Ion currents through nanometer-scale liquid-filled channels - biological ion channels, solid-state nanopores, carbon nanotubes, and two-dimensional van der Waals slits - exhibit low-frequency fluctuations whose power spectral density empirically follows Hooge's law SI(f)=αI2/(Nf)S_I(f) = \alpha I^2/(N f), or slight generalizations SI∝1/f1+aS_I \propto 1/f^{1+a}, a phenomenology documented for roughly five decades without an accepted microscopic explanation. Robin, Lizée, Yang, Emmerich, Siria and Bocquet (Faraday Discussions 246, 556, 2023) measured current fluctuations across two-dimensional channels with heights of 6-20 nm made of pristine graphite, pristine hexagonal boron nitride, and plasma-activated graphite, and proposed a formalism in which the 1/f factor emerges from non-flux-conservative fluctuations of ion density in the fluidic reservoirs flanking the channel, while deviations from Hooge's law are encoded in an in-channel spectral factor reflecting heavy-tailed ion residence times or adsorption-desorption kinetics. The formalism's microscopic grounding is incomplete (the physical origin of the power-law-distributed ion waiting times in pristine, atomically smooth channels remains unresolved in the literature), the reservoir mechanism has not been directly experimentally tested, and the decomposition's validity across channel materials, surface chemistries and geometries remains to be established. The open problem is to construct and validate a microscopic account of the low-frequency noise - deriving the 1/f factor and its material-dependent corrections from specified ion and interfacial dynamics - so that noise spectra become a quantitative probe of nanoscale ion transport.

Background

Transport of water and dissolved ions through channels with nanometer-scale cross-sections is the operating principle of biological ion channels and of artificial nanopores and nanofluidic devices. When a voltage or a salt-concentration difference drives an ionic current through such a channel, the measured current fluctuates; its noise is characterized by the power spectral density SI(f)S_I(f), the mean squared current fluctuation per unit frequency. At low frequencies (below roughly 100 Hz) the spectrum is dominated by so-called 1/f noise (also called flicker or pink noise), for which SI∝1/fS_I \propto 1/f. First observed in vacuum tubes by Johnson in 1925, 1/f noise has since been reported in metals, semiconductors, graphene, neuronal activity, and electrolyte conduction. In ionic systems it is conventionally described by Hooge's empirical law, SI(f)=αI2Nf,S_I(f) = \frac{\alpha I^2}{N f}, where II is the mean current, NN the mean number of charge carriers in the pore, and α\alpha a dimensionless constant typically of order 10−410^{-4} to 10−310^{-3}. The law is remarkably robust: it holds in micrometer-sized electrolyte-filled pores (measured by Hooge himself), in biological membranes, in solid-state silicon-nitride nanopores (Smeets et al., 2008, who confirmed the 1/N1/N scaling of the noise power), and in individual carbon nanotubes, where Secchi et al. (2016) showed that the 1/f-noise amplitude scales with the tube wall's surface charge, the effective charge decorating the confining wall. Often the exponent itself deviates: Siwy and Fuliński (2002) measured SI∝1/f1+aS_I \propto 1/f^{1+a} with aa anywhere between 0 and 1 and current-dependent, in membranes.

Despite this breadth, no microscopic explanation is accepted. Classical models interpret 1/f noise as conductance fluctuations arising from fluctuations in the number or the mobility of the charge carriers; the intermittency picture of Van der Ziel (1950) recovers Hooge's scaling only after averaging Lorentzian trapping spectra over a quenched distribution of barrier heights, i.e. by assuming strong microscopic disorder that single, atomically smooth, crystalline channels do not obviously possess. The relevant spectral structure appears at frequencies near 1 Hz, whereas most candidate microscopic processes (ion hops, mobility relaxation in water) live on picosecond-to-millisecond timescales, a severe timescale mismatch. Two recent theoretical threads reframed the question. Gravelle, Netz and Bocquet (2019) identified adsorption kinetics - the binding and release of ions at the pore wall - as a source of low-frequency noise in open nanopores. Marbach (2021) showed that the reservoirs of solution flanking even a minimal pore generate fractional noise with S∝1/f3/2S \propto 1/f^{3/2} in the number of particles inside the pore, because diffusive exchange of ions with the reservoirs produces long-lived correlations.

Robin et al. (2023) reported measurements on two-dimensional slit channels of height 6-20 nm, width 100 nm and length 1 μ\mum, fabricated by van der Waals assembly (stacking atomically thin crystals so that spacers define a slit), in three variants with similar geometry but radically different surfaces: pristine graphite and pristine hexagonal boron nitride (hBN), which are atomically smooth, and plasma-activated graphite, which carries a much higher surface charge together with nanometric roughness. In pristine channels the spectrum follows SI∝1/f1+aS_I \propto 1/f^{1+a} with aa between 0 and 0.5 at frequencies below about 100 Hz; the variance of the fluctuations is proportional to the dissipated power IΔVI \Delta V (a fluctuation-dissipation-type relation); and 1/f noise persists even when the current is driven by a salt-concentration difference rather than a voltage. Activated graphite channels show two power-law regimes, below about 2 Hz and above about 20 Hz, with an order-of-magnitude difference in amplitude, consistent with the adsorption-driven long-term memory independently measured in such channels (Robin et al., Science 379, 161, 2023). To rationalize all of this, the authors argue that current fluctuations are not flux-conservative - meaning that a fluctuation of the ion flux through the pore is not instantaneously mirrored by the current at the electrodes - because an ion emerging from the channel perturbs the charge density in the flanking reservoirs, and this perturbation is spread electrostatically over the reservoir and relaxes diffusively to the electrode over macroscopic times. Their main result factorizes the spectrum as SI(f)≃I2Nf(1+C~W(f)),S_I(f) \simeq \frac{I^2}{N f}\left(1 + \tilde C_W(f)\right), where the 1/f1/f factor originates in the reservoirs and C~W(f)\tilde C_W(f) encodes the correlation statistics of the ion-injection process WW by which the channel feeds ions into the reservoir: Poissonian (uncorrelated) passage reproduces Hooge's law; waiting times between ion passages distributed as a power law P(τ)∝1/τ1+aP(\tau) \propto 1/\tau^{1+a} produce the observed 1/f1+a1/f^{1+a} spectra; and a two-state adsorption-desorption process produces the two-regime activated-channel spectrum. The fits to the measured spectra are good, but the authors state explicitly that the physical interpretation of the required heavy-tailed waiting-time distributions on atomically smooth pristine surfaces is not clear, the reservoir picture rests on simplifying assumptions (a spatially homogeneous stochastic source, instantaneous electrostatic relaxation within the reservoir), and the framework's applicability across the wider family of systems where 1/f noise is observed - biological channels, carbon nanotubes, solid-state nanopores of various materials - is untested.

Problem Statement

Establish a microscopic, experimentally validated account of the low-frequency current noise that accompanies liquid ion transport through nanometer-scale channels - that is, of the empirical 1/f (Hooge-type) spectrum and its material-dependent deviations. The unresolved question comprises three jointly linked components: (1) Derive the 1/f factor from specified microscopic dynamics: determine whether it originates, as proposed, in the fluidic reservoirs flanking the channel through non-flux-conservative fluctuations of ion density, or from some other mechanism (for example surface-charge fluctuations or collective interfacial dynamics). The reservoir picture makes distinctive predictions - such as the dependence of the noise amplitude and of the low-frequency cutoff D/R2D/R^2 - with DD the ion diffusivity and RR a characteristic reservoir size - on the reservoir size - which offer a route to direct experimental or simulational tests. (2) Identify the physical mechanism generating the observed exponents SI∝1/f1+aS_I \propto 1/f^{1+a} with a=0a = 0-0.5 in pristine, atomically smooth channels - equivalently, the microscopic origin of the power-law (heavy-tailed) statistics of ion residence or waiting times in the channel. (3) Determine how channel material and surface properties - surface charge, roughness, adsorption sites, electronic properties, hydrodynamic slippage (the enhanced liquid flow at the wall resulting from low wall friction) - control the structure of the fluctuation spectrum, and establish (or delimit) the generality of the factorization SI(f)=(1/f)×Schannel(f)S_I(f) = (1/f) \times S_{channel}(f) across channel materials, surface chemistries and geometries, including its relation to surface-based mechanisms such as adsorption kinetics proposed in the nanopore literature. These components jointly serve one overarching objective: a predictive microscopic theory of nanochannel current noise, sufficient that measured noise spectra can be used as a quantitative characterization of nanoscale interfacial ion dynamics. A resolution requires all three components; establishing any one of them alone does not resolve the task.

References

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