Transport phenomena in nanofluidics

Reto B. Schoch, Jongyoon Han, and Philippe Renaud

Rev. Mod. Phys. 80, 839 – Published 17 July 2008

Abstract

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 microchannel-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.

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DOI:https://doi.org/10.1103/RevModPhys.80.839

Authors & Affiliations

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

^*Present address: Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA. Author to whom correspondence should be addressed. reto.schoch@a3.epfl.ch

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Vol. 80, Iss. 3 — July - September 2008

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Figure 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 410.Reuse & Permissions
Figure 2
Gouy-Chapman-Stern model of the solid-electrolyte interface, with the corresponding potential distribution $ψ$ vs the distance $z$ from the wall. The solid is illustrated with a negative surface potential $ψ_{s}$ , described by three layers in solution. The inner Helmholtz plane layer $(ψ_{i})$ consists of nonhydrated co-ions and counterions, whereas the outer Helmholtz plane layer $(ψ_{d})$ is built up of only hydrated counterions. The diffuse layer is defined beyond the outer Helmholtz plane. At the slip plane, the $ζ$ potential can be experimentally investigated, and as the distance between the outer Helmholtz plane and the slip plane is negligible in most cases, the $ζ$ potential is usually equal to $ψ_{d}$ . Adapted from 414.Reuse & Permissions
Figure 3
Volume densities $n_{+}$ and $n_{-}$ of positive and negative ions, respectively, near a negatively charged surface as a function of the dimensionless number $κ z$ . 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 215.Reuse & Permissions
Figure 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 410.Reuse & Permissions
Figure 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 147.Reuse & Permissions
Figure 6
Normalized mobility $μ_{nano} ∕ μ$ as a function of $2 λ_{D} ∕ h$ for two values of ion valence $z_{i}$ . 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 360.Reuse & Permissions
Figure 7
Complex impedance plot of 50-nm-high nanochannels measured in a $0.1 M$ $KCl$ solution for frequencies $ω$ from $20 Hz$ to $1 MHz$ . A value of $2.22 M Ω$ has been obtained for the nanochannel resistance. The inset shows the electric equivalent circuit modeling the EDL on the platinum electrodes, consisting of the nanochannel resistance $R_{nano}$ , the stray capacitance of the chip $C_{stray}$ , and the impedance of the constant phase element $Z_{CPE}$ . Adapted from 414.Reuse & Permissions
Figure 8
Schematic behavior of the nanochannel conductance as a function of the $KCl$ concentration. The impact of the characteristic surface charge density $σ_{s}$ , the width $w$ , the length $d$ , and the height $h$ of the nanochannel on its conductance is shown. Below the transition $KCl$ concentration $c_{t}$ , a conductance plateau $G_{p}$ is reached, where $c_{t}$ corresponds to 0.5 times the excess mobile counterion concentration $c_{e}$ . From 413.Reuse & Permissions
Figure 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 425.Reuse & Permissions
Figure 10
Donnan equilibrium at the microchannel-nanochannel interface. (a) Schematic drawing presenting a bulk solution (I) and an adjacent cation-selective nanochannel (II). (b) Concentrations $c_{i}^{I}$ of cations and anions in the bulk solution are equal, whereas in the permselective nanochannel the concentration of cations $c_{ca}^{II}$ is higher than the concentration of anions $c_{an}^{II}$ . (c) The Donnan potential $Φ_{D}$ is negative in the cation-selective nanochannel. From 410.Reuse & Permissions
Figure 11
Schematic representation of the membrane presenting the concentration profile and resulting potential differences for a permselective membrane, separating solution 1 from solution 2. The concentrations in solution 1 are $c_{1}^{I}$ in the bulk and $c_{1}^{II}$ at the membrane surface, and in solution 2 they are $c_{2}^{I}$ in the bulk and $c_{2}^{II}$ at the membrane surface. The wall charge concentration averaged over the channel volume is $c_{w}^{II}$ . The membrane potential $Φ_{m}$ is composed of the Donnan potential $Φ_{D_{1}}$ in solution 1, the Donnan potential $Φ_{D_{2}}$ in solution 2, and the diffusion potential $Φ_{diff}$ . Adapted from 400.Reuse & Permissions
Figure 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 $ϕ^{*}$ is proportional to $S^{*}$ . The Debye length increases with dilution, reducing the effective cross section $S_{eff}$ through which anions can diffuse. This results in a lower effective flux $ϕ_{eff}$ . Adapted from 363.Reuse & Permissions
Figure 13
Variation of the relative permeability $P_{eff} ∕ P^{*}$ 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 363.Reuse & Permissions
Figure 14
Potential and concentration profiles in a 50-nm-high nanochannel for two $KCl$ concentrations. (a) At a low ionic strength of $0.001 M$ $KCl$ , the potential remains negative in the nanochannel. The difference between the numerical solution of Eq. (6) (dotted black line) and the approximated analytical expression of Eq. (52) (gray solid line) is small. (b) Concentration profiles of charged species are determined by the Boltzmann equilibrium [Eq. (24)]. At low ionic strength, the enrichment of cationic species (black solid line, net charge $+ 1 e$ ) and the exclusion of anionic species (gray solid line, net charge $- 1 e$ ) results. Adapted from 363.Reuse & Permissions
Figure 15
Partition coefficient $β_{part}$ as a function of the solution concentration $c$ , or of the ratio of pore radius $r_{0}$ to Debye length $λ_{D}$ , for completely porous (P) and solid (S) spheres of identical size and total charge. Calculations were performed for $r_{0} = 20 nm$ , surface charge densities of walls and spheres of $0.01 C m^{- 2}$ , and a 1:1 univalent electrolyte. Dashed lines indicate that asymptotic values are approached at high ionic strength for each value of $a_{p} ∕ r_{0}$ , where $a_{p}$ is the particle radius. Adapted from 438.Reuse & Permissions
Figure 16
Apparent diffusion coefficient $D_{app}$ of three lectin proteins as a function of the $p H$ value of the solution, measured in a 50-nm-high nanochannel at an ionic strength of $10^{- 3} M$ . $D_{app}$ of the three lectin proteins shows three peaks, corresponding to the $p I$ values of the acidic band lectin $L_{a}$ , the middle band lectin $L_{m}$ , and the basic band lectin $L_{b}$ . The marks on the $p H$ axis show the respective $p I$ values of the lectins, which have been measured with isoelectric focusing. The connecting lines are for guidance only. From 411.Reuse & Permissions
Figure 17
Nanofluidic control of protein transport by a gate voltage without bias between the microchannels. When the gate voltage $V_{g}$ is negative, the positively charged protein avidin can diffuse through the nanochannel, whereas the protein is repelled when $V_{g}$ is positive. (a) Schematics and (b) corresponding fluorescence images. Adapted from 234.Reuse & Permissions
Figure 18
Schematic illustrating concentration polarization at nanochannel interfaces. Concentration profiles of salt in the diffusion boundary layer (DBL) on both sides of the cation-selective 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 455.Reuse & Permissions
Figure 19
Scheme of a typical voltage-current curve of a cation-exchange 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 $I_{\lim}$ . 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 392.Reuse & Permissions
Figure 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 $E_{n}$ results in a nonequilibrium double layer of field-induced space charges that are mobile, and its counterparts are fixed co-ions inside the nanochannel. $E_{n}$ in combination with a tangential field $E_{t}$ result in an electro-osmotic velocity of the second kind $v_{eo 2}$ , which can destroy the DBL, leading to overlimited currents through the nanochannel. Electro-osmotic and electrophoretic velocities $v_{eo}$ and $v_{ep}$ , respectively, are also shown. Adapted from 270.Reuse & Permissions
Figure 21
Preconcentration of proteins in the cathodic microchannel in front of a permselective nanochannel. (a) Fluorescence images of preconcentrated recombinant green fluorescent protein (rGFP) inside the upper microchannel show that the preconcentration zone increases in size and concentration over time (illustrated for 0, 3, 6, and $9 \min$ at $10 V$ ). The size of the depleted region is typically much larger than the enrichment zone, and hence cannot be captured in these images. (b) Fluorescence intensity of preconcentrated rGFP as a function of time for three values of the applied voltage. The linear behavior between the protein concentration and the measured fluorescence intensity (not shown) has been used to calculate a preconcentration factor greater than 600-fold after $9 \min$ at $10 V$ . Adapted from 410.Reuse & Permissions
Figure 22
Entropic trapping of DNA in a nanofilter column. (a) Cross-sectional schematic of the device consisting of alternating shallow $(t_{s})$ and deep $(t_{d})$ 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. $R_{0}$ is the radius of gyration. From 182. (c) Escape of a DNA molecule from the trap, where $x_{f}$ is the length of the DNA strand in the filter. (d) Energy $Δ ϵ$ vs $x_{f}$ . The transition state at which the strands overcome the energy barrier is when $x_{f} = x_{f, c}$ . Adapted from 183.Reuse & Permissions
Figure 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 pillar-free region. From 45.Reuse & Permissions
Figure 24
Bidirectional motion of negatively charged macromolecules in an anisotropic nanofilter array, driven by two orthogonal electric fields $E_{x}$ and $E_{y}$ . (a) Ogston sieving and (b) entropic trapping are obtained when $λ_{D} ⪡ t_{s}$ . In this regime at high ionic strength, steric interactions dictate the jump dynamics and the passage rate $P_{x}$ . (c) Electrostatic sieving becomes dominant when $λ_{D} \approx t_{s}$ . The mean drift distance $L$ plays a determinant role for the migration trajectory, with a shorter $L$ leading to a larger stream deflection angle $θ$ , where $θ$ is defined with respect to the positive $y$ axis. Silicon pillars serve as supporting structures to prevent collapse of the top ceiling. Adapted from 141.Reuse & Permissions
Figure 25
Nanopore in a metal-oxide-semiconductor capacitor membrane, in which the biosensor was simulated to have single nucleotide resolution. Left panel: Image (top view) of a nanopore sputtered into the capacitor membrane. Center panel: Schematic cross section of the novel biosensor. Right panel: Cross-sectional transmission electron microscope image of the capacitor membrane. From 162.Reuse & Permissions
Figure 26
Simultaneous current and force measurements of a DNA molecule in a nanopore at $0.1 M$ $KCl$ . Differences in the applied voltage change the bead position until the electrical force $F_{el}$ is equal to the optical force $F_{ot}$ . 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 $k_{trap}$ is the trap stiffness. From 240.Reuse & Permissions
Figure 27
Transition from current increase to current decrease for DNA translocations through a nanopore. (a) Conductance changes $Δ G$ and (b) relative conductance changes $Δ G ∕ G$ due to DNA molecule translocation for $KCl$ concentrations between $50 m M$ and $1 M$ . 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 434.Reuse & Permissions
Figure 28
Operation mechanism of nanowire sensors. (a) Schematic of a metal-oxide-semiconductor field-effect transistor, where S, D, and G are the source, drain, and gate metal electrodes, respectively. If a negative gate potential $V_{G}$ is applied, holes are accumulated in $p$ -type silicon ( $p$ -Si), leading to a conductance increase. (b) Schematic of an ion-sensitive field-effect transistor, in which a conductance increase is measurable upon binding of negative molecules to the surface of $p$ -Si. Adapted from 357.Reuse & Permissions
Figure 29
Multiplexed detection with nanowire arrays. (a) Conductance vs time data were recorded simultaneously for two $p$ -type silicon nanowires (NW). NW1 was functionalized with prostate specific antigen (PSA) antibody 1, and NW2 was modified with ethanolamine. Vertical lines correspond to the addition of (1) $9 pg {ml}^{- 1}$ PSA, (2) $1 pg {ml}^{- 1}$ PSA, (3) $10 μ g {ml}^{- 1}$ bovine serum albumin, and (4) a mixture of $1 ng {ml}^{- 1}$ PSA and $10 μ g {ml}^{- 1}$ PSA antibody 1. (b) Complementary sensing of PSA using the $p$ - and $n$ -type nanowires NW1 and NW2, respectively. Vertical lines correspond to times when PSA solutions with the following concentrations were added: (1) $0.9 ng {ml}^{- 1}$ ; (2) $0.9 ng {ml}^{- 1}$ ; (3) $9 pg {ml}^{- 1}$ , (4) $0.9 pg {ml}^{- 1}$ , and (5) $5 ng {ml}^{- 1}$ . Black arrows correspond to points where the solution was switched from protein to pure buffer solutions. From 530.Reuse & Permissions
Figure 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 308.Reuse & Permissions
Figure 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 464.Reuse & Permissions
Figure 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 410.Reuse & Permissions
Figure 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 275.Reuse & Permissions

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