Article pubs.acs.org/accounts
Voltage-Rectified Current and Fluid Flow in Conical Nanopores Published as part of the Accounts of Chemical Research special issue “Nanoelectrochemistry”. Wen-Jie Lan,† Martin A. Edwards,† Long Luo,† Rukshan T. Perera,† Xiaojian Wu,‡ Charles R. Martin,*,‡ and Henry S. White*,† †
Department of Chemistry, University of Utah, 315 South 1400 East, Salt Lake City, Utah 84112, United States Department of Chemistry, University of Florida, Gainesville, Florida 32611-7200, United States
‡
CONSPECTUS: Ion current rectification (ICR) refers to the asymmetric potential-dependent rate of the passage of solution ions through a nanopore, giving rise to electrical current− voltage characteristics that mimic those of a solid-state electrical diode. Since the discovery of ICR in quartz nanopipettes two decades ago, synthetic nanopores and nanochannels of various geometries, fabricated in membranes and on wafers, have been extensively investigated to understand fundamental aspects of ion transport in highly confined geometries. It is now generally accepted that ICR requires an asymmetric electrical double layer within the nanopore, producing an accumulation or depletion of charge-carrying ions at opposite voltage polarities. Our research groups have recently explored how the voltage-dependent ion distributions and ICR within nanopores can induce novel nanoscale flow phenomena that have applications in understanding ionics in porous materials used in energy storage devices, chemical sensing, and low-cost electrical pumping of fluids. In this Account, we review our most recent investigations on this topic, based on experiments using conical nanopores (10−300 nm tip opening) fabricated in thin glass, mica, and polymer membranes. Measurable fluid flow in nanopores can be induced either using external pressure forces, electrically via electroosmotic forces, or by a combination of these two forces. We demonstrate that pressure-driven flow can greatly alter the electrical properties of nanopores and, vice versa, that the nonlinear electrical properties of conical nanopores can impart novel and useful flow phenomena. Electroosmotic flow (EOF), which depends on the magnitude of the ion fluxes within the double layer of the nanopore, is strongly coupled to the accumulation/depletion of ions. Thus, the same underlying cause of ICR also leads to EOF rectif ication, i.e., unequal flows occurring for the same voltage but opposite polarities. EOF rectification can be used to electrically pump fluids with very precise control across membranes containing conical pores via the application of a symmetric sinusoidal voltage. The combination of pressure and asymmetric EOF can also provide a means to generate new nanopore electrical behaviors, including negative dif ferential resistance (NDR), in which the current through a conical pore decreases with increasing driving force (applied voltage), similar to solid-state tunnel diodes. NDR results from a positive feedback mechanism between the ion distributions and EOF, yielding a true bistability in both fluid flow and electrical current at a critical applied voltage. Nanoporebased NDR is extremely sensitive to the surface charge near the nanopore opening, suggesting possible applications in chemical sensing.
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INTRODUCTION An asymmetric, or non-Ohmic, current−voltage (i−V) response in a synthetic nanopore was first observed by Wei, Bard, and Feldberg in 1997 when a voltage was applied across a quartz nanopipette (20 nm radius orifice) immersed in a dilute aqueous KCl solution.1 Wei et al. correctly deduced that the i− V asymmetry, commonly called “ion current rectification” (ICR), resulted from the potential-dependent redistribution of ions within the pipet and that ICR required both (i) electric surface charge (i.e., an electrical double layer) and (ii) an asymmetry in the pore geometry in order to be observed. An example of ICR commonly observed in conical nanopores is shown in Figure 1. In the past two decades, nanopore-based ICR has received great attention within the chemical, physics, © 2016 American Chemical Society
and engineering communities, motivated largely by applications in fluidics and chemical or biological analyses.2−8 ICR is closely related to classical electrokinetic phenomena in charged porous media, e.g., electroosmotic flow (EOF) and streaming potentials,9 and as described below, the same physical models and equations used to describe these phenomena have been adapted to successfully model ICR in conical pores. What separates these classical topics from ICR is that an asymmetric distribution of charge within the pore or channel is required to observe current rectification,10 a criterion that is realized by using either a nanopore with an asymmetric Received: July 31, 2016 Published: September 30, 2016 2605
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we consider a conical nanopore with a uniform negative surface charge, as shown in Figure 1, containing an aqueous KCl solution both inside the nanopore and in the external reservoir. When a negative potential is applied to the Ag/AgCl electrode inside the nanopore, relative to an identical Ag/AgCl electrode in the bulk solution, the K+ flux is directed from the external bulk solution to the pore interior, while Cl− moves in the opposite direction. As the pore is cation-selective, Cl− ions are rejected by the glass surface because of electrostatic repulsion, and their translocation rate through the pore orifice from the pore interior to the bulk solution is reduced. A consequence of this anion rejection is an accumulation of K+ and Cl− within the pore interior, resulting in nanopore conductivity greater than that based on the bulk-solution KCl concentration. Conversely, when a positive potential is applied inside the pore interior relative to the external solution, the transport of Cl− from the external solution to the internal solution is rejected by the surface charges, and Cl− is depleted within the pore interior, thus decreasing the nanopore conductivity and the observed ionic current.14 In general, a larger conical-shaped nanopore displays less ICR than a smaller pore, while greater ICR is observed in more dilute solutions. These observations are generalized by the statement that ICR is more pronounced when the width of the double layer due to the surface charge (characterized by the Debye length) is comparable to the radius of the nanopore orifice. The Woermann model14 has been confirmed by Cervera et al.,16 Liu et al.,17 and White and Bund,18 who solved the electrostatic (Poisson) and transport (Nernst−Planck) equations by finite-element simulations that semiquantitatively captured the i−V response of conical nanopores. The simulations suggested that incomplete overlap of the double layer in nanopores can still give ICR if the double layer constitutes a significant proportion of pore cross-sectional area, in accordance with the experimental phenomena reported by Jacobson7 and Wang.19 More recently, Sa and Baker demonstrated that ICR through a conical nanopipette is altered by bringing the nanopipette probe in close proximity to a charged substrate.20,21 Rectification phenomena and their theoretical descriptions have also been reported for ionic liquids.22 Reed and co-workers also recently reported a related review of voltage-gated ion transport in engineered nanochannels.8 ICR is a strong function of the temperature because the ion distributions within a double layer are strongly temperaturedependent. Figure 1 shows the i−V response of a 35 nm radius nanopore in glass as a function of temperature from 10 to 35 °C. As shown in Figure 2, these and other data were used to compute the activation energies, EA, for K+ and Cl− ion transport in nanopores of varying size and at different KCl concentrations via Arrhenius analyses using current as a measure of rate. As anticipated, the EA values for transport within an electrically charged conical nanopore differ from the bulk-solution values because of the temperature-dependent distribution of the ions within the double layer. More interestingly, as shown in Figure 2, nanopores that display ion current rectification also display a large decrease in EA under accumulation-mode conditions (at applied negative voltages) and a large increase in EA under depletion-mode conditions (at positive voltages), in good agreement with predictions from finite-element simulations.2 This work also discusses in detail the complex relationship between the double layer ion distribution and EA.
Figure 1. Typical i−V response of a conical glass nanopore (35 nm radius pore in 0.1 mM KCl electrolyte). An electrolyte is placed both inside the nanopore and in the external reservoir, and a voltage is applied across the nanopore. The applied potential refers to the potential of the electrode in the internal solution relative to the external solution. The current flowing through the pore increases as the temperature rises. Reproduced from ref 2. Copyright 2015 American Chemical Society.
geometry and uniform surface charge (i.e., a conical nanopore) or a geometrically symmetric pore (e.g., a cylindrical nanopore) with an asymmetric charge distribution.11,12 Modern nanofabrication methods allow the synthesis of nanopores and nanochannels, either one at a time or as large ensembles, within a membrane.3−7 Furthermore, the spatial distribution of the surface charge can be precisely controlled and manipulated by a wide variety of surface chemical modifications and treatments, allowing remarkable control over ICR behavior tailored for specific applications.3 Nonlinear i−V behavior in biological ion channels has origins similar to that observed in solid-state pores.13 However, solid-state nanopores and nanochannels offer the advantage of being readily designed, fabricated, and integrated into technological systems. In this Account, we briefly review recent developments in our understanding of how low Reynolds number flow of fluid through conical nanopores is coupled to ICR and the distribution of ions within the nanopore. A brief review of the origin of ICR is first presented to acquaint unfamiliar researchers with the idea of potential-dependent ion distributions in asymmetric pore geometries. We then describe a simple experiment illustrating how pressure-driven flow alters the ion distributions and thus the ability to rectify current. Following these background concepts, the more complex fluid phenomena of rectified EOF (and pumping applications) and ionic negative differential resistance are presented to illustrate the power of coupling electrostatics and flow at the nanoscale.
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ION CURRENT RECTIFICATION The ion “accumulation/depletion” model initially proposed by Woermann,14 built upon previous works on ion-selective porous membranes,15 is generally accepted as the correct interpretation of the origin of ICR in conical nanopores. Nanopores that display a significant nonlinear i−V response, such as in Figure 1, tend to be ion-selective, i.e., the flux of anions in one direction through the orifice differs from the flux of cations in the opposite direction, as a consequence of the electric surface charge on the nanopore wall. Anions are electrostatically rejected by nanopores that have negative surface charges, and vice versa. However, ion selectivity by itself is not sufficient to generate a nonlinear i−V response. A further requirement is asymmetry, in either the pore geometry or the spatial distribution of surface charge. To illustrate ICR, 2606
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Figure 2. Activation energy, EA, of electrolyte transport through a conical glass nanopore. (a, b) Representative Arrhenius plots constructed from ionic currents measured at +0.35 V and −0.35 V for a 35 nm pore in 1 mM KCl electrolyte. (c, d) EA as a function of both KCl concentration and pore size at +0.35 V and −0.35 V, respectively. Reproduced from ref 2. Copyright 2015 American Chemical Society.
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Figure 3. (a) Influence of pressure on the ICR behavior of large (185 nm) and small (30 nm) conical nanopores (experiments). (b) Simulated nanopore conductivity as a function of the applied voltage and pressure illustrating the effect of pressure-driven flow on the local ion distributions. z = 0 corresponds to the pore orifice; negative values of z correspond to the pore interior. In the absence of pressure (0 mmHg), the conductivity in the pore is lower than the solution conductivity at a positive voltage (0.4 V) and higher at a negative voltage (−0.4 V), consistent with the accumulation/depletion model of ICR. Adapted from ref 23. Copyright 2011 American Chemical Society.
EFFECT OF PRESSURE-DRIVEN FLOW ON ION CURRENT RECTIFICATION Any fluid flow through a nanopore orifice may disrupt the equilibrium distribution of current-carrying cations and anions responsible for ICR.23 For example, applying a higher external pressure on the bulk-solution side of the nanopore results in flow of bulk solution across the nanopore orifice. Qualitatively, this bulk flow causes the fluid and ions inside the nanopore to be replaced by solution that contains ions at a concentration corresponding to the bulk value. Consider a negative voltage applied across a nanopore with a negative surface charge, which corresponds to the high-conductivity ion accumulation state. An inward flow, if sufficiently large, will result in the higher concentration of ions in the pore being replaced by bulk solution containing a lower concentration of ions, thereby reducing the current. The pressure-dependent ICR response is strongly dependent on the pore orifice size. Figure 3a shows experimental i−V responses for single nanopores in glass membranes with radii of 185 and 30 nm. The nanopores were immersed in a 0.01 M KCl solution, and pressures ranging from 0 to −160 mmHg were applied across the membrane (internal vs external solution; a negative pressure results in flow from the bulk solution into the nanopore). In the absence of applied pressure, both nanopores display significant ICR. The two nanopores, however, display remarkably different ICR behaviors when pressure is applied across the membrane. A moderate applied pressure inducing flow across the 185 nm conical nanopore eliminates ICR, resulting in a nearly Ohmic i−V response. In contrast, applied pressure has negligible effect on the highly rectified i−V response of the 30 nm pore. The relationship between the pore size and pressuremodulated rectification can be qualitatively understood by considering the effect of fluid flow on the ion distribution within the pore. The pressure-driven flow, Q, through the conical nanopore is approximated by eq 1,24
Q=
3πr 3ΔP 8η cot θ
(1)
where r is the pore orifice radius, ΔP is the pressure across the nanopore, η is the solution viscosity, and θ is the nanopore halfcone angle. Notably, the cubic dependence of Q on the nanopore orifice differs from the r4 dependence in a cylindrical pore. This relationship results in a ∼235 times greater volumetric flow rate for the 185 nm radius nanopore than for the 30 nm radius nanopore. Thus, the same pressure has a much larger effect on the flow in the larger nanopore. Our proposed mechanism is supported by finite-element simulations of the ion distributions based on simultaneous solution of the Nernst−Planck, Poisson, and Navier−Stokes equations.18,23 When a pressure is applied across the membrane of the 185 nm pore, the convective flow prevents the depletion and accumulation of the charge-carrying ions, and the conductivity approaches that of the bulk solution, as shown in Figure 3b. Under these conditions, the pore does not rectify the ion fluxes. Conversely, in the 30 nm radius pore, the ICR is independent of the applied pressure, as the flow velocity is too small to cause a redistribution of ions. Simple, reversible, and stable control over ICR in nanopores is still challenging. Compared with other chemical methods used to change ICR (e.g., selective surface modification of the 2607
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charged membranes, application of a current caused EOF in the direction of cation migration, convectively transporting phenol across the membrane. The rate of appearance of phenol in the receiving solution is a measure of the EOF velocity. These studies showed that for equal applied currents of opposite polarity, EOF from base to tip was faster than that from tip to base (Figure 5). It should be noted that for these membranes base-to-tip flow is analogous to inside-to-out flow in single nanopores.
Figure 5. Plots of the amount of phenol transported vs time from EOF experiments on a mica membrane with pyramidal nanopores. Phenol was initially present only on the base side (blue) or tip side (red), which was biased positively vs the opposite side. Reproduced from ref 25. Copyright 2010 American Chemical Society.
Figure 4. (a) Effect of pressure on the ICR rectification factor at ±0.4 V for the 185 nm radius nanopore in Figure 3. (b) Applying an alternating series of pressures shows that the effect of pressure is reversible. Reproduced from ref 23. Copyright 2011 American Chemical Society.
The larger base-to-tip flow agrees with simulations based on the accumulation/depletion model responsible for ICR.18 Baseto-tip flow corresponds to depletion of the ionic concentration within the pore and increased ionic resistance of the membrane. In this mode a higher electric field is necessary to maintain a current equivalent to that required in accumulation mode. Since the velocity of EOF increases with the electric field across the membrane,27 higher flow is observed in the base-to-tip direction (Figure 5). Additionally, lower concentrations, which cause higher EOF for the same electric field, enhance the flow rectification, which also implies that flow rectification should be observed upon application of equal potentials of opposite polarity. EOF data were obtained in mica membranes for pores with a variety of tip and base sizes (Table 1). The extent of EFR was
nanopore or changing the pH/electrolyte concentration of the contacting solution), the physical flow method is a nanofluidics trick for switching ICR on and off. Figure 4a shows a plot of the ion current rectification ratio (ric) for the 185 nm radius conical pore as a function of the applied pressure. The rectification ratio (or rectification factor) is the ratio of the currents measured at voltages corresponding to accumulation (−0.4 V) and depletion (0.4 V) states. In this example, ric decreases from ∼2 (“diodelike)” to ∼1 (“Ohmic”) as the pressure is increased from 0 to ±80 mmHg. The reversibility of switching ICR on and off with pressure is demonstrated in Figure 4b.
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ELECTROOSMOTIC FLOW RECTIFICATION IN CONICAL PORES Analogous to ICR, electroosmotic flow rectification (EFR) is the phenomenon of unequal EOF occurring upon application of voltages of equal magnitude but opposite polarity. The discovery of EFR in synthetic conical nanopores resulted from earlier investigations of ICR; the accumulation/depletion of ions that is responsible for ICR also gives rise to EFR in nanopores. Originally predicted from finite-element simulations of ICR that included the Navier−Stokes equation to account for flow,18 EFR was first experimentally demonstrated in porous mica membranes.25 To investigate EFR, the mica membranes were mounted in a permeation cell with a feed half-cell containing a neutral optical marker (phenol) that was detected spectrophotometrically as it appeared in the receiver half-cell.25 EOF was driven through the membrane by using a Pt electrode in each half-cell solution to pass a constant current through the pores. The mica pores have a pyramidal shape with a smaller tip opening at one face and a larger base opening on the opposite face. In these negatively
Table 1. EOF Velocities and Ion Current and EOF Rectification Ratios for Various Pyramidal-Pore Mica Membranes25 d (nm)
veof (mm/s)
tip
base
base-to-tip
tip-to-base
reof
ric
17 35 52 70 11
122 244 366 488 11
3.8 1.7 0.55 0.32 12
0.37 0.35 0.32 0.23 12
10.3 4.9 1.7 1.4 1.0
5.3 2.7 1.3 1.2 1.0
quantified by the rectification ratio, reof, which is the EOF velocity, veof, base-to-tip divided by veof tip-to-base. A key result of these early experiments was that reof is linearly related to the analogous ion current rectification ratio, ric, albeit with high current and low flow occurring in the ion accumulation state and low current and high flow occurring in the ion depletion state. This experimental observation is qualitatively consistent 2608
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Accounts of Chemical Research with predictions from simulations that ion accumulation and depletion are responsible for both phenomena.18 The effect of current and pore density on EFR in pyramidalpore mica membranes was investigated.26 EOF in the preferred base-to-tip direction increased linearly with current density (Figure 6). This is the expected result because the electric field
Figure 7. Unequal electroosmotic flows at opposite voltage polarities lead to a net flow over a complete voltage cycle. Reproduced from ref 31. Copyright 2016 American Chemical Society.
Likewise, the rectification ratio in ICR decreased with increasing frequency, as previously reported by Guerrette and Zhang for quartz nanopipettes,32 reinforcing the point that EFR and ICR emerge from the same underlying phenomena. The maximum pressure produced by the AC EOF pump against an external load, ΔPmax, was measured by pumping fluid into a closed-end capillary containing a trapped column of air.31 Over the voltage range studied, ΔPmax increased linearly with Vrms, and a maximum pressure of 200 kPa was obtained at 3.5 Vrms, which is over 150 times higher than the best result achieved with an existing AC EOF pumping technology (1.3 kPa).33 Operating an EOF pump in AC mode might offer important advantages relative to conventional DC-mode operation because DC mode requires that redox reactions occur at the device electrodes. For aqueous solutions, these reactions are typically water reduction and oxidation, which produce gas bubbles that can clog microfluidic channels and liberate H+ and OH− that can cause changes in solution pH. In AC operation, these undesired Faradaic redox reactions can be largely suppressed. To demonstrate that AC operation suppresses the unwanted redox reactions, the pH of both the inlet and outlet chambers was measured while driving EOF with an AC voltage of 3.5 Vrms at a frequency of 20 Hz and a constant DC voltage of 3.5 V (Figure 8). In DC operation, the pH of the outlet chamber
Figure 6. Plots of EOF velocity (veof) vs current density (Japp) for a mica membrane with pyramidal nanopores, showing smaller and plateauing flow rates for tip-to-base flow. Reproduced from ref 26. Copyright 2015 American Chemical Society.
increases with current and veof increases with electric field.27 In contrast, the tip-to-base EOF leveled off at high current densities (Figure 6) because tip-to-base EOF corresponds to accumulation mode, and it is known that veof decreases with increasing electrolyte concentration.27 Because of the leveling of the tip-to-base veof, the rectification ratio, reof, in general increased with current density. Finally, a membrane with a lower pore density, 106 cm−2, showed higher values of reof than a 107 cm−2 membrane as a result of the higher membrane resistance and hence higher electric field when the same current was applied. The maximum value obtained for the 106 cm−2 membrane was reof = 12 ± 2, while the maximum for the 107 cm−2 membrane was reof = 6 ± 1. EFR has more recently been observed in a conical glass nanopipette by Keyser and coworkers,28 and additional theoretical analyses of this phenomenon have appeared.29,30 On the basis of these observations of EFR, it was predicted that passing an alternating current (AC), instead of a direct current (DC) as discussed above, through a membrane with asymmetric pores would yield net flow in the base-to-tip direction.26 In one half-cycle, high-velocity flow occurs base to tip, whereas in the opposite half-cycle a lower flow rate would be directed tip to base (Figure 7). This prediction was experimentally verified in AC EOF studies using poly(ethylene terephthalate) (PET) membranes with conically shaped pores.31 The effect of the magnitude of the voltage waveform, Vrms, on the net volumetric flow rate, QEOF, was also investigated, and over the range studied (0.5 to 3.5 Vrms) QEOF increased linearly with Vrms. Again, this occurred because the electric field across the membrane increases with Vrms and the EOF velocity increases with electric field. QEOF decreases monotonically with increasing frequency over the range studied (20 to 5000 Hz), asymptotically approaching zero at the highest frequencies.31
Figure 8. pH of the inlet and outlet chambers vs time while driving rectified EOF at 3.5 V DC (open symbols) and 3.5 Vrms (20 Hz) (solid symbols). A conical-pore PET membrane was used. Reproduced from ref 31. Copyright 2016 American Chemical Society. 2609
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Accounts of Chemical Research increased with time as a result of water reduction and the pH of the inlet chamber decreased as a result of water oxidation.31 In addition, vigorous bubble formation was observed at the Pt electrode surfaces in DC mode. In contrast, in AC mode there was no change in pH of either solution over the time window studied, and no bubbles were observed. These results show that AC operation suppresses Faradaic electrochemistry, an important goal for AC EOF pumps.
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NEGATIVE DIFFERENTIAL RESISTANCE BASED ON EOF In the previous sections, we highlighted two distinct experimental observations: (i) pressure-engendered flow alters the equilibrium accumulation and depletion of ions within the nanopore that are responsible for current rectification, and (ii) EOF rectification is a consequence of the same accumulation and depletion of ions within the nanopore that is the basis of ICR. We have recently combined these two separate phenomena to create nanopore systems that exhibit solutionphase negative dif ferential resistance (NDR).34,35 NDR is a behavior where the conductance of an electrical device decreases suddenly over a narrow voltage window, leading to a decrease in current as the potential is increased. A well-known example of a solid-state device that exhibits NDR is the Esaki or tunnel diode used in microwave oscillators.36 As described below, conical nanopores can exhibit NDR as a result of nonlinear positive feedback between EOF and the ion distributions at the pore orifice. This positive feedback provides a strategy for developing exquisitely sensitive sensors for detecting solution ions that are reversibly adsorbed at the orifice and impact the EOF magnitude. Our investigations of nanopore-based NDR were founded on the work of Yusko and Mayer, who reported that voltageinduced EOF in borosilicate glass nanopores can generate enhanced ICR when the nanopore separates two miscible solutions of different conductivities.37 In their experiments, the nanopore separated an external low-conductivity mixed dimethyl sulfoxide/water electrolyte solution and an internal highconductivity water electrolyte solution. At negative voltages, EOF drives the low-conductivity solution into the nanopore orifice, creating a low-conductance state. Conversely, at positive potentials, EOF maintains the high-conductivity aqueous solution in the orifice region, creating a high-conductance state. Building on the work of Yusko and Mayer, we demonstrated a simple and general method to produce liquid-phase NDR behavior in a negatively charged conical nanopore.34 In our experiments, an external pressure was applied, which delicately balanced the EOF to create a situation where a change in the applied voltage resulted in a sudden change in the direction of net solution flow and a large change in the current. In a typical experiment, the internal high-conductivity solution contained 50 mM KCl and the external low-conductivity solution 5 mM KCl. A pressure of 10 mmHg (internal pressure greater than external pressure) was applied across the nanopore while the voltage was scanned slowly in the negative direction. As shown in Figure 9a, a sudden drop in current occurred between −1.0 and −1.1 V; we refer to the potential at which the change between high- and low-conductance states occurs as the “switching potential”, Vλ. For reasons analogous to the underlying causes of EOF rectification described in the previous section, NDR in the negatively charged nanopore occurs only when the applied potential is scanned negatively; a more
Figure 9. Negative differential resistance (NDR) observed in a glass conical nanopore in which the nanopore conductance switches from a high-conductance state to a low-conductance state. NDR results from a bistability in the fluid flow due to the nonlinear coupling of the double-layer structure, voltage-independent pressure flow, and rectified EOF. Adapted from ref 35. Copyright 2014 American Chemical Society.
Ohmic i−V response (not shown) is observed when the potential is scanned to positive values. A proposed explanation for NDR is based on a simple positive feedback mechanism, schematically shown in Figure 9b,c, involving the ion distribution and fluid flow near the orifice. When a positive pressure and a negative voltage are simultaneously applied across the nanopore, a net flow is established, resulting in EOF (white arrows) driving the lowerconcentration KCl solution into the nanopore while the pressure-driven flow (red arrow) pushes the higher-concentration KCl solution out of the nanopore. These two flows, in addition to the nanopore surface charge, determine the K+ and Cl− distributions within the nanopore and thus the nanopore conductivity. In our experiments, the applied pressure is held 2610
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Accounts of Chemical Research constant while the applied voltage is scanned to negative values. While the potential is scanned, the balance in flow within the nanopore shifts from an outward pressure-dominated flow at low voltages to an inward EOF-dominated flow at high voltages. The change in flow direction results in a decrease in the concentrations of K+ and Cl− near the nanopore orifice, which results in an increased thickness of the double layer, generating a more negative potential of the nanopore surface. The increased surface potential results in an increase in the EOF into the pore, further reducing the ion concentration. This positive feedback between EOF and the ion distribution results in a sudden drop in current at the switching potential. Finite-element simulations of the i−V response for a 260 nm radius nanopore are shown in Figure 10, along with the
Figure 11. (top) DC and (bottom) AC i−V responses of a nanopore showing the dependence of the NDR switching potential, Vλ, on the bulk solution pH. Adapted from ref 35. Copyright 2014 American Chemical Society.
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CONCLUSIONS AND OUTLOOK During the past 20 years, nanopores and nanochannels have provided a versatile platform for fundamental studies of transport in nanoscale spaces while also serving as a workhorse platform for electrical-based chemical sensing. In this Account, we have highlighted more recent studies of the complex coupling of the ion distributions and fluid flow within asymmetric pores, providing examples of how this complex interaction can be used to electrically pump fluids and to engineer nanopore devices with novel electrical behavior. Given the ability to design and construct nanopores and nanochannels with a range of geometries and from different materials, it is likely that EOF rectification and NDR (and other transport phenomena, e.g., osmotic rectification38 and others yet recognized) can be tailored for numerous applications in fluidics and sensing. The use of well-defined nanopores allows very precise electrical and flow measurements that can be quantitatively analyzed and modeled, allowing both the development of theory and the engineering of transport in highly confined spaces. These studies also provide a firmer understanding of fluid flow and ion transport phenomena that occur in charged porous media, especially in electrode materials and membranes used in energy storage. Electric currents in such materials arise from ion transport mechanisms that are fundamentally similar to those described in this Account but that are difficult to analyze in terms of microscopic parameters because they represent ensemble-averaged values of transport through large numbers of pores of different sizes and shapes. Investigations of transport in well-defined nanopores provides a means to unravel and understand transport in these complex systems at a very fundamental level.
Figure 10. Simulated (a) i−V and (b) volumetric flow rate demonstrating that a nonlinear positive feedback cycle causes bistability in the fluid flow and ion conductance within the nanopore. Reproduced from ref 35. Copyright 2014 American Chemical Society.
simulated volumetric flow rate at the orifice. The simulation predicts Vλ = −1.26 V for a 5 mM KCl external solution, in excellent agreement with the experimental measurement (Vλ = −1.11 V). The simulation demonstrates that at potentials positive of Vλ, the flow is directed outward from the nanopore (represented by a positive sign) and its magnitude varies linearly with the potential, a consequence of increasing EOF offsetting pressure-driven flow. As the potential passes through Vλ, a strong inward flow suddenly develops, and the lowconductivity external solution flows into the nanopore, resulting in a significant decrease in the current. Because EOF and Vλ are functions of the surface charge density, σ, any physical or chemical stimulus that changes σ will influence Vλ.35 For example, varying the pH of the external solution alters the silanol dissociation equilibrium, changing σ and causing Vλ to vary by ∼4 V/pH unit. Figure 11 shows the DC and AC i−V responses of the nanopore as a function of pH as the DC voltage is slowly scanned. This dependence of Vλ on pH is remarkably more sensitive than the 0.059 V/pH unit Nernstian response of glass electrodes. The sharp peaks observed in the AC response demonstrate the very narrow potential range over which the NDR switch from high to low conductance states occurs (
