3B) 24 . On the experimental side, voltage-gating was observed in both short pores and 2D slit-like channels (see Fig. 3B and C) 25,33 . ...
... • In terms of circuitry, we believe that gated channels discussed in Section II are the building blocks which offer some of the most promising perspectives 25,32,49 . They potentially allow to control ion transport at the molecular level, through the use of 'switches' on the surface of the pores. ...
... Voltage-gated nanofluidic channels. A Schematic design of a single-ion pump, as suggested by ref.32 . Several electrodes are placed along the surface of a nanotube, each bearing a local charge, with which an ion of opposite sign can pair up. ...
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... example, to discuss the capacitance of nanoporous systems [19][20][21] . The lattice models may be taken to the continuum limit, and the resulting path integral solutions have been used to discuss various ion-exchange phase transitions that arise in the presence of fixed discrete charges inside the channel 9,22,23 and the ionic Coulomb blockade phenomenon 13 . Such models are particularly rich theoretically, as they support a mapping to non-Hermitian quantum mechanics 24 . ...
... We first assume for simplicity that the channel is filled with water that has isotropic dielectric permittivity w = 80, and that it is embedded in an insulating medium with much lower permittivity m (for a lipid membrane 7 , m ∼ 2). The effective Coulomb interaction V (x) between two monovalent ions separated by a distance x on the channel axis can be computed exactly by solving Poisson's equation 8,12,13 . A simple approximate expression can be obtained for ...
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... New functionalities in nanoscale fluid transport have been achieved by exploiting analogies with condensed matter phenomena. The analogy between surface charge in a nanochannel and doping in a semiconductor [1] has led to the development of nanofluidic diodes [2,3] and transistors [4]; the similarity between ionic and electronic Coulomb interactions [5] allows for ionic Coulomb blockade [6][7][8][9]. More recently, it has been suggested thatbeyond mere analogies-nanofluidic transport can directly couple to electronic effects within the channel wall, as the solid-liquid interface can host fluctuation-induced electromagnetic phenomena [10,11]: energy and momentum transfer mediated by interfacial charge fluctuations. ...
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