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Nanofluidics: a pedagogical introductionSimon Gravelle

To cite this version:

Simon Gravelle. Nanofluidics: a pedagogical introduction. 2016. �hal-02375018�

Nanofluidics: a pedagogical introduction

Simon Gravelle

01 MARCH 2016

1 Generalities

1.1 What is nanofluidics?

Nanofluidics is the study of fluids confined in struc-tures of nanometric dimensions (typically 1−100 nm)[1, 2]. Fluids confined in these structures exhibit be-haviours that are not observed in larger structures,due to a high surface to bulk ratio. Strictly speak-ing, nanofluidics is not a new research field and hasbeen implicit in many disciplines [3, 4, 5, 6, 7], buthas received a name of its own only recently. Thisevolution results from recent technological progresswhich made it possible to control what occurs atthese scales. Moreover, advances have been madein observation/measurement techniques, allowing formeasurement of the small physical quantities inher-ent to nano-sized systems.

Even though nanofluidics is born in the footstep ofmicrofluidics, it would be incorrect to consider it anextension of microfluidics. Indeed, while in microflu-idics the only scale which matters is the size of thesystem, nanofluidics has to deal with a large spec-trum of characteristic lengths which induce coupledphenomena and give rise to complex fluid behaviours[8]. Moreover, since nanofluidics is at the intersectionbetween physics, chemistry and biology, it concernsa wide range of domains such as physiology, mem-brane science, thermodynamics or colloidal science.Consequently, a multidisciplinary approach is oftenneeded for nanofluidics’ research.

Some striking phenomena taking place at thenanoscale have been highlighted during the past fewyears. For example, super-fast flow in carbon nan-otubes [9, 10, 11], nonlinear eletrokinetic transport[12, 13] or slippage over smooth surfaces [14] havebeen measured. Those effects are indicators of therichness of nanofluidics. Accordingly, this field cre-ates great hopes, and the discovery of a large varietyof new interesting effects in the next decades is a

reasonable expectation. Moreover, one can noticethat most of the biological processes involving fluidsoperate at the nano-scale, which is certainly not bychance [8]. For example, the protein that regulateswater flow in human body, called aquaporin, has gotsub-nanometric dimensions [15, 16]. Aquaporins areknown to combine high water permeability and goodsalt rejection, participating for example to the highefficiency of human kidney. Biological processes in-volving fluid and taking place at the nanoscale attestof the potential applications of nanofluidics, and con-stitute a source of inspiration for future technologicaldevelopments.

Hereafter, an overview of the current state ofnanofluidics is presented. First, a brief state-of-the-art, mainly focused on nano-fabrication and mea-surement techniques is given. Then some currentapplications linked to nanofluidics are described.

1.2 State-of-the-art

Nanofluidics has emerged from the recent progressesof nanoscience and nanotechnology, such as pro-gresses made in developing nano-fabrication tech-nologies. Fabricating well-controlled channels is amajor challenge for nanofluidics, and is a necessarycondition for a systematic exploration of nanofluidicphenomena. This requires a good control of devicedimensions and surface properties (charge, roughness,etc). For example, the improvement of lithographytechniques (electron, x-beam, ion-beam, soft...) al-lows the fabrication of slit nanochannels [17]. FocusIon Beam (FIB) allows to drill nanopores in solidmembranes [18, 19]. There are also coatings and de-position/etching techniques that can be used to tunethe surface properties [20, 21]. Siria et al. were ableto manipulate a single boron nitride (BN) nanotubein order to insert it in a membrane separating elec-trolyte reservoirs and perform electric measurements[22]. Great developments of Scanning Tunneling Mi-

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croscope (STM) or Atomic Force Microscopy (AFM)allow to characterize the fabricated devices.

In parallel, the efforts invested in nanofabricationhave been combined to an improvement of measure-ment techniques. Most of them are based on themeasurement of electric currents, and have been de-veloped since the early days of physiology. But fora full understanding of nanofluidic properties, otherquantities have to be made accessible. For exam-ple, local values of a velocity field can be obtainedusing nano-Particle Image Velocimetry (nano-PIV).Surface Force Apparatus (SFA) have been used toexplore the hydrodynamic boundary condition andmeasure forces that play an important role in nanoflu-idics, such as van der Waals or electric forces.

In addition, a current challenge concerns waterflow measurements. The main difficulty is due tothe magnitude of typical flows through nanochannels:∼ 10−18 m3/s (it would take several years to growa drop of 1 nl with such a flow). In order to detecta water flow through a nanochannel, some poten-tial candidates have emerged during the last decade.One can cite, for example, Fluorescence Recovery Af-ter Photobleaching (FRAP), confocal measurementsor coulter counting measurements that have beenreported to detect respectively 7 · 10−18 m3/s [23],10−18 m3/s [24] and 10−18 m3/s [25]. However, theinconvenient of most of the existing measurementtechniques is that they are indirect and require theuse of dyes or probes.

Meanwhile, numerical progresses combined withcalculation capacity improvement allow for the theo-retical exploration of a large variety of nanofluidicproperties. For example, the friction of water on solidsurfaces can be investigated using ab initio methods[26], while molecular dynamics simulations are goodcandidates for fluid transport investigations [27, 28].

1.3 Applications

Some important applications of nanofluidics are listedhereafter.Biology – First of all, most of the biological pro-cesses that involve fluids take place at the nanoscale[15, 29, 30, 31]. For example, the transport of waterthrough biological membranes in cells is ensured byaquaporins, a protein with subnanometric dimen-sions. Aquaporins appear to have an extremely highwater permeability, while ensuring an excellent saltrejection 1. Another example of proteins with nano-metric dimensions are ion pumps and ion channels,

1Note that the shape of aquaporins is discussed in reference[32].

that ensure the flow of ions across cell membranes[33, 34]. Combined, those proteins allow the (human)kidney, which is an example of natural desalinationand separation tool, to purify water with an energycost far below current artificial desalination plants[8].

Desalination – At the same time, some of the mostused (man-made) desalination techniques, consistingin the separation of salt and water in order to producefresh water, are using nanofluidic properties [35, 36].This is the case of membrane-based techniques, suchas Reverse Osmosis (RO) [37], Forward Osmosis (FO)[38] or ElectroDialysis (ED) [39]. The improvementof the membrane technology has made it possible todesalinate with an energy consumption close to theminimum energy set by thermodynamics.

Extraction of mixing energy – Another inter-esting application of nanofluidics concerns the ex-traction of the energy of mixing from natural wa-ter resources. This so-called blue energy is the en-ergy available from the difference in salt concentra-tion between, for example, seawater and river water.Pressure-Retarded Osmosis (PRO) converts the hugepressure difference originating in the difference in saltconcentration (∼ bars) between reservoirs separatedby a semipermeable membrane into a mechanicalforce by the use of a semipermeable membrane withnanosized pores [40]. Siria et al. proposed anotherway to convert blue energy based on the generation ofan osmotic electric current using a membrane piercedwith charged nanotubes [22].

Nanofluidic circuitry – The recent emergence ofnanofluidic components benefiting of the surface ef-fects of nanofluidics leads naturally to an analogywith micro-electronics. Indeed, some nanofluidic com-ponents imitate the behaviour of over-used micro-electronic components such as the diode or the tran-sistor [12] 2. Even if a complete analogy between bothfields fails due to the physical differences betweenions and electrons, controlling/manipulating nano-flows the same way we control electric currents wouldallow for regulating, sensing, concentrating and sep-arating ions and molecules in electrolyte solutions[42] with many potential applications in medicine,such as drug delivery or lab-on-a-chip analyses.

An overview of the full complexity of nanofluidicsis highlighted in the following by the description ofsome theoretical bases. The first part provides thedefinitions of the characteristic lengths that separatethe different transport regimes and lead to a largevariety of behaviours. The second part describes

2Note that nanofluidic diodes are discussed in reference [41].

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the numerous forces that play a role in nanofluidicsand that are at the origin of the various phenomena.Finally, the third part focuses on transport responseof a membrane pierced with a slit nanochannel andsubmitted to external forcing.

2 Definitions

2.1 Characteristic lengths

The richness of nanofluidics comes from the existenceof a large number of characteristic lengths related tothe finite size of the fluid’s molecules, to electrostaticsor to the fluid dynamics. Indeed, when one or moredimensions of a nanofluidic system compares withthose characteristic lengths, new phenomena mayappear. An overview of length scales at play innanofluidics can be seen in figure 1. In what follows,a description of each length is given.

0.1 nm 1 nm 10 nm 100 nm 1000 nm

Molecular scale

Debye length

Duhkin length

Slip length (simple surface)

Slip length (micro-nano- structured surfaces)

Continuum description

Figure 1: Overview of length scales at play innanofluidics, freely inspired from refer-ence [1].

The molecular length scale is associated with thefinite size of the fluid’s molecules and its components(molecules, ions...). More precisely, it is linked totheir diameter σ, typically in the angstroms scale(1 A = 1 ·10−10 m). For example, σ ∼ 3 A for the wa-ter molecule, σ ∼ 4− 5 A for common ionic species(Na, K, Cl) [43]. This length defines a priori the ul-timate limit of the study of nanofluidic transport [1].In the vicinity of a confining wall, fluids can experi-ence some structuring and ordering at the molecularlength scale. An example of water molecules neara solid surface is shown in figure 2. This effect isexacerbated in confining pores, when there is onlyroom for a limited number of molecules. In that case,strong deviations from continuum predictions can beexpected. Another important effect related to thesize of molecules, and thus to the molecular length isosmosis; which is the phenomenon by which a solventmoves across a semipermeable membrane (permeableto the solvent, but not to the solute) separating two

solutions of different concentrations. Note that thequestion of the robustness of hydrodynamics for con-finement below one nanometer and the phenomenonof osmosis are both discussed in my thesis [44].

Figure 2: Water molecules (oxygen in red, hydrogenin white), next to a graphene sheet (ingray).

The Bjerrum length is defined considering twocharged species in a solution. It corresponds to thedistance at which the thermal energy kBT , with kBthe Boltzmann constant and T the absolute tempera-ture, is equal to the energy of electrostatic interaction.The Bjerrum length `B can be written as

`B =Z2e2

4πεkBT, (1)

with e the elementary charge, Z the valency, ε thedielectric permittivity of the medium. For two mono-valent species in water at ambient temperature, `Bis approximately equal to 0.7 nm. Depending onthe considered solution (monovalent ionic species,organic solvent with low dielectric constant, etc.), `Bcan be either large enough to be clearly dissociatedfrom the molecular length, or be of the same orderas the molecular length. The physics has to be dif-ferentiated in each case. There are physical effectswith important implications on nanofluidic transportthat are linked to the Bjerrum length. For exam-ple, for confinements below `B, one expects a largefree-energy cost to undress an ion from its hydrationlayer and make it enter the pore, with consequenceson filtering processes of charged species.The Gouy-Chapman length is constructed in thespirit of the Bjerrum length. It is defined as thedistance from a charged wall where the electrostaticinteraction of a single ion with the wall becomes ofthe order of the thermal energy. For a surface chargeΣ, it can be written as

`GC =e

2πΣ`B. (2)

For monovalent species in water and a typical surfacecharge Σ ∼ 50 mC/m2, `GC is approximately equalto 0.7 nm .

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The Debye length is the characteristic length ofthe layer that builds up near a charged surface inan ionic solution. This layer counter balances theinfluence of the electric charge, and is of main im-portance in the study of transport at the nanoscale,as will be discussed later. The Debye length can bewritten as

λD =1√

8πlBc0, (3)

with c0 the concentration in ionic species. Indeed,when a solid surface is immersed in an aqueous so-lution, it usually acquires a surface charge Σ dueto chemical reactions (dissociation of surface groupsand specific adsorption of ions in solution to thesurface [45, 46]). In response to this surface charge,the ionic species in the liquid rearrange themselvesand form a layer that screens the influence of thesurface charge. This layer of ionic species is calledthe Electrical Double Layer (EDL). Note that theDebye length is independent of the surface chargeΣ, and inversely proportional to the square root ofthe salt concentration c0. Typically λD is equal to30 nm for c0 = 10−4 M, 3 nm for c0 = 10−2 M and0.3 nm for c0 = 1 M.The Dukhin length is based on the comparisonbetween the bulk to the surface electric conductance,which links the electric current to an applied electricfield. It characterizes the channel scale below whichsurface conductance dominates over bulk conduc-tance [1]. In a channel of width h and surface chargedensity Σ, the excess in counterion concentration isce = 2Σ/he with e the elementary charge and wherethe factor 2 accounts for the two surfaces. One maydefine a Dukhin number Du = |Σ| /hc0e. A Dukhinlength `Du can then be defined as

`Du =|Σ|c0e

. (4)

For a surface with a surface charge density Σ =50 mC/m2, `Du is typically 0.5 nm for c0 = 1 M, while`Du = 5 µm for c0 = 10−4 M.The slip length is defined as the depth inside thesolid where the linear extrapolation of the velocityprofile vanishes. Unlike previous lengths, that areall related to electrostatics, the slip length comesfrom the dynamic of the fluid near a solid surface. Itcharacterizes the hydrodynamic boundary conditionof fluids at interfaces. Its expression can be derivedas follows: first, assume that the tangential force perunit area exerted by the liquid on the solid surfaceis proportional to the fluid velocity at the wall vw:σxz = λvw, with λ the friction coefficient, z thenormal to the surface, x the direction of the flow.

Then, combining this equation with the constitutiveequation for a bulk Newtonian fluid, σxz = η∂zvx,one obtains the Navier boundary condition [14]:

vw =η

λ∂zvx

∣∣∣w

= b∂zvx

∣∣∣w, (5)

where the slip length b = η/λ is defined as the ratiobetween the bulk liquid viscosity and the interfacialfriction coefficient. Accordingly, several kinds ofhydrodynamic boundary conditions can apply:

• the no-slip boundary condition supposes thatthe fluid has zero velocity relative to the bound-ary, vw = 0 at the wall, and corresponds to avanishing slip length b = 0;

• the perfect-slip boundary condition correspondsto the limit of an infinite slip length (b → ∞),or equivalently a vanishing friction coefficient(λ→ 0). It corresponds to a shear free boundarycondition. Traditionally, the perfect-slip bound-ary condition is used when the slip length b ismuch larger than the characteristic length(s) ofthe system;

• the partial-slip boundary condition concerns in-termediate slip length.

For simple liquids on smooth surfaces, slip lengths upto a few tens of nanometers have been experimentallymeasured [14].

2.2 Mathematical description of the EDL

The Electrical Double Layer (EDL) plays a funda-mental role in nanofluidics due to a large surfacearea to volume ratio. It corresponds to the layer ofionic species that counter-balances the influence of asurface charge. Numerous phenomena, that will bediscussed later, take their origin within the EDL, soa mathematical description of this layer is of mainimportance here. The conventional description isgiven hereafter.

The Gouy-Chapman theory is at the basis ofEDL’s description. It is based on the following hy-pothesis [47]:

• ions are considered as (punctual) spots,

• the dielectric permittivity of the medium is sup-posed constant in the medium,

• the charge density and the electrical potentialare seen as continuum variables,

4

• the correlations between ions as well as the ion-solvent interactions are not taken into account(mean field theory),

• only electrostatic interactions are considered.

Poisson-Boltzmann equation – Under the previ-ous hypotheses, let us write the equation that un-derlies the distribution of ions near a flat surface.Consider monovalent ions near a flat surface S, lo-cated at z = 0, with homogeneous surface chargedensity Σ and surface potential Vs, as shown in fig-ure 3. The link between the electrical potential V (z)

0 ĸz

Vs

-

---

-

--

--

-

++

-

-

-

-

--

-

-

+

+

+

+

+

++

-

-

-- +

1 2 3 4 5 6 0 1 2 3 4 5 6 7 8κ z

0

0.2

0.4

0.6

0.8

1

V /

Vs

c /

c0

2

1.5

1

0.5

0

Figure 3: Left: scheme of the studied configura-tion. Right: Debye-Hckel solution (11)for the electrical potential V in blue andlinearized Boltzmann equation (12) forthe concentration profile c± in red.

and the charge density ρe(z) at a z distance of thesurface inside the ionic solution is given by a Poissonequation:

∆V =d2V

dz2= −ρe

ε, (6)

where ε = ε0εr is the solvent permittivity (for waterat ambient temperature, εr ≈ 80). The idea is toignore the thermal fluctuations of V and ρe and toconsider their respective averaged values only. Atthe thermal equilibrium, the densities of positivec+(z) and negative c−(z) ions are governed by theBoltzmann equation:

c±(z) = c0e∓βeV (z), (7)

with β = 1/kBT and c0 the concentration in ion ofcharge ± e far from the wall. The charge densityreads ρe = e(c+ − c−) = −2ec0 sinh(βeV ). Couplingthis equation with (6), we get the Poisson-Boltzmannequation for the electrical potential V (z):

d2V

dz2=

2ec0

εsinh(βeV ). (8)

Introducing the previously described Bjerrum length`B, defined as the length at which the thermal energy

balances the electrostatic one, we can rewrite theequation 8:

βed2V

dz2= 8π`Bc0 sinh(βeV ) = κ2 sinh(βeV ), (9)

where κ = (8π`Bc0)1/2 corresponds to the inverseof the previously described Debye length λD. Thisequation describes the evolution of the electricalpotential next to a charged surface.Linearized Poisson-Boltzmann equation – Inthe general case, the Poisson-Boltzmann equationcan not be solved analytically. For small potentials(eV � kBT ), an approximate form of the Poisson-Boltzmann, the Debye-Hckel equation, can be writtenas

d2V

dz2= κ2V. (10)

Assuming that the electrical potential vanishes farfrom the surface, the solution of the Debye-Hckelequation reads

V (z) = Vse−κz, (11)

where Vs is the surface electrical potential. Equation(11) is plotted in figure 3. The electrical potentialis screened over a distance κ−1 = λD, the Debyelength, which then gives the width of the EDL. Thelinearization of the equation (7) gives

c± = ρs exp(∓βeV (z)) ≈ c0(1∓ βeV (z))

= c0(1∓ βeVs exp(−κz)). (12)

Equation (12) is plotted in figure 3 for both ± species.Non-linear Poisson-Boltzmann equation – Insome situations, a solution for the non-linear Poisson-Boltzmann equation exists. Let us consider here thecase of a single flat wall, the electrolyte is located inz > 0 and the solid wall in z < 0. A surface chargedensity Σ < 0 is located at z = 0. The electric fieldis taken to be equal to 0 inside the wall as well as farfrom the wall inside the electrolyte. At the wall, theelectrostatic boundary condition links the electricfield and the surface charge:

∂V

∂z

∣∣∣∣z=0

= −4π

εΣ. (13)

Solving the PB equation (8) with this boundarycondition (13) leads to [48]

V (z) = − 2

eβln

(1 + γe−z/λD

1− γe−z/λD

), (14)

where γ is the positive root γ0 of the equation:

γ2 +2`GCλD

γ − 1 = 0. (15)

5

For a positive surface charge Σ, the solution forV is identical, though with γ = −γ0(< 0). Thesurface potential Vs can be written as Vs =4 arctan(−z/λD)/βe.

2.3 Nanoscale forces

Now that the general ideas of the Gouy-Chapman de-scription of the EDL have been given, let us describesome of the most important forces that play a rolein nanofluidics. These forces are at the origin of thelarge range of phenomena observed in nanofluidics,and they give rise to both equilibrium or kinetic phe-nomena [2]. Note that the distinction we will makebetween forces is artificial since they all are electricalin nature [49], but it still makes sense because ofthe many different ways in which the electrical forcepresents itself.

As a side note, each system depends fundamentallyon individual forces that are applied between indi-vidual atoms. However, in practice, large systems(∼ 10 nm) can usually be described with continuumtheory, which statistically averages the single inter-actions. This is why we may speak of forces exertedby walls on particles or molecules, or between walls,or between particle or molecules.Electrostatic forces are long range interactionsacting between charged atoms or ions [49]. Two par-ticles of respective charges Q1 and Q2 at a distancer act on each other as follows:

F (r) =Q1Q2

4πε0εrr2, (16)

where εr is the dielectric permittivity of the medium.F (r) is directed along the axis defined by the positionof the two particles. Equation (16) is known asthe Coulomb law. Electrostatic forces can be eitherattractive or repulsive, depending on the sign of theproduct Q1Q2. They are, for example, at the originof the building of the Electrical Double Layer (EDL).Van der Waals forces are residual forces of electro-static origin which are always present, even betweenneutral atoms. They are relatively weak in compari-son to chemical bonding for example (see below), butthey nevertheless play a role in a large range of phe-nomena such as adhesion, surface tension or wetting.Van der Waals forces even manifest themselves atmacroscopic scales since they are at the origin of theadhesion of gecko, a decimetric reptilia, on solid sur-faces. Van der Waals forces include attractions andrepulsions between atoms, molecules and surfaces.They have three possible origins such as:

• the force between two permanent dipoles,

• the force between a permanent dipole and aninduced dipole,

• the force between two induced dipoles (Londondispersion force).

Van der Waals forces are long-range, can bringmolecules together or mutually align/orient them,and are not additive. They have to be described withthe quantum mechanical formalism, which is beyondthe scope of the present work.

The DLVO (Derjaguin, Verwey, Landau, Over-beek) theory gives a large picture of nanoscale forceswhich includes van der Waals forces and coulombicforces. However, some effects that appear at veryshort range can not be described in the frameworkof the DLVO theory. Non-DLVO forces are discussedin what follows.

Chemical or bonding forces link two or moreatoms together to form a molecule [49]. Bonds arecharacterized by the redistribution of electrons be-tween the two or more atoms. The number of cova-lent bonds that an atom can form with other atomsdepends on its position in the periodic table. Thisnumber is called the valency. For example, it is equalto one for hydrogen and two for oxygen, which leadsto water molecule H2O (H-O-H). Notice that covalentbonds are of short range (0.1− 0.2 nm) and directedat well-defined angles relative to each other. For ex-ample, they determine the way carbon atoms arrangethemselves to form diamond structure. Notice thatcovalent bonding comes from complex quantum in-teractions which are beyond the scope of the presentwork.

Repulsive steric forces appear when atoms arebrought too close together. It is associated with thecost in energy due to overlapping electron clouds(Pauli/Born repulsion). A consequence is the sizeexclusion, widely used in membranes from angstrmto micrometer [2]. It plays a role for example in aqua-porins (water channels), that offer a low resistancefor water molecules, but do not allow ions to passthrough. To pass through this channel, the ion needsto lose its water shell, which is energetically unfavor-able. Notice that, combining electrostatic forces andsteric forces, it is possible to develop K+ channelswith a high selectivity for K+ over Na+ while bothhave water shells [50].

Solvation forces (or structural forces) are relatedto the mutual force exerted by one plate on anotherwhen they are separated by a structured liquid 3.

3Liquid structuring in central in reference [51]

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Next to a solid surface, density oscillations are ex-pected. If two solid surfaces immersed in a fluid areseparated by a short distance, liquid molecules mustaccommodate the geometric constraint, leading tosolvation forces between the two surfaces, even in theabsence of attractive walls. Depending on both sur-face properties (well ordered, rough, fluid-like) andfluid properties (asymmetrically shaped molecules,with anisotropic or non pair-wise additive interactionpotential), the resulting solvation forces can be eithermonotonic or non-monotonic, repulsive or attractive.See reference [49] for more details. Notice that, in thecase of water molecules, solvation forces are calledhydration forces.

Hydrophobic forces come from interactions be-tween water and low water-soluble objects (molecules,clusters of molecules...). These substances usuallyhave long carbon chains that do not interact withwater molecules, resulting in a segregation and anapparent repulsion between water and nonpolar sub-stances. The hydrophobic effect, which results fromthe presence of hydrophobic forces, is actually anentropic effect: each water molecule can form fourhydrogen bonds in pure water, but can not form asmuch if surrounded by hydrophobic (apolar) species.Hence, apolar molecules (or clusters of molecules)will rearrange themselves in order to minimize thecontact surface with water. An example is the mix-ing of fat and water, where fat molecules tend toagglomerate and minimize the contact with water.

Non-conservative forces, such as friction or vis-cous forces, are referred as non-conservative forcesbecause they involve energy transfer from one bodyto another. Contrary to other forces, which act on abody and generate a motion according to the secondlaw of Newton, non-conservation forces have no forcelaw and arise as a reaction to motion. Inside a liquid,friction is linked to a fluid property: the viscosity,which is a property of a fluid to resist to a shear. Itcomes from collisions between neighbouring particlesthat are moving at different velocities. For example,when a fluid flows through a pipe, the particles gen-erally move quickly near the pipe’s axis and slowlynear its walls, leading to stress. The friction betweenwater molecules leads to a dissipation that has tobe overcome, for example by a pressure differencebetween the two ends of the pipe, to keep the fluidmoving.

Other forces, such as gravitational or inertial forces,are of lesser importance in nanofluidics and are notdiscussed here.

2.4 Some consequences

The previously described forces give rise to a largevariety of phenomena. As an illustration, some ofthem are presented here.

Cohesion is related to attractive forces betweenmolecules of the same substance. It is due to inter-molecular attractive forces. They can be van derWaals forces or hydrogen bonding. Cohesion is atthe origin, for example, of the tendency of liquids toresist separation.

Adhesion corresponds to attractive forces betweenunlike molecules. They are caused by forces acting be-tween two substances, which can have various origins,such as electrostatic forces (attraction due to oppo-site charges), bonding forces (sharing of electron),dispersive (van der Waals forces) etc. For example,water tends to spread on a clean glass, forming athin and uniform film over the surface. This is be-cause the adhesive forces between water and glassare strong enough to pull the water molecules out oftheir spherical formation and hold them against thesurface of the glass.

Surface tension is related to the elastic tendencyof liquids which makes them acquire the least surfacearea possible. This results from the fact that whenexposed to the surface, a molecule is in an energeti-cally unfavorable state. Indeed, the molecules at thesurface of the liquid lack about half of their cohesiveforces, compared to the inner molecules of the bulkliquid [52]. Hence a molecule at the surface has lostabout half its cohesion energy. Surface tension is ameasure of this lost of energy per surface unit. Inthe thermodynamic point of view, it is defined asthe excess of free energy due to the presence of aninterface between two bulk phases [53]. The surfacetension γ is of the order of magnitude of the bondenergy ε between molecules of the fluid divided bythe cross section area of a molecule σ2:

γ ∼ ε

σ2.

Finally one may notice that surface tension is alsopresent at liquid-liquid, liquid-solid and solid-air in-terfaces.

Wetting is the study of the spreading of a liquiddeposited on a solid (or liquid) substrate. Whena small amount of liquid is put in contact with aflat solid surface, there are two different equilibriumsituations: partial wetting, when the liquid shows afinite contact angle θ, and total wetting, in which theliquid spreads completely over the surface and whereθ is not defined. The property of the fluid to spread

7

on the surface is characterized by the spreading pa-rameter S which measures the difference between theenergy per unit area of the dry surface of the solidsubstrate and the wetted surface:

S = γSV − (γSL + γLV ) (17)

where γSV , γSL and γLV denote the free energies perunit area of respectively the solid-vapour interface,the solid-liquid interface and the liquid-vapour inter-face equal to the surface tension γ. In the case ofa positive S, the surface energy of the dry surfaceis larger than the energy of the wetted surface, sothe liquid tends to extend completely to decreasethe total surface energy, hence θ is equal to zero. Anegative S corresponds to a partial wetting situation,where the liquid does not completely spread on thesurface and forms a spherical cap, adopting an angleθ > 0. From the equilibrium of the capillary forcesat the contact line or from the work cost for movingthe contact line, one gets the Young-Dupr relation:

cos θ =γSV − γSL

γ. (18)

Capillary forces originate in the adhesion betweenthe liquid and the solid surface molecules. It isstrongly linked to the existence of a surface tension,as well as to the concept of wetting and contactangle. In certain situations, those forces pull theliquid in order to force it to spread the solid surface.Depending on the configuration, it can make theliquid fill a solid channel for example.

2.5 Transport in nanochannels

In this section, we consider various transport phe-nomena that can occur in a nanochannel separatingtwo reservoirs containing an electrolyte. The purposeis to give a simple expression of each flux as a func-tion of various driving forces (mechanical pressure,solute concentration and electrical potential gradi-ents). For the sake of simplicity, the nanochannelis chosen to be a slit (∼ 2D) and entrance effectsare not taken into account 4. The walls are perpen-dicular to z, respectively located in z = ±h/2 anddriving forces are applied along x, see figure 4. Thechannel has a length L along x and a width w alongy. The Reynolds number Re = ρvL/µ (where ρ andµ are respectively the fluid density and the dynamicviscosity and v and L are respectively the character-istic velocity and length of the flow) is assumed to

4Note that hydrodynamic entrance effects are discussed inreferences [32, 54].

be lower than one and the problem to be stationary.Hence, the governing equation for the flow is theStokes equation:

η4 ~u = ~∇p+ ~F (19)

where η is the fluid viscosity, ~u is the velocity field, pis the hydrodynamic pressure and ~F a volume force.The surface charge density is Σ, and will be differentfrom 0 if specified only. Unless otherwise stated, theheight of the channel h will be considered large incomparison to the typical range of the potential (i.e.the Debye length). Unless otherwise stated, the no-slip boundary condition will be used for the solventalong walls. The system is shown on figure 4.

Σ

h

L

p-Δp/2V-ΔV/2c-Δc/2

p+Δp/2V+ΔV/2c+Δc/2

Q, I, J

x

z

Figure 4: Sheme of the 2D channel used for thecalculations.

A flow through the membrane can occur as a con-sequence of a force near the membrane [55]. Here weconsider this force to be a mechanic pressure drop∆p, difference in solute concentration ∆c or differ-ence in electrical potential ∆V . We suppose thatthe considered forcing are weak, so equilibrium pro-files remain unmodified along z, and flows are linearfunctions of the forces operating. Hereafter we willstudy the volume flow Q, the ionic flow Ji and theelectrical current Ie resulting from ∆p, ∆c and ∆V .The phenomenological equations linking the threeflows to the three forces write:

Q = L11∆P + L12∆c+ L13∆V,

Ji = L21∆P + L22∆c+ L23∆V, (20)

Ie = L31∆P + L32∆c+ L33∆V,

where LIJ are coefficients. According to Onsager’slaw, the matrix of coefficients LIJ is symmetrical,i.e. LIJ = LJI . Finally, one assumes that in themiddle of the channel (z=0), concentration, electricalpotential and pressure evolve linearly with x.

Direct terms

The direct terms of the matrix of transport (20)correspond to the diagonal terms LII . They linkeach flux with their natural force, respectively the

8

solvent flow with the pressure gradient, the ionicflow with the salt gradient and the ionic current withthe electrical gradient. Each of them is calculatedhereafter in the previously described configuration(slit nanochannel).L11 – the hydrodynamic permeability charac-terizes the flow transport across a given structureunder a pressure gradient. Using both symmetry andimpermeability of the walls, one gets for the velocityfield: ~u = ux(z)~ux. So the Stokes equation 19 canbe written as η∂2

zux = ∂xp. A first integration of theStokes equation between 0 and z gives η∂zux = z∂xp,where we used that ∂zux|z=0 = 0 by symmetry, andthat ∂xp does not depends on z. Another integrationbetween −h/2 and z using that ∂xp = ∆p/L gives

u(z) = uw +1

η

(z2

2− h2

8

)∆p

L, (21)

where uw is the wall velocity, which depends on thehydrodynamic boundary condition (see the definitionof the slip length, subsection 2.1). Finally, L11 incase of the no-slip boundary condition (uw = 0) canbe written as

L11 =Q

∆p= − 1

12η× wh3

L. (22)

Hence the hydrodynamic permeability of a membranein the low Reynolds number regime is limited by theviscosity of the fluid, and depends strongly on thedimensions of the channel.L22 – the ionic permeability characterizes theionic flow through a membrane under a salt concen-tration gradient. From the Fick’s law of diffusion:

~j± = −D±~∇c±, (23)

where D± are diffusion coefficients of the ± speciesrespectively, one can write the total flow Ji assumingthat D+ = D− = D:

Ji =

∫S

(~j++~j−). ~dS = −wD∫ h/2

−h/2∂x(c+(x)+c−(x))dz.

(24)In a neutral channel, and assuming that c±(x) =x∆c/L+ c0, it gives

L22 =Ji∆c

= −D × S

L, (25)

where S = hw is the surface of the channel. Noticethat equation (25) describes ionic flow through amembrane under salt gradient in absence of surfacecharge. In case of the presence of a surface charge

density Σ, the nanochannel exhibits a selective perme-ability for ion diffusive transport [1]. Consequently,the concentration of counterions inside the channelis higher that the bulk concentration, while the con-centration in co-ions is lower. Therefore, ions of thesame charge as the nanochannel exhibit a lower per-meability, while ions of the opposite charge have ahigher permeability through the nanochannel. Fol-

1e-05 0.0001 0.001 0.01 0.1 1

c0 (mol/L)

0

1

2

3

4

5

6

β

Figure 5: Equation (26) for β+ (continuous) andβ− (dashed) as a function of the bulkconcentration for a negatively chargedsurface.

lowing Plesis et al., an effective nanochannel sectioncan be defined for each species S±eff = β±S where βis an exclusion/enrichment coefficient [56]:

β± =1

h

∫ h/2

−h/2e∓φ(z)dz; (26)

where φ(z) = βeV (z) with V (z) the electrical poten-tial. One can use the linearised Poisson-Boltzmannequation to calculate the ion concentration profile inthe slit. An example is shown in figure 5.

L33 – the ionic conductance characterizes theionic current through a membrane under an appliedelectrical potential difference: G = Ie/∆V . First, letus define the (bulk) conductivity of the solution κb:

κb = e(µ+c+ + µ−c−) (27)

with µ± and c± respectively the mobility and thevolume density of ± ions [45]. At high ionic strength,or for a neutral channel (Σ = 0), equation (27) canbe used directly to calculate the conductance of thechannel Gbulk = κbωh/L. However, for a non-neutralchannel (Σ 6= 0), if one looks at the ionic conductanceversus the salt concentration on a log-log scale, aconductance plateau is observed at low ionic strength.

9

This is due to the contribution to the total currentof ions of the EDL. This excess counterions concen-tration can be written as [57]:

ce =2Σ

he(28)

where the 2 accounts for the two surfaces. From thisexcess counterions concentration, one can define asurface conductance Gsurf = eµceωh/L. Then thetotal conductance is the sum of a bulk conductanceand a surface conductance:

G = Gbulk +Gsurf = µcsewh

L+ 2Σµ

w

L(29)

where we assumed that µ+ = µ− and defined cs = 2c0

with c0 the salt concentration. Finally, the ioniccurrent Ie and the voltage drop ∆V are linked asfollows:

L33 =Ie

∆V= µ (cseh+ 2Σ)× w

L. (30)

Cross terms

Additionally to the direct terms, there are crossphenomena coming from couplings between hydro-dynamics, ion diffusion and electrostatics. Usingstatistical mechanics, Onsager has shown the neces-sity of equality between the term LIJ and LJI . So inwhat follows, only three terms among the six crosscoefficients are explicitly calculated, the last threebeing deduced from Onsager’s relation.L13 / L31 – The phenomenon by which a differenceof electrical potential ∆V induces a water flow iscalled electro-osmosis (L13). Its conjugate effect iscalled streaming current (L31) and corresponds tothe generation of an electric current by a pressuredriven liquid-flow [58]. Hereafter, we will do explicitcalculations for the case of electro-osmosis (L13).

Electro-osmosis takes its origin in the ion dynamicswithin the Electrostatic Double Layer (EDL), inwhich the charge density ρe = e(ρ+ − ρ−) is non-vanishing. The dynamics of the fluid is described bythe stationary Stokes equation with a driving forcefor the fluid Fe = ρeEe, where the electric tangentialfield Ee is defined as Ee = −∂xV = −∆V/L, and isdirected along x [59]:

η∂2zux + ρeEe = 0. (31)

Using that the charge density is linked to the elec-trostatic potential of the EDL as follows:

ρe = −ε∂2V

∂z2, (32)

one finds, after a double integration of the equation(31):

ux(z) =ε

η(V (z)− ζ)Ee (33)

where we used the no slip boundary condition andwhere ζ is the zeta potential, which is the value ofthe electrostatic potential at the shear plane, i.e. theposition close to the wall where the velocity vanishes5. In the no-slip case, the zeta potential is equalto the surface potential Vs. As a remark, one cannotice that in the case of a finite slip at the wall,characterized by a slip length b, the potential ζ takesthe expression:

ζ = Vs × (1 + bκeff) (34)

where Vs is the electrostatic potential at the wall andκeff the surface screening parameter (κeff = −V ′(z =0)/Vs). In the case of a weak potential, the screeningparameter is approximately equal to the inverse ofthe Debye length λD. Note that the velocity inthe fluid results from a balance between the drivingelectric force and the viscous friction force at thesurface.

An integration of equation (33) gives the followingexpression for the total water flow:

Q = whUEO − surface correction terms, (35)

where UEO is the eletro-osmotic velocity UEO =−εζEe/η. Figure 6 shows a scheme of the velocity

λD

UEO

Figure 6: Schematic representation of the velocityprofile, equation (33) without and withsurface correction terms, respectively onthe left and on the right.

profile, with and without the surface correction terms.Finally, neglecting the surface correction terms (that

5Notice that sometimes the zeta potential is defined as Vs andone has to consider an amplified electro-osmotic mobilityto take into account the effect of slippage.

10

are of the order of λD/h� 1), one can write:

L13 =Q

∆V≈ ζε

η× wh

L. (36)

Hence electro-osmosis is caused by coulomb forceand limited by viscous dissipation.

Accordingly, the streaming current (L31), which isthe electric current generated by a pressure drivenliquid-flow can be written as

L31 =Ie

∆P=ζε

η× wh

L. (37)

L12 / L21 – The generation of a flow under a solutegradient is called chemi-osmosis (L12) [60]. Its conju-gate effect is the generation of an excess flux of saltunder a pressure drop ∆p (L21). Here the expressionof the L12 coefficient is obtained in the case of a flowgenerated by a solute gradient.

So, let us assume the existence of a salt concen-tration difference ∆c. The salt concentration in themiddle of the channel, cmid(x), is assumed to varylinearly along the axis x: cmid(x) = c0 + ∆c× x/L,where c0 is the concentration in the left reservoir (theconcentration in the right reservoir being c0 + ∆c).From the mechanical equilibrium in z together withthe Stokes equation along z, one can deduce thehydrostatic pressure profile:

p(x, z) = 2kBTcmid(x) [coshφ(x, z)− 1] + p0, (38)

where φ(x, z) = eβV (x, z). Injecting this expres-sion in the Stokes equation along x, η∂2

zux(z) −∂xp(x, z) = 0, one finds:

η∂2zux(z) = 2kBT

∆c

L(coshφ− 1) . (39)

Using Poisson-Boltzmann and assuming that λD �h, one can find that the flow is:

Q = whUCO − surface correction terms, (40)

with UCO the chemi-osmotic velocity:

UCO = −kBTη

ln(1− γ2)

2π`B

∆c

c0L, (41)

where γ = tanh(φs/4). Neglecting surface correctionterms (on the order of λD/h � 1), the coefficientL12 can be written:

L12 =Q

∆c≈ −kBT

ηc0

ln(1− γ2)

2π`B× wh

L. (42)

Chemi-osmosis causes flow towards lower electrolyteconcentration. As a complement we will discuss twointeresting cases: the case of non-equal diffusioncoefficient between + and − species, and the limitof large Debye length λD compared to the channelheight h (this regime is called osmosis).

Supplement 1 – In the case of a difference in anionand cation diffusivities, an electric field is induced,and a supplementary electro-osmotic contributionhas to be taken into account [61]. Assuming a van-ishing local current in the outer region and a sym-metric electrolyte, this diffusion-induced electric fieldis proportional to β0 = (D+ −D−)/(D+ +D−) :

ED =kBT

eβ0

d ln c

dx. (43)

So the contribution of this mechanism combined withthe previously calculated velocity (equation (41))gives the following diffusio-osmotic velocity:

UDO = −kBTη

[β0ζ

ε

e+

ln(1− γ2)

2π`B

]∆c

c0L, (44)

where we used equation (33). The first term corre-sponds to the electro-osmotic effect, the directionof the generated flow depending on the sign of theproduct β0ζ, while the second term, called the chemi-osmotic effect, causes a flow towards the lower elec-trolyte concentration. Neglecting surface correctionterms, one can write:

L12 =Q

∆c≈ −kBT

ηc0

[β0ζ

ε

e+

ln(1− γ2)

2π`B

]× wh

L.

(45)

Supplement 2 – An interesting case is the limitwhere the Debye length λD is much larger than thechannel height h. In this particular case, a constantpotential (independent of z) called Donnan potentialVD builds up in the entire channel. From the electro-chemical equilibrium one gets:

c+

c−= e−2φD , (46)

c+c− = c20, (47)

c+ − c− = −2Σ

eh, (48)

where φD = eβVD. One can introduce the Dukhinnumber Du = Σ/(ec0h). Then from equation (48),and using that cosh2(x)− sinh2(x) = 1, one gets

cosh(φD) =√

1 + Du2. (49)

11

Injecting this expression in equation 39, which hasbeen obtained from mechanical equilibrium in z to-gether with the Stokes equation, one gets

Q = − 1

12η

wh3

L×∆Π, (50)

where the osmotic pressure ∆Π can be written as

∆Π = 2kBT∆c(

1 +√

1 + Du2). (51)

Hence, when λD � h, one may write

L12 =Q

∆c= −kBT

6η

(1 +

√1 + Du2

)× wh3

L.

(52)L23 / L32 – The current generated under a salt

concentration gradient is called osmotic current andthe reciprocal effect is the generation of a salt fluxunder an electrical potential gradient. The expres-sion of L23 is here obtained in the first case, i.e. inthe case of the generation of current under a saltconcentration gradient.

The electrical current Ie can be written as

Ie = w

∫ h/2

−h/2e (j+(z)− j−(z)) dz. (53)

Two contributions to the current can be expected,a contribution from the diffusive flux of salt and acontribution from the convective flux of salt. Thefirst one can be written as

ID = w

∫ h/2

−h/2e (jD,+(z)− jD,−(z)) dz, (54)

with jD,± = −D±∇c± the diffusive flux of each ion.Assuming that D = D+ = D− and rewriting thecurrent as

ID = −ew∂x∫ h/2

−h/2(c+(z)− c−(z)) dz, (55)

ID appears to be equal to zero from the global chargeelectroneutrality

Σ + e

∫ h/2

−h/2(c+(z)− c−(z)) dz = 0. (56)

Accordingly, considering the convective part in equa-tion (53) only, the current can be written as

Ie = w

∫ h/2

−h/2e (c+(z)− c−(z))ux(z)dz, (57)

where both species ± move at the same velocity ux(z)(i.e. there is no electric field along z). Using that

λD � h (thin electric debye layers as compared tothe channel width), one writes:

Ie = 2w

∫ ∞0

e (c+(z)− c−(z))ux(z)dz. (58)

In this assumption, one expects that the entire con-tribution to the current Ie comes from the convectionof ions inside the electric double layers. From thePoisson equation (6), one gets that

c+(z)− c−(z) = −∂2φ

∂z2

1

4π`B. (59)

Moreover, we know from previous section (see equa-tion (39)) that the velocity field ux(z) under a solutegradient is solution of

η∂2ux∂z2

= 2kBT∆c

L(coshφ− 1) . (60)

Injecting equation (59) in equation (58), performingan integration by part (twice) in the spirit of [62],one gets

Ie =we

2π`B

[φ∂ux∂z

]∞0

− we

2π`B

∫ ∞0

φ∂2ux∂z2

dz. (61)

From equation (60), one get

∂ux∂z

∣∣∣∣z=0

= −2kBT

η

∆c

L

∫ ∞0

(coshφ− 1) dz. (62)

Hence, using that φ(z =∞) = 0, one gets

Ie = − we

π`B

kBT∆c

ηL

∫ ∞0

(φs − φ)× (coshφ− 1)dz.

(63)with φs the normalized surface potential. Using PBequation ∇2φ = κ2 sinhφ, we make the followingchange of variable:

dz = − dφ

κ√

2(coshφ− 1), (64)

which allows to solve the integral in equation (63).One finds [62]:

Ie = 2we

πη`B

kBT

κ

(2 sinh

φs2− φs

)∆c

L, (65)

that can be rewriten in terms of surface charge (using2 sinhφs/2 = eΣ/εkBTκ):

Ie = 2wΣkBT

2πη`B

(1− κ`GC argsinh

1

κ`GC

)∆c

Lc0,

(66)

12

where we have introduce the Gouy-Chapmann length`GC = e/(2πΣ`B). Neglecting the surface correctionterms one finds:

L23 =Ie∆c≈ 2wΣ

kBT

2πη`B

1

Lc0(67)

Reciprocally, the flux of salt under electrical gra-dient can be written as

L32 =Ji

∆V≈ 2wΣ

kBT

2πη`B

1

Lc0(68)

Various comments

We did not give an exhaustive list of phenomena thatcan occur at the nanoscale. For example thermiceffects induced by temperature difference have notbeen discussed.

Notice that in practice, it is not easy to studyeach case alone. For example, a channel with adifference of electrical potential, will be subjectedto various flux, leading to a charge accumulationat each entrances, i.e. the apparition of an inducedsalt difference. This is part of the complexity ofnanofluidics.

ACKNOWLEDGMENTS: S.G. thanks LydericBocquet, Christophe Ybert, Laurent Joly, BenjaminRotenberg, John Palmeri and Menka Stojanova forusefull comments. This research was supportedby the European Research Council programs Mi-cromegas project.

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