Transport phenomena in nanofluidics

TL;DR: In this paper, the authors investigated the transport properties of 50-nm-high 1D nanochannels on a chip and showed that they can be used for the separation and preconcentration of proteins.

Abstract: This thesis explores transport phenomena in nanochannels on a chip. Fundamental nanofluidic ionic studies form the basis for novel separation and preconcentration applications for proteomic purposes. The measurements were performed with 50-nm-high 1D nanochannels, which are easily accessible from both sides by two microchannels. Nanometer characteristic apertures were manufactured in the bonded structure of Pyrex-amorphous silicon – Pyrex, in which the thickness of the amorphous silicon layer serves as a spacer to define the height of the nanochannels. The geometry of the nanometer-sized apertures is well defined, which simplifies the modeling of the transport across them. Compared to biological pores, the present nanochannels in Pyrex offer increased stability. Fundamental characteristics of nanometer-sized apertures were obtained by impedance spectroscopy measurements of the nanochannel at different ionic strengths and pH values. A conductance plateau (on a log-log scale) was modeled and measured, establishing due to the dominance of the surface charge density in the nanochannels, which induces an excess of mobile counterions to maintain electroneutrality. The nanochannel conductance can be regulated at low ionic strengths by pH adjustment, and by an external voltage applied on the chip to change the zeta potential. This field-effect allows the regulation of ionic flow which can be exploited for the fabrication of nanofluidic devices. Fluorescence measurements confirm that 50-nm-high nanochannels show an exclusion of co-ions and an enrichment of counterions at low ionic strengths. This permselectivity is related to the increasing thickness of the electrical double layer (EDL) with decreasing salt concentrations, which results in an EDL overlap in an aperture if the height of the nanochannel and the thickness of the EDL are comparable in size. The diffusive transport of charged species and therefore the exclusion-enrichment effect was described with a simple model based on the Poisson-Boltzmann equation. The negatively charged Pyrex surface of the nanometer characteristic apertures can be inversed with chemical surface pretreatments, resulting in an exclusion of cations and an enrichment of anions. When a pressure gradient is applied across the nanochannels, charged molecules are electrostatically rejected at the entrance of the nanometer-sized apertures, which can be used for separation processes. Proteomic applications are presented such as the separation and preconcentration of proteins. The diffusion of Lectin proteins with different isoelectric points and very similar compositions were controlled by regulating the pH value of the buffer. When the proteins are neutral at their pI value, the diffusion coefficient is maximal because the biomolecules does not interact electrostatically with the charged surfaces of the nanochannel. This led to a fast separation of three Lectin proteins across the nanochannel. The pI values measured in this experiment are slightly shifted compared to the values obtained with isoelectric focusing because of reversible adsorption of proteins on the walls which affects the pH value in the nanochannel. An important application in the proteomic field is the preconcentration of biomolecules. By applying an electric field across the nanochannel, anionic and cationic analytes were preconcentrated on the cathodic side of the nanometer-sized aperture whereas on the anodic side depletion of ions was observed. This is due to concentration polarization, a complex of effects related to the formation of ionic concentration gradients in the electrolyte solution adjacent to an ion-selective interface. It was measured that the preconcentration factor increased with the net charge of the molecule, leading to a preconcentration factor of > 600 for rGFP proteins in 9 minutes. Such preconcentrations are important in micro total analysis systems to achieve increased detection signals of analytes contained in dilute solutions. Compared to cylindrical pores, our fabrication process allows the realization of nanochannels on a chip in which the exclusion-enrichment effect and a big flux across the nanometer-sized aperture can be achieved, showing the interest for possible micro total analysis system applications. The described exclusion-enrichment effect as well as concentration polarization play an important role in transport phenomena in nanofluidics. The appendix includes preliminary investigations in DNA molecule separation and fluorescence correlation spectroscopy measurements, which allows investigating the behavior of molecules in the nanochannel itself.

Summary

Introduction

• The transport of fluid in and around nanometer-sized objects with at least one characteristic dimension below 100 nm enables the occurrence of phenomena that are impossible at bigger length scales.
• Continuum models and molecular-dynamics simulations 844 B. Electrokinetic effects in nanochannels 844 1. Basic physics of electrokinetics 845 2. Comprehensive electrokinetic theory for nanochannels 846 C. Electrode-electrolyte interface 849 1.
• The roots of nanofluidics are broad, and processes on the nanometer scale have implicitly been studied for decades in chemistry, physics, biology, materials science, and many areas of engineering.
• Then the authors describe operation mechanisms and physical effects of nanopores and nanowires for label-free biomolecule detection in Sec. V.

II. ELECTROKINETIC EFFECTS

• Here the authors introduce the reader to electrostatics in liquids and electrokinetic effects, which are the most important and fundamental concepts for the description of transport in nanofluidics.
• This classical theory was established decades ago in well-established disciplines, but it cannot be omitted in the new interdisciplinary field of nanofluidics !.
• Eijkel and van den Berg, 2005a", and will therefore be summarized.

A. Electrostatics in liquids

• When considering that a solid in contact with a liquid bears a surface charge, one perceives that this parameter is of increased importance in nanochannels, since they have a high surface-to-volume ratio.
• Surface charge is caused by the dissociation of surface groups and the specific !nonelectric" adsorption of ions in solution to the surface !.
• Depending on the number and type of acid and basic groups present in solution !.
• At small distances, van der Waals forces contribute to the attractive part of the interaction, for example, between dissolved particles, whereas coagulation is prevented by repulsive or attractive electrostatic forces.

1. Electrical double layer

• Due to the fixed surface charge at the solid interface, an oppositely charged region of counterions develops in the liquid to maintain the electroneutrality of the solidliquid interface.
• He introduced the Stern layer between the inner and outer Helmholtz planes, in which the charge and potential distribution are assumed to be linear, and a diffuse layer further from the wall where the Gouy-Chapman theory is applied.
• The outermost and third layer is the diffuse layer, composed of mobile co-ions and counterions, in which resides the slip plane bearing the % potential !described hereafter".
• The % potential at this plane can be experimentally determined, and is therefore an important parameter in colloid science for determining the stability of particles, and in "TAS for de- Rev. Mod.

3. Debye-Hückel approximation

• The potential Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 decays exponentially in the diffuse layer with the characteristic distance given by the Debye length -D=,−1.
• This value corresponds to the thickness of the EDL, which increases with dilution as presented in Table I.
• For a symmetrical zi :zi electrolyte with concentration ci at 25 °C, the value of the Debye length -D !unit is meters" can be written as -D = 3.04.

5. Surface charge density

• Yates et al., 1974", and has shown consistency with experimental data obtained for oxides by potentiometric titration measurements !.
• TABLE I. EDL thicknesses -D for typical KCl concentrations at 25 °C. KCl concentration !M" Debye length -D !nm".

6. Surface conductance

• The EDL contains more ions than the bulk, which results in a surface conductance due to their motion in the electric field !.
• Urban et al., 1934; Kittaka and Morimoto, 1976", and this conductance can be measured in an ac field at sufficiently high frequency.
• Decades ago, only the diffuse part of the EDL was considered for the surface conductance K#, and values on the order of 10−9–10−8 S were reported !.
• Ions can move laterally almost unimpeded, and it is separated from the mobile part of the fluid by the slip plane.
• Lobbus et al. !2000" have reported that the contribution of the stagnant layer K#i to the total surface conductance K# cannot be neglected, and that the Dukhin number increases with dilution.

7. Continuum models and molecular-dynamics simulations

• The authors focus on nanochannels with heights between a few nanometers and 100 nm, a scale at which continuum and mean-field theories are valid.
• At such length scales, molecular-dynamics simulations are used to determine transport in nanopores for a specific period of time with known physical laws !.
• In a typical molecular-dynamics simulation, a set of molecules with initial random positions is assumed, to which are assigned initial random velocities corresponding to the Boltzmann distribution at the temperature of interest.
• It can be difficult to obtain the continuum limit from previously described simulations because simulated systems are usually limited to tens of nanometers and time scales of nanoseconds.

B. Electrokinetic effects in nanochannels

• Fluid flow in nanochannels is conveniently achieved by electrokinetic techniques or capillary forces.
• Tas et al., 2003", allowing for the spontaneous filling of such channels with liquid.
• Pressure-driven flow can also be used in nanochannels, but high pressures !.

1. Basic physics of electrokinetics

• The motion of electrically charged molecules and particles due to an applied electric field in moving substances such as water is studied in electrokinetics.
• Placing the molecule in an electric field results in its movement because ions in the EDL are not fixed to the surface.
• This concentration is dependent on the flow speed, and therefore results in a conductance change of the nanoprobe located in the EDL !Bourlon et al., 2007".
• This phenomenon is denoted the electroviscous effect and is classified as follows !.

2. Comprehensive electrokinetic theory for nanochannels

• It is important to consider aforementioned electrokinetic phenomena to develop a comprehensive transport model of molecules in nanochannels.
• The space-charge model has been simplified by Pennathur and Santiago !2005a" to describe electrokinetic transport in long nanochannels, confirmed by their experimental studies !.
• These authors also showed that the radially limited diffusion of ions by electromigration in response to the wall charge increases the dispersion of all ions relative to neutral species.

1. Charge-transfer resistance

• Modeling the electrode-electrolyte interface by capacitance does not describe the entire electric circuit.
• As the reaction proceeds, excess electrons in the metal accumulate until the potential increases, which reduces the barrier for the reverse reaction !.
• These two competing reactions eventually reach equilibrium, resulting in a zero net current across the interface.
• For operation of an electrode near its equilibrium condition, the charge-transfer overpotential $ct dominates the overall current.

2. Constant phase element

• Metz, 2003; Gawad, 2004" that the impedance deviates from the purely capacitive behavior, that is expected theoretically for smooth and clean surfaces like that of liquid mercury.
• If the electrode has a microscopic roughness due to scratches, pits, etc., the effective solution resistance varies along the surface, and the current density is inhomogeneous on a microscopic scale.
• The constant phase element obtained its name because the phase angle of ZCPE is independent of the frequency, and has a value of −!90nCPE".
• The electric field and fluid flow in a nanochannel can be substantially different from those in the connecting microchannel, resulting in large gradients at the geometric interface.

A. Electrical characterization of nanochannels

• Electrical measurements of nanochannels are performed because they do not require fluorescent labels and are integrable on a chip, which is favorable for potential applications.
• Furthermore, electrical characterizations of fluidic systems can readily be parallelized.
• Indeed, the authors see that a conductance plateau !on a log-log scale" is obtained at low ionic strength when the Debye length becomes comparable to the characteristic dimension of the nanochannel.
• This effect is explained by an excess of mobile counterions in the nanometer-sized aperture, which equilibrate the surface charge to maintain electroneutrality.
• A nanochannel can be distinguished from a nanopore by its length, as the nanochannel length is much longer than the nanometer characteristic dimension of the opening, whereas a nanopore has a length approximately within an order of magnitude of its minimal opening dimension.

1. Impedance spectroscopy

• To electrically characterize nanochannels that are interfaced with microchannels, the authors recommend the placement of electrodes as close as possible to the nanochannel openings in order to increase the sensitivity !.
• Platinum electrodes were used by Schoch et al. !2005", because this material has a low chemical reactivity and hence will not be reduced or oxidized when electric potentials are applied.
• Diffusion-limited currents cannot occur and chemical reactions are not induced because of the alternating electric field !.
• Impedance spectroscopy involves relatively simple electrical measurements that can readily be automated, the results of which may often be correlated with many complex material variables.
• The effective electric resistance of the nanochannels corresponds to the high intercept point of the extrapolated semicircle with the real axis !.

2. Nanochannel conductance

• For the electrical modeling of a nanochannel filled with a 1:1 electrolyte such as KCl, the conductivity of the bulk solution &bulk has to be considered !.
• The conductance in a nanochannel for ionic strengths higher than the excess mobile counterion concentration ce is therefore dependent on channel geometry %first term in Eq. !39"&.
• Erickson et al., 2000; Gu and Li, 2000; Kirby and Hasselbrink, 2004", which results in a higher number of attracted counterions near the surface, neutralizing the fixed surface charge.
• Ninham and Parsegian, 1971; Healy et al., 1980"; it states that the charge densities of two objects are functions of their separation distances, and has been discussed by Behrens and Borkovec !1999" and Behrens and Grier !2001".

3. Ionic current rectification

• The previously mentioned nanochannels are symmetric, and their measured currents for voltages of the same amplitude but opposing polarities have similar absolute values.
• The electric potential inside conical nanopores has been calculated with the shape of an asymmetric sawtooth !see Fig. 9", which allows ionic current rectification to be explained by the ratchet mechanism !.
• Siwy and Fulinski, 2004", strong enough to pump ions against their concentration gradient !.
• Such patterned discontinuities can be achieved in the axial direction by controlling the position of the reaction front, which is possible because the time for ion diffusion across the channel is much shorter than the transit time of ions through the channel !.
• The highest ionic current rectification ratios have been obtained when the above-described rectification effects were combined by positively and negatively patterning charged regions in conical nanopores !.

1. Donnan potential

• When a charge-selective channel is in equilibrium with an adjacent electrolyte solution, the electrochemical potentials "5 i of the permeating cation or anion i are equal on sides I and II !see Fig. 10" !.
• From these two equations and Eq. !43", the concentrations of cations and anions in the nanochannel are calculated to be !.
• The Donnan effect can be enhanced using species with high ion valence !.

2. Membrane potential

• The membrane potential has been developed in the context of ion-exchange membranes, and the authors present it here because charge-selective nanochannels have similar characteristics to permselective membranes.
• Ion- Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 exchange membranes are important tools for separation processes, which can be classified into mass separation, chemical synthesis, and energy conversion and storage processes !.
• The membrane potential can be measured directly, whereas the Donnan potential is usually calculated.

C. Exclusion-enrichment effect

• Section III.A.2 discussed the excess of counterions present inside nanochannels at low ionic strength.
• This enrichment of counterions in and exclusion of co-ions from a nanometer-sized opening due to electrostatic interactions with the surface charge is called the exclusionenrichment effect !EEE", described by Plecis et al. !2005".
• Subsequently, the authors discuss the mathematical model of this phenomenon.
• At high ionic strength, the instantaneous flux :* is proportional to the geometric cross section S*.
• For cationic species, the enrichment effect at low ionic strength increases the number of cations in the nanochannel that can be transported by diffusion.

1. Permselectivity of a nanochannel

• The permselectivity induced by the EEE in nanoporous structures was first experimentally investigated with the advent of nanoporous membranes.
• Additional transport studies found a higher permselectivity of these membranes on anions or cations when the Debye length was on the order of the pore radius.
• The first quantitative study of the EEE in a nanochannel was described by Plecis et al. !2005", who developed a simple model of the nanochannel permeability for varying ionic strength.
• The measured relative permeability Peff /P* for different charged probes is shown in Fig. 13, presenting an exclusion of anions and an enrichment of cations.

2. Model of the exclusion-enrichment effect

• At low ionic strength, the EDL thickness becomes comparable to the nanochannel height, resulting in an overlap of the diffuse parts of the EDLs in the nanometer-sized aperture.
• 52" The numerical solution of the complete PoissonBoltzmann equation %Eq. !6"& and the approximated potential presented in Eq. !52" are compared in Fig. 14!a" for different ionic strengths !.
• It is known that the DebyeHückel approximation generally overestimates the electric potential.
• At low ionic strength, where the electric potential remains high in the whole nanochannel, the exclusion of anionic species and enrichment of cationic probes becomes important.

D. Partitioning at the interface

• The transport of large molecules through porous membranes was initially examined to understand the permeability of biological structures such as glomerular capillaries in the kidney, or walls of blood capillaries !.
• Later, renewed interest in fine pores occurred when track-etch processes allowed the fabrication of nanopores with better-controlled geometries !.
• Bean et al., 1970", and models based on the hindered diffusion of macromolecules through pores had been developed !.
• Hereafter, the authors present a partitioning theory that accounts for size and electrostatic effects, and then they focus on electrostatic sieving.

1. Partition coefficient for colloids

• Smith and Deen !1983" theoretically studied electrostatic effects on the partitioning of spherical colloids between a dilute bulk solution and cylindrical pores.
• They defined the fundamental partition coefficient <part, which expresses the ratio of the macromolecule concentration in the pore cm and the bulk cm,b, <part = cm cm,b = 2+ 0 1−ap/r0 exp' =p!r/r0" kBT (!r/r0"d!r/r0" , !54" where r is the radial coordinate and =p is the potential energy of interaction between the colloid and the pore wall.
• Note that the exclusion-enrichment coefficient < assumes a point charge and accounts for electrostatic effects, compared to the partition coefficient <part, which also includes size effects on partitioning.
• <part has been calculated by solving the linearized Poisson-Boltzmann equation and determining the free energy of the system and the interaction potential energy.
• This demonstrates that completely porous spheres are electrostatically less hindered by their transport through a pore than solid spheres, because the elementary charge is distributed throughout the porous sphere, whereas for solid spheres the same number of charges reside on the sphere surface.

2. Electrostatic sieving of proteins

• In reverse osmosis, electrostatic partitioning is used to purify water by applying a pressure gradient over a nanoporous membrane, through which only pure water can pass; negatively charged molecules are rejected at the entrance of the cation-selective apertures !.
• As a protein comes close to an oppositely charged stationary surface, the protein net charge will change because of the electric field from the surface.

3. Active control of partitioning

• Partitioning can be controlled actively with field effects !.
• It has been proposed that such devices do not require EDL overlap !.
• Nishizawa et al., 1995; Chun et al., 2006", nanofluidic channels with a gate voltage enable researchers to address individual nanometer-sized openings.

E. Concentration polarization

• When an external electric field is applied through a nanochannel, electrokinetic transport is superimposed on diffusion.
• Holtzel and Tallarek, 2007", and a detailed analysis of bulk concentration polarization for a two-parallel-plate electrode model with suddenly applied electric fields was provided by Bazant et al. !2004".
• At the anodic side of a cationselective nanochannel !see Fig. 18", the concentration of ions in solution is reduced because of the lower trans- port number of cations in the solution relative to that in the nanometer-sized opening !.
• This concentration gradient leads to a diffusional flux of salt Js diff, which is oriented toward the nanochannel on the dilute side and the bulk solution on the concentrated side of the nanochannel.

1. Limiting current through nanochannels

• Concentration polarization can lead to a high salt concentration on the cathodic side of a nanochannel, and if it exceeds the solubility limits of the solution constituents, precipitation of salts may occur, resulting in an additional electric resistance.
• Phys., Vol. 80, No. 3, July–September 2008 the dimension of the DBL is largely determined by the microchannel height !.
• As a consequence, pH value shifts are obtained, with increasing pH values on the anodic side of the nanochannel and decreasing pH values on the cathodic side.

2. Electroconvection for mixing

• The reason for transport in the overlimiting current regime III !see Fig. 19" is a subject of intensive discussion in the literature !.
• Two types of electroconvection are known: the classical electro-osmosis !.
• The latter type results when tangential and normal electric fields are applied to a permselective interface leading to polarization of the double layer, a lateral pressure drop in the double layer, and a local inconsistency of the electroneutrality approximation.

3. Preconcentration of molecules

• The authors have described the ion enrichment and depletion that can be achieved on the cathodic and anodic sides of a permselective opening, respectively.
• In Sec. III.D.2, the authors mentioned that co-ions are rejected at the entrance of counterion-selective nanochannels through which a pressure gradient is applied.
• The pI of rGFP is 4.5, and, as a result, this protein has a high net charge in the experimental buffer.
• Figure 21!b" shows that the preconcentration factor increases with time and increasing magnitude of the applied voltage, indicating that concentration polarization is a dynamic positive feedback process.
• The highest preconcentration factors of a millionfold were reported by Wang, Stevens, and Han !2005", who used electrokinetic trapping in combination with electro-osmotic flow of the second kind to preconcentrate analyte molecules on the anodic side of a chargeselective nanochannel.

IV. MACROMOLECULE SEPARATION MECHANISMS USING NANOMETER-SIZED STRUCTURES

• During the past several years, significant interest has emerged in separating macromolecules, and it has been demonstrated that nanofluidics can accomplish this task by exploiting various mechanisms.
• Such microfabricated regular sieving structures hold promise as an alternative to gels or capillaries to improve biomolecule separation speed and resolution.
• Until recently, the main focus has been placed on the separation of long DNA molecules !.

A. Entropic trapping

• When a DNA molecule is in a relaxed state and in equilibrium, it has a spherical shape with a radius of gyration R0.
• If the molecule is forced into an opening with dimensions smaller than 2R0, it has to deform from its state of minimal energy.
• It has been demonstrated that longer DNA molecules move faster through the column because they have a better chance of escaping entropic traps.
• Simulations of the trapping lifetime as a function of molecular size and electric field strength have been presented by Tessier and Slater !2002" and Tessier et al. !2002".

D. Anisotropy for continuous-flow separation

• The continuous-flow operation of a separation device permits continuous and convenient harvesting of purified biomolecules into specific reservoirs, allowing for further analysis of the biomolecules of interest.
• This represents a more favorable separation scheme than systems in which an analyte gets separated into its components but is finally eluted into the same reservoir.
• Huang, Tegenfeld, Kraeft, et al., 2002", rectification of Brownian motion !.
• The separation of differently sized or charged biomolecules is obtained because macromolecules have different mean characteristic drift distances L in the deep channels between two consecutive nanofilter crossings, leading to distinct stream deflection angles C and jump passage rates Px !see Fig. 24".

V. NANOPORES AND NANOWIRES FOR LABEL-FREE BIOMOLECULE DETECTION

• There are two main types of detection methods that are commonly integrated into "TAS: fluorescent and electrical detections.
• Optical approaches are straightforward and provide direct visual proof.
• But with regard to industrial applications, a lab-on-a-chip device should be inexpensive, and electronic sensors are therefore favorable.
• The authors present promising electrical and label-free biomolecule detection methods, which are based on nanofluidic objects such as nanopores and nanowires.

A. Biomolecule translocations through nanopores

• Kasianowicz et al. !1996" demonstrated that a DNA molecule can be detected as a transient decrease in the ionic current when it passes through a nanopore, and the passage duration allows for the determination of polymer length.
• Thus, the ultimate goal of this technique is to realize rapid DNA sequencing !.
• The authors further present interesting effects of biological and synthetic nanopores, which have been observed on the way to electrical DNA sequencing.

1. Biological nanopores

• For nanopore sequencing, an opening at the nanometer scale is required, and +-hemolysin has been chosen for such investigations !.
• This transmembrane 33 kD protein is isolated from Staphylococcus aureus; it has an asymmetric structure and a diameter of approximately 1.4 nm at its narrowest point.
• At neutral pH and high ionic strength, +-hemolysin remains open, unlike most membrane channels, and allows a steady ionic current of $100 pA to pass at an applied voltage of 100 mV !Kasianowicz et al., 1996".
• The model system +-hemolysin has allowed determining the length of single-stranded DNA and RNA, which is proportional to the measured translocation duration !.

2. Synthetic nanopores

• The biological nanopore +-hemolysin cannot be redesigned at will, and it has a lower stability with buffer pH and ionic strength than a solid-state nanopore !.
• Iqbal et al., 2007", the blocked current amplitude during translocation of proteins !.
• Additionally, it has been reported that, at low electric fields, double-stranded DNA cannot permeate a pore smaller than 3 nm, while single-stranded DNA can, due to the difference in the DNA diameter.
• Figure 26 shows that the presence of DNA in the nanopore leads to an increase in the current of about 40 pA rather than a decrease, and the authors explain this phenomenon subsequently.

3. Current decrease and increase

• The principle of particle counting is based on the Coulter counter type device !.
• Micromachined Coulter counters have been fabricated !.
• Because ions in the channel have a higher self-energy compared to ions in the bulk, a nanopore represents an energy barrier that increases the blockade current.
• Interestingly, current decreases and increases result when DNA molecules pass through a nanopore, as demonstrated by Chang et al. !2004".

B. Nanowire sensors

• Similar to nanopores, nanowires are one-dimensional nanostructures that show detectable electrical changes upon interactions with molecules.
• The reader may perceive that nanowires have electrical effects similar to those of previously described nanochannels.
• Carbon nanotubes have remarkable electrical properties !.

1. Fabrication techniques

• Nanowires can be fabricated using both “top-down” and “bottom-up” methods, but the latter techniques are less well suited for bulk production, which would hinder widespread application of the technology.
• Morales and Lieber, 1998" are favored for electric transport studies in semiconductor nanowires, and these structures have been grown with diameters down to a few nanometers !.
• Cui et al., 2003; Zheng et al., 2004", and germanium/silicon core/shell heterostructures have demonstrated enhanced gate coupling !.

2. Operation mechanism

• The operation mechanism of nanowire sensors is based on the principle of a field-effect transistor, relying on a controllable conductivity between the source and the drain upon change in the gate potential.
• The most commonly used field-effect transistor is the metal-oxidesemiconductor field-effect transistor as shown in Fig. 28!a", with the semiconducting element of a p-type silicon !.
• Conductivity changes between source and drain can be induced by a gate electrode, as well as by binding molecules to the surface of a semiconductor, and these changes lead to charge concentration modulations of p-Si.
• By reducing the size of p-Si, the charge accumulation or depletion in the nanowire occurs within a significant portion of the one-dimensional structure, leading to increased detection sensitivities.
• For the selective detection of macromolecules, it is essential to control the ionic strength of the buffer.

3. Single-molecule sensitivity

• Nanowires are coated with a recognition group at their surface to achieve specific binding with biomolecules.
• In addition to chemical species, various biomolecules have been sensed down to femtomolar concentrations, such as DNA and DNA enzymatic processes !.
• Thus, multiplexing is robust against false positive signals arising from nonspecific binding or electronic noise.

4. Detection efficiency

• The surface area of a nanowire in contact with a liquid is small, which could lead to a limited overall sensitivity for the detection of analytes at low concentrations.
• Sheehan and Whitman !2005" calculated that a 10-"m-long hemicylindrical sensor with a diameter of a few tens of nanometers needs an accumulation time under static diffusion in a 1 fM analyte solution of $1 h for the first molecule, $1 day for the tenth molecule, and $1 week for the 100th molecule.
• Zheng et al. !2005" reported that electrokinetic effects lead to an enhancement in the local concentration of species, such as dielectrophoretic trapping of biomolecules !.
• Even faster detection response times have been reported by Stern, Wagner, Sigworth, et al. !2007"; they attributed this to their fluid injection system, which is not laminar and does not rely on diffusion to transport analyte molecules to the nanowires.

VI. FABRICATION OF NANOCHANNELS

• Transport through nanoporous systems has been investigated in membrane science for decades !.
• Various methods of producing nanochannels exist and are divided into the categories of top-down and bottom-up fabrication methods !.
• Fabrication technologies do not limit the realization of nanostructures, but they have to be connected to the macroscopic world.
• In fact, the interaction between water and the silica network is not limited to the surface, but can also occur behind the glass-water interface !.

A. Wet and dry etching

• 2D nanofluidic channels with the smallest dimension of about 20–25 nm have been developed !Mao and Han, 2005".
• The nanometer-sized dimension of the channel is obtained by etching a nanometer-high recess in glass or silicon, followed by a bonding process to a glass substrate to form the nanofluidic channel !see Fig. 30".
• Buffered oxide etch has a stable etch rate and has been used for precise bulk micromachining in glass, whereas reactive ion etching has been applied on silicon substrates.
• Haneveld et al. !2003" used anisotropic etching of .110/ silicon resulting in channel sidewalls that are precisely perpendicular to the wafer surface.
• It has to be considered that glass etching processes lead to an unintentional increase in surface roughness !.

B. Sacrificial layer methods

• Nanochannels can be made with a nanometer-thick sacrificial layer that is first used to define the male form of the nanochannel and is then removed in order to open the aperture !.
• Tas et al. !2002" used the fabrication of nanowires on the sidewall of a submicrometer step !.
• Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 Centimeter-long nanochannels can be fabricated with a sacrificial SiO2 layer !.
• Sacrificial layers can be used as nanometer-sized spacers between two substrates to define their separation distance.

C. High-energy beam techniques

• The focused ion beam is a powerful tool used to directly fabricate structures on the substrate down to some tens of nanometers.
• In electron-beam lithography, the pattern is directly exposed to the resist %or ice layer !.
• This technique has been used to produce monolithic nanofluidic sieving structures that are similar to two-dimensional artificial gels used for DNA molecule manipulations !.

D. Nanoimprint lithography

• A low-cost and high-throughput method for the fabrication of 2D nanofluidic channels is nanoimprint lithography, which was first reported by Chou and Krauss !1997".
• In a single step with elevated temperature and pressure, a channel template can be imprinted onto a thin polymer film cast on a glass cover slip !.
• The nanochannel dimensions are controlled by a simple relationship involving the initial polymer layer thickness and the mold pattern configuration, and the grating period can be reduced by frequency doubling !.
• Nanoimprint lithography has also been demonstrated on free-standing membranes !.

E. Bonding and sealing methods

• To achieve encapsulation of bulk-machined nanochannels, bonding and sealing has to be performed.
• Depending on the bulk material in which the nanochannels have been produced !.
• For the fabrication of silicon-on-glass wafers, anodic bonding is known to result in high bonding strength when simultaneously exposing the wafers to an elevatedtemperature treatment of about 500 °C and a dc voltage of approximately 750 V !Knowles and van Helvoort, 2006".
• Fabrication processes at lower temperatures have been developed because high temperatures are not suitable for heat-sensitive materials such as electrodes.

F. Realization of single nanopores in membranes

• The realization of single nanopores fabricated in a membrane are not the principal focus of this review, but the authors present selected techniques in this area.
• These pores are suitable for further biochemical surface modifications.
• More recently, multiple nanopores with controlled sizes between 5 and 25 nm were fabricated in a 15-nmthick nanocrystalline silicon membrane, which has remarkable mechanical properties !.

VII. CONCLUSIONS AND PERSPECTIVES

• This review shows that nanofluidics offers a variety of unique properties in nanochannels, nanofilters, nanopores, nanowires, and at the microchannel-nanochannel interface.
• In addition to the elucidated properties of nanofluidic systems, chemical and physical properties of single molecules can be investigated in nanochannels !.
• Such single-molecule interrogations have demonstrated restriction mapping of DNA molecules !.

Figures

FIG. 22. Entropic trapping of DNA in a nanofilter column. !a" Cross-sectional schematic of the device consisting of alternating shallow !ts" and deep !td" regions. Electrophoresed DNA molecules are trapped whenever they meet a thin region. !b" Top view of the device. The width w of the DNA molecule in contact with the thin region determines the trapping lifetime, and therefore the overall mobility. R0 is the radius of gyration. From Han and Craighead, 2000. !c" Escape of a DNA molecule from the trap, where xf is the length of the DNA strand in the filter. !d" Energy ;= vs xf. The transition state at which the strands overcome the energy barrier is when xf=xf,c. Adapted from Han et al., 1999.

FIG. 23. Scheme for separation of DNA strands at the interface with a nanopillar array. During the gather phase, molecules are accumulated at the interface with the pillar region. Then, a pulse !drive" inserts shorter molecules fully, but longer molecules only partially, in the constriction zone. During the recoil phase, longer strands backtrack and remain in the pillarfree region. From Cabodi, Turner, and Craighead, 2002.

FIG. 9. Voltage-current curve showing ionic current rectification of conical nanopores, explained with the ratchet mechanism. The electric potential in conical nanopores takes the form of an asymmetric tooth, which turns when the external potential V is changed. For positive voltages, the force to move ions through the pore is smaller than for negative voltages, when ions are trapped inside the pore. The amount of current transported by cations or anions can be calculated from such voltage-current curves. From Siwy and Fulinski, 2004.

FIG. 16. Apparent diffusion coefficient Dapp of three lectin proteins as a function of the pH value of the solution, measured in a 50-nm-high nanochannel at an ionic strength of 10−3M. Dapp of the three lectin proteins shows three peaks, corresponding to the pI values of the acidic band lectin La, the middle band lectin Lm, and the basic band lectin Lb. The marks on the pH axis show the respective pI values of the lectins, which have been measured with isoelectric focusing. The connecting lines are for guidance only. From Schoch et al., 2006.

FIG. 17. Nanofluidic control of protein transport by a gate voltage without bias between the microchannels. When the gate voltage Vg is negative, the positively charged protein avidin can diffuse through the nanochannel, whereas the protein is repelled when Vg is positive. !a" Schematics and !b" corresponding fluorescence images. Adapted from Karnik, Castelino, and Majumdar, 2006.

FIG. 27. Transition from current increase to current decrease for DNA translocations through a nanopore. !a" Conductance changes ;G and !b" relative conductance changes ;G /G due to DNA molecule translocation for KCl concentrations between 50 mM and 1M. Each symbol represents an individual pore, and the line is a linear fit to the data. The line in !b" has been calculated for a change in surface charge as a function of the salt concentration. The inset shows the DNA molecule translocation time as a function of KCl concentration. From Smeets, Keyser, Krapf, et al., 2006.

FIG. 1. Electrical double layer !EDL, shaded in gray" at high ionic strength, it is thin, allowing co-ions and counterions to pass through the nanochannel. At low ionic strength, the EDL thickness increases, resulting in a counterion-selective nanochannel. From Schoch, 2006.

FIG. 4. Schematic representation of the potential distribution in a nanochannel with height h in direction z when the EDLs overlap !solid line", compared to the EDL potentials if the opposite wall is not present !dashed line". Adapted from Schoch, 2006.

FIG. 26. Simultaneous current and force measurements of a DNA molecule in a nanopore at 0.1M KCl. Differences in the applied voltage change the bead position until the electrical force Fel is equal to the optical force Fot. Reversing the voltage at t=13 s drives the DNA strand out of the pore, returning the open-pore current !dashed lines". The nanopore axis is defined in the z direction, and ktrap is the trap stiffness. From Keyser, Koeleman, Van Dorp, et al., 2006.

FIG. 18. Schematic illustrating concentration polarization at nanochannel interfaces. Concentration profiles of salt in the diffusion boundary layer !DBL" on both sides of the cationselective nanochannel are presented, as well as the fluxes per area J of cations !ca", anions !an", and salt !s" in the DBL and in the nanochannel. The superscripts mig and diff refer to migration and diffusion, di and co refer to dilute and concentrate solutions, and bu and nano refer to bulk phase and nanochannel interface, respectively. Adapted from Strathmann, 2004.

FIG. 31. Scanning electron microscope images of 2D defined nanochannels fabricated by two different sacrificial layer methods. !a" Formation of a nanowire on the sidewall of a step and !b" etching of a sacrificial strip separating the substrate and the capping layer. From Tas et al., 2002.

FIG. 32. Cross section of a chip illustrating two microchannels linked by a nanochannel, whose height is defined by the thickness of the amorphous silicon !a-Si" layer. The inset shows a scanning electron microscope image of the 50-nm-high nanochannel. Adapted from Schoch, 2006.

FIG. 5. Dimensionless species velocity due to electro-osmosis as a function of the normalized nanochannel height, for an anion in !1" 60- and !2" 200-nm-high channels, and for an uncharged molecule in !3" 60- and !4" 200-nm-high channels. Adapted from Garcia et al., 2005.

FIG. 6. Normalized mobility "nano/" as a function of 2-D /h for two values of ion valence zi. Measured normalized effective mobility in a nanochannel is compared with model predictions for both 40 and 100 nm channels. The large open symbols are the data, and the smaller, closed symbols are the corresponding error bars. Adapted from Pennathur and Santiago, 2005b.

FIG. 8. Schematic behavior of the nanochannel conductance as a function of the KCl concentration. The impact of the characteristic surface charge density #s, the width w, the length d, and the height h of the nanochannel on its conductance is shown. Below the transition KCl concentration ct, a conductance plateau Gp is reached, where ct corresponds to 0.5 times the excess mobile counterion concentration ce. From Schoch and Renaud, 2005.

FIG. 30. Cross-sectional scanning electron microscope images of nanochannels with heights of 55 and 25 nm fabricated in a glass substrate and bonded with another glass cover. From Mao and Han, 2005.

FIG. 33. Transmission electron microscope image of !a" initial 60-nm-diam pore and !b" the same sample after ion beam exposure yielding a shrunken 1.8-nm-diam pore. From Li et al., 2001.

FIG. 12. Schematic of the exclusion effect on anionic transport. At high ionic strength, the Debye length is small so that anions !co-ions" can freely diffuse through the whole cross section S* of the nanochannel. The instantaneous flux without electrostatic interactions :* is proportional to S*. The Debye length increases with dilution, reducing the effective cross section Seff through which anions can diffuse. This results in a lower effective flux :eff. Adapted from Plecis et al., 2005.

FIG. 13. Variation of the relative permeability Peff /P* of a 50-nm-high nanochannel vs KCl concentration for three probe charges. Marks represent experimental measurements, whereas dotted lines are theoretical fits obtained with Eq. !53". Anions are excluded from the nanochannel when EDL overlap occurs at low KCl concentrations, in which regime cations are attracted, and therefore show enhanced transport through the nanochannel. Adapted from Plecis et al., 2005.

FIG. 19. Scheme of a typical voltage-current curve of a cationexchange membrane. In region I, the current is increasing more or less linearly with the applied voltage according to Ohm’s law. Due to concentration polarization, a limiting current is observed in region II with the magnitude Ilim. The classical theory of concentration polarization predicts a true saturation of the voltage-current curve !dashed line", but other effects lead to overlimited currents in region III. Adapted from Rubinstein and Zaltzman, 2000.

FIG. 20. Schematic of the anodic side of a cation-selective nanochannel under a high external electric field E. The normal field component to the bulk surface En results in a nonequilibrium double layer of field-induced space charges that are mobile, and its counterparts are fixed co-ions inside the nanochannel. En in combination with a tangential field Et result in an electro-osmotic velocity of the second kind veo2, which can destroy the DBL, leading to overlimited currents through the nanochannel. Electro-osmotic and electrophoretic velocities veo and vep, respectively, are also shown. Adapted from Leinweber and Tallarek, 2004.

Full text

Transport phenomena in nanofluidics
Reto B. Schoch
*
Microsystems Laboratory, STI-LMIS, École Polytechnique Fédérale de Lausanne (EPFL),
CH-1015 Lausanne, Switzerland;
Department of Electrical Engineering and Computer Science, Massachusetts Institute of
Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA;
and Department of Biological Engineering, Massachusetts Institute of Technology, 77
Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
Jongyoon Han
Department of Electrical Engineering and Computer Science, Massachusetts Institute of
Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA;
and Department of Biological Engineering, Massachusetts Institute of Technology, 77
Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
Philippe Renaud
Microsystems Laboratory, STI-LMIS, École Polytechnique Fédérale de Lausanne (EPFL),
CH-1015 Lausanne, Switzerland
!Published 17 July 2008"
The transport of fluid in and around nanometer-sized objects with at least one characteristic
dimension below
100 nm enables the occurrence of phenomena that are impossible at bigger length
scales. This research field was only recently termed nanofluidics, but it has deep roots in science and
technology. Nanofluidics has experienced considerable growth in recent years, as is confirmed by
significant scientific and practical achievements. This review focuses on the physical properties and
operational mechanisms of the most common structures, such as nanometer-sized openings and
nanowires in solution on a chip. Since the surface-to-volume ratio increases with miniaturization, this
ratio is high in nanochannels, resulting in surface-charge-governed transport, which allows ion
separation and is described by a comprehensive electrokinetic theory. The charge selectivity is most
pronounced if the Debye screening length is comparable to the smallest dimension of the nanochannel
cross section, leading to a predominantly counterion containing nanometer-sized aperture. These
unique properties contribute to the charge-based partitioning of biomolecules at the micro-
channel-nanochannel interface. Additionally, at this free-energy barrier, size-based partitioning can be
achieved when biomolecules and nanoconstrictions have similar dimensions. Furthermore, nanopores
and nanowires are rooted in interesting physical concepts, and since these structures demonstrate
sensitive, label-free, and real-time electrical detection of biomolecules, the technologies hold great
promise for the life sciences. The purpose of this review is to describe physical mechanisms on the
nanometer scale where new phenomena occur, in order to exploit these unique properties and realize
integrated sample preparation and analysis systems.
DOI: 10.1103/RevModPhys.80.839 PACS number!s": 47.61.!k, 82.39.Wj, 73.63.!b, 87.85.Rs
CONTENTS
I. Introduction 840
II. Electrokinetic Effects 841
A. Electrostatics in liquids 841
1. Electrical double layer 841
2. Potential distribution in EDL 842
3. Debye-Hückel approximation 842
4. Gouy-Chapman model 843
5. Surface charge density 843
6. Surface conductance 844
7. Continuum models and molecular-dynamics
simulations 844
B. Electrokinetic effects in nanochannels 844
1. Basic physics of electrokinetics 845
2. Comprehensive electrokinetic theory for
nanochannels 846
C. Electrode-electrolyte interface 849
1. Charge-transfer resistance 849
2. Constant phase element 850
III. Phenomena at the Microchannel-Nanochannel
Interface 850
A. Electrical characterization of nanochannels 850
1. Impedance spectroscopy 850
2. Nanochannel conductance 851
3. Ionic current rectification 853
B. Donnan equilibrium 854
*
Present address: Department of Mechanical Engineering,
Stanford University, Stanford, CA 94305, USA. Author to
whom correspondence should be addressed.
reto.schoch@a3.epfl.ch
REVIEWS OF MODERN PHYSICS, VOLUME 80, JULY–SEPTEMBER 2008
0034-6861/2008/80!3"/839!45" ©2008 The American Physical Society839

1. Donnan potential 854
2. Membrane potential 854
C. Exclusion-enrichment effect 855
1. Permselectivity of a nanochannel 855
2. Model of the exclusion-enrichment effect 857
D. Partitioning at the interface 857
1. Partition coefficient for colloids 858
2. Electrostatic sieving of proteins 858
3. Active control of partitioning 859
E. Concentration polarization 860
1. Limiting current through nanochannels 860
2. Electroconvection for mixing 861
3. Preconcentration of molecules 862
IV. Macromolecule Separation Mechanisms Using
Nanometer-Sized Structures 863
A. Entropic trapping 863
B. Ogston sieving 863
C. Entropic recoil 864
D. Anisotropy for continuous-flow separation 864
V. Nanopores and Nanowires for Label-Free Biomolecule
Detection 865
A. Biomolecule translocations through nanopores 865
1. Biological nanopores 866
2. Synthetic nanopores 866
3. Current decrease and increase 867
B. Nanowire sensors 868
1. Fabrication techniques 868
2. Operation mechanism 869
3. Single-molecule sensitivity 869
4. Detection efficiency 870
VI. Fabrication of Nanochannels 870
A. Wet and dry etching 871
B. Sacrificial layer methods 871
C. High-energy beam techniques 872
D. Nanoimprint lithography 872
E. Bonding and sealing methods 872
F. Realization of single nanopores in membranes 872
VII. Conclusions and Perspectives 873
Acknowledgments 874
Lists of Abbreviations and Symbols 874
References 875
I. INTRODUCTION
Nanofluidics is defined as the study and application of
fluid flow in and around nanometer-sized objects with at
least one characteristic dimension below 100 nm !Eijkel
and van den Berg, 2005a". At such length scales, struc-
tures have a high surface-to-volume ratio, leading to
new physical phenomena that are not observed at mac-
rofluidic or microfluidic size scales !Di Ventra et al.,
2004; Han, 2004". One significant benefit of nanofluidics
is that it presents the possibility of learning new science
using controlled regular nanostructures !Mukhopadhyay,
2006", which makes it relevant for many areas in nano-
science and nanotechnology.
The term nanofluidics has only recently been intro-
duced with the rise of micro total analysis systems
!
"
TAS"!Manz et al., 1990", which aim to integrate all
steps of biochemical analysis on one microchip !Squires
and Quake, 2005; Whitesides, 2006". The roots of nanof-
luidics are broad, and processes on the nanometer scale
have implicitly been studied for decades in chemistry,
physics, biology, materials science, and many areas of
engineering. This dynamic new field has drawn attention
in technology, biology, and medicine due to advances in
biomolecule preparation and analysis systems !Lieber,
2003", single-molecule interrogations !Craighead, 2003,
2006", and other unique modes of molecular manipula-
tion. For example, molecules can be controlled by
charge in nanochannels because of their electrostatic in-
teractions with the electrical double layer !EDL",a
shielding layer that is naturally created within the liquid
near a charged surface. Moreover, size-based filtration
and sieving can be achieved because the length scales of
biomolecules and synthetic nanometer-sized objects are
similar.
In this review, we present the following unique prop-
erties of nanofluidic systems: nanowires can be operated
as field-effect transistors to detect chemical and biologi-
cal species label-free, and transport through nanochan-
nels leads to analyte separation and new phenomena
when the EDL thickness becomes comparable to the
smallest channel opening !Fig. 1". Such charge-selective
features were first described in membrane filtration !us-
ing mainly irregular nanoporous systems", and industrial
applications have been developed. However, mem-
branes are not in the scope of this review, and the reader
is referred to the comprehensive literature of membrane
science !Helfferich, 1962; Sata, 2004; Strathmann, 2004".
We focus instead on geometrically well-defined, solid-
state nanochannels with one cross-sectional dimension
between a few nanometers and 100 nm that are manu-
factured on a chip with standard microfabrication tech-
nology processes. Compared to membranes, which yield
statistical results, single, well-designed, and controlled
nanochannels are ideal physical modeling systems to
study fluidics in a precise manner. Carbon nanotubes are
not covered in this review, and the reader is referred to
Whitby and Quirke !2007".
The authors are inspired by nature’s ultimate nano-
fluidic system, namely, transmembrane protein channels.
They have remarkable features such as high charge se-
lectivity, which was first described in studies of cell-
function-controlling potassium channels on the cell
Hi
g
hi
on
i
c strengt
h L
ow
i
on
i
c strengt
h
FIG. 1. Electrical double layer !EDL, shaded in gray" at high
ionic strength, it is thin, allowing co-ions and counterions to
pass through the nanochannel. At low ionic strength, the EDL
thickness increases, resulting in a counterion-selective
nanochannel. From Schoch, 2006.
840
Schoch, Han, and Renaud: Transport phenomena in nanofluidics
Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008

membrane. These channels are 10 000 times more per-
meable to potassium than to sodium although both at-
oms have one positive net charge !Israelachvili, 1992;
Doyle et al., 1998". At the same time, these ion channels
exhibit high transport throughput, approaching the dif-
fusional limit. Such a performance has not been
achieved by any artificial nanopore, which demonstrates
that there is much to be learned from biology.
In this review, we first present classic electrostatic and
electrokinetic theory relevant to nanofluidics in Sec. II.
Expert readers may focus directly on Sec. II.B.2, which
describes electrokinetic effects of nanochannels. When a
nanochannel is contacted, the geometric size difference
between the interfacing channel and the nanochannel
leads to a free-energy barrier, and the unique properties
of the microchannel-nanochannel interface are analyzed
in Sec. III. Section IV focuses on systems that include
nanometer-sized structures for separating biomolecules
such as DNA and proteins. Then we describe operation
mechanisms and physical effects of nanopores and nano-
wires for label-free biomolecule detection in Sec. V. The
fabrication techniques of nanometer-sized openings that
we discuss in this review are summarized in Sec. VI.
Finally, we draw conclusions and offer perspectives
about transport phenomena in nanofluidics.
II. ELECTROKINETIC EFFECTS
Here we introduce the reader to electrostatics in liq-
uids and electrokinetic effects, which are the most im-
portant and fundamental concepts for the description of
transport in nanofluidics. This classical theory was estab-
lished decades ago in well-established disciplines, but it
cannot be omitted in the new interdisciplinary field of
nanofluidics !Eijkel and van den Berg, 2005a", and will
therefore be summarized.
A. Electrostatics in liquids
When considering that a solid in contact with a liquid
bears a surface charge, one perceives that this parameter
is of increased importance in nanochannels, since they
have a high surface-to-volume ratio. Surface charge is
caused by the dissociation of surface groups and the spe-
cific !nonelectric" adsorption of ions in solution to the
surface !Perram et al., 1973; Behrens and Grier, 2001".
Depending on the number and type of acid and basic
groups present in solution !Hunter and Wright, 1971;
Davis et al., 1978; Sonnefeld et al., 1995", the solid has
either a positive or a negative surface charge density,
which is phenomenologically described by
#
s
=#
i
q
i
/A,
where q
i
=z
i
e is the net charge of ion i, z
i
is the valency
of ion i, e is the electron charge, and A is the surface
area. A typical value of high charge density and fully
ionized surfaces is
#
s
=0.3 C m
−2
, which corresponds to
one charge per $0.5 nm
2
. At a specific pH value of the
solution, the surface bears no net charge; this is known
as the point of zero charge, which is $2 for glass !Parks,
1965; Iler, 1979". We focus on glass for reasons described
in Sec. VI to do with the fabrication of nanochannels.
Surface charges result in electrostatic forces, which
are important for the description of long-range interac-
tions between molecules and surfaces in liquids, and
thus they govern transport in nanofluidic systems. At
small distances, van der Waals forces contribute to the
attractive part of the interaction, for example, between
dissolved particles, whereas coagulation is prevented by
repulsive or attractive electrostatic forces. The interplay
between van der Waals and electrostatic forces was ini-
tially described in the 1940s in the Derjaguin-Landau-
Verwey-Overbeek !DLVO" theory !Overbeek, 1952; Is-
raelachvili, 1992", which is important for the description
of colloidal stability. van der Waals forces are greatly
insensitive to variations in electrolyte concentration and
pH, which is not true for electrostatic forces.
1. Electrical double layer
Due to the fixed surface charge at the solid interface,
an oppositely charged region of counterions develops in
the liquid to maintain the electroneutrality of the solid-
liquid interface. This screening region is denoted as the
EDL because ideally it consists of opposite charges,
some of which are bound while others are mobile.
The electrical double layer was initially represented
by a simple capacitor, usually attributed to the model of
Helmholtz. Gouy and Chapman treated one layer of
charge smeared uniformly over a planar surface im-
mersed in an electrolyte solution !Overbeek, 1952".
Stern recognized that the assumptions that the electro-
lyte ions could be regarded as point charges and the
solvent could be treated as a structureless dielectric of
constant permittivity were quite unsatisfactory. He intro-
duced the Stern layer between the inner and outer
Helmholtz planes, in which the charge and potential dis-
tribution are assumed to be linear, and a diffuse layer
further from the wall where the Gouy-Chapman theory
is applied. This model is presented in Fig. 2, which is
separated into three layers !Hunter, 1981". The first layer
is at the inner Helmholtz plane and bears the potential
$
i
, where co-ions and counterions are not hydrated and
are specifically adsorbed to the surface. The second
layer is defined by the outer Helmholtz plane with po-
tential
$
d
, consisting of a layer of bound, hydrated, and
partially hydrated counterions. The outermost and third
layer is the diffuse layer, composed of mobile co-ions
and counterions, in which resides the slip plane bearing
the
%
potential !described hereafter". In most cases, the
outer Helmholtz plane and the slip plane are situated
close to each other !Bhatt et al., 2005", allowing the ap-
proximation of
$
d
with the
%
potential for practical pur-
poses.
The slip plane, or shear surface, is an imaginary plane
separating ions that are immobile at the surface from
those that are mobile in solution. The
%
potential at this
plane can be experimentally determined, and is there-
fore an important parameter in colloid science for deter-
mining the stability of particles, and in
"
TAS for de-
841
Schoch, Han, and Renaud: Transport phenomena in nanofluidics
Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008

scribing fluid flow in channels !Dittrich et al., 2006". The
%
potential is dependent on the pH and ionic strength of
the solution, showing a slight increase in its absolute
value for a decreasing number of ions in solution
!Lyklema and Overbeek, 1961; Hunter, 1981; Gu and Li,
2000".
2. Potential distribution in EDL
We now provide an overview of the potential distribu-
tion at charged interfaces; a more comprehensive treat-
ment has been given by Hunter !1981" and Overbeek
!1952". To calculate the potential distribution near a
charged surface, the electrochemical potential
"
5
i
of ion i
in a liquid phase at constant pressure and temperature
has to be considered !Israelachvili, 1992",
"
5
i
=
"
˜
i
+ z
i
F
$
=
"
˜
i
0
+ RT ln!
&
a
c
i
/c
0
" + z
i
F
$
, !1"
where
"
˜
i
is the chemical potential, F is the Faraday con-
stant,
$
is the electric potential due to the surface
charge,
"
˜
i
0
is the standard chemical potential of ion i at
constant pressure and temperature, R is the gas con-
stant, T is the temperature,
&
a
is the activity coefficient,
c
i
is the molar concentration of ion i, and c
0
is the stan-
dard molarity of 1 mol l
−1
. At equilibrium, the electro-
chemical potential of the ions must be the same every-
where %i.e., grad!
"
5
i
"=0&, and the electrical and
diffusional forces on the ion i must be balanced,
#
"
˜
i
=− z
i
F #
$
, !2"
where # = grad. Insertion of the chemical potential
"
˜
i
=
"
˜
i
0
+RT ln!
&
a
c
i
/c
0
" into Eq. !2", and its integration from
a point in the bulk solution where
$
=0 and n
i
=n
i
'
and
the bulk volume density n
i
'
=1000N
A
c
i
, leads to the
Boltzmann equation, giving the local concentration of
each type of ion in the diffuse layer,
n
i
= n
i
'
exp!− z
i
e
$
/k
B
T", !3"
with k
B
the Boltzmann constant and the conversion
e/k
B
=F/R is applied. The volume charge density
(
of all
ions present in the neighborhood of the surface is given
by
(
= e
#
i
n
i
z
i
. !4"
One further important equation is required, the fun-
damental Poisson equation, giving the net excess charge
density at a specific distance from the surface:
#
2
$
=
d
2
$
dz
2
=−
(
)
0
)
r
, !5"
where #
2
$
=div!grad
$
" and z is the surface normal di-
rection. Substituting Eqs. !3" and !4" into Eq. !5", we
obtain the complete Poisson-Boltzmann equation, which
describes how the electrostatic potential due to a distri-
bution of charged atoms varies in space,
#
2
$
=
d
2
$
dz
2
=−
e
)
0
)
r
#
i
n
i
'
z
i
exp%− z
i
e
$
!z"/k
B
T&. !6"
3. Debye-Hückel approximation
The Poisson-Boltzmann equation %Eq. !6"& is a
second-order elliptic partial differential equation, and
can be solved analytically by assuming that the surface
potential is small everywhere in the EDL !z
i
$
i
*25.7 mV at 25 °C" and by expanding the exponential
!using the relation e
−
+
=1−
+
for small
+
", which leads to
the Debye-Hückel approximation
#
2
$
=
d
2
$
dz
2
=
,
2
$
!z", !7"
where
,
=
'
e
2
#
i
n
i
'
z
i
2
)
0
)
r
k
B
T
(
1/2
. !8"
,
is called the Debye-Hückel parameter and is mainly
dependent on the bulk volume density n
i
'
. The potential
-
-
-
-
-
-
-
-
i
nner
H
e
l
m
h
o
l
tz p
l
ane
outer Helmholtz plane
slip plane
ψ
s
ψ
i
ψ
d
ζ
ψ
z
specifically adsorbed
anion, nonhydrated
nonspecifically adsorbed
cation, hydrated
water molecule
specifically adsorbed
cation, partially hydrated
surface charge
Stern layer diffuse layer (Gouy-Chapman)
+
+
+
+
+
+
+
+
FIG. 2. Gouy-Chapman-Stern model of the solid-electrolyte
interface, with the corresponding potential distribution
$
vs
the distance z from the wall. The solid is illustrated with a
negative surface potential
$
s
, described by three layers in so-
lution. The inner Helmholtz plane layer !
$
i
" consists of nonhy-
drated co-ions and counterions, whereas the outer Helmholtz
plane layer !
$
d
" is built up of only hydrated counterions. The
diffuse layer is defined beyond the outer Helmholtz plane. At
the slip plane, the
%
potential can be experimentally investi-
gated, and as the distance between the outer Helmholtz plane
and the slip plane is negligible in most cases, the
%
potential is
usually equal to
$
d
. Adapted from Schoch et al., 2005.
842
Schoch, Han, and Renaud: Transport phenomena in nanofluidics
Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008

decays exponentially in the diffuse layer with the char-
acteristic distance given by the Debye length -
D
=
,
−1
.
This value corresponds to the thickness of the EDL,
which increases with dilution as presented in Table I.
For a symmetrical z
i
:z
i
electrolyte with concentration c
i
at 25 ° C, the value of the Debye length -
D
!unit is
meters" can be written as
-
D
=
3.04 . 10
−10
z
i
)
c
i
=
2.15 . 10
−10
)
I
s
, !9"
where the ionic strength I
s
is
I
s
=
1
2
#
c
i
z
i
2
. !10"
Consider that the region of varying potential extends
to a distance of about 3-
D
before the potential has de-
cayed to about 2% of its value at the surface. Based on
the approximation z
i
$
s
*25.7 mV, the solution of the
Debye-Hückel approximation is
$
!z" =
$
s
exp!−
,
z". !11"
4. Gouy-Chapman model
As the Debye-Hückel approximation is not valid for
high surface potentials, the Poisson-Boltzmann equation
has to be solved explicitly. Analytically, this can be done
only under the assumption of a symmetrical electrolyte
where the valence of the co-ion is equal to the valence of
the counterion, leading to the Gouy-Chapman equation
tanh%z
i
$
˜
!z"/4& = tanh!z
i
$
˜
s
/4"exp!−
,
z", !12"
where
$
˜
=e
$
/k
B
T is the dimensionless potential, and at
25 ° C
$
˜
=1 for
$
=25.7 mV. For small values of
+
,
tanh
+
*
+
is valid !Taylor expansion", and Eq. !12" re-
duces to Eq. !11". The Poisson-Boltzmann equation for
high surface potentials and unsymmetrical electrolytes
can only be solved numerically.
The resulting ion distribution from the Gouy-
Chapman model is presented in Fig. 3!a", where almost
all the charge is balanced by the accumulation of coun-
terions and relatively little by the reduction of co-ions.
This is why it is possible to treat an unsymmetrical elec-
trolyte system as symmetric, as is done in the Gouy-
Chapman model, without incurring too much error.
When the Debye-Hückel approximation is used instead,
the two ion types are of equal significance, as shown in
Fig. 3!b".
5. Surface charge density
The surface charge density must balance the charge
density in the adjacent solution,
#
s
=−
+
0
'
(
dz. !13"
On substituting Eq. !5" and the integrated Eq. !6" into
Eq. !13", we obtain a relationship between the surface
charge density
#
s
and the surface potential
$
s
,
#
s
= )
0
)
r
+
0
'
d
2
$
dz
2
dz
=
'
2)
0
)
r
k
B
T
#
i
n
i
'
%exp!− z
i
e
$
s
/k
B
T" −1&
(
1/2
dz.
!14"
Since the surface potential
$
s
is difficult to determine
experimentally, the charge density of the diffuse layer
can be calculated by stopping the integration at the
shear plane. We thus achieve an equation that relates
the
%
potential to the diffuse layer charge density,
#
d
=
'
2)
0
)
r
k
B
T
#
i
n
i
'
%exp!− z
i
e
%
/k
B
T" −1&
(
1/2
dz,
!15"
which has been used, for example, in a site-binding
model of the oxide-electrolyte interface !Yates et al.,
1974", and has shown consistency with experimental
data obtained for oxides by potentiometric titration
measurements !Abendrot, 1970".
TABLE I. EDL thicknesses -
D
for typical KCl concentrations
at 25 °C.
KCl concentration !M" Debye length -
D
!nm"
10
0
0.3
10
−1
1.0
10
−2
3.1
10
−3
9.6
10
−4
30.5
10
−5
96.3
01234
n
-
n
+
n
+0
=n
-0
01234
n
-
n
+
κ z
κ z
(
a) (b)
FIG. 3. Volume densities n
+
and n
−
of positive and negative
ions, respectively, near a negatively charged surface as a func-
tion of the dimensionless number
,
z. Distribution from !a" the
Gouy-Chapman model, which shows an excess of positively
charged ions, and !b" the Debye-Hückel approximation with a
symmetrical co-ion and counterion distribution. Adapted from
Hunter, 1981.
843
Schoch, Han, and Renaud: Transport phenomena in nanofluidics
Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008

Frequently asked questions

Q1. What are the contributions in "Transport phenomena in nanofluidics" ?

Schoch et al. this paper presented a review of the properties of nanofluidics in terms of their properties in nanochannels, nanofilters, nanopores, nanowires and at the microchannel-nanochannel interface. 

Q2. What is the versatile method for creating a nanowire?

Crystallization is essential to create a nanowire, and is composed of droplet formation, nucleation, and growth, for which the vapor-liquid-solid growth method seems to be the most versatile. 

Q3. What can be investigated with a fabricated nanopore in an insulating material?

ionic strength, temperature, functionalized surface" can potentially be investigated with a fabricated nanopore in an insulating material. 

Q4. What is the effect of asymmetric pore geometries on the current?

asymmetric pore geometries or surface charge discontinuities in a nanopore lead to nonlinear, diodelike voltage-current curves at symmetric electrolyte conditions. 

Q5. What is the effect of EDL extension on the permeability of the nanochannel?

When the exclusion and enrichment of ions occurs at low ionic strength, the local concentration of charged species is no longer homogeneous in the cross section of the nanochannel, because the EDL extension increases the counterion concentration near the charged surface. 

Q6. What is the effect of EDL overlap on the nanochannel?

At low ionic strength, the EDL thickness becomes comparable to the nanochannel height, resulting in an overlap of the diffuse parts of the EDLs in the nanometer-sized aperture. 

Q7. What is the effect of reducing the size of p-Si on the detection sens?

By reducing the size of p-Si, the charge accumulation or depletion in the nanowire occurs within a significant portion of the one-dimensional structure, leading to increased detection sensitivities. 

Q8. What is the principle of the sieving of rodlike molecules in a nanofilter column?

In contrast to long DNA molecules which separate according to entropic trapping, the overall mobility of rodlike molecules in a nanofilter column decreases with increasing length. 

Q9. Why do the authors believe that the potential to design systems with enhanced fluid transport is important?

The authors believe that further knowledge of liquid slip is important for nanofluidics because of the potential to design systems with enhanced fluid transport ! 

Q10. What is the important parameter for the characterization of translocation events?

Other important parameters for the characterization of translocation events are the measured current amplitude and the characteristic dispersion of values for individual translocation durations, which have also been used to discriminate different types of polynucleotides of similar length ! 

Q11. Why is the binding of antibodies to the surface of latex colloids detectable?

The binding of antibodies to the surface of latex colloids has been reported to be detectable upon translocation through a nanopore because of the increased particle size ! 

Q12. What is the smallest single nanopore that can be achieved in a solid?

The smallest single nanopore that can be achieved in a solid is fabricated with high-energy beams, and is promising for DNA translocation measurements as presented in Sec. IV. 

Q13. Why does the translocation time depend on the length of the polymer?

the translocation time is not linearly related to DNA length, but grows as a power of the polymer length, which could be due to the hydrodynamic drag of the DNA polymer section outside the pore ! 

Q14. How did Keyser and Van Dorp measure the electrical force acting on a single DNA?

By combining ionic current measurements with optical tweezers, Keyser, Koeleman, Van Dorp, et al. !2006" determined the electrical force acting on a single DNA molecule in a solid-state nanopore. 

Q15. What are the advantages of microfabricated regular sieving structures?

Such microfabricated regular sieving structures hold promise as an alternative to gels or capillaries to improve biomolecule separation speed and resolution. 

Q16. What is the effect of coating the nanopore walls with gold?

The poorly defined nature of the chemistry and charge of the polymeric pores has limited the understanding of the current-rectification function; coating the nanopore walls with a gold layer has offered deeper insight ! 

Q17. How can the transport be controlled by changing the potential applied to the membrane surface?

Unlike metal-coated nanoporous membranes, where the transport can be controlled by changing the potential applied to the membrane surface !