Christophe Tournassat
email: c.tournassat@brgm.fr

James A Davis

Christophe Chiaberge

Sylvain Grangeon

Ian C Bourg

Modeling the Acid-Base Properties of Montmorillonite Edge Surfaces

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Clay minerals are natural fine-grained particles, ubiquitous in terrestrial weathering environments, that strongly influence the permeability, mechanics, and pore water chemistry of soils, sediments, and rocks in which they are found [START_REF] Bourg | Sealing shales versus brittle shales: A sharp threshold in the material properties and energy technology uses of fine-grained sedimentary rocks[END_REF][START_REF] Tournassat | Natural and Engineered Clay Barriers[END_REF] . In particular, these minerals control the performance of natural and engineered clay barriers used in the isolation of landfills and contaminated sites and proposed for use in the geological storage of high-level radioactive waste [START_REF] Borisover | Chapter 2 -Adsorption of inorganic and organic solutes by clay minerals[END_REF][START_REF] Bourg | Chapter 6 -Self-diffusion of water and ions in clay barriers[END_REF] . A widely examined feature of clay minerals that influences the performance of these barriers 3 is their cation-exchange capacity, the result of isomorphic substitutions in the structure [START_REF] Bourg | Ion exchange phenomena[END_REF] . In addition to this cation-exchange capacity, clay minerals, including montmorillonite (the most studied swelling clay mineral) carry oxide-type functional groups on their edge surfaces. These oxide-type functional groups, though present in much smaller quantity than the cation exchange sites, dominate the adsorption of trace metals, oxyanions, and organic molecules [START_REF] Manning | Adsorption and stability of arsenic(III) at the clay mineralwater interface[END_REF][START_REF] Schlegel | Evidence for the nucleation and epitaxial growth of Zn phyllosilicate on montmorillonite[END_REF][START_REF] Strawn | The use of XAFS to distinguish between inner-and outersphere lead adsorption complexes on montmorillonite[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] , the dissolution and growth kinetics of clay minerals [START_REF] Marty | Dissolution kinetics of synthetic Na-smectite. An integrated experimental approach[END_REF][START_REF] Yang | Kaolinite dissolution and precipitation kinetics at 22 °C and pH 4[END_REF] , and the colloidal mechanics of clay particles [START_REF] Tombácz | Surface charge heterogeneity of kaolinite in aqueous suspension in comparison with montmorillonite[END_REF] , at least in some conditions.

In the case of simple oxide minerals, surface reactivity is strongly influenced by their net proton surface charge, which varies with pH through reactions of the following type, where >SOH n is a generic surface site [START_REF] Davis | Surface ionization and complexation at the oxide/water interface: I. Computation of electrical double layer properties in simple electrolytes[END_REF] :

> SOH ⇄ > SOH ିଵ + H ା (1)
The reaction described by Eq. ( 1) gives rise to a variable net proton surface charge density (σ H ) that depends on pH and ionic strength (I, dimensionless [START_REF] Solomon | The definition and unit of ionic strength[END_REF] ). In the case of simple oxide minerals, the last few decades have seen steady advances in the characterization of their proton surface chemistry through a combination of experiments (in particular, acid-base titration and electrophoretic mobility measurements), surface complexation model (SCM) calculations, and bond-valence theory predictions of the intrinsic acidity constants of surface functional groups 15- 19 . In the case of the oxide-type functional groups on the montmorillonite edge surfaces, however, σ H remains poorly understood, as shown by the wide range of reported values of the point of zero net proton charge (p.z.n.p.c., the pH value where σ H = 0) and by the variety of models that have been used to describe σ H [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] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF] . As described below, the difficulty in characterizing σ H on montmorillonite relates to the high permanent structural charge of the solid (σ 0 ≈ -1 mmol c ⋅g -1 ) and to the presence of two different surfaces (edge and basal surfaces) with very different properties. These properties have no equivalent in simple oxide minerals, but are widespread features of lamellar structures including vernadite, a phyllomanganate that controls the fate of trace metals in certain oxidized soils and sediments [START_REF] Bargar | Structural characterization of terrestrial microbial Mn oxides from Pinal Creek, AZ[END_REF][START_REF] Manceau | Ba and Ni speciation in a nodule of binary Mn oxide phase composition from Lake Baikal[END_REF] , green rust, a layered double hydroxide that influences the mobility of iron and anions in reduced soils [START_REF] Trolard | a new mineral of the pyroaurite-iowaite group: description and crystal structure[END_REF] , and synthetic phases widely used in materials chemistry, such as synthetic layered double hydroxides [START_REF] Wang | Arsenate adsorption by Mg/Al-NO3 layered double hydroxides with varying the Mg/Al ratio[END_REF] .

The challenges associated with characterizing the proton surface reactivity of montmorillonite were summarized a decade ago by Bourg et al. [START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF] . A first challenge is that montmorillonite particles undergo significant dissolution during acid-base titration measurements. This can be rendered near-negligible in the pH range from about 4.5 to 9.5 through a careful choice of clay pretreatment, storage, and titration procedures [START_REF] Duc | Temperature effect on the acid-base behaviour of Na-montmorillonite[END_REF][START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 1. Literature review[END_REF][START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 2. Evidence from continuous potentiometric titrations[END_REF][START_REF] Duc | Coupled chemical processes at clay/electrolyte interface: A batch titration study of Na-montmorillonites[END_REF][START_REF] Tournassat | The titration of clay minerals. Part I. Discontinuous backtitration technique combined to CEC measurements[END_REF] (see below). Unfortunately, most montmorillonite acid-base titration datasets do not fit the best practices established by Duc et al. [START_REF] Duc | Temperature effect on the acid-base behaviour of Na-montmorillonite[END_REF][START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 1. Literature review[END_REF][START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 2. Evidence from continuous potentiometric titrations[END_REF][START_REF] Duc | Coupled chemical processes at clay/electrolyte interface: A batch titration study of Na-montmorillonites[END_REF] . A second challenge is that acid-base titration measurements do not directly measure the net proton surface charge density σ H but rather δσ H , the change in net proton surface charge density relative to its initial value σ H,init (ߜߪ H = ߪ H -ߪ H,init ). Conversion of ߜߪ H to σ H values requires knowledge of σ H,init or of the p.z.n.p.c. at the conditions of interest [START_REF] Sposito | On points of zero charge[END_REF] . For simple oxide surfaces, this is achieved by measuring the sum of the adsorbed ion charge densities q i of all species except H + and OH -(∆q) and applying the charge balance relation [START_REF] Sposito | On points of zero charge[END_REF] :

ߪ 0 + ߪ H + Δݍ = 0 (2)
For montmorillonite, experimental uncertainties make it impossible to accurately determine ߪ H using Eq. ( 2), because ߪ H is small compared to ߪ 0 and Δݍ [START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF][START_REF] Tournassat | The titration of clay minerals. Part I. Discontinuous backtitration technique combined to CEC measurements[END_REF] . Alternatively, the p.z.n.p.c. of oxide surfaces is sometimes determined by assuming that it coincides with the point of zero salt effect (p.z.s.e., the pH value at which acid-base titration curves carried out at different ionic strengths intersect) or with the isoelectric point (i.e.p., the pH value where the electrophoretic mobility equals zero) [START_REF] Davis | Surface complexation modeling in aqueous geochemistry[END_REF] . These alternative methods, however, are neither strictly rigorous [START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF][START_REF] Sposito | On points of zero charge[END_REF] nor applicable to montmorillonite, for which a p.z.s.e. or i.e.p. are not observed [START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 1. Literature review[END_REF][START_REF] Sondi | Electrokinetics of pure clay minerals revisited[END_REF][START_REF] Tertre | Surface chemistry of kaolinite and Na-montmorillonite in aqueous electrolyte solutions at 25 and 60ẹgreC: Experimental and modeling study[END_REF][START_REF] Tombácz | Colloidal behavior of aqueous montmorillonite suspensions: the specific role of pH in the presence of indifferent electrolytes[END_REF] . Finally, the p.z.n.p.c. of montmorillonite is sometimes assumed to correspond to the pH value obtained by adding incremental amounts of dry solid to a solution [START_REF] Avena | Proton adsorption and electrokinetics of an argentinean montmorillonite[END_REF] , but this so-called "mass titration" technique, just like a standard acid-base titration, is sensitive to the initial protonation state of the solid [START_REF] Ẑalac | Application of mass titration to the point of zero charge determination[END_REF] .

A third challenge arises when attempting to develop a SCM for montmorillonite. An important assumption of SCMs is that the ratio of activity coefficients of different surface species is a function of the surface electrostatic potential ߰. For the reaction described by Eq. ( 1), the relationship is almost always expressed with a quasi-thermodynamic equilibrium equation of the form:

ܭ ୟ = ሾ> SOH ିଵ ሿሺH ା ሻ ሾ> SOH ሿ exp ൬ ߰ܨ- ܴܶ ൰ (3)
where K a is an intrinsic equilibrium constant, (H + ) is the thermodynamic activity of H + , square brackets denote mole fractions, ߰ is the surface electrostatic potential, F is the Faraday constant (96 485 C mol -1 ሻ, R is the ideal gas constant (8.314 J mol -1 K -1 ), and T is temperature (in K).

Closure of any SCM requires a model of ߰ as a function of surface charge, ionic strength, and other conditions. For this, most studies rely on the Gouy-Chapman relation, a mean-field theory prediction based on the Poisson-Boltzmann equation (PBE) of the relationship between surface charge density and surface potential on an infinite planar surface in contact with an ideal electrolyte solution [START_REF] Davis | Surface complexation modeling in aqueous geochemistry[END_REF][START_REF] Sposito | The surface chemistry of natural particles[END_REF] . Montmorillonite consists of flake-shaped, 1-nm-thick layers such that the narrow edge surface cannot be a priori treated as an infinite planar surface [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] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF] Bourg et al. [START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF] showed that the first three challenges outlined above could be resolved using information available a decade ago. The second challenge was resolved by using an SCM to predict both σ H,init (from reported conditions of clay pre-treatment) and σ H . The first challenge required identifying experimental datasets that followed that best practices established by Duc et al. [START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 1. Literature review[END_REF][START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 2. Evidence from continuous potentiometric titrations[END_REF][START_REF] Duc | Coupled chemical processes at clay/electrolyte interface: A batch titration study of Na-montmorillonites[END_REF] and that used montmorillonite samples pre-treated by repeated washing in a solution of well-known pH and ionic strength (as required to predict σ H,init ); the available database included only two datasets, those of Baeyens and Bradbury [START_REF] Baeyens | A mechanistic description of Ni and Zn sorption on Namontmorillonite. Part I: Titration and sorption measurements[END_REF] and Duc et al. [START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 2. Evidence from continuous potentiometric titrations[END_REF] . The third challenge was resolved by solving a two-dimensional version of the PBE near the edge surface of an isolated montmorillonite particle. The fourth challenge, however, could not be satisfactorily resolved, because of large disagreements between different bond-valence model predictions of the intrinsic pK a values of edge surface sites [START_REF] Bickmore | Ab initio determination of edge surface structures for dioctahedral 2:1 phyllosilicates: Implications for acid-base reactivity[END_REF][START_REF] Tournassat | The titration of clay minerals II. Structure-based model and implications for clay reactivity[END_REF] and, also, because of insufficient knowledge of the densities of different edge surface sites [START_REF] Tournassat | The titration of clay minerals II. Structure-based model and implications for clay reactivity[END_REF] . Eventually, Bourg et al. [START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF] concluded that experimental data on the acid-base chemistry of montmorillonite edge surfaces could not be predicted from first principles using existing knowledge.

The last decade has seen a significant number of new studies of the acid-base surface chemistry of montmorillonite. Several new potentiometric titration studies have been carried out [START_REF] Bogolepov | Simulation of U(VI) and Co(II) sorption on montmorillonite. Radiochemistry[END_REF][START_REF] Gu | Modeling the adsorption of Cd(II), Cu(II), Ni(II), Pb(II) and Zn(II) onto montmorillonite[END_REF][START_REF] Korichi | Sorption of uranium (VI) on homoionic sodium smectite experimental study and surface complexation modeling[END_REF][START_REF] Kriaa | Acid-base chemistry of montmorillonitic and beidelliticmontmorillonitic smectite[END_REF][START_REF] Rozalén | Surface chemistry of K-montmorillonite: Ionic strength, temperature dependence and dissolution kinetics[END_REF][START_REF] Tan | Characterization of Lin'an montmorillonite and its application in the removal of Ni2+ from aqueous solutions[END_REF] as well as one study that attempted to determine the i.e.p. of montmorillonite edge surfaces [START_REF] Pecini | Measuring the isoelectric point of the edges of clay mineral particles: The case of montmorillonite[END_REF] .

Several studies have provided new estimates of the edge specific surface area of montmorillonite particles [START_REF] Marty | Dissolution kinetics of synthetic Na-smectite. An integrated experimental approach[END_REF][START_REF] Forestier | Textural and hydration properties of a synthetic montmorillonite compared with a natural Na-exchanged clay analogue[END_REF] . Models of ψ edge have been refined using new Poisson-Boltzmann calculations [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] and Grand Canonical Monte Carlo (GCMC) simulations [START_REF] Delhorme | Acid-base properties of 2:1 clays. I. Modeling the role of electrostatics[END_REF] . Finally, ab initio MD simulations have been used to predict the intrinsic pK a values of montmorillonite edge surface sites [START_REF] Liu | Interfacial structures and acidity of edge surfaces of ferruginous smectites[END_REF][START_REF] Liu | Surface acidity of 2:1-type dioctahedral clay minerals from first principles molecular dynamics simulations[END_REF][START_REF] Liu | Temperature dependence of interfacial structures and acidity of clay edge surfaces[END_REF][START_REF] Liu | Acidity of edge surface sites of montmorillonite and kaolinite[END_REF][START_REF] Liu | Atomic-scale structures of interfaces between phyllosilicate edges and water[END_REF][START_REF] Liu | Atomic scale structures of interfaces between kaolinite edges and water[END_REF][START_REF] Tazi | Absolute acidity of clay edge sites from ab-initio simulations[END_REF] . Here, we revisit the conclusions of Bourg et al. [START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF] in light of these advances. First, we show that new knowledge obtained during the last decade provides a more coherent picture of the proton surface chemistry of montmorillonite. Then, we develop a state-of-the-art SCM for montmorillonite edge surfaces, and we note possibilities for future improvements. This new SCM is mainly based on a combination of the existing 2D Poisson-Boltzmann calculations, which are representative of clay edges geometry, with the development of a revised thermodynamic concept for the description of the edge sites, and with the use of pK a values that have been obtained from ab initio MD simulations.

Constraints on the proton surface chemistry of montmorillonite edges 2.1. Montmorillonite edge structure

Current knowledge of the densities of different types of montmorillonite edge functional groups remains limited by uncertainty regarding edge crystallographic orientations. Crystal growth theory calculations, surface Coulomb energy considerations, atomistic simulations, and in situ observation of dissolution kinetics using atomic force microscopy suggest the that most stable edge surfaces are perpendicular to [010] and [110] (or, equivalently, [1 ത 10ሿ) [START_REF] Bickmore | In situ atomic force microscopy study of hectorite and nontronite dissolution: implications for phyllosilicate edge surface structures and dissolution mechanisms[END_REF][START_REF] Bleam | The surface Coulomb energy and proton Coulomb potentials of pyrophyllite {010},{110},{100}, and {130} edges[END_REF][START_REF] Churakov | Ab initio study of sorption on pyrophyllite: Structure and acidity of the edge sites[END_REF][START_REF] Newton | Edge structure of montmorillonite from atomistic simulations[END_REF][START_REF] Newton | Molecular dynamics simulations of pyrophyllite edge surfaces: structure, surface energies, and solvent accessibility[END_REF][START_REF] White | Analysis and implications of the edge structure of dioctahedral phyllosilicates[END_REF] , where the Miller indices are assigned according to Churakov [START_REF] Churakov | Ab initio study of sorption on pyrophyllite: Structure and acidity of the edge sites[END_REF] (Figure 1). The two edge surfaces are sometimes referred to as the B and AC edges, respectively, following the seminal study by White and Zelazny [START_REF] White | Analysis and implications of the edge structure of dioctahedral phyllosilicates[END_REF] . Recent molecular dynamics (MD) and ab initio MD simulations of hydrated clay edges suggest that the surface normal to [110] may be significantly more stable than the surface normal to [010] in the presence of liquid water [START_REF] Newton | Edge structure of montmorillonite from atomistic simulations[END_REF][START_REF] Newton | Molecular dynamics simulations of pyrophyllite edge surfaces: structure, surface energies, and solvent accessibility[END_REF][START_REF] Kwon | Structure and stability of pyrophyllite edge surfaces: Effect of temperature and water chemical potential[END_REF] .

Montmorillonite edge surfaces are routinely assumed to have the same stoichiometry and structure as the bulk crystal, with minor bond-length relaxation to accommodate over-or undercoordinated surface O atoms [START_REF] Bickmore | Ab initio determination of edge surface structures for dioctahedral 2:1 phyllosilicates: Implications for acid-base reactivity[END_REF] . Recent MD and ab initio MD simulation results reveal a more complex picture, the most well-established finding being that cations in the octahedral layer can adopt a five-fold coordination [START_REF] Liu | Atomic-scale structures of interfaces between phyllosilicate edges and water[END_REF][START_REF] Newton | Edge structure of montmorillonite from atomistic simulations[END_REF][START_REF] Hadi | Modelling CEC variations versus structural iron reduction levels in dioctahedral smectites. Existing approaches, new data and model refinements[END_REF][START_REF] Lammers | Molecular dynamics simulations of cesium adsorption on illite nanoparticles[END_REF] .

Edge specific surface area

An important parameter in studies of montmorillonite edge reactivity is the edge specific surface area, a s,edge . Unfortunately, the N 2 gas adsorption method with the Brunauer-Emmett-Teller technique (N 2 -BET) quantifies only the external surface area of crystals (stacks of montmorillonite layers) after drying, which provides no information on a s,edge [START_REF] Michot | Chapter 2.10 -Surface Area and Porosity[END_REF] . The ethylene glycol monoethyl ether (EGME) adsorption method quantifies the total specific surface area of montmorillonite layers, a s , dominated by the basal surfaces [START_REF] Srodon | Surface area and layer charge of smectite from CEC and EGME/H 2 O-retention measurements[END_REF] . In principle, a s,edge equals the difference between the measured a s value and the theoretical basal specific surface area a s,basal [≈ 750 to 780 m 2 g - 1 70 ]. Unfortunately, EGME-derived a s values show a significant dependence on experimental conditions [START_REF] Michot | Chapter 2.10 -Surface Area and Porosity[END_REF] that precludes a precise calculation of a s,edge .

Direct measurements of a s,edge for montmorillonite have been achieved using two approaches.

The first consists in evaluating edge specific surface area by statistical analysis of particle morphology from atomic force microscopy (AFM) or transmission electron microscopy (TEM) images [START_REF] Cadene | Study of individual Namontmorillonite particles size, morphology, and apparent charge[END_REF][START_REF] Nadeau | The physical dimensions of fundamental clay particles[END_REF][START_REF] Tournassat | Nanomorphology of montmorillonite particles: Estimation of the clay edge sorption site density by lowpressure gas adsorption and AFM observations[END_REF] . Alternatively, the derivative isotherms summation (DIS) method distinguishes different clay surfaces in a single gas adsorption measurement based on differences in adsorption energy [START_REF] Michot | Chapter 2.10 -Surface Area and Porosity[END_REF] . Comparison of microscopic imaging and DIS results yields satisfactory agreement (Table S-1), with a possible slight overestimation of edge surface area by the DIS method [START_REF] Reinholdt | Morphological properties of vermiculite particles in size-selected fractions obtained by sonication[END_REF] .

Edge surface electrostatic potential

As noted above, SCMs of proton adsorption on montmorillonite require a model of the edge surface electrostatic potential, ψ edge . Most modeling studies have assumed that clay edges do not develop a significant electrostatic potential (ψ edge = 0) [START_REF] Tournassat | The titration of clay minerals II. Structure-based model and implications for clay reactivity[END_REF][START_REF] Bradbury | A mechanistic description of Ni and Zn sorption on Namontmorillonite. Part II: modeling[END_REF][START_REF] Grambow | Eu(III), Am(III), Cm, Ac(III), Tc(IV), Th, Zr, and U(VI) on MX80 bentonite: an experimental approach to assess model uncertainty[END_REF][START_REF] Montavon | Study of the interaction of Ni 2+ and Cs + on MX-80 bentonite; Effect of compaction using the "capillary method[END_REF][START_REF] Tertre | Modelling Zn(II) sorption onto clayey sediments using a multi-site ion-exchange model[END_REF][START_REF] Wolthers | Reactivity of bentonite. An additive model applied to uranyl sorption[END_REF] or that ψ edge follows the Gouy-Chapman relation for an infinite planar surface [START_REF] Tertre | Surface chemistry of kaolinite and Na-montmorillonite in aqueous electrolyte solutions at 25 and 60ẹgreC: Experimental and modeling study[END_REF][START_REF] Avena | Proton adsorption and electrokinetics of an argentinean montmorillonite[END_REF][START_REF] Gu | Modeling the adsorption of Cd(II), Cu(II), Ni(II), Pb(II) and Zn(II) onto montmorillonite[END_REF][START_REF] Avena | Acid-base behavior of clay surfaces in aqueous media[END_REF][START_REF] Avena | Proton binding at clay surfaces in water[END_REF][START_REF] Barbier | Adsorption of lead and cadmium ions from aqueous solution to the montmorillonite/water interface[END_REF][START_REF] Charlet | Cation adsorption on oxides and clays: the aluminum case[END_REF][START_REF] Ikhsan | Surface complexation modeling of the sorption of Zn(II) by montmorillonite[END_REF][START_REF] Kowal-Fouchard | Use of spectroscopic techniques for uranium(VI)/montmorillonite interaction modeling[END_REF][START_REF] Madrid | Description of titration curves of mixed materials with variable and permanent surface charge by amathematical model. 1. Theory. 2. Application to mixtures of lepidocrocite and montmorillonite[END_REF][START_REF] Marcussen | Nickel sorption to goethite and montmorillonite in presence of citrate[END_REF][START_REF] Stadler | Modeling of H + and Cu 2+ adsorption on calciummontmorillonite[END_REF][START_REF] Tombácz | Surface charge heterogeneity and aggregation of clay lamellae in aqueous suspensions[END_REF][START_REF] Wanner | The acid/base chemistry of montmorillonite[END_REF][START_REF] Zachara | Edge complexation reactions of cadmium on specimen and soil-derived smectite[END_REF] . The few studies that solved the PBE near clay edges for realistic geometries, however, showed that ψ edge differs strongly from zero and from the Gouy-Chapman model prediction and is sensitive to the stacking arrangement of the clay layers [START_REF] Kraepiel | On the acid-base chemistry of permanently charged minerals[END_REF][START_REF] Chang | The electrical double layer of a disked-shaped clay mineral particle: effect of electrolyte properties and surface charge density[END_REF][START_REF] Chang | The electrical double layer of a disked-shaped clay mineral particle: effect of particle size[END_REF][START_REF] Secor | Spillover of the diffuse double layer on montmorillonite particles[END_REF] . Bourg et al. [START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF] solved a two-dimensional form of the PBE near the edge surface of an isolated montmorillonite layer (consistent with exfoliated layers in Namontmorillonite suspensions at ionic strengths lower than I~0.2) and found that ψ edge was reasonably described by:

߰ܨ ௗ ܴܶ = ܣ ଵ asinh ቀܣ ଶ ൫ܳ ௗ + ܣ ଷ ൯ቁ (4)
where Q edge (C⋅m -2 ) is the charge at the edge and A 1 (unitless), A 2 (m -2 ⋅C) and A 3 (C⋅m -2 ) are parameters that were fitted with the ψ edge values obtained from the full resolution of the 2D PB equation. For montmorillonite at 25 °C, Tournassat et al. [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] refined the values of these parameters to: A 1 = 1.4 -1.2 log I, A 2 = 11 + log I, and A 3 = -0.02 × (-log I) 1.60 . This equation can be compared with the classical Gouy-Chapman model for infinite planar surfaces [START_REF] Davis | Surface ionization and complexation at the oxide/water interface: I. Computation of electrical double layer properties in simple electrolytes[END_REF] :

ிట ோ் = 2 asinhሺܤ • ߪሻ with ܤ = ଵ ඥ଼ఌఌ బ ோ்•ଵ•ூ (5) 12
where σ is the surface charge density (in m -2 ). A key prediction of Eq. 4 is that ψ edge is negative when the edge surface charge is zero (Figure 2) because of a spillover of negative electrostatic potential from the basal surfaces [START_REF] Chang | The electrical double layer of a disked-shaped clay mineral particle: effect of electrolyte properties and surface charge density[END_REF][START_REF] Secor | Spillover of the diffuse double layer on montmorillonite particles[END_REF] . A potential limitation to the accuracy of the PBE is that it uses the mean-field theory approximation, i.e., it neglects specific ion-ion and ion-surface site interactions. On infinite flat surfaces, the PBE is nevertheless known to adequately predict the structure of the electrical double layer except at high salinities or in the presence of multivalent counterions [START_REF] Sposito | The surface chemistry of natural particles[END_REF][START_REF] Henderson | Insights from theory and simulation on the electrical double layer[END_REF][START_REF] Tinnacher | Ion adsorption and diffusion in smectite: molecular, pore, and continuum scale views[END_REF] . In the case of montmorillonite, an opportunity to verify the validity of Eq. ( 4) is provided by GCMC simulations of the acid-base chemistry of individual clay particles where water was treated as a uniform dielectric continuum [START_REF] Delhorme | Acid-base properties of 2:1 clays. I. Modeling the role of electrostatics[END_REF] . To this end, we implemented Eq. ( 4) in PHREEQC [START_REF] Parkhurst | Description of input and examples for PHREEQC Version 3-a computer program for speciation,batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF] (the modification of the source code is made available in the supporting information file; a executable file can be sent upon request) and predicted the protonation of surface sites using the same site densities and pK a values as Delhorme et al. [START_REF] Delhorme | Acid-base properties of 2:1 clays. I. Modeling the role of electrostatics[END_REF] (PHREEQC scripts and database are available in the supporting information file). An excellent agreement was found between the two predictions (Figure 3), suggesting that Eq. ( 4) is valid over the entire range of experimental conditions where montmorillonite particles are made of single layers. This condition is met in experiments carried out with montmorillonite particles dispersed in a NaCl or NaClO 4 background electrolyte at ionic strengths lower than I~0.2 [START_REF] Méring | On the hydration of montmorillonite[END_REF][START_REF] Norrish | The swelling of montmorillonite[END_REF][100][101] .

Figure 3. Comparison of the GCMC simulations of Delhorme et al. [START_REF] Delhorme | Acid-base properties of 2:1 clays. I. Modeling the role of electrostatics[END_REF] (circles and squares) with an SCM that used Eq. ( 4) (solid lines) at three different ionic strengths in NaCl salt background.

Both models used the same site densities and intrinsic pK a values. Activity coefficients of solute species in the SCM were modeled using the extended Debye-Hückel formalism. Blue symbols and lines refer to aluminol sites having a pK a of 3.75 and a site density of 4.5 nm -2 ; red symbols and lines refer to silanol sites having a pK a of 8.0 and a site density of 5.2 nm -2 . Shaded areas correspond to conditions that cannot exist, i.e., domains where the specified ionic strength is exceeded because of high H + or OH -concentrations.

Intrinsic pK a values of edge surface sites

Until recently, theoretical estimates of the intrinsic pK a values of edge surface sites were based on bond-valence theories. The resulting predicted pK a values were highly sensitive to model assumptions [START_REF] Bickmore | Ab initio determination of edge surface structures for dioctahedral 2:1 phyllosilicates: Implications for acid-base reactivity[END_REF][START_REF] Tournassat | The titration of clay minerals II. Structure-based model and implications for clay reactivity[END_REF] and did not provide accurate predictions of experimental montmorillonite titration data [START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF] . In the last decade, ab initio MD simulations have proved capable of accurately predicting the intrinsic pK a values of simple oxide surface functional groups 102-104 , including oxide-type functional groups on montmorillonite edge surfaces [START_REF] Liu | Interfacial structures and acidity of edge surfaces of ferruginous smectites[END_REF][START_REF] Liu | Surface acidity of 2:1-type dioctahedral clay minerals from first principles molecular dynamics simulations[END_REF][START_REF] Liu | Temperature dependence of interfacial structures and acidity of clay edge surfaces[END_REF][START_REF] Liu | Acidity of edge surface sites of montmorillonite and kaolinite[END_REF][START_REF] Liu | Atomic-scale structures of interfaces between phyllosilicate edges and water[END_REF][START_REF] Liu | Atomic scale structures of interfaces between kaolinite edges and water[END_REF][START_REF] Tazi | Absolute acidity of clay edge sites from ab-initio simulations[END_REF] . In particular, a series of papers by Liu et al. [START_REF] Liu | Interfacial structures and acidity of edge surfaces of ferruginous smectites[END_REF][START_REF] Liu | Surface acidity of 2:1-type dioctahedral clay minerals from first principles molecular dynamics simulations[END_REF][START_REF] Liu | Temperature dependence of interfacial structures and acidity of clay edge surfaces[END_REF][START_REF] Liu | Acidity of edge surface sites of montmorillonite and kaolinite[END_REF][START_REF] Liu | Atomic-scale structures of interfaces between phyllosilicate edges and water[END_REF] S-2).

Experimental data

Duc et al. [START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 1. Literature review[END_REF][START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 2. Evidence from continuous potentiometric titrations[END_REF][START_REF] Duc | Coupled chemical processes at clay/electrolyte interface: A batch titration study of Na-montmorillonites[END_REF] carried out a comprehensive examination of montmorillonite acid-base titration methodologies and identified a series of best practices. These included careful montmorillonite pre-treatment and storage procedures (use of a series of acid washes at pH ≈ 4, exchangeable cation homogenization at ionic strength I ≈ 1, and rinsing steps; storage in liquid water at low temperature for relatively short durations, or storage of freeze-dried samples as an acceptable alternative). For the titration experiments themselves, they noted the importance of using an inert atmosphere and limiting the experiments to the pH range ≈ 4.5-9.5. They found that a continuous titration method was acceptable if equilibration times between additions of acid or base were short (< 10 min) and if hysteresis was quantified; alternatively, a discontinuous batch titration method was acceptable if the atoms released by clay dissolution (Si, Al, Mg, Fe) were analyzed (both in solution and on the clay cation-exchange sites) to account for pH buffering by side reactions. In the alkaline domain, however, correction of the raw batch titration data for measured side reactions failed to reproduce the continuous titration curves. The authors concluded on the superiority of fast, continuous methods for quantifying the dissociable surface charge of clays.

Among the many reported studies of montmorillonite titration [START_REF] Tombácz | Surface charge heterogeneity of kaolinite in aqueous suspension in comparison with montmorillonite[END_REF][START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 2. Evidence from continuous potentiometric titrations[END_REF][START_REF] Duc | Coupled chemical processes at clay/electrolyte interface: A batch titration study of Na-montmorillonites[END_REF][START_REF] Tournassat | The titration of clay minerals. Part I. Discontinuous backtitration technique combined to CEC measurements[END_REF][START_REF] Tertre | Surface chemistry of kaolinite and Na-montmorillonite in aqueous electrolyte solutions at 25 and 60ẹgreC: Experimental and modeling study[END_REF][START_REF] Tombácz | Colloidal behavior of aqueous montmorillonite suspensions: the specific role of pH in the presence of indifferent electrolytes[END_REF][START_REF] Avena | Proton adsorption and electrokinetics of an argentinean montmorillonite[END_REF][START_REF] Baeyens | A mechanistic description of Ni and Zn sorption on Namontmorillonite. Part I: Titration and sorption measurements[END_REF][START_REF] Avena | Acid-base behavior of clay surfaces in aqueous media[END_REF][START_REF] Ikhsan | Surface complexation modeling of the sorption of Zn(II) by montmorillonite[END_REF][START_REF] Madrid | Description of titration curves of mixed materials with variable and permanent surface charge by amathematical model. 1. Theory. 2. Application to mixtures of lepidocrocite and montmorillonite[END_REF][START_REF] Stadler | Modeling of H + and Cu 2+ adsorption on calciummontmorillonite[END_REF][START_REF] Tombácz | Surface charge heterogeneity and aggregation of clay lamellae in aqueous suspensions[END_REF][START_REF] Wanner | The acid/base chemistry of montmorillonite[END_REF]105 , Bourg et al. [START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF] determined that only three studies [START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 2. Evidence from continuous potentiometric titrations[END_REF][START_REF] Duc | Coupled chemical processes at clay/electrolyte interface: A batch titration study of Na-montmorillonites[END_REF][START_REF] Baeyens | A mechanistic description of Ni and Zn sorption on Namontmorillonite. Part I: Titration and sorption measurements[END_REF] followed this set of best practices . One of these studies used the batch titration technique [START_REF] Baeyens | A mechanistic description of Ni and Zn sorption on Namontmorillonite. Part I: Titration and sorption measurements[END_REF] , and its results at alkaline pH values are therefore of unclear reliability. Of the acid-base titration studies published over the last decade [START_REF] Bogolepov | Simulation of U(VI) and Co(II) sorption on montmorillonite. Radiochemistry[END_REF][START_REF] Gu | Modeling the adsorption of Cd(II), Cu(II), Ni(II), Pb(II) and Zn(II) onto montmorillonite[END_REF][START_REF] Korichi | Sorption of uranium (VI) on homoionic sodium smectite experimental study and surface complexation modeling[END_REF][START_REF] Kriaa | Acid-base chemistry of montmorillonitic and beidelliticmontmorillonitic smectite[END_REF][START_REF] Rozalén | Surface chemistry of K-montmorillonite: Ionic strength, temperature dependence and dissolution kinetics[END_REF][START_REF] Tan | Characterization of Lin'an montmorillonite and its application in the removal of Ni2+ from aqueous solutions[END_REF] , none followed the entire established set of best practices while also reporting the pH and ionic strength of the solutions used during clay pre-treatment. The present study, therefore, focuses on the dataset from Duc et al. [START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 2. Evidence from continuous potentiometric titrations[END_REF] as the best available set of montmorillonite acid-base titration results.

An interesting indirect quantification of edge surface charge was pursued by Pecini and Avena [START_REF] Pecini | Measuring the isoelectric point of the edges of clay mineral particles: The case of montmorillonite[END_REF] , who measured the i.e.p. of montmorillonite loaded with cationic dyes with a strong affinity for the basal surface. Under the assumption that the dyes completely screen the basal surface charge but do not interact with the edge surfaces (such that the measured i.e.p. reflects edge surface properties) and within the well-established theoretical limitations associated with relating an i.e.p.

to a point of zero charge [START_REF] Sposito | On points of zero charge[END_REF] , the results of Pecini and Avena [START_REF] Pecini | Measuring the isoelectric point of the edges of clay mineral particles: The case of montmorillonite[END_REF] suggest that the point of zero charge of clay edge surfaces (the pH value where σ 0,edge + σ H = 0, where σ 0,edge corresponds to the permanent structural charge density "expressed" on the edge surface) is in the pH range 3.8-5.5, in qualitative agreement with a previous measurement of the i.e.p. of thermally treated Cu-montmorillonite for which the structural layer charge was reduced to almost zero, whereby the authors estimated that the edge surfaces are negatively charged at pH > 3.5 106 .

Towards a predictive SCM for montmorillonite edge surfaces

Reactivity of edge surface functional groups

To determine whether the new knowledge summarized above allows a predictive simulation of the acid-base chemistry of montmorillonite, we developed a generic SCM for the [010] and [110] edge surfaces in our modified version of PHREEQC that applies Eq. ( 4) for the calculation of ψ edge . Edge surface functional groups were modeled as groups of one octahedral and two tetrahedral cations together with their associated OH groups. The overall charge of a group was calculated from bond valence principle applied to the terminal oxygen atoms of the edge sites.

For example, a fully-deprotonated functional group with no isomorphic substitution on the [010] edge corresponds to the grouping of two Si-O -sites and one Al-(OH) 2 -site and is denoted >Si-Al-SiO 4 -3 (leaving the unreactive H out of the formula). It can be protonated four times with intrinsic pK a values of 8.3, 7, 7, and 3.1 as predicted by ab initio MD simulation (section 2.4 and Figure 1). This representation deviates from the classical description used in SCMs, where each surface hydroxyl is modeled as a distinct functional group with a single pK a value 107 . The advantages of the present representation are that the calculation of the number of sites affected by neighboring substitutions is straightforward and that future implementation of metal adsorption through the formation of multi-dentate surface complexes is made easier with regards to the calculation of surface site activities [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]108 . Simple cases were used to verify that the grouping of surface sites had little effect on model predictions (Fig. SI-1 in supporting information; PHREEQC script files are also available in order to test further the effect of grouping sites). The fractions of edge functional groups that include octahedral or tetrahedral substitutions were calculated from the structural formula of montmorillonite with the assumption that edge chemistry reflects that of the bulk mineral. Because of the absence of ab initio MD simulation predictions of the reactivity of Fesubstituted sites on the [110] edge surface, sites with Fe(III) for Al substitutions on this surface were assigned the same properties as sites without substitutions, while sites with Fe(II) for Al substitutions were assigned the same properties as sites with Mg for Al substitutions. Surface site densities were estimated from crystallographic considerations and from reported clay unit cell formulae: each site group, SiAlSi, AlAlSi, SiMgSi, SiFe III Si, and SiFe II Si had a density of 2.06 ×

x i nm -2 , where ݔ வୗ୧ିିୗ୧ = ቀ ୪ାାୣ ାୣ ቁ ୭ୡ୲ , ݔ வୗ୧ିୣ ିୗ୧ = ቀ ୣ ୪ାାୣ ାୣ ቁ ୭ୡ୲ , ݔ வୗ୧ିୣ ିୗ୧ = ቀ ୣ ୪ାାୣ ାୣ ቁ ୭ୡ୲ , ݔ வ୪ି୪ିୗ୧ = 2 × ቀ ୪ ୗ୧ା୪ ቁ ୲ୣ୲ , and
ݔ வୗ୧ି୪ିୗ୧ = 1 - ݔ வୗ୧ିିୗ୧ -ݔ வ୪ି୪ିୗ୧ -ݔ வୗ୧ିୣ ିୗ୧ -ݔ வୗ୧ିୣ ିୗ୧
. This calculation implicitly neglects the possible existence of edge sites with both octahedral and tetrahedral substitutions or with two tetrahedral substitutions, for which predicted pK a values are not available.

Impact of clay permanent structural charge

Proton adsorption by Na-H cation exchange on basal surfaces was taken into account by using a constant selectivity coefficient of 3.2 (log K = 0.5) in near agreement with previously published values [START_REF] Tournassat | The titration of clay minerals II. Structure-based model and implications for clay reactivity[END_REF]109 . The permanent structural charge was set to 0.9 mol c •kg -1 , yielding an overall charge density of σ 0 = -0.12 C•m -2 for the clay layers. As noted above, the edge surface charge equals σ 0,edge + σ H = 0, where σ 0,edge is the edge surface charge density resulting from nearby isomorphic substitutions. Bourg et al. [START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF] assumed that σ 0,edge = σ 0 for simplicity. Here, we modeled edge surface charge arising from near-edge isomorphic substitutions in an even simpler way: we added -1 to the valence of edge functional groups that include an isomorphic substitution of Al for Si or of Mg or Fe(II) for Al. For example, a fully-deprotonated edge functional group with no isomorphic substitutions (>Si-Al-SiO 4 -3 ), upon substitution of Mg for Al, becomes (>Si-Mg-SiO 4 -4 ).

Predicted acid-base properties of edge surfaces

According to the predictive model described above, montmorillonite edge surface charge has little dependence on the type and extent of layer structural substitutions, but it depends significantly on crystallographic orientation between the [010] and [110] edge surfaces (Figure 4, top). In the presence of 0.1 mol⋅L -1 NaCl, the [010] surface is positively charged at pH < 5, whereas the [110] surface is positively charged only at pH < 3.5, consistent with the experimental findings of Thomas et al. 106 and Pecini and Avena [START_REF] Pecini | Measuring the isoelectric point of the edges of clay mineral particles: The case of montmorillonite[END_REF] . The relative proportion of [010] and [110] edge surface orientations on montmorillonite is unfortunately unknown, which leaves one unavoidable free parameter in the presently developed SCM.

Changes in ionic strength are predicted to have a marked effect on surface charge but not on the shape of the charge vs. pH curves, especially for the surface perpendicular to [110] (Figure 4, bottom). A decrease in ionic strength results primarily in a translation of the surface charge curve towards higher pH values, in agreement with experimental results.

Comparison with potentiometric titration data

Model predictions were compared with the potentiometric titration results of Duc et al. [START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 2. Evidence from continuous potentiometric titrations[END_REF] A good agreement was found between experimental and predicted potentiometric titration data for SWy-2 montmorillonite if the specific edge surface area was set to 14 m 2 ⋅g -1 and the relative abundance of [010] and [110] edges was set to 1:1 (Figure 5). The modeled edge specific surface area is in agreement with the value of 19.2 m 2 ⋅g -1 measured on the sample used for the titration experiment (Table S-1), because the DIS method tends to overestimate the surface area [START_REF] Reinholdt | Morphological properties of vermiculite particles in size-selected fractions obtained by sonication[END_REF] . A lesser agreement was found for the lowest ionic strength (I=0.001) at pH values greater than 8, but the significance of this discrepancy is tempered by the increased experimental uncertainty at pH > 9 [START_REF] Bourg | Modeling the acid-base surface chemistry of montmorillonite[END_REF] . For MX80 montmorillonite, the agreement between experimental and predicted potentiometric titration curves was acceptable if the specific edge surface area was set to 12 m 2 ⋅g - 1 and the relative abundance of [010] and [110] edges was set to 1:1 (Figure 5) (the effect of varying the relative abundance of [010] and [110] edges can be seen on Figure 4, and it can be quantified by using the PHREEQC script files provided in the supporting information). The modeled edge specific surface area was larger than the measured value (Table S-1, from 6 to 9 m 2 ⋅g -1 , depending on the considered study). However, the values reported in Table S-1 were not measured for the same samples used in the titration experiments. For both titration datasets, the pH values corresponding to ߜߪ H = 0 were correctly predicted at all investigated ionic strengths.

A better agreement with the experimental data likely could have been achieved by adjusting the pK a values of edge surface sites in the limit of the uncertainty reported in Liu et al. [START_REF] Liu | Interfacial structures and acidity of edge surfaces of ferruginous smectites[END_REF][START_REF] Liu | Surface acidity of 2:1-type dioctahedral clay minerals from first principles molecular dynamics simulations[END_REF][START_REF] Liu | Temperature dependence of interfacial structures and acidity of clay edge surfaces[END_REF][START_REF] Liu | Acidity of edge surface sites of montmorillonite and kaolinite[END_REF][START_REF] Liu | Atomic-scale structures of interfaces between phyllosilicate edges and water[END_REF] (from ± 0.6 to ± 1.6 depending on the site under consideration), but such a refinement was not deemed justified in light of the uncertainties in the experimental data. Such exercise can be done using the files made available in the supporting information.

Figure 1 .

 1 Figure 1. Edge surface sites of a model montmorillonite particle. Inserts with red and green

Figure 2 .

 2 Figure 2. Comparison of the relationships between surface charge (horizontal axis) and surface

  explored the influence of edge crystallographic orientation and the presence of octahedral and tetrahedral substitutions on the intrinsic pK a values of edge surface groups. Their predictions are summarized in Figure 1. It is noteworthy that similar sites on the surfaces perpendicular to the [010] and [110] crystallographic directions have different reactivity according to the ab initio MD calculations. For example, silanol sites (>SiOH) have pK a values of 7.0 on the [010] edge vs. 8.3 on the [110] edge when no octahedral substitution is present. These values are significantly different from the values predicted by bond valence methods 40,41 and used by Bourg et al. 20 (Table

Figure 4 .

 4 Figure 4. Top: Predicted edge surface charge as a function of pH for montmorillonite particles immersed in a 0.1 mol⋅L -1 NaCl background electrolyte. "No", "Al" and "Mg" substitutions refer to simulations carried out with ݔ( வୗ୧ିିୗ୧ = 0; ݔ வ୪ି୪ିୗ୧ = 0), ݔ( வୗ୧ିିୗ୧ = 0; ݔ வ୪ି୪ିୗ୧ = 0.1) and ݔ( வୗ୧ିିୗ୧ = 0.2; ݔ வ୪ି୪ିୗ୧ = 0) respectively. Bottom: Influence of NaCl concentration (0.001, 0.01, and 0.1 mol⋅L -1 ) on the predictions obtained if x>Si-Al-Si = 1.

Figure 5 . 1 M

 51 Figure 5. Comparison of model predictions (lines) and potentiometric titration data (symbols[START_REF] Duc | Sensitivity of the acid-base properties of clays to the methods of preparation and measurement: 2. Evidence from continuous potentiometric titrations[END_REF][START_REF] Delhorme | Acid-base properties of 2:1 clays. I. Modeling the role of electrostatics[END_REF]

Figure 1 297x420mm

 1 Figure 1 297x420mm (300 x 300 DPI)

Figure 2 117x161mm

 2 Figure 2 117x161mm (300 x 300 DPI)

Figure 4 297x420mm

 4 Figure 4 297x420mm (300 x 300 DPI)

Figure 5 297x420mm

 5 Figure 5 297x420mm (300 x 300 DPI)

  Al 1.57 Mg 0.25 Fe III 0.09 Fe II 0.09 )Na 0.34 O 10 (OH) 2 , respectively 27 . Reported montmorillonite structural formulae can vary as a function of sample preparation and calculation method 65 . This variability adds a minor source of uncertainty to our model predictions according to Figure 4. The ݔ வୗ୧ିିୗ୧ , ݔ வୗ୧ିୣ ିୗ୧ , ݔ வୗ୧ିୣ ିୗ୧ and ݔ வ୪ି୪ିୗ୧ values were set, respectively, to 0.125,

	(Si 4 )(0.045, 0.045 and 0 for MX80 montmorillonite and to 0.125, 0.003, 0.112 and 0.13 for SWy-2
	montmorillonite.
			for the
	SWy-2 and MX80 reference montmorillonites (the dataset for SWy-2 was made available in
	Delhorme et al. 50 ). The structural formulae of SWy-2 and MX80 montmorillonites were reported
	as	(Si 3.87 Al 0.13 )(Al 1.52 Mg 0.25 Fe III 0.224 Fe II 0.006 )Na 0.39 O 10 (OH) 2	and

ACS Paragon Plus Environment Environmental Science & Technology

Acknowledgement

This work was supported by the French Radioactive Waste Management Agency (Andra) in the framework of the Andra-BRGM scientific partnership (CTEC project). J.A.D. acknowledges funding from L'Institut Carnot for his visit to the BRGM and from the U.S. Department of Energy under Contract DE-AC02-05CH11231 under the auspices of the Office of Nuclear Energy, Used Fuel Disposition program. I.C.B. was supported by the U.S. Department of Energy under Contract DE-AC02-05CH11231 through the Office of Science, Office of Basic Energy Sciences, Geosciences program.


Coriolis term on the 45 • rotated grid.

The model (Dynamics) Rectangular Box Configuration of the NEMO: 30

• × 20

• with 1 4

• resolution and 5 z levels. 120 × 80 × 5 nodes in (x, y, z) coordinates, 64 time steps per day.

∂u ∂t = v(ω + f ) Enstr.cons. - ∂(u 2 + v 2 )
A z = 1.2 × 10 -4 m 2 s ; A h = 200 m 2 s , f = 4π 86400 sin(lat) ∂u ∂z w 0 = -0.1 N m 2 cos Bπ * lat -24 • 44 • -24 • hz1ρ0 B = 2, Double gyre B = 1,
Single gyre (u ⊥ , ω) Lateral Boundary = 0 (Impermeability and Free-Slip conditions)

PACO page 2 of 14 Impermeability + free-slip boundary conditions:

d d d d d d a a a a a a a a ω ω ω ω ω ω ω ω u u u u u u u u u v v v v v v v v v v v v v
( V , n) = 0, ∂( V , τ ) ∂ n = 0
Impermeability:

u| bnd = v| bnd ,
Free-slip:

v| bnd = u| bnd = u(x + h/2, y -h/2) u| bnd = v| bnd = v(x + h/2, y -h/2) d d d d d d a a a a a a a a a a a a a u u u u u u u u u u u v v v v v v v v v v v v v 0 1 2 3 0 1 2 3 Figure: 45 • rotated grid ∂u ∂t = f v xy ∂v ∂t = -f u xy u(t) = u(0) exp(if t) ∂ ∂t            u 1 u 2 ... un v 1 v 2 ... vn            =            0 0 0 0 f /4 f /4 0 0 0 0 0 0 f /4 f /4 0 0 ... 0 0 0 0 0 0 f /4 f /4 -f /4 -f /4 0 0 0 0 0 0 -f /4 -f /4 0 0 0 0 0 0 ... 0 0 -f /4 -f /4 0 0 0 0                       u 1 u 2 ... un v 1 v 2 ... vn            PACO page 4 of 14
u u u u u u u u u u u v v v v v v v v v v v v v d d s 0 1 2 3 0 1 2 3 Figure: 45 • rotated grid ∂u ∂t = f v xy ∂v ∂t = -f u xy u(t) = u(0)e +f t/4 ∂ ∂t            u u ... un v v ... vn            =            f /4 0 0 0 f /4 f /4 0 0 0 f /4 0 0 f /4 f /4 0 0 ... 0 0 0 0 0 0 f /4 f /4 -f /4 -f /4 0 -f /4 0 0 0 0 -f /4 -f /4 0 0 0 0 0 0 ... 0 0 -f /4 -f /4 0 0 0 -f /4                       u 1 u 2 ... un v 1 v 2 ... vn            PACO page 5 of 14 ∂v ∂x ωa = α 1 v 1/2 + α 2 v 3/2 h ∂u ∂y ωa = α 1 u 1/2 + α 2 u 3/2 h ∂v ∂x ωa = α 1 v -1/2 + α 2 v 1/2 h ∂u ∂y ωa = α 1 u -1/2 + α 2 u 1/2 h ∂ω ∂x v = α 1 ω 0 + α 2 ω 1 h ∂ω ∂y u = α 1 ω 0 + α 2 ω 1 h
with the Initial Guess

α 1 = α 2 = 0 α 1 = -1, α 2 = 1 α 1 = -1, α 2 = 1 t t t t t a a a a a a a a a a a a a a a a ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω u u u u u u u u u u u u v v v v v v v v v v v v v v v v 0 1 2

Data Assimilation

The model:

x(t) = M 0,t (x(0), α) with x = (u, v, T, S, ssh) T Cost function J J = 10 -4 ( x(0) -x bgr 2 + α -α bgr 2 ) + + T t=0 t (u -u ref ) 2 + (v -v ref ) 2 + (ssh -ssh ref ) 2 dxdydt
Layout:

Joint control of the initial point x(0) (interpolation errors) and the set of α;

Artificially generated data by the same model on the aligned grid;

Data Assimilation with the sequence of assimilation windows: 10, 30, 50 days with 30 iterations made in each window;

Analysis of the solution on the 800 days interval.

Minimization is performed by M1QN3 (JC Gilbert, C.Lemarechal);

Adjoint is generated by Tapenade (Ecuador team, INRIA).

PACO page 7 of 14

Optimized Coefficients: 45 The model is closer to the reference one with optimal BC, Data assimilation allows to get the optimized position and shape of the boundary.

• rotation ∂v ∂x ωa = α 1 v -1/2 + α 2 v 1/2 h ∂u ∂y ωa = α 1 u -1/2 + α 2 u 1/2 h ∂v ∂x ωa = α 1 v 1/2 + α 2 v 3/2 h ∂u ∂y ωa = α 1 u 1/2 + α 2 u 3/2 h ∂ω ∂x v = α 1 ω 0 + α 2 ω 1 h ∂ω ∂y u = α 1 ω 0 + α 2 ω 1 h α 1 = 0, α 2 = 0 α 1 = -1.5, α 2 = 0.5 α 1 = -1, α 2 = 1 α 1 = -1, α 2 = 1 d d d a a a a a a a a a ω ω ω ω ω ω ω ω ω u u u u u u u u u v v v v v v v v v v v v v 0

BUT As well as for any adjoint parameter estimation

The control may violate the model physics;

The physical meaning of the optimal boundary is difficult to understand;

The set of α is not unique;

The problem of identifiability is not addressed yet;

The problem of stability is not even posed.

  Figure: 45 • rotated grid

  Figure: 30 • rotated grid

  Figure: 45 • rotated grid


Aurélie Lecestre

Nicolas Mallet

Mickaël Martin

Thierry Baron

Guilhem Larrieu

Fabrication of GaAs nanowires and GaAs-Si axial heterostructure nanowires on Si (100) substrate for new applications

I. INTRODUCTION

III-V nanowires-based devices represent a great interest for various applications including nanoelectronics [1], nanophotonics [2], biological sensors [3]. Integration of such nanostructures on the conventional silicon platform is a challenge. In order to make the bridge between fundamental studies on NW patterning and large scale device processing based on NWs new approaches should be developed. Traditionally, two complementary ways are used for nanowire fabrication: bottom-up growth and top-down patterning. The main advantage of bottom-up growth is the crystalline quality of nanowires and of the nanowires surface. However, the reproducibility and dimension control are difficult to reach, especially when selective area epitaxy approach is conducted to create localized NW array. Moreover, substrate orientation has an effect on nanowires orientation. In order to obtain vertical nanowires, Si (111) orientation is required [4], which is not the conventional substrate used in microelectronics (Si(100)).

The top-down approach allows nanowire patterning with better reproducibility, and dimension control accuracy. However, fine tuning of the nanowire etching is still a challenge together with the integration of III-V NW on silicon. Here, we will present for the first time a demonstration of III-V NW patterning on Si (100) based on a detail study of the nano-structuration.

II. PATTERNING FOR GAAS NANOWIRES ARRAY

A. Process steps description

The starting wafer is composed a thin GaAs layer synthetized on Si (100) substrate by MOCVD without antiphase boundaries [5,6]. The layer thickness is comprised between 200nm and 700nm. A top-down approach that couples Electron Beam Lithography (EBL) with anisotropic plasma etching was used to pattern the designed nanostructures. The EBL was carried out with a RAITH 150 writer at energy of 30 keV on a thin (90nm) negative-tone resist layer, namely hydrogen silsesquioxane (HSQ). After 1) LAAS-CNRS, Université de Toulouse, CNRS, INP, Toulouse, France.

2) Université Grenoble Alpes, CEA-MINATEC, CNRS-LTM, rue des Martyrs, F-38054 Grenoble Cedex, France.

exposure, HSQ was developed by immersion in 25% tetramethylammonium hydroxide (TMAH) for 1 min (Fig. 1a) [7] to enhance the pattern contrast. HSQ patterns were subsequently transferred to the GaAs top layer by ICP-RIE (SPTS Omega 201, et Sentech SI500) in a Cl2/N2 plasma based chemistry down to the silicon substrate (Fig. 1-b) [8].

In order to obtain axial heterostructure, a second step of plasma etching using SF6/C4F8 chemistry was conducted (Fig. 1-c).

The last step involves removing the resist and the passivation layer by wet etching with HF (Fig. 1-d).

B. PATTERNING OF GAAS BY PLASMA

In order to determinate suitable parameters to obtain vertical nanowires, several parameters have been studied, shown in the Fig. 2.

ICP power acts on plasma density (corresponding to the quantity of reactive species and ions). When the power increases, the increase of reactive species boosts the etch rate but also impacts the etching angle. At low ICP power, the dissociation energy is weak, promoting the reactive species formation to the ions formation. Therefore the chemical reaction is favored over the ionic bombardment. The chemical etching being isotropic, the etching profile is re-entering. It is necessary to apply a sufficient power level to obtain a good balance between chemical and physical etching, to tend towards vertical profiles (etching angle = 90°) as illustrated on Fig. 2. GaAs has been etched thanks to the combination of two gases: Cl2 and N2. Cl2 is a reactive gas promoting the chemical etching. N2 is a neutral gas favoring the physical etching (ions bombardment). By decreasing the Cl2 / N2 gas ratio, the reactive species quantity is reduced, therefore the etch rate decreases and the etching angle increases. In these conditions, etching profile is not vertical because physical etching is favored over chemical etching (Fig. 3). All parameters are inter-dependant, the objective is to find right parameters to obtain a good balance between physical and chemical etching (Fig. 4).

III. AXIAL HETEROSTRUCTURE NANOWIRES

To go further in the complexity of this work, we have also demonstrated the possibility to create GaAs/Si NW heterostructure based on this top-down approach.

After GaAs etching step, 50nm of HSQ resist is remaining on GaAs nanowires. The Si etching has to be selective compared to HSQ and not laterally etch the GaAs NWs. We used a process based on fluorine chemistry resulting on a Si/HSQ selectivity of 5. In this way, the top part of GaAs nanowires is not damaged (Fig. 5).

For a perfect patterning of these hetero-structured nanowires, the challenge is to master the transition between the two etching chemistries in order to not degrade the Si/GaAs region. For that purpose, the end point detection during GaAs etching is critical. The over-etch of GaAs has to be less than 3 s, otherwise a lateral etching at the bottom of nanowires is accentuated resulting in mechanical fragility of the NWs and carrier mobility scattering.

IV. CONCLUSION

For the first time, we have demonstrated the fabrication of vertical GaAs nanowires array on Si (100) substrate. Moreover, this approach allows obtain axial heterostructure nanowires GaAs-Si. This technology opens the way to new integration schemes for nanoelectronics and nanophotonics applications.

Figure 1 .

 1 Figure 1. Process flow chart: a) First step is electron beam lithography with a negative resist (HSQ). b) ICP-RIE of GaAs thin layer by stopping on silicon substrate. c) ICP-RIE of silicon substrate. d) Last step is a wet etching to remove inorganic resist.

Figure 2 .

 2 Figure 2. Influence of ICP power on etching angle and etch rate.

Figure 3 .

 3 Figure 3. Influence of gas ratio on etching angle and etch rate.

Figure 4 .

 4 Figure 4. GaAs nanowires array (diameter = 40nm).

Figure 5 .

 5 Figure 5. Heterostructured GaAs-Si nanowires array (diameter = 40nm) on Si(100) substrate.

ACKNOWLEDGMENT

The authors thank F. Carcenac and P. Dubreuil for technical assistance. This work was supported in LAAS-CNRS micro and nanotechnologies platform member of the French RENATECH network.