Nanofluidic diodesw

TUTORIAL REVIEW

www.rsc.org/csr | Chemical Society Reviews

Nanofluidic diodesw Li-Jing Chengz and L. Jay Guo* Received 6th July 2009 First published as an Advance Article on the web 13th October 2009 DOI: 10.1039/b822554k Ionic rectifying effect is a unique ion transport phenomenon observed in certain types of nanofluidic devices and cannot be implemented in microfluidics. Analogous to a diode in solid-state electronics, these diode-like nanofluidic devices can be used to turn the ionic flow on and off depending on the polarity of the applied electric-field. In this tutorial review, we summarize recent advances in the experimental and theoretical studies of ion current rectification in several types of nanofluidic devices. We also present a unified model to elucidate the physical mechanism behind the asymmetric ion transport behavior in nanofluidics.

1. Introduction Ion transport through nanometer-sized channels has been widely studied due to the desire and interest to understand the activity of biological ion channels in physiological processes,1 and the prospect of exploiting the property in biomedical and chemical applications, such as molecule delivery and sensing.2,3 Fabricated on solid substrates, artificial nanochannels or nanopores provide robust and controllable means for more versatile applications4–7 and can be easily integrated with many existing microfluidic devices. The general benefit of small channel size is the capability of handling attoliter-scale samples, resulting in minimal usage of reagent, Department of Electrical Engineering and Computer Science, The University of Michigan, Ann Arbor, Michigan, USA. E-mail: [email protected] w Part of the themed issue: From microfluidic application to nanofluidic phenomena. z Current address: Institute of Physics, Academia Sinica, Taiwan. E-mail: [email protected]

Li-Jing Cheng obtained his PhD in 2008 in Electrical Engineering from the University of Michigan, Ann Arbor, USA. He is presently working as a post-doctoral researcher at Academia Sinica, Taiwan. His current research focuses on single-molecule studies of DNA–protein interaction via nanofluidics and magnetic tweezers.

Li-Jing Cheng

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The Royal Society of Chemistry 2010

precise quantity control and efficient process. In addition to these benefits, nanochannels with suitable channel size and reservoir bath concentrations offer selectivity to charged ions and molecules. The charge selectivity in such devices derives from the fact that when the critical size of nanochannels is comparable to the Debye length, the concentration of the counter-ions in nanochannels can be enhanced while that of co-ions is diminished due to the electrostatic interaction between the ions and the charged channel walls. In this regime, the fluidic channel favors the passage of counter-ions and the ion conductance through the nanochannel is governed by the surface charge rather than the bath ion concentration in the reservoirs.8–10 These unique properties open up the possibility to selectively deliver specific types and controllable amounts of molecules or ions through the nanochannels by electrokinetic transport. A nanofluidic device which can conduct ion current preferentially in one direction and inhibits the current flow in the opposite direction is of great interest since it may open a

L. Jay Guo received his MS and PhD from the University of Minnesota in 1995 and 1997, respectively. He was a research associate at Princeton University from 1998–1999. He joined the Department of Electrical Engineering and Computer Science at the University of Michigan, Ann Arbor in 1999, and is currently an associate professor of Electrical Engineering and Computer Science, Applied Physics, and Macromolecular L. Jay Guo Science and Engineering. His research areas include nanofabrication technology, nanofluidic devices and transport, organic photovoltaics, polymer waveguide and surface plasmonic nanostructures with applications in biosensors and light sources, nanoimprint lithography with applications in polymer based electronic and photonic devices. Chem. Soc. Rev., 2010, 39, 923–938 | 923

new way to handle molecular or ion species in fluid, and may even shed light on the mechanism of biological ionic channels. So far most of the flow controls in microfluidic devices are based on mechanical elements, such as valves and pumps. Examples are pneumatic valves actuated by gas pressure11,12 and electrostatic valves which utilize electrostatic force to deform the channel structure. These mechanical switches rely on moving structures to open or close the fluidic flow and are usually relatively bulky and with slow response time. Since most biomolecules of interest are charged, it would be intriguing to explore the possibility of using regulated electrical interactions to switch their transport in a microfluidic environment without any moving parts. By doing so, we may be able to enhance the precision control of flow of ionic species and increase the integration density. In a sense, the approach is analogous to the integrated electronic circuits: voltage or field-controlled electronic devices fabricated on the same substrate to control the flow of electrical current and to achieve multitude of functions. To make the idea reality, different types of electrical control functions should be realized; among them the rectifying effect is one of the essential functions to be developed. Similar to the function of a diode in electronic circuits, ionic rectification allows us to turn off or on the ionic flow by simply switching the polarity of the voltage bias along the device. In this review, we present an overview of the state-of-the-art nanofluidic technologies that have been developed to produce a rectifying effect. In addition, we will elucidate the common physics behind the ionic rectification behaviors observed in different types of nanofluidic channels reported to date. 1.1 Analogous electrokinetic basis between nanofluidic and semiconductor devices About twelve decades ago, Nernst and Plank formulated the basic equations in ionic contexts describing the diffusion of charged particles and their migration in a self-consistent electric field. Sixty years later, these equations were adopted by van Roosbroeck to treat the transport of electrons and holes in semiconductors.13 It was found that electrons in solidstate materials transport in a similar way as the mobile ions in electrolyte solutions. In fact, ions and electrons resemble each other in many ways. From the aspect of transport kinetics, both ions and electrons can be basically treated using the Drude model which describes them in the context of kinetic theory for a neutral dilute gas and, hence, accounts for their ohmic behaviors. Also, they both flow by diffusion and drift mechanisms. The Einstein relations can be applied to both ions in a dilute electrolyte solution and the electrons in most materials, except degenerately doped semiconductors, because they obey Maxwell–Boltzmann statistics. Likewise, Debye– Hu¨ckel theory can handle both in terms of the screening effect of the electric field from the individual charged particles. Additionally, Reiss14 and Shockley15 pointed out that electrons and holes in semiconductors have a strong similarity to the excess and deficiency of protons in water, corresponding to acidic and basic solutions. From a material point of view, it is known that electrons as majority carriers are found chiefly in an n-type semiconductor where they neutralize the positively charged donors, and holes 924 | Chem. Soc. Rev., 2010, 39, 923–938

are similarly found in p-type semiconductors. Correspondingly, in nanochannels, the positively and negatively charged channel surfaces play similar roles as the donor and acceptor doping species in n- and p-type semiconductor materials, respectively. The nanochannels with negative surface charges have enhanced cation concentrations, while those containing positive surface charges have anions as majority charge carriers. When two nanochannels with different surface charge polarities connect, the majority counter-ions from both sides diffuse across the junction and leave a fixed surface charge behind. The exposed surface charges generate a Donnan potential, which is equivalent to a built-in potential in a semiconductor system. They describe the same physics, which is to suppress the diffusion of mobile ions across the space charge region until the system reaches equilibrium. Such equilibrium, characterized by an unequal distribution of diffusible ions between the two parts, is specifically called a Donnan equilibrium. The phenomenon can also be observed in a nano-/micro-channel interface as will be discussed later. While the semiconductor and nanofluidic systems share some similar attributes, they are also different in several aspects. First, most importantly, in fluidic systems most of the cations and anions of interest do not recombine as electrons and holes do in semiconductors, or even as proton and hydroxyl ions do in aqueous solutions. In addition, the mobilities of ions in solutions are very small (104 cm2 V1 s1) compared to the corresponding values for electrons and holes in typical semiconductors (few hundreds to thousands cm2 V1 s1). The doping level in semiconductor materials is usually high enough and dopants completely ionized so that it is appropriate to assume equal concentrations of the fixed charge and the majority carriers, which could be several orders of magnitude greater than that of minority carrier. However, the counter-ion concentration in nanochannels is not solely determined by the surface charge density on the channel walls. The geometry and the bulk concentration of the solutions outside the nanochannels are involved as well. To achieve a high counter-ion to co-ion ratio, a nanochannel should possess high surface charge density, along with small critical dimensions and low ion concentration in bulk solutions.

2. Nanofluidic devices with rectifying effect Several types of nanofluidic devices have been reported to produce ion current rectification. The rectifying effect, exhibited as strong asymmetric ion conduction when switching the voltage biases, is found to be the result of symmetry-breaking in channel geometry, surface charge distribution, bath concentrations, or a combination of these aspects. We first present the experimental observation of each aspect. 2.1 Asymmetric channel geometry A conical nanopore is a nanofluidic device that utilizes asymmetric geometry to rectify ion currents. Prepared by irradiating heavy ions (Bi or U ions) on a 12 mm thick polymer film such as polyethylene terephthalate (PET) or polyimide foils and subsequent chemical etching, the single conical nanopore has a large opening diameter of about 600 nm on one side of the polymer membrane and a small opening This journal is

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Fig. 1 SEM images of the etched side of conical nanopores having larger openings, on (A) a PET film irradiated by Bi ions, and (B) a polyimide film by U ions followed with chemical etching from one side. The small pore opening is below the SEM resolution. (Reproduced with permission from Surf. Sci., 2003, 532, 1061.22)

diameter ranging from 2 to 15 nm on the other side.16–22 The scanning electron microscopy (SEM) images in Fig. 1 show the resulting nanopores on PET and polyimide foils. The chemical etching process produces negatively charged carboxyl groups on the surfaces of this polymer, making the pore cation selective. The conical pore shape and its negative surface charge offer some interesting properties such as rectification of ionic current under symmetric electrolyte concentrations, and the capability of pumping ions against concentration gradients, driven by harmonic electric-field oscillation.18 Fig. 2 shows the typical current–voltage characteristics of a single conical PET nanopore measured under symmetric (i.e. same) KCl concentrations in both sides.23 The I–V curves show that with the pore’s tip side grounded, an applied negative potential on the anode gives higher ion conductance than a positive potential. It was found that the size of the pore tip must be smaller than 15 nm in order to rectify the ion

current when filled with high KCl concentrations. With such small size, the pore tip opening is comparable to the Debye length and hence becomes very ion-selective. The rectification becomes stronger at lower KCl concentration. The rectifying factor (or degree of rectification), which is defined as the ratio of forward-bias current to reverse bias current, can be as high as 4 in a 0.1 M KCl solution at pH 7.22 Beside geometry, it is found that the rectifying effect depends on the ionic strength and the pH value of the electrolyte. Because the isoelectric point (pI) of the carboxylated pore surface is about 3, the rectifying effect functions at neutral or basic pH condition at which the carboxyl surfaces are deprotonated, leaving pore surfaces with excess negative charges. At pH 3, the pore surface charge becomes almost zero and the device exhibits linear I–V characteristics. This is a proof that surface charge is essential for conical nanopores to produce ion current rectification. The sign of the surface charge in the nanopores can also be adjusted by surface grafting. The I–V curves in Fig. 3 shows that the conical nanopores with opposite signs of surface charge display contrary rectification polarity. The ionic rectifying effect was also observed in other nanofluidic devices based on an asymmetric channel geometry. Wei et al. pulled nanopipettes directly from clean quartz tubes using a microprocessor-controlled CO2 laser-based puller, to shrink the opening of the glass capillaries on one side to about 20 nm in radius, resulting in glass capillaries with different size of openings.24 With similar channel structure to conical nanopores and negative surface charges, the nanopipettes exhibit asymmetric ion conductance. In another approach, Li and Chen proposed a novel method to fabricate asymmetric nanopores.25 They exposed a pore of diameter about 100 nm

Fig. 2 Rectifying properties of a single conical nanopore in a PET membrane with the voltage applied on the base side with the tip side grounded. (A) I–V characteristics obtained under symmetric ion concentration at pH 8 and 3 M KCl (K), 1 M KCl (&) and 0.1 M KCl (,). (B) I–V curves recorded under symmetric 0.1 M KCl at pH 8 (’) and pH 3 (n). (Reproduced with permission from Adv. Funct. Mater., 2006, 16, 735.23)

Fig. 3 (Left) I–V curves of a single gold conical nanopore modified with 2-mercaptopropionic acid to form negative surface charges measured in 0.1 M KF solution at pH 6.6 (&) and pH 3.5 (K). (Right) I–V curves of a single gold conical nanopore modified with mercaptoethylammonium to form positive surface charges obtained in 0.1 M KF, pH 6.6. (Reproduced with permission from J. Am. Chem. Soc., 2004, 126, 10850.21)

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to an argon beam at room temperature to gradually close the pore on one side. Diameters of the pores as small as 1.8 nm were achieved. The asymmetric pore size yields rectification of ion current. 2.2

Asymmetric bath concentrations

A rectifying effect can also be produced in a homogeneous nanofluidic channel containing ion concentration gradients along the channel.26 The asymmetric ion conductance can be observed under the condition that only one side of channel has electric-double layer overlap, due to the low bath electrolyte concentration, while the other side does not. In this circumstance, the charged channel walls provide asymmetric electrostatic impact on the ions between the two sides of the nanochannel. Such an asymmetric electrostatic impact of a nanochannel on ions induces ion current rectification. The ionic rectifying effect in conical nanopores shares the same physical basis except that it is produced by different approach. The experiment was performed on planar nanochannels of two thicknesses, 20 and 4 nm, which were formed by removing amorphous silicon sacrificial structures embedded in silica layers by gaseous XeF2 etching. The images in Fig. 4 show that the channel height can be well controlled by the thickness

of the sacrificial layers. During measurement, one bath with low KCl concentration (termed CL) is fixed at 0.1 mM and the other side with high concentration (termed CH) has a final concentrations between 0.1 mM to 1 M. It is found that three regimes of I–V characteristics were obtained. They are the symmetric regime (i.e. ohmic behavior), rectifying regime, and weakened rectifying regime. The I–V characteristics corresponding to different CH with fixed CL = 0.1 mM are plotted in Fig. 5(Ia–e) and (IIa–e). Fig. 5(If) and (IIf) summarize the ion conductance under forward and reverse biases, and rectifying factor IF/IR, where IF and IR were the currents measured at 5 and 5 V bias voltage, respectively. In the symmetric regime, in which CH is lower than 10 mM (for 4-nm channel) or 1 mM (for 20-nm channel), both IF and IR are almost identical and the nanochannels showed the ohmic behavior in conducting ionic currents. Although the bath concentration is asymmetric, the current level is almost equal to that obtained in the case of symmetric concentration (i.e. CL = CH = 0.1 mM). In the second regime, the current becomes asymmetric with respect to forward and reverse voltage biases. As shown in Fig. 5(f), the rectifying effect is maximized at CH = 0.1 M with IF/IR B 3.5. However, in the third regime when CH = 1 M, both IF and IR increased, yielding weaker asymmetric I–V characteristics. The results show that to produce ion current rectification, only one of the baths should have its ion concentration low enough to allow the EDLs to overlap at that side of nanochannel. The bath concentration in the other side should be high enough to avoid EDL overlap. However, the concentration in CH side cannot be too high; otherwise the surface charge has negligible effect on the ion conductance. 2.3 Asymmetric surface charge distribution

Fig. 4 (a) Phase-contrast microscopic top-view image of the device. Five 60 mm-long nanochannels connect between two microfluidic channels connecting the nanochannels and microchannels. The transparent bar lying across the nanochannels is a PDMS wall separating two microchannels. (b) Cross-section SEM image which shows that each nanochannel is 2.5 mm wide and about 20 nm thick. (Reproduced with permission from Nano Lett., 2007, 7, 3165.26)

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The idea of a nanofluidic diode consisting of opposite surface charges on either half of the nanochannel was proposed by Daiguji and his co-workers.27 The asymmetric surface charge distribution in nanochannels gives strong rectification of ion current. The fundamental of this device structure is essentially the same as the widely-studied bipolar membranes in electrochemistry. Bipolar membranes comprised of a negatively charged cation exchange membrane and a positively charged anion exchange membrane presents ion rectification28 and have been used for decades in many applications such as electrodialysis and chemical separation.29,30 Possessing similar properties but much smaller size, nanofluidic diodes may be integrated on a fluidic chip to perform pH control and chemical separation process in a miniaturized scale. Several experimental results of the bipolar nanofluidic diode have been reported. The key challenge to creating this type of nanofluidic diode is to define different surface charges on the channel walls. Karnik and his co-workers developed diffusionlimited patterning (DLP) to pattern the cationic protein avidin inside biotinylated nanofluidic channels.31 The nanofluidic channels were fabricated by the removal of 30 nm thick silicon sacrificial channel patterns in a silica layer. To form an asymmetric surface charge distribution, the whole surface of the 30 nm thick nanofluidic channel was first covered by biotin. Avidin was then introduced from one of the channel This journal is

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Fig. 5 Measured I–V characteristics of 4-nm thick nanochannels (Ia–e), and 20-nm thick nanochannels (IIa–e) under various asymmetric concentrations (CL8CH). The forward-biased and reverse-biased conductances and the rectifying factor IF/IR are summarized in (If) and (IIf) for 4-nm and 20-nm thick nanochannels, respectively. The concentration in the left side (CL) is fixed at 0.1 mM while the right side (CH) varies from 0.1 mM to 1 M. The channel width is 2.5 mm 5 and length, 60 mm. (Reproduced with permission from Nano Lett., 2007, 7, 3165.26)

openings until half of the nanochannel length was covered with the avidin protein molecules. Since the avidin coated and biotinylated surfaces contain positive and neutral surface charges, respectively, surface charge discontinuity is generated

Fig. 6 (a) Schematic diagram of a nanofluidic diode consisting of a positively charged surface and neutral surface in either half of the channel. The positive charge is produced by avidin while the neutral charge is produced by biotin moieties. (b) Epifluorescence image showing fluorescently labeled avidin in half the nanofluidic diode. (Reproduced with permission from Nano Lett., 2007, 7, 547.32) (c) and (d) Schematic diagram and microscopic image of a solid-oxide nanofluidic diode composed by Al2O3 and SiO2 nanochannels. (Reproduced with permission from ACS Nano, 2009, 3, 575.35)

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along the nanochannels. Fig. 6(a) and (b) show a schematic diagram and epifluorescence image, respectively, of fluorescently labeled avidin in a nanofluidic diode. The ion rectification of the nanofluidic diode is found to be dependent on electrolyte concentration. The current rectification is prominent at intermediate concentrations of 1 mM and 10 mM KCl. The effect becomes weaker at higher concentration because the surface charge effects become negligible. Linear I–V characteristics were also observed at low KCl concentrations. The authors suggest the possible reasons to be the enhanced water dissociation at reverse bias or the polarization effect at channel entrances leading to an increase of series resistance out of the nanochannels.32 Vlassiouk et al. reported a nanofluidic diode created by modifying the surface chemistry of the conical nanopores in PET membranes.33 The surface of the nanopores were modified to have one half of the nanopore covered by carboxyl groups and the other half by amino groups. The dependence of rectifying behavior on the pH values and electrolyte concentrations were studied. It was found that the maximum rectifying factor occurs at pH 4–7 at which the surface charge densities were maximized. Furthermore, the rectifying factor reaches about 120 at 10 mM KCl but decreases by two orders of magnitude when the concentration increases from 10 mM to 1 M. Because the weakened rectifications at higher concentration results from the change of ion transport behavior from electrophoresis to electroosmosis, the authors conclude that it is electrophoresis that depletes the ions from the nanofluidic diode at reversed bias. Chem. Soc. Rev., 2010, 39, 923–938 | 927

Apart from the aforementioned surface chemical modification approach, a heterogeneous surface charge distribution in the nanochannels can be achieved by using different solid oxide materials which have different isoelectric points (pI). We have demonstrated this principle by constructing the first solid oxide heterogeneous nanofluidic devices using nickel oxide and silicon dioxide.34 Recently, we developed more robust bipolar nanofluidic channels composed of a negativelycharged silicon dioxide nanochannel and a positively-charged aluminium oxide nanochannel connected in series as illustrated in Fig. 6(c).35 The pI of a SiO2 surface is about 3, while that of Al2O3 and NiO surfaces are about 9 and 11, respectively.36,37 Therefore, positively and negatively charged surfaces can be generated in aqueous solution at physiological pH, and with improved stability. The effective surface charges on these solid oxide materials are measured to be of the same order of magnitude (1–5 mC m2).35 By employing different solid oxides having distinct pI values in a nanofluidic device, we can tailor the surface charge distributions in the nanochannel.

The fabrication method used to make heterogeneous nanochannels is similar to that used in making homogenous SiO2 nanochannels, which is based on a sub-20 nm thick sacrificial layer approach to precisely define the height of the nanochannel.26 The difference here is that two additional Al2O3 thin films were patterned by photolithography to sandwich half of the sacrificial layer in order to form a nanochannel comprising of two sections made of different oxide surfaces. A microscopic image of the fabricated device is shown in Fig. 6(d). Even without further chemical treatments, such nanofluidic devices possess relatively robust physical and chemical properties. The I–V characteristics of the SiO2–Al2O3 nanofluidic diode was measured at varied bath KCl concentrations ranging from 10 mM to 1 M. In the measurement, as illustrated in Fig. 6(c), a voltage bias Vd was applied to the bath that connects to SiO2 nanochannel while the Al2O3 side was grounded. The experimental I–V curves presented in Fig. 7(a)–(f) (empty circles) display strong rectification effect at every concentration

Fig. 7 I–V characteristics of a nanofluidic diode (equivalent channel width = 2.5 mm 10) measured at different bath KCl concentrations: (a) 0.01 mM, (b) 0.1 mM, (c) 1 mM, (d) 10 mM, (e) 100 mM and (f) 1 M. The open circles are experimental data and the dotted curves are the calculated result based on a simplified 1D transport model (discussed below). (g) Log–log plot of the experimental forward-conductances (open squares), reverse-conductances (open circles) and the rectifying factors (shaded bars) at different bath concentrations. The rectifying factor is defined by IF/IR under the bias of 1 or 1 V, at different bath concentrations. The solid and dashed curves are the theoretical conductances calculated by 2D PNP equations. (Reproduced with permission from ACS Nano, 2009, 3, 575.35)

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except at 1 M. Fig. 7(g) summarizes the channel conductances under biases of 1 and 1 V (empty squares and circles, respectively) and the associated rectifying factors obtained at varied bath concentrations (shaded bars). The rectifying factor reaches its maximum, 321, when the bath KCl concentration is 1 mM. It is also important to note that the rectifying factor can be even greater if the currents are measured at higher voltage biases. We found that above 1 mM the rectifying factor degrades significantly with the increase of bath ion concentration. Below 1 mM, the rectifying factors decrease with ion concentration but are still relatively high: about 200 for 0.1 mM and decreases to 30 for 0.01 mM. Such high rectification behavior has not observed in prior work.32–34 The reason for this may be that our photolithographic approach can produce more abrupt junctions in surface charge distribution that can not be achieved by surface chemistry modification via diffusion. The abrupt junction gives a more efficient control of ion accumulation and depletion in the junction region. It is worthwhile to point out that a biological form of nanofluidic diode was also implemented by modifying a nonrectifying biological porin, OmpF, into a molecular diode. It was achieved by mutating the protein to create a channel structure which contains spatially separated selectivity filters enabled by opposite charges.38 A rectifying factor of greater than 5 was observed in this protein p–n junction ion channel. Another way to obtain asymmetric surface charge distribution in the porin is to adjust the pH values in the two reservoirs. Under highly asymmetric pH conditions, the porin presents diode-like I–V characteristics.39 These interesting results also imply that the rectifying effect can be achieved in a channel structure as short as the order of a Debye length.

3. Theory of rectifying effect in nanofluidic devices 3.1 Qualitative interpretations of ion rectification in the nanofluidic devices In the previous section, we have introduced various types of nanofluidic devices capable of rectifying ionic currents. Evidently, we can find that it is the breaking of symmetry in device geometry, ion concentration or channel surface charge, that causes the rectification effect. But why do these nanofluidic devices with different configurations behave in such a similar manner? It can be envisaged that they must share some common physics. Theoretical calculations of electrical potential profiles and ion distributions obtained by solving Poisson–Nernst–Planck (PNP) equations or electrodiffusion equations can be applied to explain the phenomena. However, before delving into much quantitative details, we would like to give the readers a qualitative understanding of the mechanism behind the rectifying effect based on a unified model. On the basis of this model, we will see that the ionic rectification resulting from the accumulation or depletion of ions in response to different bias polarities is due to the asymmetric cation/anion ratios established by certain forms of symmetry breaking in nanofluidic devices. We interpreted the phenomena by analyzing the drift current of K+ and Cl ions near the entrances of the nanochannel right after the external voltage is applied and the space This journal is

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charge that causes a non-uniform electric potential has built up.26 Under this condition the ion concentrations are still nearly identical to those at equilibrium and the diffusion fluxes are negligible. Electroneutrality is assumed to be preserved in such a transient phase when ionic concentrations change over time and reach steady-state values. This method is an extension of the analysis proposed by Pu et al. in explaining the ion-enrichment and ion-depletion effects at nanochannel–microchannel interfaces with low ion concentration.40 Instead of being applied to the micro/nanochannel interface, the analysis is expanded here to explain the similar effect of ion-enrichment or depletion developed inside the nanochannel that brings about the consequent ion current rectification.26 From a mathematical point view, one can treat the analysis as a way to predict qualitatively the steady-state solution to an electrodiffusion equation, i.e., whether the ion concentrations in the nanofluidic device increase or decrease under an external electric-field, by examining the initial conditions of drift currents at time zero at which the applied electric-field just builds up. As mentioned before, the rectifying effect stems from the accumulation and depletion of ions in the channel in response to different bias polarities. We would like to make an educated guess on how the ion concentration changes under an applied electric field based on the cation/anion ratios. To gain an intuitive understanding, we use some simple diagrams to explicate the idea. First of all, let us consider a nanofluidic device illustrated in Fig. 8(a) which has asymmetric cation/ anion ratios between the two sides of channel. The solid and the empty sticks beside a box placed in the channel center represent cation and anion concentrations. The scale of the sticks depicts their quantity. Here we have similar cation and anion concentrations in the left side of the channel (cL+ = cL), while in the right side the concentration of cation is much greater than that of anion (cR+ c cR). The subscripts denote left (L) and right (R); while the superscripts represent cation (+) and anion (). It is known that at location x of the nanochannel, the current J of an ion species i flowing through a cross-section area of A driven by an electric-field E can be expressed as Ji(x) = mici(x)NAE(x)A(x), where m, c and NA are electrophoretic mobility, ion concentration and Avogadro’s number, respectively. Because the ions do not redistribute immediately right after the external electric-field builds up, the ion current can be seen to be proportional to the ion concentration at equilibrium. It is worth noticing that the exact amount of E(x)A(x) is not important in this analysis since it is ‘‘self-adjusted’’ to follow the conservation of total ion currents, or Kirchhoff’s current law, i.e., JL+ JL = JR+ JR. To present the relation graphically, the sum of the two sticks in each side of channel should be equal in length. When an electric field is applied to the right, the cations are driven to the right side whereas the anions, to the left side as indicated by the arrow in Fig. 8(b). The amount of the currents is determined by the same ion concentrations shown in Fig. 8(a). It can be found that under such bias polarity, the amount of the cation flux flowing out of the channel is larger than that flowing into the channel, i.e., JL+ o JR+. Moving to an opposite direction, the anion flux flowing outward to the left is greater than that flowing inward from the right, Chem. Soc. Rev., 2010, 39, 923–938 | 929

Thus, the cation/anion ratio, which is equal to c+/ c, can be expressed as eqn (1): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 0 2 2 2ss 103 cþ @f þ f þ 4cb A a¼ ¼ ; where f ¼ 2cb qNA h c

Fig. 8 Diagram of cation (solid arrows) and anion (empty arrows) fluxes in a nanofluidic device containing asymmetric cation/anion ratios under zero bias (a), and right after the build-up of a positive (b) and a negative (c) external electric field. The consequent ion depletion or accumulation taking place in the nanochannel at steady state can be estimated by comparing the amount of ion fluxes drifting inward and outward.

i.e., JL 4 JR. For both types of ions, the outward currents are larger than the inward currents. In such condition, when the system reaches steady state, the ions in the channel are depleted, resulting in low ion conduction. On the contrary, reversing the direction of the applied electric field turns over the arrows of ion fluxes as shown in Fig. 8(c). In this case, more ions are driven into the channel while less are taken out. Such an initial condition leads to an accumulation of cations and anions in the channel and hence an increase of ion conductance at the steady-state regime. Therefore, rectification of ion current is established. Now we have demonstrated that it is the asymmetric cation/ anion ratios at the two sides of nanochannel that generate rectification of ion current. Later we will see how the asymmetric cation/anion ratios in a nanofluidic device can be physically produced. Before we can start, we have to know what parameters determine the cation/anion ratio in a nanochannel. The following is the derivation of cation/anion ratio. In the Donnan equilibrium, the averaged cation concentration c+ and anion concentration c can be calculated by solving the equation of electroneutrality c+ c + f = 0, combined with the law of mass action, c+c = cb2, where cb represents the bulk ion concentration outside the nanochannel, and f is the fixed charge concentration of the nanochannel. The sign of f depends on the polarity of the fixed surface charge in the nanochannels. The results give:41 cþ ¼ f =2 þ c ¼ f =2 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðf =2Þ2 þ c2b

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðf =2Þ2 þ c2b

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ð1Þ

where the fixed charge concentration f is a function of surface charge density ss (C m2), and channel height h. Eqn (1) indicates that a decrease of cb, or increase of f yields a greater cation/anion ratio a. A unipolar solution of counter-ions is created in the nanochannel when cb o f/2 because the surface charge of nanochannels governs the ion concentrations. When cb 4 f/2, the cation/anion ratio decreases as the surface charge has less influence on the ions in nanochannels. The parameters in eqn (1) also tell us that the asymmetric cation/anion ratios can be created by several means, including asymmetric bath concentrations cb, asymmetric channel geometry h and asymmetric channel surface charge ss, or even the combination of them. The unified model based on the analysis of asymmetric cation/anion current is summarized in Fig. 9 to interpret the ionic rectifying effect observed in three different types of nanofluidic diodes. We can find that the origin of all of the three nanofluidic diodes features a symmetry-breaking in the cation/anion ratios. It can be seen that with having opposite polarity of counter-ions, a biased bipolar nanochannel can accumulate or deplete ions much more efficiently than the other two types of nanochannel. This explains why bipolar nanochannels give higher rectifying factor than conical nanopores or nanochannels with asymmetric ion concentrations. It is worth noting that the analysis can also be applied on the nanochannel–microchannel interface to explain ion polarization. For simplicity, in the analysis we only consider the case where the nanofluidic devices which contain monovalent salts and the electrokinetic mobility of the cation and anion have almost equal value (KCl in this case). However, one can extend the idea to examine the system involving asymmetric ion mobility or ion valence. 3.2 Quantitative description of ion rectification by solving Poisson–Nernst–Planck (PNP) equations To understand how the rectifying effect takes place in nanofluidic devices in more detail, the ion concentration distributions and the electric potential profiles in these three types of rectifying nanofluidic devices have been theoretically studied and reported in several reports. Several simplified models have been developed to describe the effect; however, most of them derive from the fundamental physical model based on Poisson–Nernst–Planck (PNP) equations. The total electric potential denoted by V satisfies the Poisson equation (2): X re0 er rV ¼ F zi ci ð2Þ i

where e0, er, and F are the permittivity of vacuum, the relative permittivity of the solution and the Faraday number, while the summation is carried over all the ions present in the solution with ci (mol m3) being the concentration of ion species i (e.g. K+ or Cl). The ionic flux of the ionic species i contributed by the diffusion current due to concentration gradients and the This journal is

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Fig. 9 Interpretation of ionic rectification in different types of nanofluidic devices based on the analysis of asymmetric ion currents building up right after the external electric fields are applied. The solid (dashed) lines represent the cation (anion) concentration profiles in nanochannels. The solid (empty) arrows symbolize cation (anion) fluxes. For each ion species, if the inward current is greater than outward current, ions will accumulate in the channel when the system reaches the steady state. On the contrary, if there is more outward current than inward current, ion depletion takes place in the channel. The superscripts of a current flux, J, denote ion species; in the subscripts, n and B indicate nanochannel and bath regions, whereas L and R are left and right.

drift current induced by potential gradients is defined by the Nernst–Planck equation (3): Ji = (Dirci + zimiFcirV)

(3)

where Di, mi and zi are the diffusivity, electrophoretic mobility and the valance number of ion species i. Since we are interested in the steady-state solution, the flux should satisfy the timeindependent continuity equation (4) when the system reaches a stationary regime: rJi = 0

(4)

The combination of these three equations together with the appropriate boundary conditions allows the calculation of electric potential distribution and the ion fluxes in a system with nonequilibrium conditions. Conical nanopores. Cervera et al. computed ion concentrations and electrical potential profiles in a synthetic conical nanopore filled with 0.1 M KCl responding to different voltage biases.42 A set of two-dimensional PNP equations was solved in This journal is

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spherical coordinates with the origin at the cone apex. The nanopore is 12 mm long and the radii of the openings at the tip and the base side are 5 and 300 nm, respectively. Since the pore opening angle is set to a small angle of 1.41, the ionic fluxes were assumed to have only radial components. Besides, the access resistance is neglected; thus Donnan equilibrium is assumed at both pore entrances. Fig. 10 shows the calculated results at different voltages applied across the nanopore with negative surface charges. The ion profiles indicates that at zero bias, the concentration of counter-ion, K+, is higher than that of Cl in the nanopore, especially close to the pore tip, due to the strong electrostatic interactions between charged walls and ions. When an electric field is applied from the pore tip to the base, both K+ and Cl ion concentrations increase throughout the nanopore and accumulate even more at the tip region. The accumulation of ions results in the increase of ion conductance. On the contrary, when an external electric field is applied to an opposite direction, both types of ions in the nanopore are depleted. As a result, the ion conductance decreases. The calculated I–V curves in Fig. 10 (right) reproduce the Chem. Soc. Rev., 2010, 39, 923–938 | 931

Fig. 10 (Left) Average ion concentration distributions and electrical potential profiles in a conical nanopore calculated by PNP equations: (top) zero bias, (middle) 0.5 V, and (bottom) 0.5 V, applied on the tip side with the base side grounded. (Right) Experimental data (empty and filled circles) and theoretical results for negatively (a) and positively (b) charged nanopores with varied surface charge densities s. (Reproduced with permission from Europhys. Lett., 2005, 71, 35.42)

experimental trends measured from nanopores with varied surface charge densities and opposite charge polarities. It should be noted that the polarity of the rectifying effect is determined by the polarity of the surface charge. It is the asymmetric impact of electrostatic interaction between channel walls and ions that produces the ionic rectification. If there is no surface charge in the nanopore device, the rectifying effect will disappear despite its asymmetric geometry. Concentration gradient in homogeneous nanochannels. We reported the theoretical study of the electrokinetics of a homogeneous nanofluidic channel containing ion concentration gradients by solving two-dimensional PNP equations.26 The system is composed by a 10 mm long, 20 nm thick homogeneous nanochannel connected with two 2 mm size square reservoirs. The ion concentrations in the two reservoirs are defined as CL and CH for low-concentration and highconcentration baths, respectively. The CL side has KCl concentration fixed at 0.1 mM and the CH side has KCl concentration present in a final concentration between 932 | Chem. Soc. Rev., 2010, 39, 923–938

1 mM to 1 M. Fig. 11 is the calculated average ion concentration and potential profiles along the device under different electrical biases. The results show that at low concentration, (i.e. CH = 1 mM at which the Debye length mD B 10 nm is comparable to the channel height), the concentration of K+ ions is enhanced while that of Cl ions is suppressed compared with the bath concentrations (Fig. 11(a)). In this case, the concentration gradient in the nanochannel is less remarkable because the surface charge has greater impact on the ion concentrations. When the CH reservoir is biased, both K+ and Cl ion concentrations slightly increase but the profiles remain flat in the channel. Instead, the applied voltages induce accumulation or depletion of ions outside the nanochannel at both channel accesses. With such low ion concentrations in both sides, EDL overlap occurs throughout the nanochannel despite of the asymmetry in the bath concentrations. Once again, the ion concentrations in the nanochannel are mostly controlled by the surface charge rather than by the external voltages in the baths. Therefore, either positive or negative bias leads to almost the same ion concentration level and This journal is

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Fig. 11 Calculated average ion concentration and potential profiles along a 20 nm thick, 10 mm long nanochannel placed between two KCl reservoirs containing different concentrations CL (left) and CH (right) as illustrated in the schematics. In this system, the CL side is electrically grounded and its concentration is fixed to 0.1 mM, while the final concentrations of the CH side is set to be (a) 1 mM, (b) 10 mM, (c) 0.1 M or (d) 1 M. Vd is biased at 5, 0 or 5 V as shown in the columns from left to right. EDLO stands for EDL overlap. Letters A or D with indications denote the locations of the accumulation or depletion of ions. (Reproduced with permission from Nano Lett., 2007, 7, 3165.26)

potential gradient, which explains the measured symmetric I–V characteristics. As CH is increased to 10 or 100 mM, the ion concentration profile, as shown in Fig. 11(b) and (c), either buckles up or slumps depending on the polarities of the applied voltage. A negative bias at CH elevates both K+ and Cl concentrations in the nanochannel leading to high channel conductance. On the other hand, when CH is positively biased, both K+ and Cl are depleted to a low concentration, resulting in extension of EDL-overlap along the nanochannel and, hence, low channel conductance. It is this different ion distribution that produces the rectifying effect observed in this regime. With high concentration in CH, 1 M for example, the amount of ions is large enough to shield the surface charges in the nanochannels. In this case, almost no EDL overlap occurs throughout the entire channel except its very left end. As can be seen in Fig. 11(d), the accumulation or depletion of This journal is

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both K+ and Cl ions are relatively less manifested and a large, constant ion gradient along the channel is established. The reason for the weakened accumulation when CH is negatively biased is that the potential drop in the nanochannel is alleviated due to the access resistance produced by the depletion of ions around the channel access in the CL bath. The higher the CH, the greater the voltage drops across the access resistance in the CL bath. In this case, the measured conductance no longer represents the intrinsic property of the nanochannel. On the other hand, at positive bias, the amount of ion depletion is comparably small as the background ion concentration in the nanochannel increases with high CH. Voltage biases of different polarities show less influence on ion distribution; therefore, the asymmetric I–V characteristic is not observable. Such induced access resistance should also exist in the nanopores, leading to weakened rectification under very high bath concentration. Chem. Soc. Rev., 2010, 39, 923–938 | 933

Bipolar nanochannels. Calculation of two-dimensional PNP equations for a nanofluidic diode having asymmetric surface charges was first reported by Daiguji et al.27 The simulation was performed for a 30 nm-thick, 5 mm-long nanochannel straddled by two 1 mm square size reservoirs. To understand the pressure profiles in the nanochannel, a Navier–Stokes equation was also included in the calculation. It is assumed that the left and right halves of the channel have surface charge densities of 2 and 2 mC m2 and that the channel contains 5 mM KCl aqueous solution. The resulting averaged ion concentration, electrical potential and pressure profiles along the channel are plotted in Fig. 12. It shows that at forward bias, i.e. a 5 V applied to the negatively charged nanochannel with the other side grounded, both the K+ counter-ions in the negative channel and the Cl counter-ions in the positive channel migrate toward the junction driven by the electric field. As a result, at steady state, both types of ions pile up around the nanochannel junction. The high ion concentration leads to increased channel conductance. At reverse bias, i.e. with 5 V applied to the negatively charged nanochannel with the other side grounded, the counter-ions on both sides are extracted out of the channel leaving a depletion zone at the junction. As a result of the ion depletion, channel conductance is reduced significantly. Because the flux is constant throughout the channel, the potential drops mostly in the low ion concentration regions. This explains why the

potential changes gradually along a forward-biased channel while the potential drops abruptly across the junction of nanochannel at reverse bias. Several current–voltage models of the bipolar nanofluidic diode have been reported in the literature recently.43–45 Based on a 1D Nernst–Planck equation, the I–V curves for a bipolar nanochannel with a length of L for each half of the channel (total length of the device is 2L), and a bath concentration of cb can be described by eqn (5):35 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F f 1 2 coth1 1 þ ð2cb =f Þ2 þ 2JL=fFD Vd ¼2 sinh RT 2cb qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 1 þ ð2cb =f Þ2 þ 1 þ ð2cb =f Þ2 þ2JL=fFD ð5Þ where F is the Faraday number and J is the current density responding to the applied voltage Vd; the diffusion coefficients D of cations and anions are assumed to be the same.46 The first and second terms of the equation are contributed by the Donnan potentials across the channel entrances and the junction of the heterogeneous channels, respectively, while the last term is derived from the sum of the voltage drops along the positive and negative nanochannels. Except for some minor difference in the mathematical forms, eqn (5) is basically identical to that reported in other literature to describe the I–V

Fig. 12 Electric potential (along the channel center), ion concentration (averaged over the channel height) and pressure profiles (along channel center) for (a) a forward bias of 5 V and (b) a reverse bias of 5 V. The surface charge densities at the left and right halves of the channel are assumed to be 2 and 2 mC m2, respectively. The bulk concentration is 5 mM. The schematics of the device on the bottom shows that ions accumulate at the junction under forward bias but deplete from the junction under reverse bias. (Reproduced with permission from Nano Lett., 2005, 5, 2274.27)

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relation of bipolar membranes.47,48 The model is found to agree with most of the experimental I–V curves under various bath ion concentrations as shown in Fig. 7. The I–V model has the following features: (1) At large forward biases, the equation approximates to eqn (6): JF ¼ JðVd 0Þ 2 ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f F 2 F 4 1 þ ð2cb =f Þ Vd þ Vd ð6Þ ¼ DF 8L RT RT (2) At low bath ion concentrations (cb { f/2), the forward current is proportional to the fixed charge density f of the nanochannels but independent of bath ion concentrations cb (eqn (7)): 2 ! f F 1 F Vd þ Vd JF ðcb f =2Þ ¼ DF ð7Þ 2L RT 4 RT (3) At high bath ion concentrations (cb c f/2), both f and cb affect the forward ion current density giving the relation (8): 2 ! 1 F f F JF ðcb f =2Þ ¼ DF ð8Þ cb Vd þ Vd L RT 8 RT Because of the high cb, if Vd is not so large, eqn (8) reduces to a linear form, independent of fixed charge density f, JF (cb c f/2) = (DF2/LRT)cbVd. Eqn (6)–(8) suggest that in the forward conduction regime, the current is a quadratic function of the forward bias, which has also been identified in previous literature.27 However, if the fixed charge density is great enough that the concentration of the minority ions in the nanochannel is negligible, the I–V curve will have an exponential relationship as in the case of semiconductor devices and some ion exchange membranes.49,50 (4) Under reverse biases (Vd { 0), the second term of eqn (5), i.e. the junction voltage, dominates the total voltage. The current saturates to: Jsat ¼

2DF 2 c Lf b

ð9Þ

Eqn (7)–(9) well describe the observed trend of the forward and part of the reverse ion conductance at different bath concentration regimes as summarized in Fig. 7(g) of the previous section. 3.3 Comparison of rectifying effects in nanofluidic diodes and semiconductor diodes As mentioned in the beginning of the article, ionic and electronic systems share many similarities, and it would be interesting to compare the physics behind the current rectification in a nanofluidic and a semiconductor diode. When an n- and a p-type semiconductor is put together, a positive and a negative space charge region is established at the junction, and the mobile charges are depleted. The resultant built-in potential due to the formation of space charge impedes the diffusion of majority carriers from both types of semiconductor to cross the junction. In a nanochannel p–n diode similar situation takes place as discussed earlier. This journal is

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While both nanofluidic diodes and semiconductor diodes exhibit like rectifying I–V characteristics, they work differently in some ways. For instance, the forward-bias current in semiconductor diodes is dominated by the diffusion of charge carriers having enough energy to overcome the built-in potential in the space-charge region. Once across the junction, these charge carriers becomes the minority carriers, which will recombine with the majority carriers within about a diffusion length from the edge of depletion or space-charge region. The result limits the potential drop in the depletion region, shielding the rest of the device from the effect of an electric field. But in a nanofluidic diode, there is no combination of cations and anions in the solution. Therefore, when the majority cations flow from the negative channel across the junction driven by a forward bias, they keep flowing through the positive channel. Meanwhile, the majority-anion concentrations in the positive channel must have a corresponding change in their distributions throughout the entire channel to maintain electroneutrality. In this case, the applied electric field affects the entire device, and drift current dominates the forward-bias ion flow.

4. Other diode-based nanofluidic devices p–n Junction diodes are the elementary building blocks of most semiconductor devices. They can be found in bipolar junction transistors, field-effect transistors or thyristors. The idea of constructing some new devices based on combination or rearrangement of diodes is found to be applicable in nanofluidics. 4.1 Nanofluidic diac A double junction diode in the form of a SiO2–Al2O3–SiO2 nanochannel has been developed with equal length of 20 mm in each segment as illustrated in Fig. 13(a).35 Having similar characteristics of a semiconductor pnp diode (or diac, as named after a diode for alternating current), the nanofluidic pnp diode has been proposed theoretically27 and demonstrated experimentally51 to have its current saturating under both positive and negative biases. The result is the consequence of the fact that either polarity of the applied bias can cause one of the two p–n junctions to be reverse biased and hence saturates the ion current. It can be seen in the ion concentration and potential profiles calculated using 2D PNP equations (Fig. 14) that almost all the applied voltage drops across the reverse-biased p–n junction yielding a high electric field in a restricted space. The I–V curve of a pnp diode measured by sweeping the voltage forward and backward is shown in Fig. 13(b). Two hysteresis loops can be seen in the saturation regimes. By sweeping the current and acquiring the voltage, a distinctive breakdown behavior was observed in the I–V curve in Fig. 13(c) which resembles the DC behavior of a semiconductor diac.52 This I–V characteristic is composed by three regimes: (1) saturation regime, (2) negative-resistance regime, and (3) breakdown regime. The saturation regime and breakdown regime show a large difference in channel conductance and are about 8.1 and 92.7 pS, respectively. The negative-resistance regime can disappear after a high-current, long-term operation, but the breakdown was found to be reproducible as long as the current level is not too high. Similar breakdown behaviors in Chem. Soc. Rev., 2010, 39, 923–938 | 935

Fig. 13 (a) Schematic of a nanofluidic diac consisting of SiO2–Al2O3–SiO2 nanochannels. Each segment is 20 mm long, 2.5 mm 5 wide. I–V characteristic of the device with 1 mM of KCl concentration measured by sweeping voltage and acquiring current (b), and by sweeping current and acquiring voltage (c), with the arrows indicating the sweeping directions. Three conduction regimes are observed in (c): (1) saturation regime, (2) negative-resistance regime and (3) breakdown regime. The dashed lines extending from regime (1) are the anticipated I–V behavior. (Reproduced with permission from ACS Nano, 2009, 3, 575.35)

Fig. 14 Ion concentration distributions and potential profiles calculated by 2D PNP equations in a 20-nm thick nanofluidic diac filled with 2 mM KCl solution in both sides under the bias of Vd = 2 V. The surface charge densities of SiO2 and Al2O3 are 4 and 4 mC m2, respectively. Each segment of the heterogeneous nanochannel is 5 mm long. The inset shows the absolute value of the electric field distribution near the depletion junction with the peak B30 MV m1 at x = 2.5 mm. (Reproduced with permission from ACS Nano, 2009, 3, 575.35)

ionic devices have been studied in field-enhanced water dissociation in bipolar ion-exchange membranes29,30,49,51,53 or the punch-through in biological membranes.54 The punch-through behavior tends to take place in thin membranes that are a few tens of nanometer thick, and therefore is not likely the case in our 60 mm-long nanochannels, because it requires a much larger voltage to deplete all the K and Cl ions in the nanochannels. Therefore, we ascribe the drastic increase of ion currents in the breakdown regime (segment (3) of the I–V curve in Fig. 13(c)) to the dissociation of water molecules to positive hydronium and negative hydroxyl ions under the high electric field at the reverse-biased junction (e.g. the left SiO2–Al2O3 junction of Fig. 14). The electric field is about 2 107 V m1 which is close to the result solved by 2P PNP equations as shown in the inset of Fig. 14. With the high electric field and the induced lowering of the dielectric constant, the water dissociation constant can increase according 936 | Chem. Soc. Rev., 2010, 39, 923–938

to the second Wien effect.49,55 Correspondingly, a similar breakdown effect is known in a non-heavily doped semiconductor p–n junction, where the behavior represented by sudden increase in the reverse current is caused by the electron–hole pair generation through impact ionization under high electric field.55 It is worth pointing out that not every fabricated heterogeneous nanochannel exhibits an apparent breakdown effect. We found that the alignment accuracy of the top and bottom Al2O3 patterns during the device fabrication may affect whether the breakdown behavior can be observed. The misalignment of the two patterns creates a gradual junction having overlapped SiO2 and Al2O3 surfaces instead of an abrupt boundary. As a result, the electric field at the junction is significantly weakened under the reverse bias. 4.2 Nanofluidic triode Here we also demonstrate an ionic switching device implemented by the same technology. Fig. 15 is a microscopic image of a three-terminal double junction nanofluidic device, an npn triode, composed by a SiO2 nanochannel sandwiched by two Al2O3 nanochannels.35 This triode device has an additional SiO2 nanochannel in the middle branching out to a microchannel. The I–V characteristics measured between the three sets of two terminals with 50 mM KCl concentration shown in Fig. 15 indicates nanofluidic diodes between terminals A and B and between terminals B and C, while the channel across A and C having a low ion conductance is an npn diac. The rectification factors of the two diodes are greater than 30. The nanofluidic triode can be used as a switch for the regulation of ion flow. Consider a circuit built by the nanofluidic triode in Fig. 16(a); the nanofluidic triode is biased to produce a constant ionic current IB flowing through the terminal B. By adjusting the corresponding voltage bias across A and C, VAC, the ionic current IB can be directed to terminal A or C serving as a single-pole, double-throw switch as illustrated in Fig. 16(c). Fig. 16(b) relates the ion current flowing out of terminal A, IA, and the applied voltage VAC at varied IB values. When terminal B is open (IB = 0), the measured IA is essentially a diac behavior which can be treated This journal is

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Fig. 15 (Top) Microscopic image and the corresponding schematics of a nanofluidic triode, which is a three-terminal double junction nanofluidic device composed by Al2O3 and SiO2 nanochannels, connecting to three separate PDMS microchannels (rendered as the gray areas in the schematics). The scale bar in the microscopic image is 50 mm. (Bottom) I–V characteristics of the nanofluidic triode measured across every two terminals with a KCl concentration of 50 mM. (Reproduced with permission from ACS Nano, 2009, 3, 575.35)

as a leakage current, IAO, across the two junctions from terminal C to A. The leakage current IAO here is relatively high because the high ion concentration we used makes it difficult to turn off one of the junctions at reverse bias. Otherwise, IAO can be reduced to a few pA by lowering the ion concentrations as shown in the diac in Fig. 13, or by increasing the surface charge density in nanochannels. It was found that with a given positive ion current IB, the ion currents IA saturates to a high current level when biased at a large negative VAC. In this regime (1), the nanofluidic junction between terminal A and B turns on whereas the other side turns off. The ion current at terminal A is contributed by the entire ion current IB migrating from B, along with the leakage current IAO from C, i.e. IA = IB + IAO. When VAC shifts from negative to positive, the AB junction turns off while the BC junction turns on. In this transition regime (2), there exists a range of bias in which both junctions slightly turn on. For

instance, at zero VAC, IB can flow to both terminals A and C with equal amount leading to IA = IB/2. In regime (3), a large positive VAC bias turns off the AB junction but turns on the BC junction. IB flows to terminal C. What is measured in terminal A is just the leakage current IAO flowing from A to C regardless of different IB applied into the device. The nanofluidic triode device can be operated in such a way that the direction of ion flow is switchable at a channel intersection. Such function requires two-dimensional nanofluidic networks which can not realistically be achieved in nanopore-based devices. The demonstration of the nanofluidic switch inspires us to consider the possibility of creating a nanofluidic bipolar junction transistor (BJT), which is a three-terminal device similar to the proposed structure but has a short base area (the middle section of the device), to amplify the input IB current, i.e. IA/IB c 1. In a semiconductor BJT the base region must be made shorter than the diffusion length of the carriers (electrons or holes), so that they can diffuse across it in much less time than the minority carrier life time of the semiconductor, to minimize the percentage of carrier recombination before reaching the other junction. Unlike the electron–hole system in semiconductor materials, cations and anions do not recombine (except protons and hydroxide ions). Therefore, when a nanofluidic p–n junction is forward-biased, both ions can accumulate at the p–n junction, at significantly larger levels than the background concentrations at equilibrium, resulting in the domination of drift current. Because of the dissimilar charge transport mechanisms, the nanofluidic triode should operate in a different way to resemble the function of a semiconductor BJT. The nanofluidic triode can, for example, have the AB junction forward-biased whereas the BC junction reverse-biased such that the cation current injecting from terminal A migrates fast enough across the middle base section toward terminal C to reduce the possibility of being caught by the electric field from terminal B. The way it operates is essentially more close to the function of a vacuum tube triode. This device behavior, however, may be very difficult to achieve in an ionic system because the mobility of ions in electrolyte solutions is very low, about seven orders of magnitude less than that of the electrons or holes in semiconducting materials.

Fig. 16 (a) Schematic of a nanofluidic triode operated as an ionic switch. (b) IA–VAC relationships of the device with varied IB under a KCl concentration of 50 mM. (c) Switching conditions at corresponding operation regimes indicated in (b). (Reproduced with permission from ACS Nano, 2009, 3, 575.35)

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Such low mobility makes it difficult for a nanofluidic device to realize the same function with comparable device dimensions.

5. Conclusion Ionic rectification is a unique effect observed in nanofluidic devices and cannot be implemented in microfluidic devices. The rectifying phenomenon relies on the asymmetric electrostatic interactions between the mobile ions and the fixed surface charges in nanochannels. It can be achieved in the charged nanochannels which have asymmetric geometries, asymmetric bath concentrations or asymmetric surface charge distributions. Based on the PNP model, it is found that the rectifying phenomenon is attributed by the accumulation or depletion of ions inside nanochannels and, hence, the difference in conductances in response to different bias polarities. We elucidate that the basis of the ionic rectifying effect in the nanofluidic devices is to produce asymmetric cation/anion ratios at the two entrances of the nanochannels. The condition produces the asymmetric ion currents from the two sides of the nanochannel right after the applied electric field builds up, and consequently allows the accumulation and depletion of ions in nanochannels to be controllable by the applied potential. Rectifying nanofluidic devices may be used for separation and detection of charged molecules. The water dissociation effect under a reverse-biased diode could be used to control the local pH values within a microfluidic system.

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