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Menghua Zhao, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Jean Comtet

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[Defect-modulated ionic friction at hBN/water interfaces](https://mdr.nims.go.jp/datasets/16a5324b-7df9-4583-9316-6863882b9f37)

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Defect-modulated ionic friction at hBN/water interfacescommunicationsmaterials ArticleA Nature Portfolio journalhttps://doi.org/10.1038/s43246-025-00910-3Defect-modulated ionic friction at hBN/water interfacesCheck for updatesMenghua Zhao 1,2, Kenji Watanabe 3, Takashi Taniguchi 4 & Jean Comtet 1Charge transport at solid/liquid interfaces is vital to energy conversion, electrochemistry, andbiological activities. These buried interfaces are the locus where continuum approaches break down,and molecular details become of utmost importance, with traditional ensemble-averaged studiesgiving an incomplete picture of the dynamics.Here,webuild upon recently developed single-moleculemicroscopy optofluidic platform, to investigate the statistics of single charge transport at aqueoushexagonal Boron Nitride interfaces, demonstrating the microscopic origin of its non-Gaussiancharacter and the control of transport by irradiation-induced surface defects. By increasing irradiationof the hBN crystals, wemodulate themorphological distribution of adsorption sites, leading to a slow-down of interfacial charge transport, akin to an increasing frictional interaction. Charge hoppingdisplacements feature exponentially-decaying arms, strongly departing from Gaussian distributions.2D Brownian dynamics simulations evidence that these exponential tails originate from molecularjumps between trapping sites, allowing a consistent match between statistical distributions and theeffective diffusion coefficient. Our study highlights the key yet overlooked role of defects in regulatinginterfacial charge transport, with relevance for energy applications.Ionic transport at the solid/liquid interface is ubiquitous both in naturalsettings and industrial applications, and is crucial for a large number ofdomains ranging from biology1,2, blue energy conversion3,4, filtration5,6, up toelectrochemical and catalytic reactivity7. Interfacial transport accordinglystands at the crossroad where continuum descriptions meet with the mole-cular granularity of the interface. Molecular details related to the solid surfacecan accordingly have massive impacts on interfacial transport. Versatileexperimental toolboxes, including vibrational sum frequency generationspectroscopy8,9, electrokinetic detection10,11 and atomic force microscopy12,13have been developed to study the nanoscale ion and liquid transport at theinterface, revealing new physical phenomena, such as fast flow in carbonnanotubes14, high selectivity of biological ion channels5,6 or giant energyconversion rate3,15. However, despite substantial progress in recent years, themolecular processes underlying ionic transport at interfaces remain elusivedue in part to experimental challenges associated with the direct observationsof ion and solvent transport at the molecular scale16–18. Conventional tech-niques, including the aforementioned ones, mostly employ indirect mea-surements, detecting ensemble quantities that are temporally and spatiallyaveraged. Those macroscopic fluxes lose precise information related to thetemporal and spatial details of single-moleculedynamics19. Inparticular,whilenon-Gaussian processes at liquid–solid interface have long been recognizedfor single particle transport20–23, the generalizability of these observations tosingle-charge and single-ion dynamics, as well as the microscopic origin ofsuch peculiar transport, remains poorly understood. Finally, recentadvancement innanofluidicshas furtherpushed theconfinementof the liquidtransport down to one single molecule size5,6,18, a situation where continuumapproaches are clearly breaking down16,24,25, calling for an urgent under-standing of the interfacial ion transport at the single molecular scale16,17,25,26.Following the recent discovery of defect-induced solid-state emittersin hexagonal Boron Nitride (hBN) materials27, visualization of interfacialcharge transport has recently become a reality28–30. The emergence offluorescent sensors based on quantum emission of hBN defects opens newroutes to study the interfacial transport at the single molecular scale in alabel-free way29,30. Thanks to its large band gap of ~6 eV, the hBN can hostperturbed energy states created by surface defects that can be opticallyactivated by an external light excitation at room temperature31. Thispeculiar propertyhas recentlybeen shown togrant thehBNunprecedentedability for direct label-free readout of charge and ionic exchanges with thesolvent30,32,33 or biomolecule adsorption34. Emergent studies have beencarried out with this technique to reveal the proton transport pathway30,physiochemical adsorption32,35 and electrochemical kinetics36 at theinterface.1Laboratory of Soft Matter Science and Engineering, ESPCI Paris, PSL University, Sorbonne Université, CNRS, Paris, France. 2School of Information Mechanicsand Sensing Engineering, Xidian University, Xi’an, China. 3Research Center for Electronic and Optical Materials, National Institute for Materials Science,Tsukuba, Japan. 4Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan. e-mail: jean.comtet@espci.frCommunications Materials |           (2025) 6:200 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s43246-025-00910-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s43246-025-00910-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s43246-025-00910-3&domain=pdfhttp://orcid.org/0000-0002-7320-8692http://orcid.org/0000-0002-7320-8692http://orcid.org/0000-0002-7320-8692http://orcid.org/0000-0002-7320-8692http://orcid.org/0000-0002-7320-8692http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-2389-3879http://orcid.org/0000-0003-2389-3879http://orcid.org/0000-0003-2389-3879http://orcid.org/0000-0003-2389-3879http://orcid.org/0000-0003-2389-3879mailto:jean.comtet@espci.frwww.nature.com/commsmatFollowing these preliminary achievements, it nowappears of particularinterest to understand the role of the structural properties of the solidsurface, such as the presence and distribution of defects—inevitable in real-world materials—on interfacial charge transport and ionic friction. Defectshave indeed been suggested to play a central role in determining interfacialmolecular transport in nanofluidics24,37–41, with reports of the influence ofdefect size24, reactivity38 and chemical structure39 on the solid/liquid friction.However, most of these studies stay at the conceptual level in the context ofsimulations, lacking convincing direct experimental proofs.Here,weuse single-molecule tracking techniques to study the statisticalproperties of single proton transport on the aqueous hBN interface andfurther address their control with surface defects. By tuning the defectivestate through mild plasma treatments, we first evidence two morphologiesof adsorption sites. Focusing on charge dynamics, we report a generic non-Gaussian yet Fickian process for single charge diffusion and subsequentlydemonstrate the strong role of surface defects in setting transport properties.To rationalize these observations, we perform ad hoc simulations ofBrownian particles undergoing free 2D Brownian diffusion in a defectivelandscape. Adsorption at defect sites is modelled explicitly and treated asemission events, allowing for a direct comparison with our experimentaldistributions. Remarkably, both our experiments and Brownian dynamicssimulations evidence an exponential scaling for the elementary jumps. Suchpeculiar distributions originate from the hopping of charge between thesurface defects, highlighting the molecular origin of this non-Gaussiansurface transport process. The diffusion coefficient on the plasma-modifieddefect state is accordingly strongly set by such adsorption events, allowingusto consistently match these statistical distributions with the effective diffu-sion coefficient by a scaling law arising from the desorption-mediatedtransport. Our study sheds light on the molecular origin of the non-Gaussian interfacial distribution for ionic transport and provides newinsights on the role played by surface defects in the interfacial chargetransport: the greater the density of surface defects, the lower the interfacialdiffusion coefficient and the higher the effective frictional interactions felt bythe surface trapped charges. Our study finally suggests that a simple tuningof the defect density offers an alternativeway of regulating liquid and chargetransport in the applications of battery design, clean energy conversion anddesalination.Results and discussionExperimental methodology for single proton trackingTo access interfacial molecular transport at the single-charge level, weharvest the quantum fluorescent emission originating from hBN surfacedefects27 and their peculiar interactions with liquids28–32. Atomically smoothhBN multi-layers are first mechanically exfoliated from bulk crystals ontransparent glass coverslips. Subsequent air plasma etching is performedusing a plasma cleaner (Femto Science Cute, Femto Science Inc., Korea)with the following settings: gas time 15 s, base pressure 0.15 Torr, processpressure 0.5 Torr, plasma power 10 W, generation frequency 50 kHz andpurge time 15 s. This treatment induces fluorescent surface defects that areactivated in contact with aqueous solutions28,29,35 once excited in wide-fieldby a green laser (561 nm), with fluorescent emission at ~585 nm29. Fluor-escent emission is subsequently imaged on a low-noise EMCCD camera viaan×100 oil-immersionmicroscope objective, shown inFig. 1a. The incidentexcitation light is removed from the light path by dichroic and emissionfilters, allowing only the emitted light to be effectively captured by thecamera sensors (see Methods). Based on previously reported pH depen-dence of the fluorescence emission and ab-initio molecular dynamicssimulations30, a potentialmechanismresponsible for the quantumemission,is presented in the inset of Fig. 1a, where the non-emissive negativelycharged boron monovacancy defect, V�B , can react with surrounding sol-vated aqueous protons, leading to the transition towards a fluorescentprotonated defect VBH. By spatially and temporally tracking the correlatedemission events, we are able to dynamically probe proton charges walkingon hBN surfaces.The typical fluorescence signal consists of diffraction-limited spotsresulting fromthe activationof single defects, as shown inFig. 1b.To localizethe position of these emitters below the diffraction limit, we fit their pointspreading function (PSF) by integrated 2D Gaussians, allowing for thesubsequent localization of their position with nanometric resolution42. Thelocalization uncertainty is estimated by σLOC � σPSF=ffiffiffinp, with σPSF thestandard deviation of theGaussianfit andn the numberof photons receivedby the camera. The camera exposure time,Δt, was set at 20ms during all therecordings if not specifically specified. The trajectories were tracked viaspatially and temporally correlating neighboring defects with themaximumdisplacement,Δl=1 μm, allowing the detection of diffusion coefficient up toΔl2/(4Δt) ≈ 10−11 m2s−1. All experiments were performed using Milli-Qpurifiedwater for a total duration of 10,000 frames, with approximately tensof thousands of localizations and several thousand trajectories recorded foreach observation.The causal nexus for our trajectory analysis is legitimizedon thebasis ofthe large difference between the local hopping distances over which weevidence correlatedmotion (atmost 1 μm) and the typical distance betweenactivefluorescent sites present at each frame (larger than5 μm).As such, ourassertion regarding the observation of trajectories and spatiotemporal cor-relations is further supported by a shuffling analysis of the recorded imagesequences (see Supplementary Fig. 1), which demonstrates that thetrajectory-like correlations observed in the original data vanish when thetemporal order of frames is randomized.We further argue that transport between defects occurs through bidi-mensional diffusionof protons at the interface, consistentwith thepresence ofa physisorption energy well in the vicinity of the aqueous hBN surface, asevidenced by previous simulations30,43. Experimentally, we note that (1) themeandistance travelledby aproton in solutionduring a single frameexposureisffiffiffiffiffiffiffiffiffiffiffiffiffiffiDbulkΔtp � 100 μm, a distance far greater than the lateral dimensions ofthe hBN flakes (~10–20 μm) and (2) themean distance between free protonsclose to thehBNsurface isof theorderof180 nmatpHof5. If aprotonwere toFig. 1 | Single-molecule imaging of optically active defects. a Schematics of thesingle-molecule fluorescence microscopy setup. hBN flakes with optically activedefects (red emission point) are exfoliated onto the glass coverslip and put in contactwith water. The fluorescent quantum emission from hBN defects is triggered byexcitation through a 561 nm laser and projected onto an EMCCD camera. The insetillustrates the envisioned acid-base charge transition between non-emissive defectsin the negatively charged V�B form and fluorescent defects in the protonated VBHform30. b Principle for emitter super localization, showing a wide field image of ahBN flake with the flake edge indicated by the grey line, and a diffraction-limitedwhite spot identified as single-molecule emission. As shown in the inset, the 2Dspatial intensity profile can be approximated by aGaussian Point Spread Function ofwidth σPSF, allowing for a final uncertainty in the center position as σLOC � σPSF=ffiffiffinpwith n the number of emitted photons.https://doi.org/10.1038/s43246-025-00910-3 ArticleCommunications Materials |           (2025) 6:200 2www.nature.com/commsmatleave the interface after desorption from a first defect, it would thus becomeinstantaneously indistinguishable comparedwith the other nearby proton.Asthis indistinguishability would be in contradiction with our observations oftrajectories, the observed correlations in the activation of nearby sites in oursystem thus imply restricted exchanges between the surface and the bulk inour experimental system, characterizedby 2D interfacial charge transport.Yetwe acknowledge that obtaining direct experimental evidence regarding 2Dversus 3D transport would requiremeasuring the distribution of time it takesforprotons togo fromone site to theother, comparing themwithfirst-passagetimes for 2D or 3D diffusion9,44,45. Unfortunately, measuring such quantitieswould require temporal resolutions of microseconds for the monitoring ofemitter dynamics30,35, which is out of reach of current experimental approa-ches at the single-molecule scale.A qualitative description of active defect distribution and protondynamicsWe first report in Fig. 2 on the effect of plasma treatment on the spatialdistribution of active defects. The subsequent activation and localization offluorescent defect sites over a large number of frames can be used to createsuper-resolvedmaps inwhich each point corresponds to the position of oneoptically active site, equivalent to one proton charge adsorption event. Weshow in Fig. 2a such images reconstructed from 10,000 frames for twoplasma treatment durations, tplasma, of respectively 5 s (i) and 40 s (ii). Asshownby the zoom-ins of Fig. 2a, the spatial distribution of defect activationis highly heterogeneous, with high intensity spots corresponding to highdensities of localization events, coinciding with sparser localizations. Suchobservations are consistent with previous reports in which a similar plasmatreatmentwas applied28,29,35.Here,we refine thesequalitative observationsbyperforming quantitative clustering analysis, using an inter-localizationthreshold distance Dl of 20 nm (more details in Supplementary Fig. 2),allowing us to disentangle clusters from sparse localizations. Results of thisclustering approach are shown in Fig. 2b, where clustering localizations areshown as red dots, while the sparse localizations appear in blue. Fig. 2creveals the detailed structure of single clusters (additional examples inSupplementary Fig. 2). Clusters are of either one single strong adsorptionzone (ii) ormultiple strong adsorption zones (i). The cluster consists of tensto hundreds of adsorption events, resulting from persistent adsorptionsfrom alternative charges on various trajectories at different time frames(Supplementary Fig. 3). To characterize whether these sites actually corre-spond to groups of defects, or result from localization uncertainties of asingle defect active overmany frames, we computed their FullWidth atHalfMaximum, FWHM characterizing the spatial spread of these clusters. Asshown in Fig. 2c (additional examples in Supplementary Fig. 2), the FWHMcan reach up to 200 nm, a value much higher than the localization uncer-tainty expected for a single defect, pointing to the fact that clusters docorrespond to dense regions carrying distinct defects. Note, however, thatdue to the finite localization precision of our setup, we are unable to accessthe actual defect distribution inside clusters.Based on this clustering methodology, we can analyze in more detailthe effect of plasma treatment on the defect density and clustermorphology.We first report in Fig. 2d, the distribution of cluster area for various tplasma.This distribution follows an approximate power-law distribution, with ascaling ~1.8, characterizing the large spread in cluster size. We hypothesizethat such broad distribution is a signature of a complex cluster creationregime, proceeding from local nucleation and growth at highly damagedregions.Despite the apparent independence of the cluster size on irradiationtime demonstrated in Fig. 2d,we evidence in Fig. 2e, f a clear effect of plasmaduration on cluster density and localization density, which monotonicallyincreases up to tplasma = 15 s. The saturation at high tplasma might be asso-ciated with a shift from emissive to non-emissive defects upon continuousirradiation through an exact mechanism which remains to be properlyaddressed. It should be noted that varying the critical length used to definethe cluster results in slight alterations to the reported cluster area, but doesnot affect the aforementioned trends.Wenow turn in Fig. 3 to the discussion of the protondynamics on suchdefective landscapes.A remarkable feature of the defect activation dynamicsis the subsequent time-correlated activationofnearbydefect sites, consistentwith the transport of a single activating species along nearby defect sites.Fig. 2 | Spatial localization of emissive defects.aHeat map of active emissive sites on hBN flakes withtplasma of 5 s (i) and 40 s (ii), respectively. Visualizationof localization events is based on a density function, ρ,using the averaged shiftedhistograms58with thebin sizeof 114 nm× 114 nm. Insert images, 2 μm×2 μm, arethe zoom-in of the region of interest (ROI) randomlyselected. b Localization details of the ROI on (a). Redand blue dots correspond to the adsorption eventsinside the cluster island and the sparse zone, respec-tively. The solid lines indicate the traveling trajectory ofsingle protons with associated trajectories labeled fromI to V. c Distributions of localization events in therectangle marked in (b). The scale bar close to thedistribution curve corresponds to the number of loca-lization events within each bin of width 10 nm. Thesolid lines areGaussianfittings to the clusterswith theirFWHM indicated by arrows. d Ensemble distributionprobability of the cluster area at various tplasma from 5 sand 40 s, defined as the ratio of the number of clusterswithin a given area bin to the total number of clusters.The dashed line shows a power law with the powerexponent of−1.8. Bin size: 2000 nm2. eCluster densityon flakes at different tplasma. Each data point corre-sponds to the observation on a single hBN flake. Theblack solid square is the average of all observations ateach tplasma. f Localizations density inside clusters(trapped, red color) and outside clusters (mobile, blue)on fakes at various tplasma. Each data point correspondsto the observation on a single hBNflake. The bold solidpoint is the average of all observations at each tplasma.40167 nm73 nmaCluster area (nm2)20104ytilibaborP5 μmTrappedVIVIIITrappedMobileIII115 nm4010-410510210-2Plasma duration (s)0 40515200 40103Cluster density (μm- 2)Loc. density (μm-2)Plasma duration (s)1065 s40 s10 s15 s20 s(i) (i)(i)(ii)(ii)b cd e f(ii)s 5 :amsa lPs 0 4 :amsalP08172534200 nm200 nm400 nm400 nmρ (nm-2)https://doi.org/10.1038/s43246-025-00910-3 ArticleCommunications Materials |           (2025) 6:200 3www.nature.com/commsmatGiven the strong dependence of the distribution of activated defects onplasma treatment, as shown in Fig. 2, it is intriguing to ask whether a similardependencemay be observed on the interfacial charge dynamics. To furtherelucidate this point, we performed single-molecule tracking on surfacessubjected to different plasma treatments. Figure 3a depicts ensemble mapsof one half of trajectories longer than four frames, for two distinct plasmairradiation time: 5 seconds (top panel) and 10 seconds (bottom panel). Oneconcurrent feature is that proton charge trajectories exhibit intermittentkinetics withmany of them shifting between themobile state (hopping) andthe immobile state (trapping), as demonstrated by the trajectories (1) and(2) in the insert of Fig. 3a. Meanwhile, individual trajectories show sig-nificant heterogeneity: some remain almost immobile (trajectory (2)), whileothers exhibit high mobility (trajectory (1)). We clearly observe a largenumber of mobile trajectories for tplasma = 5 s in Fig. 3a, while that is not thecase for tplasma = 10 s. A detailed correlation between the dynamical trajec-tories and the localizations is present in Fig. 2b. Two types of trajectorymodes are revealed at short tplasma of 5 s in Fig. 2b(i). Single protons eithertravel in an intermittent way in the sparse zone (Hopping mode) or isentirely adsorpted onto the clusters (Trapping mode). Those observationsdo not stick for the cases at longer tplasma where more andmore charges arelocally trapped and eventually the majority become trapped at extremetplasma. As manifested by the Fig. 2b(ii), protons are strongly adsorbed ontocluster sites and exhibit limited mobility, demonstrated by the typical tra-jectories III-V where protonsmostly stroll inside cluster sites (trajectory III,adsorption) and rarely jump between cluster sites (trajectories IV-V,hopping).To address the observations above in a more quantitative way, wefollow a traditional route by using the square displacementSD(t) = 〈∣r(t)− r(t = 0)∣2〉 to account for the dynamics at the singlemoleculescale. As evidenced in Fig. 3b, the squared displacement calculated onindividual trajectories shows pronounced fluctuations associated with thesuccessive transitions between trapping and hopping states. In order tooffset the largediscrepanciesoriginating fromsingle trajectories,we resort totime and ensemble-averaged mean square displacement (MSD) to accountfor the collective diffusion behavior. As illustrated in the temporal evolutionof theMSD presented in the inset of Fig. 3c, fluctuations observed in indi-vidual trajectories are effectively averaged out, resulting in a relativelyFickian diffusion process withMSD= 4Dt for short time t < 0.5−1 s.Comparing treatment for 5 and 10 s’ irradiation, we do observe a significantdifference in the ensemble-averaged diffusion coefficient, withD = 8.7 × 10−14 m2s−1 and D = 1.2 × 10−14 m2s−1 for the blue and red curve,respectively. This decrease in protonmobility can be equivalently seen as anincrease in the frictional interaction of the protons with the surface, char-acterized by a friction coefficient ξ = kBT/D, equal respectively toξ = 4.7 × 10−8 kg s−1 and ξ = 3.4 × 10−7 kg s−1.We can subsequently probe the overall evolution of interfacialmobilityas a function of plasma treatment, the results of which are presented inFig. 3c, reported both as an averaged diffusion coefficient (left axis) and aneffective friction coefficient (right axis). The scattered data points wereobtained from tens of independent experiments performed on differentflakes, with the surface treatment following the same protocols. As evi-denced here, we observe a large reduction of surface mobility uponincreasing plasma duration, which may be attributed to an elevated trapdensity that impedes proton jumping.Non-Gaussian yet Fickian transport processFollowing the aforementioned observations, we propose an underlyingmechanism responsible for surface-regulated proton transport. We firstexamine the statistical distributions of the jumping displacement and showthat the proton dynamics on the hBN surface is a Fickian process, while itsjumping displacement shows a non-Gaussian distribution characterized bya central Gaussian and a side exponential decay.We focus here on a flake submitted to a short-time plasma treat-ment, resulting in relatively large mobility, and we imaged for 180,000frames to increase trajectory statistics. To analyze in more detail thestatistical properties of the observed randomwalks, we plot in Fig. 4a thedistribution of hopping displacement at various observation time steps,Δt. A central Gaussian-like distribution, with a constant standarddeviation σ typically ~20 nm, is revealed consistently for all Δt. Theapparent independence of this central distribution onΔt implies that it isassociated with time-invariant events which are likely due to the localtrappings evidenced in Figs. 2 and 3. We anticipate that the shape of thiscentral peak stems from a combination of localization errors of thesingle-molecule detection technique and jiggling motions of the trappedcharges on the local clusters. Opposite to this behavior at short dis-placement, the long-tail arms of these distributions show a pronounced5 μmat plasma=5 sD (m2 s-1)Plasma duration (s)10 20 4010-1510-13cbt (s)0 0.510-13 x10-13t (s)0 0.510SD(m2 )20 421300 nmTrapping221x 10-14t (s)0.1 0.4101MSD (m2 )Hopping300 nmt plasma=10s10-810-5ξ(kgs-1 )10-710-610-14xFig. 3 | Effect of the surface defects on proton dynamics. a Summary of one-half ofthe ensemble trajectories longer than four frames superimposed together from10,000 frames' recording at two tplasma of 5 s (top) and 10 s (bottom). The insert:zoom-in of the trajectories 1 and 2 corresponding to the ones highlighted in blue andorange, respectively. Trapping and hopping events are defined by a critical dis-placement between the adjacent two hopping events, below which the “Trapping”event is defined and above which the “Hopping” event is defined. The circle on thetrajectory indicates the localization uncertainty; dr, dx and dy denote the displace-ment, x-displacement and y-displacement between the adjacent emission events,respectively, with jdrj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijdxj2 þ jdyj2p. b Single trajectory square displacement forthe cases on (a) at tplasma of 5 s and 10 s, respectively. t denotes the time and t = 0indicates the beginning of each trajectory. c Diffusion coefficient D and frictioncoefficient ξ of protons at aqueous hBN interfaces from ensemble MSD as a functionof tplasma. Each data point comes from independent measurement from one singleflake. The insert: ensemble trajectory mean square displacement for the cases on (a)at tplasma of 5 s and 10 s, respectively. The averaged diffusion coefficientD is extractedfrom the linear fit for the initial 0.4 seconds.https://doi.org/10.1038/s43246-025-00910-3 ArticleCommunications Materials |           (2025) 6:200 4www.nature.com/commsmatevolution, exhibiting a clear widening for increasingΔt. This behavior is aclear signature of a diffusive-like transport process, where the distanceexplored by jumping particles becomes longer as the observation timeincreases. Phenomenologically, this distribution can be approximated byan ad hoc exponentially decaying function, with the probability densityP(∣dx∣) varying asPðjdxjÞ � exp � jdxjλ� �ð1ÞNote that for clarity, the distribution in Fig. 4a is shown by normalizing theprobability density to a value of unity at the center distribution.Toprobe thetransport process, we plot in Fig. 4b, the evolution of this exponential decaylength, λ, onΔt, which grows as λ � ffiffiffiffiffiΔtp, asserting its diffusive-like nature,while the central width σ remains constant over the observation time. Theseobservations indicate that charge dynamics on the aqueous hBN surface arecharacterized by a highly non-Gaussian yet Fickian process. Such processesare reminiscent of similar behaviors observed in other systems, includingcolloidal beads in entangled actin suspensions20,21 and polymer chains’surface dynamics at low molecular weight46,47. A traditional description forsuch exponential distributions is to interpret the non-Gaussian diffusion asbeing due to the convolution of Gaussian series, which represents thenormal modes of microscopic fluctuations20,21. Nevertheless, an exactrelation between the observed non-Gaussian process and its microscopicorigin is still lacking. We propose below a simpler interpretation for thisempirical observation, rooted in the microscopic spatial distribution ofdefects.We decipher the proton charge dynamics in the framework of thecontinuous-time randomwalk (CTRW), in which particles are described asalternating between trapped periods, characterized by a random adsorptiontime τ, followed by long-distance jumps. In order to quantify the adsorptiontime distribution, we define a critical jumping displacement, lc = 80 nm, tofind the residential events (∣dr∣ < lc) and the hopping events (∣dr∣ > lc), fromwhich we statistically estimate residential time distributions from theensemble trajectories, shown in Fig. 4c. The selection of lc is based on thelocalization uncertainty andmean cluster size, and itmakes negligible effecton their distributions46. Due to our definition based on a critical jumpingthreshold and to the spatial repartition of the active site in the form of acluster, adsorption events will characterize not only adsorption at single-defect sites but also at clusters.A situationwheredesorption is characterizedby a single energybarrier,would lead to an exponential distribution of the residence time21,29,35,46,47.However, as shown in Fig. 4c, the distribution of adsorption time τ ratherfollows a power-law scaling, as ψ(τ) ~ τ−α with the exponent α≃ 2.5, sug-gesting the presence of a broad distribution of desorption energies21. Theheterogeneity in defect structure evidenced in Fig. 2 suggests that such adistribution might originate at least in part from spatial heterogeneity anddefect-to-defect or cluster-to-cluster variations. However, non-exponentialdistributions in desorption times might also arise at the single-emitterlevel48, due e.g., to complex charge escape processes or fluctuations in theFig. 4 | Non-Gaussian yet Fickian diffusion process. a Histogram showing thenormalized distribution of jumping displacement for increasing sampling time, Δt,from 21.3 ms (green) to 213.0 ms (orange) as integer multiples (n) of the initialsampling time (τexp) with Δt ¼ nτexp. Red and black dotted lines indicate theGaussian fitting for the central small displacement and the exponential fitting for theside long displacement, respectively. Exponential decay length, λ, is a function of thesampling timeΔt. Binwidth: 20 nm. bLog–log plot of λ and σ as a function ofΔt. Thedotted line is a power lawfitting to λwith a slope of 0.52, close to the Fickian diffusionpower 0.5. The red square is the width, σ, of the central Gaussian, showing noevolution with the sampling time. c Distribution of τ at different tplasma. Bin size:21.3 ms. ψ is shifted by a factor of 0.5 for each tplasma from 5 s to 40 s for visual clarity.d Schematic of the Brownian random walker problem. The particle exhibitsintermittent Brownian random walking between defects. When encountering adefect, the particle is transiently adsorbed for a random period τ. fm is the molecularfrequency. e Example trajectory of one Brownian dynamics simulation run. lD, Ra,and dr are the mean inter-defect distance, adsorption radius and surface displace-ment, respectively. The red spot marks the adsorption event (not to scale).f Simulations: distribution of particle displacement at various Δt [2–1000 ms],represented by different colors. 〈τ〉, lD and bin size are 40 ms, 100 nm and 20 nm.g Dimensionless log–log plot of λ, normalized by the mean inter-defect distance lD,as a function of Δt, normalized by 〈τ〉. The dotted line is a power fitting to simu-lations with a slope of 0.5, reminiscent of the Fickian diffusion power law. The twoinsert schematics show the proton trajectories at two regimes separated by thecondition on jumping frequency Δt〈τ〉−1~1.https://doi.org/10.1038/s43246-025-00910-3 ArticleCommunications Materials |           (2025) 6:200 5www.nature.com/commsmatlocal environment. Disentangling the respective contributions of theseintrinsic and extrinsic heterogeneities might be a valuable direction forfuture studies. Interestingly, the value of α obtained here is exactly the sameas that in other systems where the size of particles differs from ours byseveral orders’ magnitude46,47,49, suggesting the universality of this peculiarpower-law distribution, whose origin still remains to be uncovered.To elucidate the microscopic origin of the exponential tails of thejumping displacement, and its evolution with observation time, we con-ducted 2DBrowniandynamics simulations in the framework of theCTRW,illustrated inFig. 4d: Protons explore freely the 2D space at the aqueoushBNsurface, obeying a standard Brownian dynamics with Gaussian distributedelemental steps. Once the simulated proton meets an active defect site, it istemporarily trapped and ceases to move for a random adsorption timebefore its next walking step. To simplify the simulations, we neglect site-to-site heterogeneities and draw the adsorption time at each defect site from aneffective power-law distribution, as discussed above. Proton displacement istracked according to such adsorption events, shown by the red dots inFig. 4e. More details can be found in the Methods section: Browniandynamics simulations.We report in Fig. 4f the resulting simulated hopping displacementdistribution by varying sampling time Δt (more extensive results in Sup-plementary Fig. 4). Remarkably, we recover the main features as in ourexperiments shown inFig. 4a,with a bimodal distribution characterized by acentral peak, corresponding to local adsorptionand localizationuncertainty,and long distance exponentially decaying tails, associated with inter-defecttransport.Wenote that the central peak in our simulations does not directlyreflect the confinedmotionof protonsbecause experimentally it comes fromthe localization uncertainty and possibly confined dynamics insideclusters46,49–51.Regarding the exponential arms, they are in high synchronizationwiththe experimental observations, exhibitingwideningwhen increasingΔt.Wenote that our simulation is in stark contrast to the majority of previoussimulations, where a power/exponential law distribution of the elementaljumping displacement is pre-assumed46,47,49,51. In our case, the single protonjumps at molecular frequency, and the associated displacement obeys a freeGaussian distributionduring transport betweendefects, naturally giving riseto an exponential distribution due to the interaction with the defects. Theobserved localization events correspond to the adsorption on the activedefect site, and the non-Gaussian displacement revealed here is stronglyreminiscent of the empirical displacement pre-defined elsewhere46,47,49,51.Such a simulation provides evidence that, in the context of our system, theexponential distribution can originate from the intermittent hoppingbetween surface adsorption sites. However, a direct interpretation of suchdisplacement distributions in other contexts should consider the specificityof each systems.Focusing on the exponential decay length, we show the evolution of thetypical decay length λwith the sampling time in Fig. 4g.Note that this lengthis normalized by the mean inter-defect distance, lD, defined from the defectdensity ρ as lD= ρ−1/2, a parameterwhose effectwewill address inmoredetailin the next section.Notably, two regimes are evidenced in Fig. 4g, dependingon the relative value of the observation time Δt with the mean adsorptiontime 〈τ〉. We argue that this transition originates from the limited adsorp-tion of protons at the defect sites. As depicted by the schematic insert on theright, for Δt > 〈τ〉, protons can jump between various adsorption sitesduring theobservation timeΔt, leading toadiffusive-like feature fromwhichwe recover the Fickian behavior λ � ffiffiffiffiffiΔtp, in the same fashion that isexperimentally evidenced in Fig. 4b. The dimensionless plot of the expo-nential decay length on the sampling time in experiments agrees well withsimulations, when λ takes the value of 140 nm, a reasonable value for theinter-defect distance28, which validates our simulation protocols and furtherstrengthen our claim of the molecular origin of the exponential distribu-tions. As depicted by the insert schematic on the left, for Δt < 〈τ〉, mostprotons are either immobile between each frame or visit at most one site,corresponding to the shaded geometry-limited regime in Fig. 4g. As aconsequence, the hopping displacement becomes independent of Δt andcorresponds to the distribution of distance to reach the first encountereddefect.Defect-modulated transport and ionic frictionIn light of the displacement distributions discussed above, we proceed toexamine in more detail the dependence of hopping statistics on irradiationtime tplasma, which sets the local defect landscape. We report in Fig. 5a thetypical evolution of the hopping displacement for distinct tplasma at a fixedsampling time Δt = 20ms. Increasing tplasma from 5 to 40 s leads to a clearnarrowing of the distribution, consistent with a reduced probability for longdisplacement, and a reduction of the diffusion coefficient D, evidencedin Fig. 3c.Alongside the qualitative analysis made in Fig. 3a, another inter-pretation of the above findings is to inspect the adsorption site morphologyfrom the accumulated emission events, summarized in Fig. 2. The density ofclusters increases with tplasma, indicating a greater prevalence of the trappedtrajectories, as shown in Fig. 2e. Moreover, the ratio of the localizationdensity corresponding to the immobile state to that of the mobile stateincreases with an increase in tplasma, as illustrated in Fig. 2f. In addition, themean adsorption period increases with plasma duration, effectively slowingdown the proton transport as shown in Fig. 5b. These findings suggest thatprotons exhibit reduced mobility on flakes at larger tplasma that port moredefects.To clarify this behavior in a quantitative way, we carried out additionalBrownian dynamics simulations, probing specifically the role of spatialdefect distribution, characterized by the mean inter-defect distance, onjumping displacement. To seek a simple mechanism, we assume that theplasma treatment only modifies the defect morphology but not the otherphysicochemical properties of the surface. As shown in Fig. 5c, uponincreasing lD (going from black to red distributions), we observe a clearc d-500 0 500dx (nm)10-110-3Ga-500 0 500dx (nm)10-110-3GlD10 20 404060Plasma duration (s)<>(ms)b20 nm70 nm200 nmdxlD-1-10 0 1010-110-4Gl D2 <>-1 (m2 s-1)10-1410-1210-1310-15D (m2s-1)10-13TrappingregimeHopping regime10-5 10-8ττξ (kg s-1)Fig. 5 | Proton dynamics modulated by surface defects. a Distributions of protonjumping displacement on the aqueous hBN surfaces treated by plasma at differenttplasma.bExperimental averaged adsorption time 〈τ〉 at different tplasma for each flake.c Simulations: distribution of the proton displacement between each frame at variousinter-defect distances [20–200 nm], indicated by different colors. The observationwindow and the adsorption time are set to be 20 ms and 40 ms, respectively. Insert:distribution of the proton displacement normalized by the inter-defect distance.d Plot of the diffusion coefficient D and friction coefficient ξ as a function of theaveraged inter-defect distance lD and the averaged adsorption time 〈τ〉. The scatteredpoints are from experiments, and the semi-transparent and solid data points cor-respond to the trapping regime and the hopping regime. The black dotted line is thescaling relation from the Brownian dynamics simulations, expected from thedesorption-mediated model30,44,52, where the hopping dominates the displacementdistributions. The red dotted line is estimated from the converging distributions ofthe displacement and represents the trapping regime where most protons are locallytrapped at the adsorption sites.https://doi.org/10.1038/s43246-025-00910-3 ArticleCommunications Materials |           (2025) 6:200 6www.nature.com/commsmatbroadening of the exponential arms, consistent with longer jumping dis-placement. This effect is in accordance with our experimental observationsin Fig. 5a: shorter plasma induces fewer defects and hence larger lD, whichbroadens the displacement distribution. Furthermore, the distributions atvarious lD collapse onto the same master curve once the displacement isnormalized by lD as shown by the insert on Fig. 5c, implying the stronginfluence of the surface defects, whose estimationwill be discussed in detailsin the next. Interestingly, we recover an empirical scaling for the ensemblediffusion,D � l2Dhτi�1 expectedby thedesorption-mediated transport30,44,52,53, with a thorough report of the effect of the 〈τ〉 and lD in SupplementaryFigs. 4–6.Wenote that this relationholds regardless of the values assigned tolD and 〈τ〉, demonstrating the robust nature of ourmolecular interpretation.Once the above semi-empirical relation is injected into the Fickianequation, λ � ffiffiffiffiffiffiffiffiffiDΔtp, we obtain the following dimensionless relation,λ=lD � Δt=hτi� �1=2, that is precisely reproduced by our simulations andcorresponds to the experimental observations in Fig. 4g. Coming back to thedistribution of hopping displacements, they should similarly collapse ontothe samemaster curve when normalized by lD at fixed Δt and 〈τ〉, a processthat is exactly reproduced by our simulations shown by the insert in Fig. 5c.We now comment on the experimental estimation of two pivotalparameters, as the typical inter-defect distance lD and the average adsorptiontime 〈τ〉. τ is detectedwith the same protocols as described in the last section,and its average 〈τ〉 is subsequently statistically summarized in Fig. 5b. Eachdata point represents the independent experimental result obtained fromonesingleflake. The broad dispersion of the experimental data is attributed to theinherent heterogeneity of each flake. It is yet evident that 〈τ〉 shows a slightincrease with the duration of the plasma exposure. This finding is in accor-dance with the report on the cumulant photoluminescent sites in Fig. 2e, f,which reveals an increase in the number of clusters and immobile events forcases involving a longer tplasma. When tplasma is longer than 15 s, most of theprotons are thus locally trapped, and it appears that 〈τ〉 reaches a saturation.Coming back to the question of the defect density and inter-defectdistance, estimating such values is notoriously difficult. Previous attemptsinclude direct measurements through Transmission ElectronMicroscopy54or tracking of cumulant photoluminescent sites28. Those methods eitherchange the surface state during their implementation or require longtracking with complex approaches to correct spatial drifting. To seek asimple, reliable approach, we propose here to extrapolate a typical inter-defect distance lD based on the exponential decay length λ, thanks to thepeculiarity of the exponential distribution of jumping displacement, bytaking lD≃ λ, when 〈τ〉 ≈Δt, as is the case in our experiments (see theSupplementary: Section 6). Such estimation has the merit of avoiding anexhaustive (and sometimes ambiguous) exploration of all the surfacedefects, as is necessary for the cumulant tracking28.We revert to the ensemble diffusion coefficient and compare itwithourexperimental estimation of lD and 〈τ〉. As shown in Fig. 5d, we find areasonable concordance between these experimentally defined parametersand the simple scaling law D � l2Dhτi�1 for the larger mobility samples(black dotted line). However, for the lower diffusion coefficients corre-sponding to the longer plasma treatments, we observe a breakdown of thisscaling, with l2Dhτi�1 becoming independent of D. This breakdown isconcomitant with the convergence of the hopping distribution in Fig. 5atowards an exponential distribution of characteristic length λc ≈ 40 nm atlarge plasma treatment times, equivalently shown by the red dotted line inFig. 5d. In this limit, the majority of protons are locally adsorbed on thecluster sites with very rare inter-cluster transport. Such confined motionleads to a strong drop of the effective diffusion coefficient (Fig. 3c), while theexponential jumping distributions converge towards finite values due to theuncertainty in the localization precision combined with confined jumpsinside clusters.Finally, we comment on the interfacial friction associated with protontransport observed in our experiments. As discussed earlier, the ensemblediffusion coefficientD can also be related to an effective friction coefficient ξ,which characterizes the effective dissipative frictional force, F = ξv, experi-enced by a charged particle moving at a drift velocity v, under linearresponse theory. These two quantities are related via the fluctuation-dissipation relation, D ~ kBT/ξ. Interestingly, the lowering of the diffusioncoefficient at high defect density is thus equivalently expressed as anincreasing frictional interaction felt by single charges with the surface. Theobserved phenomenology of a discontinuous transport limited by escapefrom deep traps bears an analogy to the discontinuous stick-slip motionencountered in single-asperity sliding innanotribology systems55, evidencedhere at the single molecule scale. Yet a key difference would lie in the broaddistributionof trapping energies and trapping time evidenced in our system,which would be expected to give rise to richer andmore complex transportbehaviours.ConclusionsIn this work, we resorted to a single-molecule localization technique touncover the key role played by surface defects in regulating single chargetransport at the aqueous hBN interface. Probing the local distribution ofplasma-induced defects through super-resolution mapping reveals thepresence of two types ofmorphologies for the adsorption sites, with sparselydistributed and weakly adsorbing sites coexisting with dense and stronglyadsorbing defect clusters. We then focus on the study of single chargedynamics in such a defective landscape by harvesting the spatio-temporalcorrelations in nearby defect activation to track the hopping motion of asingle charge between defects. We observe that longer plasma treatmentimpedes themobility, whichwe ascribe at a qualitative level to an increase indefect density leading to increasing frictional interactions. To furtherquantify transport, we analyze in more detail the observed trajectories,reminiscent of continuous-time randomwalk processes, and the associateddistribution of hopping displacements in which a central adsorption peakcoexists with an exponentially decaying arm associated with inter-defecttransport. To elucidate the origin of this bimodal distribution, we resort toBrowniandynamics simulations of freely diffusing 2DBrownianparticles ina defective landscape composed of homogeneously distributed adsorbingsites. This simple modeling approach evidences that the long-distanceexponentiallydecayingarmsarisenaturallydue to transport betweennearbydefects and scale with the inter-defect distance, while the central Gaussianpeak can be ascribed to the charge adsorption at the strong trapping sites.We further present a simple scaling law relating the interfacial ensemblediffusion coefficient to the mean defect adsorption period and inter-defectdistance, which compares favorably with our experimental data in the highmobility regime associatedwith short plasma treatments.Wefinally link theensemble diffusion coefficient to interfacial ionic friction, demonstratingthe control of molecular-scale dissipation at the nanoscale. We stress thatour analysis throughout employs no free fitting parameters, reinforcing ourclaim for the molecular origin of the exponentially decaying single chargejumps between defect sites and the control of mobility and interfacial fric-tion, with surface defects.Our findings open up new insights on the role of defects in interfacialproton transport, an overlooked parameter of importance for a variety offields involving ionic transport at solid/liquid interfaces. These advancesfurther establish a new step towards the understanding of collective particletransport at aqueous interfaces from single-molecule dynamics and opennew avenues for investigating atomistic friction and dissipative effects atsolid-liquid boundaries. Building upon the current findings and our pre-vious results30,35, future work could explore in greater depth the interplaybetween pH, defect activity, and proton diffusion dynamics. Although thepresent study focuses exclusively on protons, the broader framework ofdefect and adsorption modulated interfacial transport developed here maybe applicable to other ionic species-particularly those exhibiting strongsurface interactions. Extending thismethodology to exogenous ions, such asmetal cations, would represent an exciting direction33.MethodsSingle-molecule microscopy setupSingle-molecule tracking experiments were performed using a custom-made single-molecule localization microscope, build around an invertedhttps://doi.org/10.1038/s43246-025-00910-3 ArticleCommunications Materials |           (2025) 6:200 7www.nature.com/commsmatmicroscope (IX83, Olympus). The hBN flake was mounted on a custom-designed holder compatible with a high-numerical-aperture oil-immersionobjective (OlympusTIRFM100×,NA1.5,UPLAPO100XOHR)andexcitedwith a 561 nm laser (LCX-561L-100-CSB-PPF, Oxxius). The excitationpower measured at back focal plane of the objective was around 10mW,resulting in a power density of approximately 0.5 kW.cm−2. Photo-luminescence signal emitted from the excited sample was collected throughthe same oil-immersion objective. Fluorescent emission was separated andfiltered using dichroic and emission filters (Semrock Di03-R405/488/561/635 and FF01-446/523/600/677). The resulting emission signal was pro-jected onto an EMCCD camera (iXon Ultra, Andor, with an EM gain of150), following magnification by a factor 1.4X, leading to a projected pixelsize of 114.3 nm.Brownian dynamics simulationsSingle protons are assumed to explore freely the 2D space at the aqueoushBN surface, obeying a standard Brownian motion with Gaussian dis-tributed elemental steps. The proton jumps at amolecular frequency thatweset to 1010 Hz with the bulk diffusion coefficient ~10−8 m2s−1 56,57. Weconsider randomly distributed defects, with fixed density ρ, from which wedefine an interdefect distance lD = ρ−1/2. Defects are modeled by anadsorbing boundary with an interaction radius Ra ≈ 1 nm, which char-acterizes the typical range of interactions between the diffusing charge andthe surface defect. We note that this interaction radius is not far from theBjerrum length of 0.7 nm characterizing the typical range of electrostaticinteractions in our experimental conditions, which would presumably beinvolved in the trapping of a proton to the negatively charged V�B vacancy.In a typical simulation run, the Brownian particle starts close to a randomdefect site (at a distance 2Ra). The particle diffuses freely on the surface untilit encounters a defect. An adsorption event takes place when the Brownianparticle enters within Ra of a nearby defect site, following which the particlebecomes locally trapped at the defect site for a random adsorption time τ.We take this trapping period as obeying the power law ψ(τ) ~ τ−2.5 with itsmean value measured from experiments, before it takes the next hopping.We can simulate the effective random walks as a succession of trappingevents with key adjustable simulation parameters listed in Table 1, coveringthe scope of our experimental observations. Other numerical parameters,including the molecular jumping frequency and the attraction radius, showalmost negligible effect on the proton dynamics (Supplementary Fig. 8).Data availabilityThe authors declare that the data supporting the findings of this study areavailable within the paper and its supplementary information files.Received: 12 March 2025; Accepted: 30 July 2025;References1. Tajkhorshid, E. et al. 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The curious case of the hydrated proton.Acc.Chem. Res. 45, 101–109 (2012).58. Scott,D.W.Averagedshiftedhistograms:effectivenonparametricdensityestimators in several dimensions. Ann. Stat. 13, 1024–1040 (1985).AcknowledgementsM.Z. and J.C. acknowledge funding from the ANR (grant ‘GUACAmole’ ANR-22-CE06-0003-01), and the Ile-de-France Region in the framework of ‘DIMRespore’ and ‘DIMMaTerRE’. This project has also received financial supportfrom the CNRS through the MITI interdisciplinary programs and from theCarnot Institute ’IPGGMicrofluidique’. M.Z. acknowledges further supportfrom the National Natural Science Foundation of China (grant No.: 12388101),K.W. and T.T. acknowledge support from the JSPS KAKENHI (grant No.:21H05233 and 23H02052), the CREST (grant No.: JPMJCR24A5), JST andWorld Premier International Research Center Initiative (WPI), MEXT, Japan.Author contributionsM.Z. and J.C. designed the research. M.Z. performed the experiments andprocessed the data. M.Z. and J.C. analyzed the data and wrote the paper.M.Z. conducted the simulations with the help of J.C., K.W., and T.T.contributed thematerials. J.C. supervised the project. All authors discussedthe results and commented on the manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s43246-025-00910-3.Correspondence and requests for materials should be addressed toJean Comtet.Peer review information Communications Materials thanks HiroakiYoshida, Yecun Wu and the other, anonymous, reviewer(s) for theircontribution to the peer review of this work. Primary Handling Editors: Jet-Sing Lee. 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Toview a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2025https://doi.org/10.1038/s43246-025-00910-3 ArticleCommunications Materials |           (2025) 6:200 9https://arxiv.org/abs/2407.01934https://arxiv.org/abs/2407.01934https://arxiv.org/abs/2407.01934https://arxiv.org/abs/2409.18702https://arxiv.org/abs/2409.18702https://arxiv.org/abs/2409.18702https://doi.org/10.1038/s43246-025-00910-3http://www.nature.com/reprintshttp://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/commsmat Defect-modulated ionic friction at hBN/water interfaces Results and discussion Experimental methodology for single proton tracking A qualitative description of active defect distribution and proton dynamics Non-Gaussian yet Fickian transport process Defect-modulated transport and ionic friction Conclusions Methods Single-molecule microscopy setup Brownian dynamics simulations Data availability References Acknowledgements Author contributions Competing interests Additional information