Nanofluidics: a Pedagogical Introduction Simon Gravelle

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Nanofluidics: a pedagogical introduction Simon Gravelle

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

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Nanoﬂuidics: a pedagogical introduction

Simon Gravelle

01 MARCH 2016

1 Generalities reasonable expectation. Moreover, one can notice that most of the biological processes involving fluids 1.1 What is nanofluidics? operate at the nano-scale, which is certainly not by chance [8]. For example, the protein that regulates Nanofluidics is the study of fluids confined in struc- water flow in human body, called aquaporin, has got tures of nanometric dimensions (typically 1−100 nm) sub-nanometric dimensions [15, 16]. Aquaporins are [1, 2]. Fluids confined in these structures exhibit be- known to combine high water permeability and good haviours that are not observed in larger structures, salt rejection, participating for example to the high due to a high surface to bulk ratio. Strictly speak- efficiency of human kidney. Biological processes in- ing, nanofluidics is not a new research field and has volving fluid and taking place at the nanoscale attest been implicit in many disciplines [3, 4, 5, 6, 7], but of the potential applications of nanofluidics, and con- has received a name of its own only recently. This stitute a source of inspiration for future technological evolution results from recent technological progress developments. which made it possible to control what occurs at Hereafter, an overview of the current state of these scales. Moreover, advances have been made nanofluidics is presented. First, a brief state-of-the- in observation/measurement techniques, allowing for art, mainly focused on nano-fabrication and mea- measurement of the small physical quantities inher- surement techniques is given. Then some current ent to nano-sized systems. applications linked to nanofluidics are described. Even though nanofluidics is born in the footstep of microfluidics, it would be incorrect to consider it an 1.2 State-of-the-art extension of microfluidics. Indeed, while in microfluidics the only scale which matters is the size of the Nanofluidics has emerged from the recent progresses system, nanofluidics has to deal with a large spec- of nanoscience and nanotechnology, such as pro- trum of characteristic lengths which induce coupled gresses made in developing nano-fabrication tech- phenomena and give rise to complex fluid behaviours nologies. Fabricating well-controlled channels is a [8]. Moreover, since nanofluidics is at the intersection major challenge for nanofluidics, and is a necessary between physics, chemistry and biology, it concerns condition for a systematic exploration of nanofluidic a wide range of domains such as physiology, mem- phenomena. This requires a good control of device brane science, thermodynamics or colloidal science. dimensions and surface properties (charge, roughness, Consequently, a multidisciplinary approach is often etc). For example, the improvement of lithography needed for nanofluidics’ research. techniques (electron, x-beam, ion-beam, soft...) al- Some striking phenomena taking place at the lows the fabrication of slit nanochannels [17]. Focus nanoscale have been highlighted during the past few Ion Beam (FIB) allows to drill nanopores in solid years. For example, super-fast flow in carbon nan- membranes [18, 19]. There are also coatings and de- otubes [9, 10, 11], nonlinear eletrokinetic transport position/etching techniques that can be used to tune [12, 13] or slippage over smooth surfaces [14] have the surface properties [20, 21]. Siria et al. were able been measured. Those effects are indicators of the to manipulate a single boron nitride (BN) nanotube richness of nanofluidics. Accordingly, this field cre- in order to insert it in a membrane separating elec- ates great hopes, and the discovery of a large variety trolyte reservoirs and perform electric measurements of new interesting effects in the next decades is a [22]. Great developments of Scanning Tunneling Mi-

1 croscope (STM) or Atomic Force Microscopy (AFM) that ensure the flow of ions across cell membranes allow to characterize the fabricated devices. [33, 34]. Combined, those proteins allow the (human) In parallel, the efforts invested in nanofabrication kidney, which is an example of natural desalination have been combined to an improvement of measure- and separation tool, to purify water with an energy ment techniques. Most of them are based on the cost far below current artificial desalination plants measurement of electric currents, and have been de- [8]. veloped since the early days of physiology. But for Desalination – At the same time, some of the most a full understanding of nanofluidic properties, other used (man-made) desalination techniques, consisting quantities have to be made accessible. For exam- in the separation of salt and water in order to produce ple, local values of a velocity field can be obtained fresh water, are using nanofluidic properties [35, 36]. using nano-Particle Image Velocimetry (nano-PIV). This is the case of membrane-based techniques, such Surface Force Apparatus (SFA) have been used to as Reverse Osmosis (RO) [37], Forward Osmosis (FO) explore the hydrodynamic boundary condition and [38] or ElectroDialysis (ED) [39]. The improvement measure forces that play an important role in nanoflu- of the membrane technology has made it possible to idics, such as van der Waals or electric forces. desalinate with an energy consumption close to the In addition, a current challenge concerns water minimum energy set by thermodynamics. flow measurements. The main difficulty is due to Extraction of mixing energy – Another inter- the magnitude of typical flows through nanochannels: esting application of nanofluidics concerns the ex- ∼ 10−18 m3/s (it would take several years to grow traction of the energy of mixing from natural wa- a drop of 1 nl with such a flow). In order to detect ter resources. This so-called blue energy is the en- a water flow through a nanochannel, some poten- ergy available from the difference in salt concentra- tial candidates have emerged during the last decade. tion between, for example, seawater and river water. One can cite, for example, Fluorescence Recovery Af- Pressure-Retarded Osmosis (PRO) converts the huge ter Photobleaching (FRAP), confocal measurements pressure difference originating in the difference in salt or coulter counting measurements that have been concentration (∼ bars) between reservoirs separated reported to detect respectively 7 · 10−18 m3/s [23], by a semipermeable membrane into a mechanical 10−18 m3/s [24] and 10−18 m3/s [25]. However, the force by the use of a semipermeable membrane with inconvenient of most of the existing measurement nanosized pores [40]. Siria et al. proposed another techniques is that they are indirect and require the way to convert blue energy based on the generation of use of dyes or probes. an osmotic electric current using a membrane pierced Meanwhile, numerical progresses combined with with charged nanotubes [22]. calculation capacity improvement allow for the theo- Nanofluidic circuitry – The recent emergence of retical exploration of a large variety of nanofluidic nanofluidic components benefiting of the surface ef- properties. For example, the friction of water on solid fects of nanofluidics leads naturally to an analogy surfaces can be investigated using ab initio methods with micro-electronics. Indeed, some nanofluidic com- [26], while molecular dynamics simulations are good ponents imitate the behaviour of over-used micro- candidates for fluid transport investigations [27, 28]. electronic components such as the diode or the transistor [12] 2. Even if a complete analogy between both 1.3 Applications fields fails due to the physical differences between ions and electrons, controlling/manipulating nano- Some important applications of nanofluidics are listed flows the same way we control electric currents would hereafter. allow for regulating, sensing, concentrating and sep- Biology – First of all, most of the biological pro- arating ions and molecules in electrolyte solutions cesses that involve fluids take place at the nanoscale [42] with many potential applications in medicine, [15, 29, 30, 31]. For example, the transport of water such as drug delivery or lab-on-a-chip analyses. through biological membranes in cells is ensured by An overview of the full complexity of nanofluidics aquaporins, a protein with subnanometric dimen- is highlighted in the following by the description of sions. Aquaporins appear to have an extremely high some theoretical bases. The first part provides the water permeability, while ensuring an excellent salt definitions of the characteristic lengths that separate rejection 1. Another example of proteins with nano- the different transport regimes and lead to a large metric dimensions are ion pumps and ion channels, variety of behaviours. The second part describes 1Note that the shape of aquaporins is discussed in reference [32]. 2Note that nanofluidic diodes are discussed in reference [41].

2 the numerous forces that play a role in nanofluidics solutions of different concentrations. Note that the and that are at the origin of the various phenomena. question of the robustness of hydrodynamics for con- Finally, the third part focuses on transport response finement below one nanometer and the phenomenon of a membrane pierced with a slit nanochannel and of osmosis are both discussed in my thesis [44]. submitted to external forcing.

2 Deﬁnitions

2.1 Characteristic lengths The richness of nanofluidics comes from the existence of a large number of characteristic lengths related to the finite size of the fluid’s molecules, to electrostatics or to the fluid dynamics. Indeed, when one or more Figure 2: Water molecules (oxygen in red, hydrogen dimensions of a nanofluidic system compares with in white), next to a graphene sheet (in those characteristic lengths, new phenomena may gray). appear. An overview of length scales at play in nanofluidics can be seen in figure 1. In what follows, The Bjerrum length is defined considering two a description of each length is given. charged species in a solution. It corresponds to the distance at which the thermal energy kBT , with kB 0.1 nm 1 nm 10 nm 100 nm 1000 nm the Boltzmann constant and T the absolute temperature, is equal to the energy of electrostatic interaction. Molecular Continuum description scale The Bjerrum length `B can be written as 2 2 Slip length Slip length (micro-nano- Z e (simple surface) structured surfaces) `B = , (1) 4πkBT Debye length with e the elementary charge, Z the valency, the Duhkin length dielectric permittivity of the medium. For two monovalent species in water at ambient temperature, `B Figure 1: Overview of length scales at play in is approximately equal to 0.7 nm. Depending on nanofluidics, freely inspired from refer- the considered solution (monovalent ionic species, ence [1]. organic solvent with low dielectric constant, etc.), `B can be either large enough to be clearly dissociated The molecular length scale is associated with the from the molecular length, or be of the same order finite size of the fluid’s molecules and its components as the molecular length. The physics has to be dif- (molecules, ions...). More precisely, it is linked to ferentiated in each case. There are physical effects their diameter σ, typically in the angstroms scale with important implications on nanofluidic transport (1 A˚ = 1·10−10 m). For example, σ ∼ 3 A˚ for the wa- that are linked to the Bjerrum length. For exam- ter molecule, σ ∼ 4 − 5 A˚ for common ionic species ple, for confinements below `B, one expects a large (Na, K, Cl) [43]. This length defines a priori the ul- free-energy cost to undress an ion from its hydration timate limit of the study of nanofluidic transport [1]. layer and make it enter the pore, with consequences In the vicinity of a confining wall, fluids can experi- on filtering processes of charged species. ence some structuring and ordering at the molecular The Gouy-Chapman length is constructed in the length scale. An example of water molecules near spirit of the Bjerrum length. It is defined as the a solid surface is shown in figure 2. This effect is distance from a charged wall where the electrostatic exacerbated in confining pores, when there is only interaction of a single ion with the wall becomes of room for a limited number of molecules. In that case, the order of the thermal energy. For a surface charge strong deviations from continuum predictions can be Σ, it can be written as expected. Another important effect related to the e `GC = . (2) size of molecules, and thus to the molecular length is 2πΣ`B osmosis; which is the phenomenon by which a solvent For monovalent species in water and a typical surface 2 moves across a semipermeable membrane (permeable charge Σ ∼ 50 mC/m , `GC is approximately equal to the solvent, but not to the solute) separating two to 0.7 nm .

3 The Debye length is the characteristic length of Then, combining this equation with the constitutive the layer that builds up near a charged surface in equation for a bulk Newtonian fluid, σxz = η∂zvx, an ionic solution. This layer counter balances the one obtains the Navier boundary condition [14]: influence of the electric charge, and is of main im- η portance in the study of transport at the nanoscale, vw = ∂zvx = b∂zvx , (5) as will be discussed later. The Debye length can be λ w w written as 1 where the slip length b = η/λ is defined as the ratio λD = √ , (3) between the bulk liquid viscosity and the interfacial 8πl c B 0 friction coefficient. Accordingly, several kinds of with c0 the concentration in ionic species. Indeed, hydrodynamic boundary conditions can apply: when a solid surface is immersed in an aqueous solution, it usually acquires a surface charge Σ due • the no-slip boundary condition supposes that to chemical reactions (dissociation of surface groups the fluid has zero velocity relative to the bound- and specific adsorption of ions in solution to the ary, vw = 0 at the wall, and corresponds to a surface [45, 46]). In response to this surface charge, vanishing slip length b = 0; the ionic species in the liquid rearrange themselves and form a layer that screens the influence of the • the perfect-slip boundary condition corresponds surface charge. This layer of ionic species is called to the limit of an infinite slip length (b → ∞), the Electrical Double Layer (EDL). Note that the or equivalently a vanishing friction coefficient Debye length is independent of the surface charge (λ → 0). It corresponds to a shear free boundary Σ, and inversely proportional to the square root of condition. Traditionally, the perfect-slip bound- the salt concentration c0. Typically λD is equal to ary condition is used when the slip length b is −4 −2 30 nm for c0 = 10 M, 3 nm for c0 = 10 M and much larger than the characteristic length(s) of 0.3 nm for c0 = 1 M. the system; The Dukhin length is based on the comparison between the bulk to the surface electric conductance, • the partial-slip boundary condition concerns in- which links the electric current to an applied electric termediate slip length. field. It characterizes the channel scale below which surface conductance dominates over bulk conduc- For simple liquids on smooth surfaces, slip lengths up tance [1]. In a channel of width h and surface charge to a few tens of nanometers have been experimentally density Σ, the excess in counterion concentration is measured [14]. ce = 2Σ/he with e the elementary charge and where the factor 2 accounts for the two surfaces. One may 2.2 Mathematical description of the EDL define a Dukhin number Du = |Σ| /hc e. A Dukhin 0 The Electrical Double Layer (EDL) plays a funda- length ` can then be defined as Du mental role in nanofluidics due to a large surface |Σ| area to volume ratio. It corresponds to the layer of `Du = . (4) c0e ionic species that counter-balances the influence of a surface charge. Numerous phenomena, that will be For a surface with a surface charge density Σ = 2 discussed later, take their origin within the EDL, so 50 mC/m , `Du is typically 0.5 nm for c0 = 1 M, while −4 a mathematical description of this layer is of main `Du = 5 µm for c0 = 10 M. importance here. The conventional description is The slip length is defined as the depth inside the given hereafter. solid where the linear extrapolation of the velocity The Gouy-Chapman theory is at the basis of profile vanishes. Unlike previous lengths, that are EDL’s description. It is based on the following hy- all related to electrostatics, the slip length comes pothesis [47]: from the dynamic of the fluid near a solid surface. It characterizes the hydrodynamic boundary condition • ions are considered as (punctual) spots, of fluids at interfaces. Its expression can be derived as follows: first, assume that the tangential force per • the dielectric permittivity of the medium is sup- unit area exerted by the liquid on the solid surface posed constant in the medium, is proportional to the fluid velocity at the wall vw: σxz = λvw, with λ the friction coefficient, z the • the charge density and the electrical potential normal to the surface, x the direction of the flow. are seen as continuum variables,

4 • the correlations between ions as well as the ion- balances the electrostatic one, we can rewrite the solvent interactions are not taken into account equation 8: (mean field theory), 2 d V 2 βe = 8π`Bc0 sinh(βeV ) = κ sinh(βeV ), (9) • only electrostatic interactions are considered. dz2 1/2 where κ = (8π` c0) corresponds to the inverse Poisson-Boltzmann equation – Under the previ- B of the previously described Debye length λ . This ous hypotheses, let us write the equation that un- D equation describes the evolution of the electrical derlies the distribution of ions near a flat surface. potential next to a charged surface. Consider monovalent ions near a flat surface S, lo- Linearized Poisson-Boltzmann equation – In cated at z = 0, with homogeneous surface charge the general case, the Poisson-Boltzmann equation density Σ and surface potential V , as shown in fig- s can not be solved analytically. For small potentials ure 3. The link between the electrical potential V (z) (eV kBT ), an approximate form of the Poisson-

- 1 2 Boltzmann, the Debye-Hckel equation, can be written + - + - - 0.8 as - - - 1.5 2 + d V 2 - + = κ V. (10) + - s 0.6 2 0 Vs - - - - V 1 c dz / / + c - V 0.4 Assuming that the electrical potential vanishes far - + - - 0.5 0.2 from the surface, the solution of the Debye-Hckel - - + - + + equation reads ĸz 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 κ z −κz V (z) = Vse , (11) Figure 3: Left: scheme of the studied configura- where V is the surface electrical potential. Equation tion. Right: Debye-Hckel solution (11) s (11) is plotted in figure 3. The electrical potential for the electrical potential V in blue and is screened over a distance κ−1 = λ , the Debye linearized Boltzmann equation (12) for D length, which then gives the width of the EDL. The the concentration profile c in red. ± linearization of the equation (7) gives and the charge density ρe(z) at a z distance of the c± = ρs exp(∓βeV (z)) ≈ c0(1 ∓ βeV (z)) surface inside the ionic solution is given by a Poisson = c0(1 ∓ βeVs exp(−κz)). (12) equation: d2V ρ Equation (12) is plotted in figure 3 for both ± species. ∆V = = − e , (6) dz2 Non-linear Poisson-Boltzmann equation – In some situations, a solution for the non-linear Poisson- where = is the solvent permittivity (for water 0 r Boltzmann equation exists. Let us consider here the at ambient temperature, ≈ 80). The idea is to r case of a single flat wall, the electrolyte is located in ignore the thermal fluctuations of V and ρ and to e z > 0 and the solid wall in z < 0. A surface charge consider their respective averaged values only. At density Σ < 0 is located at z = 0. The electric field the thermal equilibrium, the densities of positive is taken to be equal to 0 inside the wall as well as far c (z) and negative c (z) ions are governed by the + − from the wall inside the electrolyte. At the wall, the Boltzmann equation: electrostatic boundary condition links the electric ∓βeV (z) field and the surface charge: c±(z) = c0e , (7)

∂V 4π = − Σ. (13) with β = 1/kBT and c0 the concentration in ion of ∂z charge ± e far from the wall. The charge density z=0 reads ρe = e(c+ − c−) = −2ec0 sinh(βeV ). Coupling Solving the PB equation (8) with this boundary this equation with (6), we get the Poisson-Boltzmann condition (13) leads to [48] equation for the electrical potential V (z): ! 2 1 + γe−z/λD V (z) = − ln , (14) d2V 2ec eβ 1 − γe−z/λD = 0 sinh(βeV ). (8) dz2 where γ is the positive root γ0 of the equation: Introducing the previously described Bjerrum length 2 2`GC ` , deﬁned as the length at which the thermal energy γ + γ − 1 = 0. (15) B λD

5 For a positive surface charge Σ, the solution for • the force between a permanent dipole and an V is identical, though with γ = −γ0(< 0). The induced dipole, surface potential Vs can be written as Vs = 4 arctan(−z/λD)/βe. • the force between two induced dipoles (London dispersion force). 2.3 Nanoscale forces Van der Waals forces are long-range, can bring Now that the general ideas of the Gouy-Chapman de- molecules together or mutually align/orient them, scription of the EDL have been given, let us describe and are not additive. They have to be described with some of the most important forces that play a role the quantum mechanical formalism, which is beyond in nanofluidics. These forces are at the origin of the the scope of the present work. large range of phenomena observed in nanofluidics, The DLVO (Derjaguin, Verwey, Landau, Over- and they give rise to both equilibrium or kinetic phe- beek) theory gives a large picture of nanoscale forces nomena [2]. Note that the distinction we will make which includes van der Waals forces and coulombic between forces is artificial since they all are electrical forces. However, some effects that appear at very in nature [49], but it still makes sense because of short range can not be described in the framework the many different ways in which the electrical force of the DLVO theory. Non-DLVO forces are discussed presents itself. in what follows. As a side note, each system depends fundamentally Chemical or bonding forces link two or more on individual forces that are applied between indi- atoms together to form a molecule [49]. Bonds are vidual atoms. However, in practice, large systems characterized by the redistribution of electrons be- (∼ 10 nm) can usually be described with continuum tween the two or more atoms. The number of cova- theory, which statistically averages the single inter- lent bonds that an atom can form with other atoms actions. This is why we may speak of forces exerted depends on its position in the periodic table. This by walls on particles or molecules, or between walls, number is called the valency. For example, it is equal or between particle or molecules. to one for hydrogen and two for oxygen, which leads Electrostatic forces are long range interactions to water molecule H O (H-O-H). Notice that covalent acting between charged atoms or ions [49]. Two par- 2 bonds are of short range (0.1 − 0.2 nm) and directed ticles of respective charges Q and Q at a distance 1 2 at well-defined angles relative to each other. For ex- r act on each other as follows: ample, they determine the way carbon atoms arrange Q1Q2 themselves to form diamond structure. Notice that F (r) = 2 , (16) 4π0rr covalent bonding comes from complex quantum interactions which are beyond the scope of the present where r is the dielectric permittivity of the medium. F (r) is directed along the axis defined by the position work. of the two particles. Equation (16) is known as Repulsive steric forces appear when atoms are the Coulomb law. Electrostatic forces can be either brought too close together. It is associated with the attractive or repulsive, depending on the sign of the cost in energy due to overlapping electron clouds (Pauli/Born repulsion). A consequence is the size product Q1Q2. They are, for example, at the origin of the building of the Electrical Double Layer (EDL). exclusion, widely used in membranes from angstrm Van der Waals forces are residual forces of electro- to micrometer [2]. It plays a role for example in aqua- static origin which are always present, even between porins (water channels), that offer a low resistance neutral atoms. They are relatively weak in compari- for water molecules, but do not allow ions to pass son to chemical bonding for example (see below), but through. To pass through this channel, the ion needs they nevertheless play a role in a large range of phe- to lose its water shell, which is energetically unfavor- nomena such as adhesion, surface tension or wetting. able. Notice that, combining electrostatic forces and + Van der Waals forces even manifest themselves at steric forces, it is possible to develop K channels + + macroscopic scales since they are at the origin of the with a high selectivity for K over Na while both adhesion of gecko, a decimetric reptilia, on solid sur- have water shells [50]. faces. Van der Waals forces include attractions and Solvation forces (or structural forces) are related repulsions between atoms, molecules and surfaces. to the mutual force exerted by one plate on another 3 They have three possible origins such as: when they are separated by a structured liquid .

• the force between two permanent dipoles, 3Liquid structuring in central in reference [51]

6 Next to a solid surface, density oscillations are ex- 2.4 Some consequences pected. If two solid surfaces immersed in a fluid are separated by a short distance, liquid molecules must The previously described forces give rise to a large accommodate the geometric constraint, leading to variety of phenomena. As an illustration, some of solvation forces between the two surfaces, even in the them are presented here. absence of attractive walls. Depending on both sur- Cohesion is related to attractive forces between face properties (well ordered, rough, fluid-like) and molecules of the same substance. It is due to inter- fluid properties (asymmetrically shaped molecules, molecular attractive forces. They can be van der with anisotropic or non pair-wise additive interaction Waals forces or hydrogen bonding. Cohesion is at potential), the resulting solvation forces can be either the origin, for example, of the tendency of liquids to monotonic or non-monotonic, repulsive or attractive. resist separation. See reference [49] for more details. Notice that, in the Adhesion corresponds to attractive forces between case of water molecules, solvation forces are called unlike molecules. They are caused by forces acting be- hydration forces. tween two substances, which can have various origins, Hydrophobic forces come from interactions be- such as electrostatic forces (attraction due to oppo- tween water and low water-soluble objects (molecules, site charges), bonding forces (sharing of electron), clusters of molecules...). These substances usually dispersive (van der Waals forces) etc. For example, have long carbon chains that do not interact with water tends to spread on a clean glass, forming a water molecules, resulting in a segregation and an thin and uniform film over the surface. This is be- apparent repulsion between water and nonpolar sub- cause the adhesive forces between water and glass stances. The hydrophobic effect, which results from are strong enough to pull the water molecules out of the presence of hydrophobic forces, is actually an their spherical formation and hold them against the entropic effect: each water molecule can form four surface of the glass. hydrogen bonds in pure water, but can not form as Surface tension is related to the elastic tendency much if surrounded by hydrophobic (apolar) species. of liquids which makes them acquire the least surface Hence, apolar molecules (or clusters of molecules) area possible. This results from the fact that when will rearrange themselves in order to minimize the exposed to the surface, a molecule is in an energeti- contact surface with water. An example is the mix- cally unfavorable state. Indeed, the molecules at the ing of fat and water, where fat molecules tend to surface of the liquid lack about half of their cohesive agglomerate and minimize the contact with water. forces, compared to the inner molecules of the bulk liquid [52]. Hence a molecule at the surface has lost Non-conservative forces, such as friction or vis- about half its cohesion energy. Surface tension is a cous forces, are referred as non-conservative forces measure of this lost of energy per surface unit. In because they involve energy transfer from one body the thermodynamic point of view, it is defined as to another. Contrary to other forces, which act on a the excess of free energy due to the presence of an body and generate a motion according to the second interface between two bulk phases [53]. The surface law of Newton, non-conservation forces have no force tension γ is of the order of magnitude of the bond law and arise as a reaction to motion. Inside a liquid, energy between molecules of the fluid divided by friction is linked to a fluid property: the viscosity, the cross section area of a molecule σ2: which is a property of a fluid to resist to a shear. It comes from collisions between neighbouring particles γ ∼ . that are moving at different velocities. For example, σ2 when a fluid flows through a pipe, the particles gen- erally move quickly near the pipe’s axis and slowly Finally one may notice that surface tension is also near its walls, leading to stress. The friction between present at liquid-liquid, liquid-solid and solid-air in- water molecules leads to a dissipation that has to terfaces. be overcome, for example by a pressure difference Wetting is the study of the spreading of a liquid between the two ends of the pipe, to keep the fluid deposited on a solid (or liquid) substrate. When moving. a small amount of liquid is put in contact with a Other forces, such as gravitational or inertial forces, flat solid surface, there are two different equilibrium are of lesser importance in nanofluidics and are not situations: partial wetting, when the liquid shows a discussed here. finite contact angle θ, and total wetting, in which the liquid spreads completely over the surface and where θ is not defined. The property of the fluid to spread

7 on the surface is characterized by the spreading pa- be lower than one and the problem to be stationary. rameter S which measures the difference between the Hence, the governing equation for the flow is the energy per unit area of the dry surface of the solid Stokes equation: substrate and the wetted surface: η 4 ~u = ∇~ p + F~ (19) S = γSV − (γSL + γLV ) (17) where η is the fluid viscosity, ~u is the velocity field, p ~ where γSV , γSL and γLV denote the free energies per is the hydrodynamic pressure and F a volume force. unit area of respectively the solid-vapour interface, The surface charge density is Σ, and will be different the solid-liquid interface and the liquid-vapour inter- from 0 if specified only. Unless otherwise stated, the face equal to the surface tension γ. In the case of height of the channel h will be considered large in a positive S, the surface energy of the dry surface comparison to the typical range of the potential (i.e. is larger than the energy of the wetted surface, so the Debye length). Unless otherwise stated, the no- the liquid tends to extend completely to decrease slip boundary condition will be used for the solvent the total surface energy, hence θ is equal to zero. A along walls. The system is shown on figure 4. negative S corresponds to a partial wetting situation, where the liquid does not completely spread on the p-Δp/2 p+Δp/2 surface and forms a spherical cap, adopting an angle V-ΔV/2 Σ V+ΔV/2 c-Δc/2 c+Δc/2 θ > 0. From the equilibrium of the capillary forces h Q, I, J at the contact line or from the work cost for moving z the contact line, one gets the Young-Dupr relation: L γ − γ x cos θ = SV SL . (18) γ Figure 4: Sheme of the 2D channel used for the calculations. Capillary forces originate in the adhesion between the liquid and the solid surface molecules. It is A flow through the membrane can occur as a con- strongly linked to the existence of a surface tension, sequence of a force near the membrane [55]. Here we as well as to the concept of wetting and contact consider this force to be a mechanic pressure drop angle. In certain situations, those forces pull the ∆p, difference in solute concentration ∆c or differ- liquid in order to force it to spread the solid surface. ence in electrical potential ∆V . We suppose that Depending on the configuration, it can make the the considered forcing are weak, so equilibrium pro- liquid fill a solid channel for example. files remain unmodified along z, and flows are linear functions of the forces operating. Hereafter we will 2.5 Transport in nanochannels study the volume flow Q, the ionic flow Ji and the In this section, we consider various transport phe- electrical current Ie resulting from ∆p, ∆c and ∆V . nomena that can occur in a nanochannel separating The phenomenological equations linking the three two reservoirs containing an electrolyte. The purpose flows to the three forces write: is to give a simple expression of each flux as a func- Q = L ∆P + L ∆c + L ∆V, tion of various driving forces (mechanical pressure, 11 12 13 solute concentration and electrical potential gradi- Ji = L21∆P + L22∆c + L23∆V, (20) ents). For the sake of simplicity, the nanochannel Ie = L31∆P + L32∆c + L33∆V, is chosen to be a slit (∼ 2D) and entrance effects are not taken into account 4. The walls are perpen- where LIJ are coefficients. According to Onsager’s dicular to z, respectively located in z = ±h/2 and law, the matrix of coefficients LIJ is symmetrical, driving forces are applied along x, see figure 4. The i.e. LIJ = LJI . Finally, one assumes that in the channel has a length L along x and a width w along middle of the channel (z=0), concentration, electrical y. The Reynolds number Re = ρvL/µ (where ρ and potential and pressure evolve linearly with x. µ are respectively the fluid density and the dynamic viscosity and v and L are respectively the character- Direct terms istic velocity and length of the flow) is assumed to The direct terms of the matrix of transport (20) 4Note that hydrodynamic entrance effects are discussed in correspond to the diagonal terms LII . They link references [32, 54]. each flux with their natural force, respectively the

8 solvent flow with the pressure gradient, the ionic density Σ, the nanochannel exhibits a selective perme- flow with the salt gradient and the ionic current with ability for ion diffusive transport [1]. Consequently, the electrical gradient. Each of them is calculated the concentration of counterions inside the channel hereafter in the previously described configuration is higher that the bulk concentration, while the con- (slit nanochannel). centration in co-ions is lower. Therefore, ions of the L11 – the hydrodynamic permeability charac- same charge as the nanochannel exhibit a lower per- terizes the flow transport across a given structure meability, while ions of the opposite charge have a under a pressure gradient. Using both symmetry and higher permeability through the nanochannel. Fol- impermeability of the walls, one gets for the velocity 6 field: ~u = ux(z)~ux. So the Stokes equation 19 can 2 be written as η∂z ux = ∂xp. A first integration of the 5 Stokes equation between 0 and z gives η∂zux = z∂xp, where we used that ∂zux|z=0 = 0 by symmetry, and 4 that ∂xp does not depends on z. Another integration 3 between −h/2 and z using that ∂xp = ∆p/L gives β

1 z2 h2 ∆p 2 u(z) = u + − , (21) w η 2 8 L 1 where uw is the wall velocity, which depends on the 0 hydrodynamic boundary condition (see the deﬁnition 1e-05 0.0001 0.001 0.01 0.1 1 of the slip length, subsection 2.1). Finally, L11 in c0 (mol/L) case of the no-slip boundary condition (uw = 0) can be written as Figure 5: Equation (26) for β+ (continuous) and β− (dashed) as a function of the bulk 3 Q 1 wh concentration for a negatively charged L11 = = − × . (22) ∆p 12η L surface.

Hence the hydrodynamic permeability of a membrane lowing Plesis et al., an effective nanochannel section in the low Reynolds number regime is limited by the ± ± can be defined for each species Seff = β S where β viscosity of the fluid, and depends strongly on the is an exclusion/enrichment coefficient [56]: dimensions of the channel. L22 – the ionic permeability characterizes the Z h/2 1 ∓φ(z) ionic flow through a membrane under a salt concen- β± = e dz; (26) h −h/2 tration gradient. From the Fick’s law of diffusion: where φ(z) = βeV (z) with V (z) the electrical poten- ~ ~ j± = −D±∇c±, (23) tial. One can use the linearised Poisson-Boltzmann equation to calculate the ion concentration profile in where D± are diffusion coefficients of the ± species the slit. An example is shown in figure 5. respectively, one can write the total flow Ji assuming that D+ = D− = D: L33 – the ionic conductance characterizes the ionic current through a membrane under an applied Z Z h/2 electrical potential difference: G = Ie/∆V . First, let Ji = (~j++~j−).d~S = −wD ∂x(c+(x)+c−(x))dz. S −h/2 us define the (bulk) conductivity of the solution κb: (24) κ = e(µ c + µ c ) (27) In a neutral channel, and assuming that c±(x) = b + + − − x∆c/L + c0, it gives with µ± and c± respectively the mobility and the J S volume density of ± ions [45]. At high ionic strength, L = i = −D × , (25) 22 ∆c L or for a neutral channel (Σ = 0), equation (27) can be used directly to calculate the conductance of the where S = hw is the surface of the channel. Notice channel Gbulk = κbωh/L. However, for a non-neutral that equation (25) describes ionic flow through a channel (Σ 6= 0), if one looks at the ionic conductance membrane under salt gradient in absence of surface versus the salt concentration on a log-log scale, a charge. In case of the presence of a surface charge conductance plateau is observed at low ionic strength.

9 This is due to the contribution to the total current one finds, after a double integration of the equation of ions of the EDL. This excess counterions concen- (31): tration can be written as [57]: u (z) = (V (z) − ζ)E (33) x η e 2Σ ce = (28) where we used the no slip boundary condition and he where ζ is the zeta potential, which is the value of where the 2 accounts for the two surfaces. From this the electrostatic potential at the shear plane, i.e. the excess counterions concentration, one can define a position close to the wall where the velocity vanishes surface conductance Gsurf = eµceωh/L. Then the 5. In the no-slip case, the zeta potential is equal total conductance is the sum of a bulk conductance to the surface potential Vs. As a remark, one can and a surface conductance: notice that in the case of a finite slip at the wall, wh w characterized by a slip length b, the potential ζ takes G = G + G = µc e + 2Σµ (29) bulk surf s L L the expression: where we assumed that µ+ = µ− and defined cs = 2c0 ζ = Vs × (1 + bκeff) (34) with c0 the salt concentration. Finally, the ionic current Ie and the voltage drop ∆V are linked as where Vs is the electrostatic potential at the wall and 0 follows: κeff the surface screening parameter (κeff = −V (z = 0)/V ). In the case of a weak potential, the screening I w s L = e = µ (c eh + 2Σ) × . (30) parameter is approximately equal to the inverse of 33 ∆V s L the Debye length λD. Note that the velocity in the fluid results from a balance between the driving Cross terms electric force and the viscous friction force at the Additionally to the direct terms, there are cross surface. phenomena coming from couplings between hydro- An integration of equation (33) gives the following dynamics, ion diffusion and electrostatics. Using expression for the total water flow: statistical mechanics, Onsager has shown the neces- Q = whU − surface correction terms, (35) sity of equality between the term LIJ and LJI . So in EO what follows, only three terms among the six cross coefficients are explicitly calculated, the last three where UEO is the eletro-osmotic velocity UEO = being deduced from Onsager’s relation. −ζEe/η. Figure 6 shows a scheme of the velocity L13 /L31 – The phenomenon by which a difference of electrical potential ∆V induces a water flow is called electro-osmosis (L13). Its conjugate effect is called streaming current (L31) and corresponds to the generation of an electric current by a pressure driven liquid-flow [58]. Hereafter, we will do explicit UEO calculations for the case of electro-osmosis (L13). Electro-osmosis takes its origin in the ion dynamics within the Electrostatic Double Layer (EDL), in λD which the charge density ρe = e(ρ+ − ρ−) is non- vanishing. The dynamics of the fluid is described by the stationary Stokes equation with a driving force Figure 6: Schematic representation of the velocity for the fluid F = ρ E , where the electric tangential e e e profile, equation (33) without and with field E is defined as E = −∂ V = −∆V/L, and is e e x surface correction terms, respectively on directed along x [59]: the left and on the right. 2 η∂z ux + ρeEe = 0. (31) profile, with and without the surface correction terms. Using that the charge density is linked to the elec- Finally, neglecting the surface correction terms (that trostatic potential of the EDL as follows: 5 Notice that sometimes the zeta potential is defined as Vs and ∂2V one has to consider an amplified electro-osmotic mobility ρ = − , (32) e ∂z2 to take into account the effect of slippage.

10 are of the order of λD/h 1), one can write: Chemi-osmosis causes flow towards lower electrolyte concentration. As a complement we will discuss two Q ζ wh interesting cases: the case of non-equal diffusion L = ≈ × . (36) 13 ∆V η L coefficient between + and − species, and the limit of large Debye length λD compared to the channel Hence electro-osmosis is caused by coulomb force height h (this regime is called osmosis). and limited by viscous dissipation. Supplement 1 – In the case of a difference in anion Accordingly, the streaming current (L31), which is and cation diffusivities, an electric field is induced, the electric current generated by a pressure driven and a supplementary electro-osmotic contribution liquid-flow can be written as has to be taken into account [61]. Assuming a vanishing local current in the outer region and a sym- Ie ζ wh metric electrolyte, this diffusion-induced electric field L31 = = × . (37) ∆P η L is proportional to β0 = (D+ − D−)/(D+ + D−):

L12 /L21 – The generation of a flow under a solute D kBT d ln c E = β0 . (43) gradient is called chemi-osmosis (L12)[60]. Its conju- e dx gate effect is the generation of an excess flux of salt So the contribution of this mechanism combined with under a pressure drop ∆p (L ). Here the expression 21 the previously calculated velocity (equation (41)) of the L coefficient is obtained in the case of a flow 12 gives the following diffusio-osmotic velocity: generated by a solute gradient. 2 So, let us assume the existence of a salt concen- kBT ln(1 − γ ) ∆c tration difference ∆c. The salt concentration in the UDO = − β0ζ + , (44) η e 2π`B c0L middle of the channel, cmid(x), is assumed to vary linearly along the axis x: cmid(x) = c0 + ∆c × x/L, where we used equation (33). The first term corre- where c0 is the concentration in the left reservoir (the sponds to the electro-osmotic effect, the direction concentration in the right reservoir being c0 + ∆c). of the generated flow depending on the sign of the From the mechanical equilibrium in z together with product β0ζ, while the second term, called the chemi- the Stokes equation along z, one can deduce the osmotic effect, causes a flow towards the lower elec- hydrostatic pressure profile: trolyte concentration. Neglecting surface correction terms, one can write: p(x, z) = 2kBT cmid(x) [cosh φ(x, z) − 1] + p0, (38)

2 where φ(x, z) = eβV (x, z). Injecting this expres- Q kBT ln(1 − γ ) wh 2 L = ≈ − β ζ + × . sion in the Stokes equation along x, η∂ ux(z) − 12 0 z ∆c ηc0 e 2π`B L ∂ p(x, z) = 0, one finds: x (45) ∆c Supplement 2 – An interesting case is the limit η∂2u (z) = 2k T (cosh φ − 1) . (39) z x B L where the Debye length λD is much larger than the channel height h. In this particular case, a constant Using Poisson-Boltzmann and assuming that λD potential (independent of z) called Donnan potential h, one can find that the flow is: VD builds up in the entire channel. From the electro- chemical equilibrium one gets: Q = whUCO − surface correction terms, (40) c+ = e−2φD , (46) with UCO the chemi-osmotic velocity: c−

k T ln(1 − γ2) ∆c 2 U = − B , (41) c+c− = c0, (47) CO η 2π` c L B 0 2Σ c+ − c− = − , (48) where γ = tanh(φs/4). Neglecting surface correction eh terms (on the order of λD/h 1), the coeﬃcient where φD = eβVD. One can introduce the Dukhin L12 can be written: number Du = Σ/(ec0h). Then from equation (48), and using that cosh2(x) − sinh2(x) = 1, one gets 2 Q kBT ln(1 − γ ) wh L12 = ≈ − × . (42) p 2 ∆c ηc0 2π`B L cosh(φD) = 1 + Du . (49)

11 Injecting this expression in equation 39, which has λD h (thin electric debye layers as compared to been obtained from mechanical equilibrium in z to- the channel width), one writes: gether with the Stokes equation, one gets Z ∞ 1 wh3 Ie = 2w e (c+(z) − c−(z)) ux(z)dz. (58) Q = − × ∆Π, (50) 0 12η L In this assumption, one expects that the entire con- where the osmotic pressure ∆Π can be written as tribution to the current Ie comes from the convection

p 2 of ions inside the electric double layers. From the ∆Π = 2kBT ∆c 1 + 1 + Du . (51) Poisson equation (6), one gets that

Hence, when λD h, one may write ∂2φ 1 c+(z) − c−(z) = − 2 . (59) 3 ∂z 4π`B Q kBT p 2 wh L12 = = − 1 + 1 + Du × . ∆c 6η L Moreover, we know from previous section (see equa- (52) tion (39)) that the velocity field ux(z) under a solute L23 /L32 – The current generated under a salt gradient is solution of concentration gradient is called osmotic current and 2 the reciprocal effect is the generation of a salt flux ∂ ux ∆c η = 2kBT (cosh φ − 1) . (60) under an electrical potential gradient. The expres- ∂z2 L sion of L23 is here obtained in the first case, i.e. in Injecting equation (59) in equation (58), performing the case of the generation of current under a salt an integration by part (twice) in the spirit of [62], concentration gradient. one gets The electrical current Ie can be written as we ∂u ∞ we Z ∞ ∂2u Z h/2 x x Ie = φ − φ 2 dz. (61) Ie = w e (j+(z) − j−(z)) dz. (53) 2π`B ∂z 0 2π`B 0 ∂z −h/2 From equation (60), one get Two contributions to the current can be expected, a contribution from the diffusive flux of salt and a Z ∞ ∂ux 2kBT ∆c contribution from the convective flux of salt. The = − (cosh φ − 1) dz. (62) ∂z η L 0 first one can be written as z=0 Z h/2 Hence, using that φ(z = ∞) = 0, one gets I = w e (j (z) − j (z)) dz, (54) D D,+ D,− ∞ −h/2 we k T ∆c Z I = − B (φ − φ) × (cosh φ − 1)dz. e π` ηL s with j = −D ∇c the diffusive flux of each ion. B 0 D,± ± ± (63) Assuming that D = D = D and rewriting the + − with φ the normalized surface potential. Using PB current as s equation ∇2φ = κ2 sinh φ, we make the following Z h/2 change of variable: ID = −ew∂x (c+(z) − c−(z)) dz, (55) −h/2 dφ dz = − p , (64) ID appears to be equal to zero from the global charge κ 2(cosh φ − 1) electroneutrality which allows to solve the integral in equation (63). Z h/2 One finds [62]: Σ + e (c+(z) − c−(z)) dz = 0. (56) −h/2 e kBT φs ∆c Ie = 2w 2 sinh − φs , (65) Accordingly, considering the convective part in equa- πη`B κ 2 L tion (53) only, the current can be written as that can be rewriten in terms of surface charge (using Z h/2 2 sinh φs/2 = eΣ/kBT κ): Ie = w e (c+(z) − c−(z)) ux(z)dz, (57) −h/2 kBT 1 ∆c Ie = 2wΣ 1 − κ`GC argsinh , where both species ± move at the same velocity ux(z) 2πη`B κ`GC Lc0 (i.e. there is no electric field along z). Using that (66)

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