FIGURA 2.3 COMPORTAMIENTO DEL EMPLEO SECTORIAL DURANTE EL PERÍODO DE 1992 AL

When a solution comes into contact with a charged surface, an EDL forms (Figure 1.2).3 The Stern layer closer to the solid surface consists of counter ions, cations in case of negatively charged surfaces, and can be considered as immobile. This layer is held strongly by electrostatic interactions to neutralize the surface and counter ion concentration is higher than in the bulk solution. The second layer, the diffuse layer, contains co-ions and counter-ions. The higher concentration of counter ions in the Stern layer compared to the adjacent diffuse layer leads to the evolution of the zeta potential (!) at the shear plane. The ion distribution between the EDL and bulk solution is described by the famous Boltzmann distribution equation;

c_! =!!_!!!"# −!_!!!"

!" Equation 1-3

Where (c_!) is the ionic concentration in the EDL and (!_!) is the bulk ion concentration.

The screening length of the EDL, the Debye length (λD), represents the local balance between the electromigration towards the surface and diffusion away from it4 and can be defined based on the Poisson-Boltzmann theory in terms of the bulk ionic strength (!_!).

!_! =! !

!!!!!! Equation 1-4

From Equation 1-4 it is clear that higher ionic strength solutions, regardless of the surface charge density, will result in a more compact EDL with narrow λD, which

5 is the case when handling most biological samples (λD = 1 nm for an aqueous solution of a monovalent symmetric electrolyte !_! = 0.1 M). At very low ionic strength buffers, the λD may reach few tens of nanometers (λD = 30 nm for !_! = 10-4 _M).

The EDL plays a major role in ion transport through nanochannels under an applied electric field, as it is the region where volume electric forces (fe) take place. The magnitude of these forces is defined by the charge density (!_!) and the applied electric field (!_!).

!_! =!!_!!_! Equation 1-5

The Poisson-Boltzmann equation quantify the local ionic concentration ratio between the electroneutral bulk and a location inside the EDL at a certain potential relative to the bulk as described in the following equations

!! =!!!!! Equation 1-6

!_! =!!_!!_! Equation 1-7

In nanochannels, this leads to interesting transport behaviour especially when the nanochannel dimensions approach twice the Debye length and EDL overlap occurs. Examples include permselectivity of the nanochannels and formation of ion enrichment and depletion zones.

6 Figure 1.2 Schematic model of the electrical double layer at a solid/ liquid interface. ψ is the electrical potential. A surface with negative charges is considered. These charges are shielded by the Stern layer and the diffuse layer. The Stern layer is formed by adsorbed immobile ions. The mobile diffuse layer is located outside the shear plane. The zeta potential ! is at the shear plane. The Stern layer and the diffuse layer form the electrical double layer. "Reprinted with permission from Abgrall, P.; Nguyen, N. T., Nanofluidic Devices and Their Applications. Analytical Chemistry 2008, 80 (7), 2326-2341. Copyright (2008) American Chemical Society."

7 1.1.1.3 Dukhin Length (!_!") and Dukhin Number (Du)

The excess counter ion near the solid surface of the nanochannel leads to surface conductivity different from the bulk conductivity inside the nanochannel determined by the bulk ion concentration (Co). The Dukhin length is defined as the surface to bulk conductivity ratio

!!" =!_!!!

! Equation 1-8

Accordingly, for a nanochannel height (h), the Dukhin number (Du), a dimensionless term, is defined as

!"=! !! !!!=!

!!"

! Equation 1-9

Large values (>1) indicate higher nanochannel ion selectivity due to surface dominated ion transport even in the absence of EDL overlap.5 While both the Dukhin length and the Debye length increase with decreasing the ionic strength, the Dukhin number takes into account the nanochannel dimensions and the entrance effects under non-equilibrium conditions. When the EDL almost overlap, the Dukhin number becomes very large extending outside the nanochanel and consequently defines the electric field lines at the nanochannel entrance. Dukhin numbers can be used to predict the kind of ion concentration polarization (ICP), propagating or non- propagating, and the extent of micromixing near nano-/microchannel junctions.

Values for !!" can vary from 0.5 nm for Co = 1 M to 5 µm for Co = 10-4 M, for a surface charge density eΣ ≈ 50 mC/m2 _{(≈ 0.3 e/nm}2_{). The surface charge for} glass is at most in the 10 mC/m2 range (≈ 0.06 e/nm2). Increasing the surface charge density will result in larger !_!", which explains the enhanced conductance and ion transport through more highly charged hydrogel nanojunctions.

8 1.1.1.4 Slip Lengths (b) and Surface Friction

Navier length or slip length (b) describes the friction of the fluid at the interface and affect ion transport at charged surfaces. Slip length for water strongly depends on wettability and shows higher values for hydrophobic surfaces than hydrophilic ones which are negative and located at the plane of shear, few tens of nanometres as compared to sub-nanometre.6 Interestingly, lengths in the micrometre range can be observed at nano-/micro-interfaces and enhanced fluid transport occurs when the pore size approaches b.

1.1.2 Electro-osmotic Flow (EOF) and Electrophoresis

As described before under section 1.1.1.2 Debye Length Scale, the EDL for negatively charged surfaces comprises a dense layer of counter ions, cations. Under the influence of an applied electric field, cations near the solid surface will migrate towards the cathode while dragging water molecules creating a normal or cathodic EOF. In contrast to microchannels, the profile of EOF in nanochannels is no longer flat as the geometry approaches the !!.7 The direction of the EOF is determined by the type of the surface charge, positive or negative, but its magnitude is the net result of many factors.

Hydrophobic surfaces exhibit low liquid-solid friction with high slip length (b) values. The large slippage results in high EOF velocity (!_!") by amplifying the surface zeta potential (!) which inturn is related to the electrostatic surface potential (!_!). This relation can be described by the following equations, where ! the dielectric constant and ! the fluid viscosity,6

9 !_!" =!!"

! Equation 1-10

!=!!_! 1+_!!

! Equation 1-11

Equation 1-11 explains why hydrophobic surfaces may show much higher zeta potential (!) than hydrophilic surfaces with similar surface electrostatic potential !!.