applied sciences

Article Ion Current Rectification in Extra-Long Nanofunnels

Diego Repetto, Elena Angeli * , Denise Pezzuoli, Patrizia Guida, Giuseppe Firpo and Luca Repetto

Department of Physics, University of Genoa, via Dodecaneso 33, 16146 Genoa, Italy; [email protected] (D.R.); [email protected] (D.P.); [email protected] (P.G.); giuseppe.fi[email protected] (G.F.); [email protected] (L.R.) * Correspondence: [email protected]



Received: 29 April 2020; Accepted: 25 May 2020; Published: 28 May 2020


Abstract: Nanofluidic systems offer new functionalities for the development of high sensitivity biosensors, but many of the interesting electrokinetic phenomena taking place inside or in the proximity of nanostructures are still not fully characterized. Here, to better understand the accumulation phenomena observed in fluidic systems with asymmetric nanostructures, we study the distribution of the ion concentration inside a long (more than 90 µm) micrometric funnel terminating with a nanochannel. We show numerical simulations, based on the finite element method, and analyze how the ion distribution changes depending on the average concentration of the working solutions. We also report on the effect of surface charge on the ion distribution inside a long funnel and analyze how the phenomena of ion current rectification depend on the applied voltage and on the working solution concentration. Our results can be used in the design and implementation of high-performance concentrators, which, if combined with high sensitivity detectors, could drive the development of a new class of miniaturized biosensors characterized by an improved sensitivity.

Keywords: nanofunnel; FEM simulation; ionic current rectification; micro-nano structure interface

1. Introduction Nanofluidics has been attracting the attention of a wide scientific community for a long time [1,2]. Nevertheless, its potential has not yet been fully exploited [3]. This interest is motivated by the fact that nanofluidic systems offer functionalities in terms of control of ionic and molecular transport that occur only at a nanoscale [4,5]. Solid-state nanopores [6] and nanochannels [7] mimicking biological counterparts [8] have been investigated and exploited in many applications [9], e.g., the detection of molecules of biomedical interest, such as proteins [10], nucleic acids [11], viruses [12–14], and nanovesicles [15] or for developing new components of ionic circuits [16], such as transistors [17], diodes [18,19], and ion pumps [20–22]. A large number of nanofluidic devices exploit electric fields for handling fluids and nano-objects near or through functional nanostructures; for this reason, understanding electrokinetic phenomena occurring at the nanoscale is of paramount importance. Many researchers focused their efforts on modeling, either by analytical or numerical methods, the electrokinetic behavior of fluids in nanoconfinement conditions or at micro-nanointerfaces [23,24], but a comprehensive understanding of all the phenomena taking place and how they influence each other is still far off. This lack of understanding is due to the complexity of the problem, which involves coupled nonlinear fluid flow, charged species transport, and a dynamic evolution of the electric field in a multi-scale space spanning from nanometers to centimeters, as stated by Han and coworkers [25]. In addition to symmetric geometries, nanopores and nanochannels with asymmetric characteristics have also been studied due to their interesting functionalities [26,27], especially when electric fields are applied. In particular, electrostatic interactions between molecules and ions in aqueous solutions, and the electric double layer

Appl. Sci. 2020, 10, 3749; doi:10.3390/app10113749 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 3749 2 of 13

(EDL), which forms on a nanostructure’s walls, can be exploited for creating ion-selective devices [28]. Concentration polarization (CP) [29] was observed at micro-nano interfaces [30], in nanochannels [17], and in ion current rectification (ICR) in nanostructures [31] with asymmetric surface charge distribution [32,33], geometry [20,26,27,33–40], or both [41]. Concerning conical nanochannels, many different parameters, such as length, cone angle, distribution of surface charge, size and shape, were analyzed to understand how they affect CP and ICR [34,42–44]. In particular, CP has been exploited to concentrate diluted analytes, such as proteins [29,45], to improve the sensitivity of biosensors. The challenging idea of combining accumulation capabilities with biological and chemical selectivity boosted the research of new methods and techniques for functionalizing the surface and/or immobilizing biomolecules. Nanochannel/nanopore surface functionalization was used for tuning the ion/molecule selectivity of the structure [46] and also for varying the rectification characteristics [46,47]. For example, Vlassiouk et al. [48] anchored biomolecules onto a conical nanopore to fabricate a biosensor for avidin, streptavidin, and immunoassays. Although research activities focused on the exploitation of CP for biosensing have been very intensive in recent years, as demonstrated by three recent reviews [29,45,49], the main disadvantage of CP-based accumulation systems is still the inability to precisely control the location of the preconcentrated biomolecule plug, as stated by Park and Yossifon [50]. In fact, by controlling the position of the plug it would be possible to overlap target biomolecules with probe molecules anchored on the surface to enhance the detection sensitivity and promote the recognition mechanisms. With this idea in mind, we developed an immunosensor [51] to accumulate target biomolecules (e.g., antigens) inside a long funnel functionalized with probe molecules (e.g., antibodies). Exploiting the high surface-to-volume ratio of this structure, we succeeded in detecting, by fluorescence measurements, probe-target molecule interactions even for very dilute solutions (nearly 1 pg/mL). Moreover, we exploited the same geometry to implement ion current rectifiers [52] and studied how analytes accumulate inside the long funnel. The promising results we obtained in both fields (immunosensing and rectifiers) motivated us to further investigate the electrokinetic phenomena occurring inside such long structures in order to further improve the combination of accumulation and detection in the same area of a device. Here, we study the distribution of ions inside an extra-long (nearly 90 µm) funnel terminating with an 80 nm wide and nearly 1 µm long nanochannel. Our investigation was carried out by numerically solving the Poisson–Nernst-Planck equations in order to characterize the electrokinetic behavior of the structure for both positive and negative voltages and to estimate the rectification ratio (RR) in the range ( 10, 10 V). − 2. Methods It is well established that the distribution of the electrical potential and the flux of each ion species due to a gradient in concentration (diffusion) and electric potential (drift) through a micro-nano fluidic device is governed by the Poisson–Nernst-Planck (PNP) equations. We solved these equations by the finite element method. Details on the implementation in the COMSOL Multiphysics® Software and on the mesh structure used in the simulations are provided in the Supplementary Materials. In the following, we provide details on the geometry that we analyzed and on the imposed boundary conditions.

2.1. Funnel and Nanochannel Geometry Figure1 shows the geometry of the device used in the numerical finite element simulations. The 3D geometry of the system (reservoir + funnel + nanochannel + reservoir) was simplified to reduce computational effort by using a 2D axis-symmetric geometry. Two reservoirs (5 µm 20 µm), cis- on the left and trans- on the right, are connected by a × funnel-shaped microchannel, 90.32 µm long and 6 µm wide at the base (cis-side). At the other extremity, the funnel ends with a rectangular nanochannel (1 µm 80 nm) linked to the trans-reservoir. The total × distance between the two reservoirs (funnel + nanochannel) is therefore 91.32 µm. This geometry was inspired by the structure used by Pezzuoli et al. [51,52]. Appl. Sci. 2020, 10, 3749 3 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 14

Figure 1. Sketch of the system geometry used for the FEM calculations. The red numbers correspond Figureto walls 1. withSketch the of same the system boundary geometry conditions. used for the FEM calculations. The red numbers correspond to walls with the same boundary conditions. 2.2. Boundary Conditions Two reservoirs (5 µm × 20 µm), cis- on the left and trans- on the right, are connected by a funnel- We set a constant surface charge of σ = 1 mC/m2 on the walls of the funnel and the nanochannel shaped microchannel, 90.32 µm long and 6 µm− wide at the base (cis-side). At the other extremity, the (labeled 4 and 5 in Figure1). By changing the boundary conditions on wall 1 in the Poisson equation, funnel ends with a rectangular nanochannel (1 µm × 80 nm) linked to the trans-reservoir. The total we performed simulations for different applied voltages (V) between the cis- and trans- reservoirs. distance between the two reservoirs (funnel + nanochannel) is therefore 91.32 µm. This geometry was The trans-reservoir was set as grounded, while the potential for the cis- one was varied in a range of inspired by the structure used by Pezzuoli et al. [51,52]. between 10 and +10 V. We studied the system filled with electrolytic solutions of KCl with different − molarities (1–10 5 M). In the Nernst–Planck equations for both ion species, K+ and Cl , we set the same 2.2. Boundary Conditions− − concentration CBulk for boundaries 1 and 2 in both reservoirs. By solving the PNP equation system, + 2 we couldWe set model a constant the distribution surface charge of K of andσ = − Cl1 m− ionsC/m in on the the funnel walls of and the in funnel the nanochannel and the nanochannel as well as (labeledcalculate 4 theand ionic 5 in currentFigure 1) in. By the changing device as the a function boundary of Vcondition and CBulks on. In wall principle, 1 in the the Poisson set of equationsequation, wethat performed describes thesimulations physics of for the different system shouldapplied include voltages the (V) Navier–Stokes between the (NS)cis- and equation. trans- reservoirs Without it,. Thwee aretrans neglecting-reservoir the was eff ectset ofas electroosmosis,grounded, while which the potential is a reasonable for the assumption cis- one was taking varied into in account a range theof betweendevice dimensions −10 and +10 and V.the We surface studied charge the system density filled [53, 54with]. To electrolytic demonstrate solutions that this of is KCl also wit anh acceptable different molaritiesapproximation (1–10 for−5 M). our In system, the Nernst we– performedPlanck equations numerical for both simulations ion species with, K a+ completeand Cl−, we PNP-NS set the same set of concentrationequations. The C resultsBulk for forboundar potassiumies 1 andand chlorine2 in both concentrations reservoirs. By along solving diff theerent PNP sections equation of the system, device wewere could the same,model withinthe distribution an error ofof 0.1%,K+ and of Cl those− ions calculated in the funnel by PNP and equationsin the nanochannel (see Supplementary as well as 5 calculateMaterials) the for ionic all of current the six in C Bulkthe devicevalues as (1–10 a function− M) studied. of V and CBulk. In principle, the set of equations that describes the physics of the system should include the Navier–Stokes (NS) equation. Without it, we3. Resultsare neglecting and Discussion the effect of electroosmosis, which is a reasonable assumption taking into account the device dimensions and the surface charge density [53,54]. To demonstrate that this is also an 3.1. Accumulation acceptable approximation for our system, we performed numerical simulations with a complete PNP-ByNS varying set of equations. the boundary The conditions results for for potassium the Poisson and equation chlorine at wall concentrations 1 of the cis-reservoir along different (wall 1 sectionsof the trans of the-reservoir device iswere grounded, the same i.e.,, within V = 0 V),an weerror calculated of 0.1%, C(Kof those+) and calculated C(Cl-), the by concentrationPNP equations of + −5 (seeK and Supplementary Cl− ions, respectively, Materials) insidefor all andof the at six the C interfaceBulk values of (1 the–10 structure M) studied as a. function of the voltage applied. Positive currents (I > 0) flow from the wide opening towards the narrow opening of the 3.nanochannel. Results and WithDiscussion 1 M concentration in the two reservoirs (CBulk), an ion accumulation occurs in the funnel for negative voltages, with a peak increase in the concentration of about 0.8% in the proximity 3.1.of the Accumulation nanochannel entrance (Figure2). On the other hand, an ion depletion (same “intensity” of the accumulation)By varying occurs the boundary for positive condition voltages.s for Interestingly, the Poisson atequation CBulk = at1 Mwall the 1 accumulationof the cis-reservoir (depletion) (wall 1is of independent the trans-reservoir of the sign is grounded, of ion charge i.e., V (Figure = 0 V),2 a,b).we calculated C(K+) and C(Cl-), the concentration of K+ andThis Cl− phenomenonions, respectively, is due inside to the and presence at the interface of surface of the charges structure on the as a walls function of the of the funnel voltage and appliednanochannel. Positive and, currents consequently, (퐼 > 0 to) flow the formationfrom the wide of an Electricopening Double towards Layer the (EDL),narrow whose opening thickness, of the nanochannel.called the Debye With length, 1 M concentrationλD, can be estimated in the two as followsreservoirs (CBulk), an ion accumulation occurs in the funnel for negative voltages, with a peak increaser in the concentration of about 0.8% in the proximity εk T of the nanochannel entrance (Figure 2). Onλ the= other hand,B an ion depletion (same “intensity” of the(1) D 2C eN accumulation) occurs for positive voltages. Interestingly,bulk atA CBulk = 1 M the accumulation (depletion) is independent of the sign of ion charge (Figure 2a, b). where kB is the Boltzmann constant, T the absolute temperature, CBulk the ionic concentration of the solution, NA the Avogadro number, ε the permittivity of free space, and e the elementary charge. Appl. Sci. 2020, 10, 3749 4 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 14

+ FigureFigure 2. 2.K K+ (a(a)) and and Cl Cl−− ((bb)) ion ion concentration concentration along the longitudinal section section of of the the funnel funnel and and 3 3 nanochannelnanochannel as as a functiona function of of the the applied applied voltage voltage between between thethe reservoirsreservoirs (C Bulk == 1 1M M = =10103 mol/mmol/3m). ).

2 Indeed,This phenomenon simulations performed is due to the considering presence no of surface surface charge, charges i.e., onσ the= 0 walls C/m , of present the funnel a constant and ionnanochannel concentration and, along consequently, the entire deviceto the formation (not shown of a here).n Electric However, Double the Layer accumulation (EDL), whose effect thickness, at a high concentrationcalled the Debye (CBulk length= 1 M), λD is, can almost be estimated negligible, as follows since the EDL is about 0.3 nm, as evidenced by the constant concentration along the transversal section in the middle of the nanochannel for both ions at 휀푘퐵푇 any applied voltage (Figure3). Only very close휆 = to√ the walls (distance less than 2 nm) does an increase(1) + 퐷 2퐶 푒푁 inAppl. the Sci. K 20ion20, 10 concentration, x FOR PEER REVIEW (a decrease for Cl−) occur.푏푢푙푘 퐴 5 of 14

where kB is the Boltzmann constant, T the absolute temperature, CBulk the ionic concentration of the solution, NA the Avogadro number, ε the permittivity of free space, and e the elementary charge. Indeed, simulations performed considering no surface charge, i.e., σ = 0 C/m2, present a constant ion concentration along the entire device (not shown here). However, the accumulation effect at a high concentration (CBulk = 1 M) is almost negligible, since the EDL is about 0.3 nm, as evidenced by the constant concentration along the transversal section in the middle of the nanochannel for both ions at any applied voltage (Figure 3). Only very close to the walls (distance less than 2 nm) does an increase in the K+ ion concentration (a decrease for Cl−) occur.

+ FigureFigure 3. 3. KK+ ((a) and and Cl Cl−− (b) ion concentration concentration along along the the transversal transversal section section at at the the middle middle of of the the nanochannel as a function of the applied voltage between the reservoirs (C = 1 M = 103 mol/m3). nanochannel as a function of the applied voltage between the reservoirs (CBulkBulk = 1 M = 103 mol/m3).

Figure 4 shows C(K+) e C(Cl−) along the longitudinal section of the nanochannel. The graphs clearly indicate that there is no difference in the concentration of the two species at any applied voltage, since all curves are superimposed. The accumulation effect is strongly dependent on CBulk. By decreasing by 5 orders of magnitude the ion concentration (CBulk = 10−5 M), the behavior drastically changes (Figure 5): (i) the accumulation effect in the funnel is much stronger with the maximum concentration, which is 50 times larger than CBulk; (ii) the independency from the ion charge sign is broken for positive applied voltages.

Figure 4. K+ (red line) and Cl− (dashed blue line) ion concentration along the longitudinal section of

the nanochannel as a function of the applied voltage between the reservoirs (CBulk = 1 M = 103 mol/m3). Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 14

Appl. Sci.Figure2020, 10 3., 3749K+ (a) and Cl− (b) ion concentration along the transversal section at the middle of the 5 of 13

nanochannel as a function of the applied voltage between the reservoirs (CBulk = 1 M = 103 mol/m3).

+ FigureFigure4 4shows shows C(K C(K+)) ee C(ClC(Cl−−)) along along the the longitudinal longitudinal section section of of the the nanochannel. nanochannel. The The graphs graphs clearlyclearlyindicate indicate that that there there is is no no di differencefference in the concentration of of the the two two species species at at any any applied applied voltage, since all curves are superimposed. The accumulation effect is strongly dependent on C . voltage, since all curves are superimposed. The accumulation effect is strongly dependent on CBulkBulk. 5 ByBy decreasing decreasing by by 5 5 orders orders of of magnitude magnitude the the ion ion concentration concentration (C (CBulkBulk == 1010−5− M),M), the the behavior behavior drastically drastically changeschanges (Figure (Figure5 ):5): (i) (i) the the accumulation accumulation eeffectffect in in the the funnel funnel is is much much stronger stronger with with the the maximum maximum concentration,concentration, whichwhich isis 5050 timestimes largerlarger thanthan CCBulkBulk; ;(ii) (ii) the the independency independency from from the the ion ion charge charge sign sign is is brokenbroken for for positive positive applied applied voltages.voltages.

+ FigureFigure 4. 4.K K+ (red(red line)line) andand ClCl− (dash(dasheded blue blue line) line) ion ion concentration concentration along along the the longitudinal longitudinal section section of of 3 3 Appl. theSci.the 20 nanochannel nanochannel20, 10, x FOR PEER as as a a functionREVIEW function of of the the applied applied voltage voltage between between the the reservoirs reservoirs (C (CBulkBulk == 11 M M == 101036 mol/m ofmol 14/ m3). ).

+ Figure 5. K (a) and Cl− (b) ion concentration along the longitudinal section of the funnel and nanochannel + − Figure 5. K (a) and Cl (b) ion concentration along the longitudinal section5 of the2 funnel3 and as a function of the applied voltage between the reservoirs (CBulk = 10− M = 10− mol/m ). nanochannel as a function of the applied voltage between the reservoirs (CBulk = 10−5 M = 10−2 mol/m3). 5 At CBulk = −510− M, both ions are accumulated in the funnel for negative voltages, while for At CBulk = 10 M, both ions are accumulated in the funnel for negative voltages, while+ for “zero” “zero”and positive and positive voltages voltages,, depletion depletionof both ions of occurs both.ions Accumulation occurs. Accumulation of K+ ions for “zero” of K andions positive for “zero” and positivevoltages voltages occurs only occurs in theonly nanochannel, in the nanochannel, whereas whereasC(Cl−) is C(Cl very − low.) is veryA zoom low.-in A of zoom-in the ion of the ion concentrationconcentration as as a a function function of ofthe the position position along along the longitudinal the longitudinal section section of the nano of thechannel nanochannel shows shows thisthis clearlyclearly (Fig (Figureure 6).6).

Figure 6. K+ (red line) and Cl− (blue line) ion concentration along the longitudinal section of the nanochannel for different applied voltages: (a) -10 V, (b) -5 V, (c) -1 V, (d) 10 V, (e) 5 V and (f) 1 V,

between the reservoirs (CBulk = 10−5 M =10−2 mol/m3).

The independence of the concentration distribution from the sign of the charge is progressively broken by varying the applied voltage from −10 (Figure 6a) up to +10 V (Figure 6d). At −10 V, both C(K+) and C(Cl−) are up to 40 times larger than CBulk in the left half of the nanochannel, while both concentrations decrease to a very low value in the right half, towards the nanochannel end. This is due to the ion depletion area, which forms in the anodic reservoir around the nanochannel end. At Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 14

Figure 5. K+ (a) and Cl− (b) ion concentration along the longitudinal section of the funnel and

nanochannel as a function of the applied voltage between the reservoirs (CBulk = 10−5 M = 10−2 mol/m3).

At CBulk = 10−5 M, both ions are accumulated in the funnel for negative voltages, while for “zero” and positive voltages, depletion of both ions occurs. Accumulation of K+ ions for “zero” and positive voltages occurs only in the nanochannel, whereas C(Cl−) is very low. A zoom-in of the ion Appl.concentration Sci. 2020, 10, 3749as a function of the position along the longitudinal section of the nanochannel shows6 of 13 this clearly (Figure 6).

+ FigureFigure 6. 6. KK+ (red(red line) line) and and Cl Cl−− (blue line) line) ion ion concentration concentration along along the the longitudinal longitudinal section section of of the the nanochannelnanochannel forfor didifferentfferent appliedapplied voltages:voltages: ( a) - -1010 V, (b) - -55 V, (c)) -1-1 V, ((dd)) 1010 V,V, ((ee)) 55 VV andand ((ff)) 11 V,V, 5 2 3 betweenbetween thethe reservoirs reservoirs (C (CBulkBulk = 1010−5− MM =10=10−2 −mol/mmol/3m). ).

The independence of the concentration distribution from the sign of the charge is progressively The independence of the concentration distribution from the sign of the charge is progressively broken by varying the applied voltage from 10 (Figure6a) up to +10 V (Figure6d). At 10 V, broken by varying the applied voltage from −10 (Figure 6a) up to +10 V (Figure 6d). At −10 V, both + − − both C(K ) and C(Cl−) are up to 40 times larger than CBulk in the left half of the nanochannel, while C(K+) and C(Cl−) are up to 40 times larger than CBulk in the left half of the nanochannel, while both bothconcentrations concentrations decrease decrease to a to very a very low low value value in inthe the right right half, half, towards towards the the nanochannel nanochannel end. ThisThis isis duedue to to the the ion ion depletion depletion area, area, which which forms forms in thein the anodic anodic reservoir reservoir around around the nanochannelthe nanochannel end. end. At low At CBulk and high electric potential difference between the reservoirs, the depletion area propagates into the channel, forcing the ions to accumulate in the left half of the nanochannel. By decreasing the absolute value of the applied voltage, the ion distribution becomes charge sign-dependent; while K+ ions are uniformly distributed in the channel and reach a plateau for “zero” and positive voltages, Cl− are progressively dropped out (Figure6d–f). The dependency on the ion charge sign at a positive applied voltage (+1 V) is clearly visible in the colored plot of Figure7: C(K +) increases in the nanochannel, while for Cl− ions, the enrichment area in the cathodic side does not penetrate into the nanochannel. As we will also see in the following, in the nanochannel C(K+) is constant along longitudinal sections, whereas it increases towards the negatively charged surfaces. The asymmetric response with respect to the applied voltage suggests the use of the device as a rectifier at low CBulk, with an “forward state” for negative voltages and a “reverse state” for positive ones. Later on, we will further discuss this point by calculating the ionic current through the device as a function of V and CBulk. Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 14 low CBulk and high electric potential difference between the reservoirs, the depletion area propagates into the channel, forcing the ions to accumulate in the left half of the nanochannel. By decreasing the absolute value of the applied voltage, the ion distribution becomes charge sign-dependent; while K+ ions are uniformly distributed in the channel and reach a plateau for “zero” and positive voltages, Cl− are progressively dropped out (Figure 6d–f). The dependency on the ion charge sign at a positive applied voltage (+1 V) is clearly visible in the colored plot of Figure 7: C(K+) increases in the nanochannel, while for Cl− ions, the enrichment area in the cathodic side does not penetrate into the nanochannel. As we will also see in the following, in the nanochannel C(K+) is constant along longitudinal sections, whereas it increases towards the negatively charged surfaces. The asymmetric response with respect to the applied voltage suggests the use of the device as a rectifier at low CBulk, with an “forward state” for negative voltages and a “reverse state” for positive ones. Later on, we willAppl. further Sci. 2020 discuss, 10, 3749 this point by calculating the ionic current through the device as a function of7 of V 13 and CBulk.

Figure 7. K+ (+a) and Cl− (b) concentrations in the nanochannel at CBulk 10−5 M5 (10−2 mol/m2 3), with3 +1 V Figure 7. K (a) and Cl− (b) concentrations in the nanochannel at CBulk 10− M (10− mol/m ), with +1 appliedV applied between between the thereservoirs. reservoirs.

TheseThese results results are are mainly mainly due due to to the the large large va valuelue of of the the Debye Debye length length for for low low concentration. concentration. In In fact, fact, forfor C CBulkBulk = =1 M,1 M, λDλ D= =0.30.3 nm, nm, th thisis is isnegligible negligible compared compared with with the the nanochannel nanochannel half half-width-width (40 (40 nm), nm), −5 5 whereaswhereas at at CBulk CBulk = 10= 10M,− λDM, = 96λD nm,= 96this nm, is even this larger is even than larger the width than theof the width nanochannel of the nanochannel itself. The increaseitself. The in increasethe Debye in the length Debye when length the when ionic the strength ionic strength of the of solutionthe solution dec decreasesreases determines determines an an −3 3 overlappingoverlapping of of the the two two EDLs EDLs in in the the nanochannel nanochannel for for a aconcentration concentration lower lower than than 10 10− M.M. −55 ++ FigFigureure 88 clearlyclearly shows that,that, at halfhalf thethe lengthlength ofof thethe nanochannel nanochannel for for C CBulkBulk = 1010− M,M, the the K K andand − ClCl −concentrationsconcentrations along along with with the the transversal transversal section section change change continuously continuously from from the the center center to to the the walls walls + + − ofof the the channel, with with an an increase increase and and decrease decrease by a factorby a factor 2 for C(K 2 for) and C(K C(Cl) and−), respectively. C(Cl ), respectively. Moreover, Moreover,at such a low at such concentration, a low concentration, the profile thedepends profile strongly depends on strongly the applied on the voltage. applied The vol dependencetage. The dependenceof the ion accumulation of the ion accumulation/depletion/depletion on CBulk is on well CBulk described is well described by the graphs by the in graphs Figure 9in, in Fig whichure 9, thein whichpercentage the percentage increase in increase the ion concentration,in the ion concentration, Ci/CBulk, is plotted Ci/CBulk along, is plotted the funnel along for the di ff funnelerentC forBulk differentand applied CBulk voltagesand applied V. The voltage accumulations V. The ac ecumulationffect for negative effect for voltage negative ( 1 V)voltage is strongly (−1 V) enhancedis strongly at − enhancedlow CBulk at(magenta low CBulk and(magenta blue curves),and blue but curves), almost but independent almost independent of the ion of charge the ion sign charge at any sign C Bulkat 5 −5 + + − any(Figure CBulk9 (Figa, b).ure Only 9a, atb). very Only low at very C Bulk low(10 C− BulkM) (10 is C(K M) )is larger C(K ) thanlarger C(Cl than−) atC(Cl the) funnelat the funnel end. The end. ion depletion for positive voltage (+1 V) also increases at low CBulk, but in this case, the effect is charge + sign dependent. While the concentration of K ions is well below CBulk in the funnel, it increases in the nanochannel by a factor of 18 with respect to CBulk (Figure9c); there are only a few Cl − ions in both the funnel and the nanochannel (Figure9d). A small increase in the Cl − ion concentration only occurs in the cathodic reservoir around the nanochannel end (enrichment area). These results are also visible in the plots in Figure7 for a voltage applied of +1 V. The ion accumulation effect in + the whole funnel is shown in the plot in Figure 10. Here, the K (Figure 10a) and Cl− (Figure 10b) concentrations in the nanofluidic device at C = 10 3 M are plotted with 10 V applied between Bulk − − the reservoirs. It is clear that the accumulation effect is independent of the ion charge sign. It should be noted that the colored scale of the two plots is different. Curves with a similar trend are reported by Liu [43] and Pietschmann [35]. The maximum concentration of positive ions is higher than the Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 14

The ion depletion for positive voltage (+1 V) also increases at low CBulk, but in this case, the effect is charge sign dependent. While the concentration of K+ ions is well below CBulk in the funnel, it increases in the nanochannel by a factor of 18 with respect to CBulk (Figure 9c); there are only a few Cl− ions in both the funnel and the nanochannel (Figure 9d). A small increase in the Cl− ion concentration only occurs in the cathodic reservoir around the nanochannel end (enrichment area). These results are also visible in the plots in Figure 7 for a voltage applied of +1 V. The ion accumulation effect in the whole funnel is shown in the plot in Figure 10. Here, the K+ (Figure 10a) and Cl− (Figure 10b) concentrations in the nanofluidic device at CBulk = 10−3 M are plotted with −10 V applied between the reservoirs. It is clearAppl. that Sci. 2020the, accumulation10, 3749 effect is independent of the ion charge sign. It should be noted that the8 of 13 colored scale of the two plots is different. Curves with a similar trend are reported by Liu [43] and Pietschmann [35]. The maximum concentration of positive ions is higher than the one of negative one of negative ions and is reached at the negatively charged walls of the funnel and nanochannel ions and is reached at the negatively charged walls of the funnel and nanochannel (not visible in this (not visible in this plot). In addition to CBulk, the accumulation (for negative voltages)/depletion (for plot). In addition to CBulk, the accumulation (for negative voltages)/depletion (for positive ones) positive ones) phenomena are strongly dependent on the surface charge on the walls of the funnel and phenomena are strongly dependent on the surface charge on the walls of the funnel and nanochannel. nanochannel. As mentioned before, with no surface charge, the ion concentration is constant and equal As mentioned before, with no surface charge, the ion concentration is constant and equal to CBulk in to C in the whole device (see Supplementary Materials). the wholeBulk device (see Supplementary Materials).

+ FigureFigure 8. 8. KK+ (a(a) ) and and Cl Cl−− ((bb)) ion ion concentration concentration along along the transversalthe transversal section section at the middle at the ofmiddle the nanochannel of the 5 2 3 Appl.nanochannelas Sci. a 20 function20, 10 ,as x of FORa thefunction PEER applied REVIEW of the voltage applied between voltage the between reservoirs the (C reservoirsBulk = 10− (CMBulk= =10 10−−5 molM =/ m10−2). mol/m3). 9 of 14

Figure 9. C/CBulk (%) for K+ (a,c) and Cl− (b,d) along the longitudinal section of the funnel and Figure 9. C/CBulk (%) for K (a,c) and Cl− (b,d) along the longitudinal section of the funnel and nanochannel,nanochannel, calculated calculated for for di differentfferent C CBulk with −11 V V ( (aa,,bb)) and and +1+1 V V ( (cc,,dd)) applied applied between between the the Bulk − reservoirs. (e) Zoom-in of the graph in (c). Note the different scale on the ordinate axis. CBulk: 1 M reservoirs. (e) Zoom-in of the graph in (c). Note the different scale on the ordinate axis. CBulk: 1 M (black line), 10−11 M (dark yellow), 10−22 M (red), 10−3 3M (green), 10−4 M4 (blue), and 10−5 M5 (magenta). (black line), 10− M (dark yellow), 10− M (red), 10− M (green), 10− M (blue), and 10− M (magenta).

Figure 10. K+ (a) and Cl− (b) concentrations in the nanofluidic device at CBulk 10−3 M (1 mol/m3), with - 10 V applied between the reservoirs. Ion accumulation in the funnel for both ionic charges is clearly visible. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 14

Figure 9. C/CBulk (%) for K+ (a,c) and Cl− (b,d) along the longitudinal section of the funnel and nanochannel, calculated for different CBulk with −1 V (a,b) and +1 V (c,d) applied between the Appl.reservoirs. Sci. 2020, 10 ,(e 3749) Zoom-in of the graph in (c). Note the different scale on the ordinate axis. CBulk: 1 M 9 of 13 (black line), 10−1 M (dark yellow), 10−2 M (red), 10−3 M (green), 10−4 M (blue), and 10−5 M (magenta).

Figure 10. K+ (+a) and Cl− (b) concentrations in the nanofluidic device at CBulk 10−3 M (1 mol/m3 3), with 3- Figure 10. K (a) and Cl− (b) concentrations in the nanofluidic device at CBulk 10− M (1 mol/m ), 10with V applied -10 V applied between between the reservoirs. the reservoirs. Ion accumulation Ion accumulation in the funnel in the for funnel both forionic both charges ionic chargesis clearly is visible.clearly visible.

3.2. Rectification

As previously mentioned, at low CBulk the asymmetric response of the device for positive and negative applied voltages suggests the possibility to use it as a current rectifier. In order to estimate this effect, we consider the current density J at the point (x,y) due to the contributions of both ion species to be dCK dCCl FDKCK dV FDClCCl dV J(x, y) = DK + D (2) − dx Cl dx − RT dx − RT dx

where Di is the diffusion coefficient and Ci is the ionic concentration of each ion species, F is the Faraday constant, and V is the electrical potential. The current which flows through section x of the device is thus: Z y0 I(x) = F J(x, y)dy (3) y0 − where the integral is calculated along the channel width, i.e., from y to y . Since we performed 2D − 0 0 simulations, the current is calculated as the amount of charge per second flowing through a segment instead of a surface and is measured in A/m. For a discrete set of data, Equation (3) can be rewritten as X I(x) = F∆y J(x, y) (4) y

where ∆y is the sampling interval for Ci, dCi/dx and dV/dx along the y-direction. Figure 11 shows the plot of the current as a function of the applied voltage for different molarity of the KCl solutions. The currents have been normalized considering the concentration value in the 1 reservoirs. At high CBulk (1 and 10− M, black and dark yellow squares), the plot is symmetric with 2 respect to the origin (i.e., no rectification). At CBulk = 10− M (red squares), the rectification effect starts 3 to be visible, whereas at 10− M (green squares), the device clearly exhibits the characteristic behavior of a diode and the current flows through the device only for negative voltage. The rectification also occurs at 10 4 M (blue squares), although at high voltage ( 10 V) the effect is less pronounced than expected, − − Appl. Sci. 2020, 10, 3749 10 of 13

5 while it decreases at 10− M (magenta squares). The general trend of the ion current rectification ratio

(ICR = I(VBias < 0) /I(VBias > 0)) is shown in Figure 12. It is clear that the ICR increases by decreasing 4 the solution concentration (from 1 M to 10− M). Below a certain concentration, the ICR also depends 3 on the applied voltage, while at CBulk = 1 M, the ICR is about 1 at any VBias, at CBulk = 10− M, the ICR 5 is ca. 2.4 at 1 V, 11 at 5 V, and 20 at 10 V. Due to the limiting-current effect, at 10− M the ICR drops to a value of ca. 5, independently from the applied voltage. Maxima of the rectification ratio as a function of the concentration were reported in both experimental and theoretical works [38,39,44,55–58]. A similar behavior was observed by our group in devices with similar geometry [27]. According to Kovarik et al. [57], maxima occur at the bulk concentration at which the pore conductivity starts to differ significantly from the bulk, because of the contribution of surface-associated counterions to the current through the pore. For this bulk concentration, the conductivity of the pore in the “forward state” is significantly higher than in the “reverse state”, resulting in a strong ICR. For lower concentrations, Appl.the conductivity Sci. 2020, 10, x FOR values PEER of REVIEW the pore in the “forward” and “reverse” states are both dominated11 by of the 14 currentAppl. Sci. arising2020, 10, fromx FOR thePEER surface REVIEW charge, thus the ICR is weakened. 11 of 14

Figure 11. Current I normalized with respect to CBulk as a function of the applied voltage V for different Figure 11. Current I normalized with respect−1 to CBulk as a function−2 of the applied−3 voltage V for−4 different valuesFigure of 11. C CurrentBulk: 1 M I (black normalized squares), with 10 respect M (dark to C Bulkyellow), as a function 10 M of(red), the applied10 M (green), voltage 10V for M different (blue), values of C : 1 M (black squares), 10 1 M (dark yellow), 10 2 M (red), 10 3 M (green), 10 4 M (blue), −5 Bulk −−1 −−2 −−3 −−4 andvalues 10 of5 M C (magenta).Bulk: 1 M (black squares), 10 M (dark yellow), 10 M (red), 10 M (green), 10 M (blue), and 10− M (magenta). and 10−5 M (magenta).

Figure 12. Current I normalized with respect to C as a function of the applied voltage V, for different Figure 12. Current I normalized with respect toBulk CBulk as a function of the applied voltage V, for 1 2 3 values of CBulk: 1 M (black squares), 10− M (pink),−1 10− M (blue),−2 and 10− M (red).−3 differentFigure 12. values Current of C Bulk I normalized: 1 M (black withsquares), respect 10 toM C(pink),Bulk as 10 a function M (blue), of and the 10 applied M (red). voltage V, for

different values of CBulk: 1 M (black squares), 10−1 M (pink), 10−2 M (blue), and 10−3 M (red). Interestingly, Liu et al. [43] and Pietschmann et al. [35] calculated ion current rectification for 4. Conclusions conical structures depending on length (up to 12 µm). They found that if a conical pore is too short, 4. Conclusions Numerical simulations performed on nearly 90 μm long nanofunnel devices showed the versatileNumerical behavior simulations and functional performed potentialities on nearly of such 90 μm nanofluidic long nanofun structuresnel devices. Depending showed on the the appliedversatile voltage behavior and and the ionic functional strength, potentialities large ion accumulation of such nanofluidic occurs along structures a wide. Dependingarea of the funnel, on the suggestingapplied voltage the useand theof ionic the device strength, in large the fieldion accumulation of high sensitivity occurs along nanosensors. a wide area Moreover, of the funnel, the asymmetricsuggesting response the use forof thepositive device and in negative the field ions of confers high sensitivityto the device nanosensors. the capability Moreover, of rectifying the ionicasymmetric currents, response with a forrectification positive andratio negative of up to ions 30 atconfers CBulk =to 10 the−4 Mdevice and theV = capability5 V. These of s rectifyingtructures representionic currents, valuable with tools a rectification for many fields ratio of of application, up to 30 at includingCBulk = 10 −4ionic M and circuits. V = 5Thus, V. These the simulation structures strategyrepresent that valuable we have tools reported for many here fields is useful of application, for guiding including researchers ionic in circuits.tuning the Thus, features the simulation of extra- longstrategy nanofunnel that we shave and reported in selecting here isthe useful proper for guiding operating researchers conditions in dependingtuning the features on the of desired extra- electrokineticlong nanofunnel behavior.s and in selecting the proper operating conditions depending on the desired Supplementaryelectrokinetic behavior. Materials: The following are available online at www.mdpi.com/xxx/s1, description of the equationsSupplementary used in Materials:the COMSO TheL Multiphysics following are model, available Figure online S1: sketch at www.mdpi.com/xxx/s1 of system geometry ,used description in Comsol of for the theequations finite element used in calculations,the COMSO LFigure Multiphysics S2: description model, of Figure the mesh S1: sketch used forof system simulations geometry, Figure used S3: in comparison Comsol for betweenthe finite theelement results calculations, obtained byFigure PNP S 2 and: description PNP-NS of equation the mesh systems used for, Figuresimulations S4: simulations, Figure S3: comparison performed consideringbetween the neutral results walls obtained (no surface by PNP charge and density). PNP-NS equation systems, Figure S4: simulations performed

Authorconsidering Contributions: neutral walls Conceptualization (no surface charge, D density)..R., D.P. and E.A.; methodology, D.R. and E.A.; software, D.R.; formalAuthor analysis, Contributions: D.R., D.P. Conceptualization and E.A.; data ,curation, D.R., D.P. D.R and.; writing E.A.; methodology,—original draft D . preparation,R. and E.A.; D.R. software, and E.A. D.R;.; formal analysis, D.R., D.P. and E.A.; data curation, D.R.; writing—original draft preparation, D.R. and E.A.; Appl. Sci. 2020, 10, 3749 11 of 13 the mechanism of ion-enrichment and depletion is too weak for a significant current rectification effect to occur. Thus, a long funnel grants strong rectification effects.

4. Conclusions Numerical simulations performed on nearly 90 µm long nanofunnel devices showed the versatile behavior and functional potentialities of such nanofluidic structures. Depending on the applied voltage and the ionic strength, large ion accumulation occurs along a wide area of the funnel, suggesting the use of the device in the field of high sensitivity nanosensors. Moreover, the asymmetric response for positive and negative ions confers to the device the capability of rectifying ionic currents, with a 4 rectification ratio of up to 30 at CBulk = 10− M and V = 5 V. These structures represent valuable tools for many fields of application, including ionic circuits. Thus, the simulation strategy that we have reported here is useful for guiding researchers in tuning the features of extra-long nanofunnels and in selecting the proper operating conditions depending on the desired electrokinetic behavior.

Supplementary Materials: The following are available online at http://www.mdpi.com/2076-3417/10/11/3749/s1, description of the equations used in the COMSOL Multiphysics® model, Figure S1: sketch of system geometry used in Comsol for the finite element calculations, Figure S2: description of the mesh used for simulations, Figure S3: comparison between the results obtained by PNP and PNP-NS equation systems, Figure S4: simulations performed considering neutral walls (no surface charge density). Author Contributions: Conceptualization, D.R., D.P. and E.A.; methodology, D.R. and E.A.; software, D.R.; formal analysis, D.R., D.P. and E.A.; data curation, D.R.; writing—original draft preparation, D.R. and E.A.; writing—review and editing, D.P., P.G. G.F. and L.R.; supervision, L.R. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

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