Philippe Leroy

Alexis Maineult
email: alexis.maineult@upmc.fr

Exploring the electrical potential inside cylinders beyond the Debye-Hückel approximation: a computer code to solve the Poisson-Boltzmann equation for multivalent electrolytes

Keywords: Electrical Properties, hydrogeophysics, non-linear differential equations, numerical modelling

Exploring the electrical potential inside cylinders beyond the Debye-Hückel approximation: a computer code to solve the Poisson-Boltzmann equation for multivalent electrolytes

Philippe Leroy, Alexis

INTRODUCTION

Many research areas in Earth sciences, physics, chemistry, and medicine are focused on investigating the electrochemical properties of cylinders [START_REF] Bocquet | Nanofluidics, from bulk to interfaces[END_REF][START_REF] Daiguji | Ion transport in nanofluidic channels[END_REF][START_REF] Schoch | Transport phenomena in nanofluidics[END_REF]. For instance, in hydrology, the electrical potential on the internal wall of cylinder can be considered in pore network models simulating water flow and solute transport [START_REF] Li | Solute dispersion under electric and pressure driven flows; pore scale processes[END_REF][START_REF] Obliger | Pore network model of electrokinetic transport through charged porous media[END_REF][START_REF] Xiong | Review of pore network modelling of porous media: Experimental characterisations, network constructions and applications to reactive transport[END_REF]. It is also used in rock physics to simulate the electrical properties of porous media to interpret, for instance, their self-potential and induced polarization (IP) response [START_REF] Bernabé | Streaming potential in heterogeneous networks[END_REF][START_REF] Jougnot | Derivation of Soil-Specific Streaming Potential Electrical Parameters from Hydrodynamic Characteristics of Partially Saturated Soils[END_REF]Bücker & Hördt, 2013a). In hydrogeophysics, researchers are actually interested in relating the permeability to the IP response of rocks and, to date, there is a debate about whether it is the grain or pore size that controls it [START_REF] Revil | Determination of permeability from spectral induced polarization in granular media[END_REF][START_REF] Revil | Is it the grain size or the characteristic pore size that controls the induced polarization relaxation time of clean sands and sandstones?[END_REF][START_REF] Revil | Predicting permeability from the characteristic relaxation time and intrinsic formation factor of complex conductivity spectra[END_REF][START_REF] Kruschwitz | Study into the correlation of dominant pore throat size and SIP relaxation frequency[END_REF][START_REF] Weller | Induced polarization and pore radius -A discussion[END_REF]. Knowing the electrochemical properties of cylinders is essential to model the induced polarization of rocks using the membrane polarization model and to better assess the effect of pore size on IP (Bücker & Hördt 2013a,b;[START_REF] Bairlein | Temperature dependence of spectral induced polarization data: experimental results and membrane polarization theory[END_REF][START_REF] Hördt | The dependence of induced polarization on fluid salinity and pH, studied with an extended model of membrane polarization[END_REF][START_REF] Hördt | Geometrical constraints for membrane polarization[END_REF][START_REF] Chuprinko | Influence of mineral composition on spectral induced polarization in sediments[END_REF].

In electrokinetics and more generally in hydrogeophysics, most transport models use the zeta potential and the linearized Poisson-Boltzmann equation, based on the Debye-Hückel (D-H) approximation, to model the electrochemical properties of pores [START_REF] Bernabé | Streaming potential in heterogeneous networks[END_REF][START_REF] Pride | Governing Equations for the Coupled Electromagnetics and Acoustics of Porous-Media[END_REF][START_REF] Conlisk | The Debye-Hückel approximation: Its use in describing electroosmotic flow in micro-and nanochannels[END_REF]; [START_REF] Schoch | Transport phenomena in nanofluidics[END_REF][START_REF] Jougnot | Derivation of Soil-Specific Streaming Potential Electrical Parameters from Hydrodynamic Characteristics of Partially Saturated Soils[END_REF]Bücker & Hördt 2013a). The D-H approximation is accurate for surface electrical potentials, considered equal to zeta potentials, of magnitude inferior or equal to 25.7 mV at a temperature of 25°C [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF][START_REF] Lyklema | Fundamentals of Interface and Colloid Science[END_REF]. Because zeta potentials on mineral surfaces are often much larger, it would be better to use the Poisson-Boltzmann equation without any approximation regarding the surface electrical potential to compute the electrical potential at the interface between mineral and water (Leroy & Revil 2004;[START_REF] Sverjensky | Prediction of surface charge on oxides in salt solutions: Revisions for 1 : 1 (M+L-) electrolytes[END_REF][START_REF] Vinogradov | Measurement of streaming potential coupling coefficient in sandstones saturated with natural and artificial brines at high salinity[END_REF][START_REF] Li | Influence of surface conductivity on the apparent zeta potential of calcite[END_REF]. For instance, amorphous silica and the basal surface of montmorillonite, two minerals that are common in the subsurface, exhibit negative zeta potentials that can easily exceed 25.7 mV in magnitude [START_REF] Rasmusson | The dynamic mobility and dielectric response of sodium bentonite[END_REF][START_REF] Sonnefeld | Determination of electric double layer parameters for spherical silica particles under application of the triple layer model using surface charge density data and results of electrokinetic sonic amplitude measurements[END_REF][START_REF] Leroy | Influence of surface conductivity on the apparent zeta potential of amorphous silica nanoparticles[END_REF][START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF].

As far as we know, there is no code available in the literature to solve numerically the Poisson-Boltzmann equation inside cylinders. We propose a short and easy-tohandle Matlab® code to model the electrochemical properties of cylinders. After a brief presentation of the P-B equation and associated boundary conditions, electrical potential profiles inside amorphous silica and montmorillonite nanotubes, computed from the full and linearized P-B equation, are showed, compared and discussed. Two different electrolytes representative of natural waters, NaCl and CaCl 2 , are considered, and zeta potentials were computed from well-established surface complexation models describing Stern and diffuse layers at the interface between mineral and water (Leroy et al. 2013[START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF].

THE POISSON-BOLTZMANN EQUATION INSIDE A CYLINDER

Let us consider a cylinder saturated with an aqueous electrolyte and having its internal wall electrically charged. The distribution of the electrical potential  (V) in the direction normal to the charged surface is usually computed using the Poisson-Boltzmann (P-B) equation [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF][START_REF] Lyklema | Fundamentals of Interface and Colloid Science[END_REF][START_REF] Leroy | Predicting the surface tension of aqueous 1:1 electrolyte solutions at high salinity[END_REF][START_REF] Leroy | A double layer model of the gas bubble/water interface[END_REF]. The input parameters of this equation are the chemical composition of the bulk electrolyte and the surface electrical potential, often equated with the zeta potential (), or the surface charge density [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF][START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF][START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF]. The zeta potential is the surface electrical potential at the plane of shear or slip plane and can be inferred from electrokinetic experiments for instance [START_REF] Lyklema | Fundamentals of Interface and Colloid Science[END_REF]Leroy & Revil 2004;[START_REF] Delgado | Measurement and interpretation of electrokinetic phenomena[END_REF][START_REF] Heuser | Electro-osmotic flow in clays and its potential for reducing clogging in mechanical tunnel driving[END_REF].

The P-B equation can be written as (Appendix A):

2 A 1 1000 N exp M i ii i q qC kT             , (1)
where N A is Avogadro's number ( 6.02210 23 mol -1 ),  is the dielectric permittivity of the electrolyte (F m -1 ), and M is the number of ionic species. The quantity q i is the ion charge (C), q i = es i where e is the elementary charge (~1.602×10 -19 C) and s i is the signed ion valence (s i =  z i with z i the ion valence, "+" standing for cations and "" for anions). The quantity C i is the concentration of ion i (mol L -1 , M), k is Boltzmann's constant ( 1.38110 -23 J K -1 ) and T is the temperature (K). Note that the factor 1000

converts C i , expressed in mol L -1 , into mol m -3 .
Eq. ( 1) can be linearized using the first two terms of the power series of exp(x) = 1x where x = q i  / kT, which works well in the case of small electrical potentials satisfying the condition || << kT / |q i |. This yields:

2 A 1 1000 N 1 M i ii i q qC kT             . ( 2
)
Considering the electroneutrality condition in the bulk electrolyte, i.e.:

1 0 M ii i qC    , (3)
Eq. ( 2) becomes the Poisson-Boltzmann equation with the Debye-Hückel (D-H) approximation:

22

    , (4)
where  =  -1 is the Debye length (m), which is given by:

2 A 1 1000 N M ii i kT qC      .
(5)

Here we consider an infinite cylinder of radius r 0 filled with saline water, with an electrical potential set to  on its internal wall. In radial coordinates, Eq. (1) writes:

      2 A 2 1 1000 N 1 exp M i ii i d r d r q r qC dr r dr kT              . ( 6
)
The boundary conditions are:

  0 r   , (7)
and for symmetry reasons:

0 0 r d dr    . ( 8
)
Solving Eqs. ( 6) to (8) allows us to obtain the electrical potential profile () r  . For numerical reasons, it is more convenient to solve a dimensionless equation. For that purpose, we replaced in Eq. ( 6) the quantity q i by es iby doing so, the electronic charge is taken out of the summation. Eq. ( 6) thus becomes:

      2 A 2 1 1000 N 1 exp M i ii i d r d r s e r e sC dr r dr kT              . ( 9
)
We define the following quantities:

r R   , ( 10
)
which is the dimensionless radius;

    er R kT   , (11)
which is the dimensionless electrical potential; and

e kT    , ( 12
)
which is the dimensionless zeta potential.

Eq. ( 9), ( 7) and ( 8) then become:

  2 1 2 2 1 exp 1 M i i i i M ii i s C s dd dR R dR sC          , ( 13
)   0 R   , ( 14
) 0 0 R d dR    . ( 15
)
Eqs. ( 13) to (15) were rewritten in a form usable by the Matlab® procedure bvp4c, which solves ordinary differential equations given their boundary conditions using the collocation method. The code, which computes () R  and then  

r  , is presented in Appendix B.
Finally, let us consider the Poisson-Boltzmann equation in radial coordinates with the Debye-Hückel approximation (from Eq. ( 4)):

      2 2 2 11 d r d r r dr r dr     . ( 16
)
The solution for the boundary conditions ( 7) and ( 8) is given by (e.g., [START_REF] Bernabé | Streaming potential in heterogeneous networks[END_REF]:

  0 0 0 J ( ) J ( ) r i r r i     , ( 17
)
where 0 J is the zero-order modified Bessel function of the first kind and i the imaginary number.

APPLICATIONS TO AMORPHOUS SILICA AND MONTMORILLONITE

NANOTUBES

We used our code to compute the electrical potential profiles inside cylinders assuming amorphous silica or montmorillonite surfaces. These two minerals are of particular interest because silicon dioxide is often used in nanofluidics [START_REF] Schoch | Transport phenomena in nanofluidics[END_REF][START_REF] Bocquet | Nanofluidics, from bulk to interfaces[END_REF][START_REF] Daiguji | Ion transport in nanofluidic channels[END_REF]. It is also the major compound of quartz sand and sandstones, which are very common in the subsurface [START_REF] Leroy | Complex conductivity of water-saturated packs of glass beads[END_REF][START_REF] Vinogradov | Measurement of streaming potential coupling coefficient in sandstones saturated with natural and artificial brines at high salinity[END_REF][START_REF] Revil | Predicting permeability from the characteristic relaxation time and intrinsic formation factor of complex conductivity spectra[END_REF], and one of the main constituents of cement [START_REF] Labbez | Surface charge density and electrokinetic potential of highly charged minerals: Experiments and Monte Carlo simulations on calcium silicate hydrate[END_REF][START_REF] Grangeon | On the nature of structural disorder in calcium silicate hydrates with a calcium/silicon ratio similar to tobermorite[END_REF][START_REF] Lerouge | In situ interactions between Opalinus Clay and Low Alkali Concrete[END_REF]. Montmorillonite is often found in sedimentary soils and rocks, and is the major compound of bentonite, which is used for instance for the ground storage of domestic wastes and underground storage of highly radioactive and long-lived nuclear wastes [START_REF] Malusis | Flow and transport through clay membrane barriers[END_REF][START_REF] Rotenberg | Modelling water and ion diffusion in clays: A multiscale approach[END_REF][START_REF] Tournassat | Modeling specific pH dependent sorption of divalent metals on montmorillonite surfaces. A review of pitfalls, recent achievements and current challenges[END_REF][START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF][START_REF] Leroy | Spectral induced polarization of Namontmorillonite dispersions[END_REF].

We consider cylindrical pores of diameters 10, 100, and 1000 nm, and containing 1:1, NaCl, or 2:1, CaCl 2 , salts of concentrations 1, 10 and 100 mM at a temperature of 25°C (mM stands for 10 -3 mol L -1 ). Pore diameters and chemical compositions of the bulk electrolyte were chosen to encompass values typical for nanopores containing monovalent and multivalent electrolytes [START_REF] Dufreche | Models for electrokinetic phenomena in montmorillonite[END_REF][START_REF] Dufreche | Molecular hydrodynamics for electro-osmosis in clays: from Kubo to Smoluchowski[END_REF][START_REF] Wang | Electrochemical charge of silica surfaces at high ionic strength in narrow channels[END_REF]Wang et al. 2010). These salts were also chosen because they contain some major ions found in natural waters [START_REF] Mccleskey | Electrical Conductivity of Electrolytes Found In Natural Waters from (5 to 90) degrees C[END_REF]. Electrochemical properties of amorphous silica were investigated as a function of pH because of the presence of silanol surface sites exchanging protons with surrounding electrolyte, responsible for its negative surface charge and zeta potential [START_REF] Sonnefeld | Determination of electric double layer parameters for spherical silica particles under application of the triple layer model using surface charge density data and results of electrokinetic sonic amplitude measurements[END_REF]. Those of the basal surface of montmorillonite were considered independent on pH because of its permanent negative surface charge and zeta potential due to ion substitution in the crystal [START_REF] Tournassat | Influence of montmorillonite tactoid size on Na-Ca cation exchange reactions[END_REF].

Surface complexation models

Zeta potentials were computed using the surface complexation models of [START_REF] Leroy | Influence of surface conductivity on the apparent zeta potential of amorphous silica nanoparticles[END_REF] for amorphous silica and [START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF] for the basal surface of montmorillonite, considering the additional adsorption of calcium ions in the Stern layer. They were not measured because of the high uncertainties associated with interpretation of electrokinetic measurements into zeta potentials [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF][START_REF] Lyklema | Fundamentals of Interface and Colloid Science[END_REF][START_REF] Delgado | Measurement and interpretation of electrokinetic phenomena[END_REF]). In the surface complexation models of Leroy et al., the charged surface does not electrostatically interact with another charged surface.

Therefore, one limitation of our approach is that the zeta potentials are computed using one plane surface in contact with an infinite electrolyte where electroneutrality occurs in the bulk part. To properly consider interacting diffuse layers when the pore is completely filled by them, it would be better to develop a P-B code considering as input parameter the surface charge density rather than the zeta potential. Nevertheless, actual geophysical codes are more familiar with the zeta potential than the surface charge density [START_REF] Vinogradov | Measurement of streaming potential coupling coefficient in sandstones saturated with natural and artificial brines at high salinity[END_REF][START_REF] Jougnot | Derivation of Soil-Specific Streaming Potential Electrical Parameters from Hydrodynamic Characteristics of Partially Saturated Soils[END_REF]Bücker & Hördt 2013a).

Therefore, developing a P-B code considering as input surface charge density rather than zeta potential will be carried out in the future when geophysical codes will also consider this input parameter.

Electrostatic surface complexation models describe electrical potential and ion distributions at the mineral/water interface [START_REF] Gouy | Sur la constitution de la charge électrique a surface d'un électrolyte[END_REF][START_REF] Chapman | A contribution to the theory of electrocapillarity[END_REF]. Their input parameters typically are the temperature, chemical composition of the bulk electrolyte, total density of surface groups, equilibrium adsorption constants of the protons and counter-ions, and capacitance(s) [START_REF] Hiemstra | Multisite proton adsorption modelling at the solid/solution interface of (hydr)oxides: a new approach. II. Application to various important (hydr)oxides[END_REF]). In the triple layer model (TLM)

or extended Stern model (Fig. 1), the "0-plane" corresponds to the surface of the mineral where protonation and deprotonation reactions occur, the "-plane" is located at the center of the compact Stern layer made of adsorbed counter-ions, and the "d-plane" delimits the onset of the diffuse layer containing counter-ions and co-ions [START_REF] Yates | Site-binding Model of the Electrical Double Layer at the Oxide/Water interface[END_REF]. The basic Stern model (BSM) is a simplified TLM where the "-plane" coincides with the "d-plane" (no capacitance necessary to describe the electrical potential between these two planes, [START_REF] Westall | A comparison of electrostatic models for the oxide/solution interface[END_REF]. In these two models, the zeta potential, , which is the electrical potential at the shear plane, is traditionally assumed to be located slightly further way from or at the "d-plane" because no water flow is considered between the mineral surface and the onset of the diffuse layer [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF][START_REF] Lyklema | Electrokinetics: The properties of the stagnant layer unraveled[END_REF]. Besides zeta potential measurements, and tritium experiments and crystallographic considerations for the total surface site densities, one way to calibrate the parameters of the surface complexation model (the adsorption equilibrium constants and capacitance(s)) is to use surface charge density measurements from acid-base potentiometric titration and cation exchange capacity (CEC) [START_REF] Tournassat | The titration of clay minerals I. Discontinuous backtitration technique combined with CEC measurements[END_REF][START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF][START_REF] Leroy | Influence of surface conductivity on the apparent zeta potential of amorphous silica nanoparticles[END_REF][START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF].

Amorphous silica

The negative surface charge density of amorphous silica arises from the deprotonation of the silanol surface sites >SiOH to form >SiO  surface species. It attracts cations in the Stern layer. Surface charge density measurements of amorphous silica nanoparticles, Degussa Aerosil 380 of average size 7 nm, in contact with NaCl or CaCl 2 electrolytes [START_REF] Dove | Surface charge density on silica in alkali and alkaline earth chloride electrolyte solutions[END_REF], were used to adjust the adsorption equilibrium constants of the Na + and Ca 2+ counter-ions in the Stern layer (Table 1). For that purpose, the following cost function, based on the least-square method [START_REF] Caceci | Fitting curves to data. The Simplex algorithm is the answer[END_REF], was minimized for each electrolyte using the gradient method:

  2 00 1 ( ) ( ) P i cal obs i R w Q i Q i    , ( 18
)
where P is the number of surface charge density measurements for each electrolyte, w i is a weighting coefficient (we took w i = 1), and Q 0cal and Q 0obs are the calculated and measured surface charge density, respectively (C m -2 ). The other parameters of the BSM, which are the total surface site density  S , the equilibrium adsorption constant of the H + ion and the C 1 capacitance, were set following the work of [START_REF] Hiemstra | Multisite proton adsorption modelling at the solid/solution interface of (hydr)oxides: a new approach. II. Application to various important (hydr)oxides[END_REF]:  S = 4.6 sites nm -2 , logK Si,H = 7.5 and C 1 = 3.3 F m -2 (Table 2). The detailed procedure to compute the surface charge density is explained in [START_REF] Leroy | Influence of surface conductivity on the apparent zeta potential of amorphous silica nanoparticles[END_REF]. It was calculated as a function of the computed surface sites densities i  using [START_REF] Davis | Surface Ionization and Complexation at Oxide-Water Interface .1. Computation of Electrical Double-Layer Properties in Simple Electrolytes[END_REF]Leroy & Revil 2004):

  0 SiO SiO M cal Qe            , ( 19
)
where

SiO    and SiO M   
are respectively the surface site densities of deprotonated silanols and adsorbed counter-ions in the Stern layer (sites m -2 ), which were computed with Matlab using the approach of [START_REF] Leroy | Influence of surface conductivity on the apparent zeta potential of amorphous silica nanoparticles[END_REF].

Table 1. Ion adsorption reactions on amorphous silica.

Surface complexation reactions "0-plane" "-plane" Adsorption constants

Si 7.5 1 Log(Na + adsorption constant) LogK Si,Na -3 2 Log(Ca 2+ adsorption constant) LogK Si,Ca -0.5 2 Total surface site density  S (nm -2 ) 4.6 1 Inner capacitance C 1 (F m -2 ) 3.3 1 1 From [START_REF] Hiemstra | Multisite proton adsorption modelling at the solid/solution interface of (hydr)oxides: a new approach. II. Application to various important (hydr)oxides[END_REF]. 2 Optimized according to the surface charge density measurements of [START_REF] Dove | Surface charge density on silica in alkali and alkaline earth chloride electrolyte solutions[END_REF].

O H Si O H      +1 0 K Si,H Si O Na Si O Na          0 +1 K Si,Na 22 Si O Ca Si O Ca          0 +2 K Si,Ca
Predicted surface charge densities of amorphous silica nanoparticles are showed in Fig. 2a. Our surface complexation model reproduces very well increasing magnitude of the negative surface charge density when pH and salinity increases, observed especially in the case of CaCl 2 electrolyte because of the adsorption constant of Ca 2+ ions higher than for Na + ions in the Stern layer (Table 2). Negative zeta potentials predicted for the chemical conditions of electrical potential calculations inside cylinders are presented in

Basal surface of montmorillonite

On the contrary to amorphous silica, we do not consider hydroxyl surface sites exchanging protons with the surrounding electrolyte for montmorillonite because we focus on the electrochemical properties of the basal surface, the dominating surface in terms of specific surface area and electrochemical properties [START_REF] Leroy | Modeling the composition of the pore water in a clay-rock geological formation (Callovo-Oxfordian, France)[END_REF][START_REF] Leroy | Spectral induced polarization of Namontmorillonite dispersions[END_REF][START_REF] Tournassat | Influence of montmorillonite tactoid size on Na-Ca cation exchange reactions[END_REF]. The negative and permanent surface charge of the basal surface of montmorillonite originates from the isomorphic substitutions inside the crystal, for instance, the replacement of Fe 3+ by Mg 2+ or Fe 2+ ions in the octahedral sheet [START_REF] Grim | Applied Clay Mineralogy[END_REF]. It is responsible for the presence of >X  surface sites that adsorb counter-ions, H + , Na + or Ca 2+ ions here, in the Stern layer (Table 3). The surface charge density of montmorillonite (Mt) was calculated according to the measured CEC (meq g -1 ), which approximately corresponds to the negative structural charge, divided by the total specific surface area of the basal surface, SSA b (m 2 g -1 ), which can be given by crystallography or measured, using the following equation [START_REF] Leroy | Modeling the composition of the pore water in a clay-rock geological formation (Callovo-Oxfordian, France)[END_REF][START_REF] Okay | Spectral induced polarization of clay-sand mixtures: Experiments and modeling[END_REF]

: A 0 3 N CEC 10 b e Q SSA  . ( 20
)
The Mt surface charge density, Q 0 , was set to 0.15 C m -2 according to Eq. ( 20), using a measured CEC of 1.24 meq g -1 and a specific surface area of the basal surface of 750 m 2 g -1 [START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF]. The C 2 capacitance was set to 5.5 F m -2 and no C 1 capacitance was considered because no surface complexation reaction was assumed to occur at the "0-plane" [START_REF] Leroy | Modeling the composition of the pore water in a clay-rock geological formation (Callovo-Oxfordian, France)[END_REF][START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF]. Adsorption equilibrium constants of sodium and calcium ions in the Stern layer were directly taken from the works of [START_REF] Gaucher | A robust model for pore-water chemistry of clayrock[END_REF], [START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF], and [START_REF] Tournassat | Modeling the Acid-Base Properties of Montmorillonite Edge Surfaces[END_REF] (Table 4).

Table 3. Ion adsorption reactions on the basal surface of montmorillonite. Surface complexation reactions "0-plane" "-plane" Adsorption constants 5.5 1 1 From Eq. (20). 2 From [START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF] according to electrophoretic mobility measurements and molecular dynamics simulations [START_REF] Sondi | Electrokinetics of pure clay minerals revisited[END_REF][START_REF] Tournassat | Comparison of molecular dynamics simulations with triple layer and modified Gouy-Chapman models in a 0.1 M NaCl-montmorillonite system[END_REF][START_REF] Bourg | Molecular dynamics simulations of the electrical double layer on smectite surfaces contacting concentrated mixed electrolyte (NaCl-CaCl 2 ) solutions[END_REF]. 3 From [START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF] and [START_REF] Tournassat | Modeling the Acid-Base Properties of Montmorillonite Edge Surfaces[END_REF] using K Mt,H = K Mt,Na K Mt,NaH with K Mt,NaH = 10 0.5 for the exchange of adsorbed Na + by H + ion. 4 From [START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF] and [START_REF] Gaucher | A robust model for pore-water chemistry of clayrock[END_REF] using K Mt,Ca = K Mt,Na ²K Mt,NaCa with K Mt,NaCa = 10 0.6 for the exchange of adsorbed Na + by Ca 2+ ion. Note that >X -surface site represents the negative surface site arising from isomorphic substitutions in the crystal of montmorillonite; 2>X -means that there are two >X - surface sites.

X Na X Na          0 +1 K Mt,Na X H X H          0 +1 K Mt,H 2 2 2 X Ca X Ca      0 +2 K Mt,Ca
Finally, the approach of [START_REF] Leroy | Modeling the composition of the pore water in a clay-rock geological formation (Callovo-Oxfordian, France)[END_REF] was used to compute the negative zeta potential of the basal surface of montmorillonite at pH = 7 (neutral condition) as a function of NaCl or CaCl 2 concentration in the salinity range 1 to 100 mM (more details on the calculation procedure is given in [START_REF] Leroy | Modeling the composition of the pore water in a clay-rock geological formation (Callovo-Oxfordian, France)[END_REF]) (Fig. 3). As in the case of amorphous silica, predicted zeta potential increases in magnitude when salinity decreases, especially for NaCl electrolyte because of weaker adsorption of Na + than Ca 2+ ions in the Stern layer (Table 4). Computed zeta potentials also largely exceed 25.7 mV.

Electrical potential profiles

The main output of the surface complexation modelling is the -potential (as function of pH and salinity) associated to the considered system. Its values for amorphous silica (pH = 7 and pH = 9) and the basal surface of montmorillonite (pH = 7) in contact with NaCl or CaCl 2 electrolyte are reported in Table 5. The so-determined values of the -potential are then used as input to solve the full P-B equation (Eqs. 12-15). We took two different pH for amorphous silica because of its surface charge becoming more negative when pH increases (Fig. 2a).

Table 5. Zeta potentials () of amorphous silica and basal surface of montmorillonite computed from our surface complexation models.

Salinities (mM)

NaCl

CaCl We also defined another parameter, , to characterize the thickness of fully developed diffuse layer, , relative to the cylinder radius:

0 r   , (21)
where  = 5 [START_REF] Manciu | Specific ion effects via ion hydration: I. Surface tension[END_REF][START_REF] Leroy | Predicting the surface tension of aqueous 1:1 electrolyte solutions at high salinity[END_REF][START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF]. Debye lengths, , computed as a function of salt concentration (Eq. 5) and ratios between diffuse layer thickness and r 0 (Eq. 21) are reported in Table 6. Our calculations show that overlapping diffuse layers may appear for salinities of 1 and 10 mM NaCl and CaCl 2 and 100 mM NaCl when r 0 = 5 nm and for 1 mM NaCl when r 0 = 50 nm because  ~ 1 and  > 1. Results of the modelling for a cylinder of radius r 0 = 5 nm are presented in Fig. 4 (amorphous silica) and Fig. 5 (montmorillonite). The linearized P-B equation significantly overestimates the electrical potential computed from the full P-B equation, except when the salinity is high (100 mM), because of the high zeta potential magnitudes and interacting diffuse layers. Observed effect increases when salinity decreases and pH increases for amorphous silica, and is particularly strong when calcium ion is in solution because of the approximation  << kT/(2e)  12.8 mV for Ca 2+ (Debye-Hückel approximation of the Poisson-Boltzmann equation, Eq. 2). Note that the Matlab computations for the full P-B equation were checked using a partial differential equations (PDE) solver based on the finite-element method (the Electrostatics module of COMSOL Multiphysics™ 3.5). Figure 5. Electrical potential profiles in montmorillonite nanotubes computed from our code (full P-B equation) (solid lines) and the linearized P-B equation (dashed lines) (r 0 = 5 nm).

When the radius of the cylinder increases from 5 to 50 nm, the linearized P-B equation only significantly overestimates the electrical potential computed from the full P-B equation when the salinity of the electrolyte is very low (1 mM) because of the high magnitude of the zeta potential (Fig. 6 and Fig. 7). At the center of the pore, the electrical potential is not significantly overestimated because the electroneutrality condition nearly occurs (1 mM NaCl) or applies (other salinities). When the radius of the cylinder is 500 nm, the electroneutrality condition is always respected in the bulk electrolyte at the center of the pore. Nevertheless, the electrical potential calculated from the full P-B equation is still overestimated by the linearized P-B equation when zeta potential is high, especially for CaCl 2 electrolyte (Fig. 8 and Fig. 9). Figure 9. Electrical potential profiles in montmorillonite nanotubes computed from our code (full P-B equation) (solid lines) and the linearized P-B equation (dashed lines) (r 0 = 500 nm).

Our calculations show that the linearized P-B equation may overestimate the electrical potential predicted by the full P-B equation for highly charged nanocylinders.

For such nanopores, the P-B equation may also have some shortcomings such as not considering the fine size of the ion, specific ion-surface interactions, attractive ion-ion interactions and ion-ion correlations [START_REF] Borukhov | Steric effects in electrolytes: A modified Poisson-Boltzmann equation[END_REF][START_REF] Wernersson | Charge Inversion and Ion-Ion Correlation Effects at the Mercury/Aqueous MgSO4 Interface: Toward the Solution of a Long-Standing Issue[END_REF][START_REF] Bonthuis | Beyond the Continuum: How Molecular Solvent Structure Affects Electrostatics and Hydrodynamics at Solid-Electrolyte Interfaces[END_REF]. Despite these shortcomings, the P-B equation is still widely used because it can describe the electrostatic behaviors of most charged systems when surfaces are separated by a distance superior to 5 nm [START_REF] Ruiz-Cabello | Accurate Predictions of Forces in the Presence of Multivalent Ions by Poisson-Boltzmann Theory[END_REF].

Its relative simplicity makes it more useable than more elaborate theories, which need more parameters and do not necessarily agree with each other [START_REF] Vlachy | Ionic effects beyond Poisson-Boltzmann theory[END_REF][START_REF] Grochowski | Continuum molecular electrostatics, salt effects, and counterion binding-A review of the Poisson-Boltzmann theory and its modifications[END_REF][START_REF] Ben-Yaakov | Beyond standard Poisson-Boltzmann theory: ion-specific interactions in aqueous solutions[END_REF][START_REF] López-García | Influence of the finite size and effective permittivity of ions on the equilibrium double layer around colloidal particles in aqueous electrolyte solution[END_REF]. There is still not a universal theory supplanting the P-B equation to describe ion and water distribution at interfaces. Furthermore, our improved electrical potential profiles compared to those from the linearized P-B equation must be considered with caution because of the assumptions inherent to water flow and solute transport modelling at the macroscopic (laboratory) scale. Considering for instance pores as cylinders and the upscaling procedure of the model predictions to the laboratory scale may also bring some significant uncertainties that render the improvement considering the full P-B equation more negligible when simulating the electrical properties of charged materials at the macroscopic scale.

CONCLUSIONS AND PROSPECTS

We have developed a short Matlab® code to numerically solve the Poisson-Boltzmann equation inside a charged cylinder filled by an electrolyte containing any type of ions. The input parameters of our code are the zeta potential and the chemical composition of the bulk electrolyte. Unlike the linearized P-B equation with the Debye-Hückel approximation, our code is not limited to low zeta potentials of magnitude inferior to 25.7 mV at a temperature of 25 °C.

We applied our code to simulate the electrochemical properties of amorphous silica and montmorillonite nanotubes containing a NaCl or CaCl 2 aqueous electrolyte. For that purpose, the zeta potentials were computed using well-established surface complexation models that consider the Stern and diffuse layer at the surface of the mineral. We show that the electrical potential inferred from the full P-B equation is overestimated by the linearized P-B equation when the zeta potentials are high (above 25.7 mV at a temperature of 25 °C) and the cylinder radius goes to the nm range where diffuse layers may overlap, especially for CaCl 2 electrolyte. Diffuse layers overlap when the cylinder radius is smaller or equal to approximately five Debye lengths, which correspond for instance to approximately 50, 15 nm and 30, 10 nm for respectively 1, 10 mM NaCl and CaCl 2 at a temperature of 25 °C. Underground, these conditions typically occur for highly charged nanopores containing a dilute aqueous electrolyte (ionic strength < 0.1 M) such as those of clays and low-pH cements.

As a perspective, it would be very interesting to combine our P-B code with transport codes in electrokinetics and hydrogeophysics that consider the electrochemical properties of charged cylinders [START_REF] Daiguji | Ion transport in nanofluidic channels[END_REF][START_REF] Jougnot | Derivation of Soil-Specific Streaming Potential Electrical Parameters from Hydrodynamic Characteristics of Partially Saturated Soils[END_REF]Bücker & Hördt 2013a). Compared to the use of the linearized P-B equation, this combination would improve, for instance, the simulation of electro-osmosis in highly charged nanopores in nanofluidics and of the self-potential and complex conductivity response of clayey rocks and low-pH cements in hydrogeophysics. In addition, to better describe interacting diffuse layers, it would be relevant to modify our P-B code to consider as input parameter the surface charge density instead of the zeta potential.

ACKNOWLEDGMENTS

Poisson equation for the electrostatic potential . [START_REF] Boltzmann | Wissenschaftliche Abhandlungen[END_REF] showed that it is possible to relate the ion concentration to the electrostatic potential at the interface when the medium is in or close to thermodynamic equilibrium. In that case, the chemical potentials µ (J) of the ion and solvent are the same at the interface and in the bulk solution [START_REF] Lyklema | Fundamentals of Interface and Colloid Science[END_REF]Revil & Leroy 2004).

This yields for the chemical potentials of the ion: where the superscript R for the chemical potentials corresponds to the reference (standard) state, which is the point of zero charge for the interface [START_REF] Sverjensky | Prediction of surface charge on oxides in salt solutions: Revisions for 1 : 1 (M+L-) electrolytes[END_REF] and the unit molar concentration of the ion at hypothetical infinite dilution for the bulk solution [START_REF] Lyklema | Fundamentals of Interface and Colloid Science[END_REF]. The quantity i a is the activity coefficient of the ion (dimensionless), which is written as: The input parameters are the radius r 0 (µm), the zeta-potential  (mV), the number of ionic species M, the sign of the charge times the valence s i and the concentration i C

(mol L -1 ) for each ion type. Note that the water temperature can also be changed in the program for a temperature between 2 and 87 °C. The dielectric permittivity of water is calculated according to the measurements reported in Lide (1990) at a pressure of one bar. In addition, to correctly solve the P-B equation, the bulk electrolyte must be electroneutral (the chemical composition must satisfy Eq. ( 3)).

The function f_PBE_solve is: Note that the initial solution (defined using bvpinit) is a vector of size 1000. This size

Figure 1 .

 1 Figure1. Sketch of the surface complexation models to compute the zeta potential () for amorphous silica (BSM, a., modified from[START_REF] Leroy | Influence of surface conductivity on the apparent zeta potential of amorphous silica nanoparticles[END_REF]) and montmorillonite (TLM, b., modified from[START_REF] Leroy | The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects[END_REF]. The symbols , Q, and C respectively represent the electrical potential, surface charge density, and capacitance. Note the different types of surface sites, with the silanols >Si-OH for amorphous silica and >X - resulting from isomorphic substitutions in the solid for montmorillonite.

Fig. 2b .

 2b Fig. 2b. Calculated zeta potential increases in magnitude when pH increases and salinity decreases, especially for NaCl electrolyte because of weaker adsorption of Na + than Ca 2+ ions in the Stern layer. Computed zeta potentials of amorphous silica also largely

Figure 2

 2 Figure 2. a: Surface charge density of amorphous silica nanoparticles (Degussa Aerosil 380 of average size 7 nm) versus pH in aqueous solutions containing NaCl or CaCl 2 electrolyte. The measurements from acid-base potentiometric titrations of Dove & Craven (2005) are represented by the symbols and the results of the modelling by the solid lines. b: Zeta potential of amorphous silica nanoparticles predicted by the basic Stern model versus pH at different NaCl and CaCl 2 concentrations. The gray shaded area represents electrical potentials where the linearized P-B equation is applicable (||  kT/e = 25.7 mV at a temperature of 298.15 K, Eq. 2).

Figure 3 .

 3 Figure 3. Zeta potential of the basal surface of montmorillonite predicted by the extended Stern model versus concentration of NaCl or CaCl 2 electrolyte at pH = 7. The gray shaded area represents electrical potentials where the linearized P-B equation is applicable (||  kT /e = 25.7 mV at a temperature of 298.15 K, Eq. 2).

Figure 6 .Figure 7 .

 67 Figure 6. Electrical potential profiles in amorphous silica nanotubes computed from our code (full P-B equation) (solid lines) and the linearized P-B equation (dashed lines) (r 0 = 50 nm).

  figure (1) plot(r*1e6,potential*1000,'r-','LineWidth',2) xlabel('radial distance (µm)') ylabel('potential (mV)') box on

Table 2 .

 2 Parameters of the basic Stern model for amorphous silica.

	Parameters	Values
	Log(H + adsorption constant) LogK Si,H

Table 4 .

 4 Parameters of the extended Stern model for the basal surface of montmorillonite.

	Parameters	Values
	Surface charge density Q 0 (C m -2 ) Log(Na + adsorption constant) LogK Mt,Na Log(H + adsorption constant) LogK Mt,H	0.15 1 0.05 2 0.55 3
	Log(Ca 2+ adsorption constant) LogK Mt,Ca Outer capacitance C 2 (F m -2 )	0.7 4

Table 6 .

 6 Debye lengths and ratios of thickness of diffuse layer to cylinder radius computed as a function of salt concentration (T = 298.15 K) (Eqs. 5 and 21).

	Salt concentration (mM)	1	10	100
		NaCl	(nm)	9.61	3.04	0.96
		CaCl	2	(nm)	5.55	1.76	0.56

* Diffuse layer overlapping.

The authors sincerely thank the Associate Editor, Gary Egbert, and the two reviewers, Andreas Hördt and Matthias Bücker, for helping them to significantly improve the quality of the manuscript. The work of P. Leroy was funded by the Horizon 2020 CEBAMA (Contract Number: 662147), Andra UP-Transfert (coordinated by Pierre Henocq) and NEEDS (Zetaclay and TRANSREAC) projects. A. Maineult acknowledges the support of the Russian Science Foundation (Grant No. 17-17-01160 'Physical-chemical models of Induced Polarization and Self-Potential with application to pumping test experiments'). Philippe Leroy warmly thanks Francis Claret for his support and Christophe Tournassat for fruitful discussions.

Appendix A: origins of the Poisson-Boltzmann equation

The P-B equation is derived from the works of [START_REF] Poisson | Mémoire Sur La Théorie Du Magnétisme[END_REF] for the electrical potential and [START_REF] Boltzmann | Wissenschaftliche Abhandlungen[END_REF] for the ion distributions at the interface between two phases such as mineral and water. In electrostatics, Poisson's equation is derived from the differential form of Gauss' law, which describes the divergence of the dielectric displacement D due to the volumetric charge density  of the medium (C m -3 ) [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF]:

The dielectric displacement is:

where E is the electrical field (V m -1 ), which is defined by:

Eq. ( A2) shows that the effect of the electrical field on the dielectric displacement is weighted by the dielectric permittivity, which measures the capacity of a solvent to affect the electrical field strength, i.e. the capacity of the molecular dipoles to align themselves to cancel part of the electrical field.

The Poisson equation is obtained by combining Eqs. (A1) to (A3) and assuming that the dielectric permittivity of the solvent is constant [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF]. This yields:

The volumetric charge density is:

where i C  is the ion concentration, for instance, at the interface between two phases such as mineral and water. Eq. (A5) shows that it is necessary to know i C  to solve the

Appendix B: the Matlab® code

Our numerical solution of the P-B equation is based on the Matlab® bvp4c function for solving boundary value problems. For implementation, Eq. ( 13) is rewritten as:

The term with the sums has to be defined as a singular term in bvp4c. The boundary conditions ( 14) and ( 15) become:

The main program is: should sometimes be increased if the code is unable to compute the Jacobian properly.

The function f_PBEmts is:


Ahmed Wahrani Khirech

Jean-François Largeau
email: jean-francois.largeau@icam.fr

Mohand Tazerout

1-Context and aims

► A waste : mainly in Europe grass clippings represent a residues and so cost for disposing. ► Conventional valorisation : leave on lawn, compost or in anaerobic digestion ► Aim of the work : study others ways for energy valorisation by dry processes: Biomass energy (energy storage) and fuel&char by pyrolysis

Guo, F., Dong, Y., Lv, Z., Fan, P., Yang, S., and Dong, L. (2015). Pyrolysis kinetics of biomass (herb residue) under isothermal condition in a micro fluidized bed. Energy Convers. Manag. 93,[367][368][369][370][371][372][373][374][375][376].

►

  and methods -Process for energy storage -► Shredders : improve the moisture diffusion ► Pellet : pellet press developed by Icam (Ø 6mm, L≈15mm) ► Log : hydraulic press logs of grass + starch or paper (Ø 50mm, L=150mm) ► Drying : ambient air/ tower with forced ventilation 5-Conclusions and Perspectives ► Dry&shape process defined to stabilize, store and handling for energy storage (thermal energy) ► Kinetic degradation studied : parameters for dry process valorization identified ► Work in progress : identification of conversion potential into fuel and char by pyrolysis ► Calorimetric bomb : PARR model 6200 ► TGA system :SETARAM Setsys-Evo ► Pyrolyser : batch process developed by Icam ► Kinetic method : Arrhenius law with FWO model ► Efficiency : Ratio between thermal energy of log/pellet prod and energy spent and >15 ► Mechanic handling : ok with/without starch, ok with paper (if paper < 4%wt) Slow pyrolysis at 600°C : Yields: 69% char, 21%liquid,10%gas 3b-Materials and methods -Energy valorisation -

Pellet press 4a-Results -Biomass characteristics, energy potential -

		Grass -this study -	Herbs residues Guo&Al	Douglas wood
	%H	50-80	N/A	N/A
	LHV (kWh)	4,6	N/A	5,5
	C wt%	74	42	48,1
	H wt%	6,6	6,2	5,5
	O wt%	17,5	47,4	45,8
	N wt%	0,4	1,86	0,4
	S wt%	1,5	0,15	0
	H/C	0,09	0,15	0,11
	O/C	0,24	1,13	0,95
	Ashes (%)	7,0	10,8	0,25
			Grass -this study-	Herbs residues Guo&Al
		A [s -1 ]	1,06E+06 1,00E+06
		Ea [J/mol]	146	105-130


C Peucheret

M E Chaibi

L Bramerie

K Hassan

M Gay

D Erasme

Novel Integrated Sources for Short-reach Transmission Over Dispersive Channel

Keywords:

We review our recent results on Si or III-V integrated devices in novel transmitter architectures for IM-DD dispersion-uncompensated links. Methods relying on chirp engineering or SSB generation are demonstrated for multi-and single-carrier modulation formats.

Introduction

Optical transmission systems employing intensity modulation (IM) and direct detection (DD) have recently been the object of renewed interest following the development of data-center interconnects (DCIs), where considerations of cost, footprint, and energy efficiency are critical. While photonic integration should be able to fulfil these requirements, new constraints are being introduced as a result of the evolution of data center networks towards more distributed architectures, with target link lengths of 10-120 km. In this segment, the used of dense wavelengthdivision multiplexing (DWDM) in the C-band is being preferred due to the availability of legacy components [START_REF] Nagarajan | Silicon photonics-based 100 Gbit/s, PAM4, DWDM data center interconnects[END_REF]. The optical links therefore become dispersion-limited and proper mitigation techniques need to be implemented in order to avoid the use of costly and bulky optical dispersion compensation modules (DCMs). In some scenarios, the inherent latency of some digital signal processing (DSP) functionalities, including some forward error correction (FEC) schemes, also needs to be avoided.

Over the past few years, several technical solutions have been proposed and demonstrated for such dispersionlimited links employing IM and DD, fueling the debate between proponents of multi-carrier solutions employing orthogonal frequency division multiplexing (OFDM) and single-carrier approaches, typically making use of 4-level pulse amplitude modulation (PAM-4). Our work contributes to this on-going debate by proposing and experimentally demonstrating new approaches for transmission over a dispersive fiber link using either OFDM or PAM-4 modulation. The proposed techniques either rely on single-sideband (SSB) modulation, or on directly-or externally-modulated laser frequency chirping properties. In particular, we show that the effect of fading dips, which are inherent to DD systems, can be circumvented by optimized SSB filtering of a directly-modulated OFDM signal using silicon micro-ring resonators (MRRs) in series [START_REF] Chaibi | Directly-modulated IM/DD OFDM transmission over 100-km SSMF using SSB filtering with two silicon micro-ring resonators[END_REF]. We also propose a novel transmitter structure exploiting the frequency chirping properties of two ring resonator modulators (RRMs) in series that have opposite frequency detunings with respect to a continuous wave (CW) laser frequency [START_REF] Chaibi | Multiple-band OFDM transmission exploiting the frequency chirping properties of silicon ring-resonator modulators[END_REF]. Finally, we demonstrate the first generation of PAM-4 SSB signals using a dual electro-absorption modulated laser (D-EML) [START_REF] Chaibi | Dispersion-uncompensated transmission of NRZ and PAM-4 single-sideband signals using D-EML[END_REF].

Multi-carrier transmission

For analogue modulation, a well know-effect of dispersion is the existence of frequency-dips in the small-signal frequency response of the channel, measured from the modulator (or directly-modulated laser -DML) driving signal input to the photodiode output, when double sideband (DSB) modulation is used. This effect, which limits the usable bandwidth when OFDM is employed, is obviously not present for SSB modulation. Optical filtering by MRRs in series has been shown to offer a good compromise between sideband suppression ratio and required guard-band in order to suppress one of the modulated sidebands while preserving the carrier. It was shown in [START_REF] Chaibi | Directly-modulated IM/DD OFDM transmission over 100-km SSMF using SSB filtering with two silicon micro-ring resonators[END_REF] that a series of two silicon MRRs was sufficient to efficiently suppress one of the sidebands of a 14.7-Gbps OFDM signal made from 111 out of 128 subcarriers modulated with 8-state quadrature amplitude modulation (QAM), and transmit it over 100-km standard single-mode fiber (SSMF), as illustrated in Fig. 1. Thanks to the tremendous progress of heterogeneous integration of III-V semiconductor devices on silicon, for instance through die or wafer bonding, the scheme is furthermore potentially integrable.

An alternative approach, where the passive MRRs need to be substituted with carrier depletion-based RRMs, is illustrated in Fig. 2(a) [START_REF] Chaibi | Multiple-band OFDM transmission exploiting the frequency chirping properties of silicon ring-resonator modulators[END_REF]. It relies on the observation that, under small signal modulation, moving from a positive to a negative frequency detuning () of the RRM resonance with respect to the frequency of the CW laser induces a

Single-carrier transmission

The generation of PAM-4 SSB signals has also recently been given consideration, essentially using complex inphase and quadrature or dual-drive Mach-Zehnder modulators. In [START_REF] Chaibi | Dispersion-uncompensated transmission of NRZ and PAM-4 single-sideband signals using D-EML[END_REF], we have demonstrated PAM-2 and PAM-4 SSB signal generation relying on joint frequency modulation (FM) and intensity modulation (IM) in the DFB and electro-absorption modulation (EAM) sections, respectively, of a D-EML. Indeed, it can be shown that an SSB signal can be obtained when a proper relation between the IM and FM modulation indices is satisfied, as well as with a proper phase shift, at all frequencies, between the IM and FM driving signals [START_REF] Kim | EML-based optical single sideband transmitter[END_REF]. The implementation is however challenging for broadband modulation. We used a monolithically integrated D-EML (Fig. 3(a)) [START_REF] Erasme | The dual-electroabsorption modulated laser, a flexible solution for amplified and dispersion uncompensated networks over standard fiber[END_REF] together with a simple pre-emphasis scheme to compensate for the residual IM introduced by the DFB section and correct its frequency response. The resulting SSB spectrum is shown in Fig. 3(b) for PAM-2 modulation at 10.7 Gbps. Figure 3(c) shows the results of BER measurements for PAM-4 at 10.7 Gbd in back-to-back as well as after 20-km and 50-km SSMF for both DSB and SSB modulation, respectively. DSB modulation outperforms SSB modulation in the back-to-back configuration with about 5-dB power penalty at a BER of 10 -9 , which is linked to an extinction ratio degradation that is inherent to the pre-emphasis technique. After 20-km SSMF propagation, the BER performance of the DSB signal is dramatically degraded with the occurrence of BER floors at a BER level of 10 -4 .

On the other hand, SSB signals provide acceptable performance even after 50-km SSMF with the occurrence of BER floors at a level as low as 10 -7 . The eye diagrams of Fig. 3

Conclusion

The use of new integrated transmitter architectures for dispersion-tolerant transmission of single-or multi-carrier modulation in the C-band has been presented. SSB OFDM signals have been generated thanks to optimized filtering with MRRs in series, while a monolithically integrated D-EML has been used to generate SSB PAM-4. A new architecture exploiting the frequency chirping behavior of RRMs has also been presented for multicarrier modulation. All these schemes are potentially integrable and scalable in bandwidth.

Fig. 2 (Fig. 2 (

 22 Fig. 2 (a) Principle of OFDM SSB signal transmission using a DML and passive MRRs in series. (b) Optical spectrum of DSB and SSB OFDM signals at 14.7 Gbps. (c) EVM performance for SSB and DSB signals back-to-back as well as after 100-km SSMF. ~ jump between the intensity modulation and the chirp-induced phase modulation. This, in turn, affects the smallsignal frequency responses of the channel, which are different for  detunings. An interesting configuration is when two complementary frequency responses exist, as shown in Fig. 2(b) for a 100-km SSMF link. In this case, the subcarriers of the OFDM signal can be separated into two groups of sub-bands, each group being applied to one single RRM according to its individual frequency response. An example of such distribution based on the errorvector magnitude (EVM) of the individual subcarriers is represented in Fig. 2(c) for quadrature phase-shift keying (QPSK) modulation at an overall bit rate of 15 Gbps (250 modulated subcarriers out of 256). Here 97% of the subcarriers have EVM values below -8.35 dB, corresponding to a bit-error-ratio (BER) of 4.4510 -3 . EVMs averaged over all subcarriers are represented in Fig. 2(d) in back-to-back configuration, after 100-km SSMF when all subcarriers are applied to a single RRM with either positive or negative detuning, as well as when the subcarriers are distributed between the two RRMs. The graph clearly illustrates the potential of the scheme where distributing the subcarriers between the two RRMs results in performance close to back-to-back, and below the FEC threshold.

Fig. 3 (

 3 Fig. 3 (a) Monolithically integrated D-EML (III-V Lab). (b) Optical spectrum of a PAM-2 DSB signal generated by applying 10.7 Gbps data to the electro-absorption section of the D-EML. (c) Comparison of BER curves for PAM-4 DSB and SSB signals in back-to-back configuration and after transmission over SSMF. Corresponding eye diagrams for PAM-4 (d) DSB back-to-back, (e) DSB after 20-km and (f) SSB after 50-km.