Philippe Leroy
email: p.leroy@brgm.fr

Christophe Tournassat

Mohamed Bizi

The influence of surface conductivity on the apparent zeta potential of TiO 2 nanoparticles

Keywords: zeta potential, electrophoretic mobility, TiO 2, nanoparticle, surface conductivity, extended Stern model

Zeta potential is a physico-chemical parameter of particular importance to describe ion adsorption and electrostatic interactions between charged particles. Nevertheless, this fundamental parameter is ill-constrained, because its experimental interpretation is complex, particularly for very small and charged TiO 2 nanoparticles. The excess of electrical charge at the interface is responsible for surface conductance, which can significantly lower the electrophoretic measurements, and hence the apparent zeta potential. Consequently, the intrinsic zeta potential can have a larger amplitude, even in the case of simple 1:1 electrolytes like NaCl and KCl. Surface conductance of TiO 2 nanoparticles immersed in a NaCl solution is estimated using a surface complexation model, and this parameter and particle size are incorporated into Henry's model in order to determine a constrained value of the zeta potential from electrophoresis. Interior conductivity of the agglomerates is calculated using a differential self-consistent model.

The amplitude of estimated zeta potential is greater than that derived from the von Smoluchowski equation and corresponds to the electric potential at the Outer Helmholtz Plane calculated by our surface complexation model. Consequently, the shear plane may be located close to the OHP, contradicting the assumption of the presence of a stagnant diffuse layer at the TiO 2 /water interface.

Introduction

Titanium dioxide is widely used as TiO 2 nanoparticles and has a large variety of potential applications in, for example, the biomedical, optical, and electronic fields [1,2]. Due to their small size, nanoparticles have a very high surface area to volume ratio and are thus of great scientific interest as they are a bridge between bulk materials and atomic or molecular structures. The properties of materials change as their size decreases to nanoscale and the proportion of surface atoms becomes significant. One of these properties, the surface ionization of titanium dioxide nanoparticles in contact with an electrolytic solution, has been studied extensively [1,[START_REF] Foissy | Analysis of adsorption of ions and polyacrylic acids in aqueous dispersions of titane dioxide[END_REF][START_REF] Foissy | [END_REF][5][6][7][8][9][10]. It is well known that the complexation reactions at the surface of an oxide mineral are strongly influenced by the development of the surface charge. The primary surface charge is determined by the reactions of protons with the surface. Surface complexation reactions between the surface sites and the ions from the bulk electrolyte at the Stern and in the diffuse layer neutralize the surface charge [START_REF] Lyklema | Fundamentals of interface and colloid science: Liquid-fluid interfaces[END_REF]. Surface charge properties are primarily determined using proton titration data. These data can be modeled using various electrostatic models such as the diffuse double layer, basic Stern, triple layer or CD-MUSIC models [1,[START_REF] Foissy | Analysis of adsorption of ions and polyacrylic acids in aqueous dispersions of titane dioxide[END_REF][START_REF] Foissy | [END_REF][5][6][7][8][9][10]. Most sophisticated models are able to reproduce data for a wide range of experimental conditions but rely on the fitting of a large number of parameters whose physical significance is not always easy to justify. Moreover, the uniqueness of a set of parameters is not always obvious. On the other hand, less sophisticated models rely on the fitting of fewer parameters but often fail to reproduce the data under all of the experimental conditions studied. At present, there is no consensus on which is the "best model" to represent charged surfaces at oxide-water interfaces.

Of all the physico-chemical parameters characterizing the solid/water interface, the zeta potential is particularly important. It is the potential at the supposed slipping plane that separates the stationary and mobile phases in tangential flow of the liquid with respect to the surface. For example, in the case of a particle undergoing electrophoresis, because of the electrostatic interactions between the applied electric field and the hydrated counter-ions in the diffuse layer, the interface develops a surface of shear [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF].

The electric potential at the slipping plane is of particular interest if we wish to estimate the critical coagulation concentration when studying nanoparticle agglomeration, for example [5,[START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF][START_REF] Derjaguin | [END_REF]. The zeta potential is also a key parameter for the study of the transport properties of electrically charged materials like oxide and clay minerals [14][15][16]. It usually makes it possible to optimize the parameters of the electrostatic surface complexation models while assimilating the position of the shear-plane to the position of the head end of the diffuse layer [10,17]. This interpretation has, however, been challenged by recent studies on titanium dioxide nanoparticles that show that titration data cannot be reproduced together with zeta potential values without having a shearplane position that changes as a function of the ionic strength. Bourikas et al. [9] and Panagiotou et al. [1] have shown that the shear plane position (in log scale) is linearly dependant on the log value of the ionic strength. This physical model leads, however, to a surface representation in which the shear-plane can be as far as 210 Å from the surface at an ionic strength of 10 -4 M (Figure 32 of [1]), the entire volume between the oxide surface and this shear-plane being considered to be "stagnant". Moreover, this physical representation of the oxide surface rules out any use of the zeta potential as a constraint for surface charge models. The nature of the physical property that causes the separation between the "stagnant" diffuse layer and the "mobile" diffuse layer is, however, not given. Molecular dynamics studies and X-ray measurements on oxide and aluminosilicate surfaces tend to show that diffuse layer water properties (density, mobility, molecule orientation) are very similar to those of bulk water [18][19][20].

Electrophoretic mobility measurements are usually used to determine the zeta potential.

However, zeta potential estimations can be erroneous due to the uncertainty concerning the value of the conversion factor used [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF]. This could explain the "observed" shift of the shear-plane position as a function of the ionic strength in surface complexation models. Numerous authors [18][19][20][21][22][23][24][25][26] emphasize that the anomalous surface conductivity of particles might explain the low zeta potential values (in amplitude) determined from electrophoresis compared to values estimated by surface complexation models at the OHP and electrical conductivity measurements. They also state that the lateral motion of adsorbed counterions at the Stern layer must not be disregarded for some materials having a large excess of electrical charge at their surface (like clay minerals or latex suspensions). Little work has been done to characterize the surface conductance due to the Stern layer for titanium dioxide nanoparticles [27]. In addition, the surface conductivity of TiO 2 nanoparticles may be strong due to the small size of the elementary particles.

The aim of this work is to study the possible influence of TiO 2 nanoparticle surface conductivity on its electrophoretic mobility in the hope that this will lead to zeta potential values in agreement with those predicted by electrostatic surface complexation models (without considering large variations of the distance between the outer boundary of the compact layer and the inner boundary of the diffuse layer). Surface speciation models are needed to calculate the surface conductance of TiO 2 . For this reason, the model developed recently by Panagiotou et al. [1] and alternative models are critically evaluated in the second chapter of this study. In the third chapter, we discuss the theories used to convert electrophoretic mobility to zeta potential and calculate surface conductance due to the double layer. In the fourth chapter, we describe our modeling strategy and use the proposed methodology to estimate the "intrinsic" zeta potential of titanium dioxide P25 in a NaCl electrolyte from electrophoresis.

Electrostatic models for titanium oxides

Panagiotou et al. model (2008)

Panagiotou et al. [1] proposed a Triple Plane Model (TPM, [6]) for the titanium oxide (P25) solid-solution interface in NaNO 3 and KNO 3 electrolyte solutions. This model is based on a state-of-the-art description of the TiO 2 surface properties with regards to protonation-deprotonation processes using the recent ab-initio calculations and DFT developments for this material. Two main surface functional groups were found to be responsible for the surface reactivity:

Ti 2 O -0.57 + H + ⇌ Ti 2 OH +0.43 log K 1 , (1)
TiO -0.35 + H + ⇌ TiOH +0.65 log K 2 .
(2) Total surface site density was fixed at a value of 5.6 sites nm -2 obtained from crystallographic considerations. Total surface area was 50 m 2 g -1 according to BET measurements.

The surface ionization model was then combined with a triple plane model, and the predictions were used together with potentiometric titration, microelectrophoresis and streaming potential experiments to describe the electrochemical properties of the TiO 2 surface. Both surface sites were considered to behave similarly with a given cation (Na + and K + ) or anion (NO 3 -). This simplification enabled the authors to reduce the number of adjustable parameters for their model:

Ti 2 O -0.57 + Na + ⇌ Ti 2 O -0.57 -Na + log K Na , (3)
TiO -0.35 + Na + ⇌ TiO -0.35 -Na + log K Na , (4) Ti 2 OH +0.43 + NO 3 -⇌Ti 2 OH +0.43 -NO 3 - log K NO3 , (5)
TiOH +0.65 + NO 3 -⇌TiOH +0.65 -NO 3 - log K NO3 . (6)
In the triple plane model, the charge of sorbing cations and anions is not attributed to only one electrostatic plane but is distributed over the three planes (0, 1, 2), thus adding two additional fitting parameters (Δz 0 and Δz 1 or Δz 1 and Δz 2 ) for each sorbed species at each surface site (Fig. 1). Again, both surface sites were considered to behave similarly with a given cation or anion. The parameters of this model are given for a NaNO 3 electrolyte background in Table 1.

We reproduced their model using PHREEQC v2.17 [START_REF] Parkhurst | User's guide to PHREEQC (Version 2) -A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF]. We were not able to reproduce their titration curves using the parameters in the reference publication. The tabulated protonation/deprotonation constants (log K 1 and log K 2 ) had to be changed slightly in order to obtain results in full agreement with the data. Hereafter, we refer to this modified model as "Reference model" (Table 1). Surface charge predictions are given in Fig. 2 together with surface potential at the 2-plane, which is considered to be the head end of the diffuse layer. This parameter can be compared to the zeta potential. 1) give surface charge results nearly identical to those of the TPM reference model in NaNO 3 electrolyte (Fig. 2) although the BSM requires only five fitting parameters instead of ten for TPM. This raises the question of the uniqueness of the TPM parameter set. In principle, this problem can be overcome by fitting a large range of data obtained under the same experimental conditions but testing different electrolyte types. Panagiotou et al. [1] provided titration data in a KNO 3 electrolyte background and were able to fit them adequately only by attributing an association constant (log K K = -1.1) and a charge distribution (Δz 1 K = 0, Δz 2 K = 1) for K + at the surface, while keeping the other parameters constant. A good agreement can also be obtained with the BSM model (Fig. 3) adding only an association constant (log K K = -1.7) for K + . There is a discrepancy between the titration predictions of the two models at 0.017 M, but the differences increase with salinity, and are therefore greater and significant for a salinity of 0.3 M and pH > 9.5. Consequently, the addition of five fitting parameters for the triple plane model relies on a very restricted subset of experimental data points. symbols depict the results of alternative models (Table 1).

The right sides of Fig. 2 and Fig. 3 show that the choice of a given model influences the prediction of the electrical potential at the head end of the diffuse layer. For these two models, however, predicted potentials are much higher in amplitude than reported zeta potentials from electrophoresis [START_REF] Foissy | Analysis of adsorption of ions and polyacrylic acids in aqueous dispersions of titane dioxide[END_REF], corroborating the hypothesis of a shear plane located in the diffuse layer.

As an alternative model, an Extended Stern Model (ESM) can reproduce perfectly the reference TPM for surface charge but diminishs the potential at the head end of the diffuse layer. One set of this model's parameters, amongst those tested successfully, is given in Table 1. Corresponding titration and potential curves are shown in Fig. 2 and Fig. 3 for NaNO 3 and KNO 3 electrolyte backgrounds, respectively.

According to Panagiotou et al. [1], the high capacitance values of the reference TPM are in agreement with theoretical and experimental studies concerning the location of the first two water overlayers and the electrolyte counterions at the rutile surface, taken here as a good analogue of the anatase surface [START_REF] Tilocca | [END_REF][30][31][32][33][34]. Predota and Vlcek and Predota et al. [31,33] showed that the first hydration layer at the rutile surface is ~1.8 Å from the surface terminal oxygen atoms and that Na + cations are located at ~1-1.8 Å. These distances are in agreement with the capacitance value C 1 = 3.2 F m -2 that locates 1-plane at d 1 = 1.7 Å from the 0-plane, while considering the following equation with the mean relative dielectric constant, ε r,1 , equal to 60 (ε 0 is the dielectric constant of the vacuum,

8.85 × 10 -12 C V -1 m -1 ) : ) 2 ( 1 ) 2 ( 1 , 0 ) 2 ( 1 d C r ε ε = , ( 8
)
where C 2 , d 2 and ε r,2 are the capacitance, distance and the mean relative dielectric constant between the 1-and 2-planes, respectively. The proposed ESM model has a lower capacitance C 1 = 2.5 F m -2 . It could therefore be considered to not be in agreement with above results. However, parameter ε r,1 is ill-defined. Taking a value of ε r,1 = 47 enables us to arrive at the same distance d 1 with the ESM model. Consequently, available data at the molecular level combined with titration data cannot be used to determine which is the best representative model since both ε r,1 values are reasonable [35]. We were, however, not able to achieve a good fit of titration data with models having a C 1 value lower than 2.5 F m -2 . This seems to confirm the presence of a layer with a high capacitance and Na sorption close to the surface. On the contrary, C 2 could be set at a value as low as 1 F m -2 in the ESM, in marked contrast with the 4.2 F m -2 value of the reference TPM. The position of the second plane is also subject to discussion. Panagiotou et al. [1] considered that the second plane ends the compact layer at a distance of 3.4 Å from the surface. However, molecular dynamics calculations show significant water density oscillations up to 12 Å from the surface associated with continuous viscosity changes [33]. As a consequence, the choice of a precise location for the second plane cannot be easily justified. Moreover, the calculation of the capacitance value for the region between the 1-and 2-planes relies heavily on the modeler's choice for ε r,2 .

Theoretical calculations indicate that the greatest portion of sorbed Na + forms bidentate complexes. We tried to roughly incorporate this information by considering the following equilibria in another alternative model (the "Bidentate" model in Table 1):

2 Ti 2 O -0.57 + Na + ⇌ (Ti 2 O -0.57 ) 2 -Na + log K Na , (9)
2TiO -0.35 + Na + ⇌ (TiO -0.35 ) 2 -Na + log K Na . ( 10
)
The charge of counter Na + was attributed partly to the 0-plane and partly to the 1-plane, since Na + was also shown to be partly dehydrated at the surface and engaged in inner sphere complexes [34]. Fig. 2 again shows that this model can match almost perfectly the reference TPM titration predictions. Note also that, depending on the chosen surface complexation model, i.e. TPM, BSM, ESM, or Bidentate, there may be different locations and then definitions of the "stagnant layer" (which is not necessarily a monolayer of sorbed counterions). For example, in the case of a NaNO 3 electrolyte, the stagnant layer is located between 1 and 2-planes for TPM, at the 1-plane for BSM, at the 1-plane for ESM and between the 0 and 1-planes for Bidentate.

Implications for mapping the "titanium dioxide/electrolyte solution" interface

This analysis shows that titration data combined with surface complexation models such as TPM or ESM cannot yield a unique and unambiguous set of interfacial parameters.

Theoretical calculations and modeling at the molecular scale can help us validate the model likelihood but uncertainties remain.

This suggests that the relationship observed between the position of the shear plane and the ionic strength (Eq. ( 7)) is model dependant. Fig. 2 and Fig. 3 show that the reference TPM and the BSM predict nearly identical potentials at the head end of the diffuse layer, in agreement with the fact that the reference TPM is an improvement on the BSM published earlier by Bourikas et al. [9]. However, the proposed ESM and "Bidentate" models predict lower potentials (in absolute value) than the reference model. Predicted potentials remain, however, higher in absolute value than commonly reported zeta potential from electrophoresis. Consequently, the question of the position of the shear plane in relation to the head end of the diffuse layer remains.

Zeta potential calculations from electrophoresis are also model-dependant. In the following chapter, several models to convert electrophoretic measurements to zeta potentials are briefly reviewed.

From the electrophoretic mobility to the zeta potential

Electrokinetic theories

The most well-known and widely used theory of electrophoresis was developed by von Smoluchowski [START_REF] Smoluchowski | Handbuch der Elektrizitat und des Magnetismus[END_REF][START_REF] Graetz | [END_REF]. He studied the movement of the liquid adjacent to a flat, electrically charged surface under the influence of an electric field applied parallel to the interface. Von Smoluchowski [START_REF] Smoluchowski | Handbuch der Elektrizitat und des Magnetismus[END_REF]

ζ η ε μ = e , ( 11
)
where η is the dynamic viscosity (in Pa s; η = 0.895×10 -3 Pa s at T = 298 K) and ε the dielectric permittivity of water (ε = ε 0 ε r = 78.3×8.85×10 -12 F m -1 at T = 298 K). The von Smoluchowski equation is valid only if the thickness of the diffuse layer is insignificant compared to the size of the particle, i.e. for thin double layer, κa >> 1, where κ is the inverse of the Debye length (in m -1 ) and a the particle radius (in m). The inverse of the Debye length is given by:

RT c z F i b i i ε κ ∑ = 2 2 , ( 12
) RT I F ε κ × × = 2 2 , ( 13
)
where F is the Faraday constant (96485 C mol -1 ), R the gas constant (8.314 J mol -1 K -1 ), c i the ions concentrations (in mol m -3 ), z i their valency, and I the ionic strength. The symbol "b" refers to the bulk ions.

For small spherical particles having a thick double layer, κa << 1, the applied electric field is not influenced by the presence of the particle and the effect of the retardation force due to the double layer on the migration of the particle is negligible. Hückel [38] considered that the main retardation force is the frictional resistance of the medium. He supposed that the electrical conductivity of the particle is the same as that of the surrounding medium. Therefore, the electric field is not distorted by the particle. He wrote the following equation, which is valid for κa << 1:

ζ η ε μ 3 2 = e . ( 14
)
Henry [START_REF] Henry | Proc. R. Soc. Lond. A[END_REF][START_REF] Henry | [END_REF] revisited Hückel's theory by considering that the conductivity of the particle is different from that of the surrounding medium. In this case, the applied electric field will be distorted so that the isopotential values can be around the particle surface. According to Henry [START_REF] Henry | [END_REF], the particle's conductivity alters the shape of the potential distribution of the applied field in the liquid, modifies the fluid motion within the electrical double layer, and therefore changes the stresses of the fluid exerted on the particle. Consequently, this conductivity leads to the mutual distortion of the applied field and the field of the double layer, and hence slows the electrophoretic motion. For spherical particles with arbitrary double-layer thickness, Henry [START_REF] Henry | [END_REF] wrote:

( ) ζ κ η ε μ s p e K K a F , , 3 2 = , (15)
[ ]

1 ) ( 2 1 ) , , ( - + = a f K K a F s p κ λ κ , ( 16
)
s p s p K K K K 2 2 2 1 + + - - = λ , ( 17
) b p p K σ σ = , ( 18
) b s b s s a K σ σ σ Σ = = , (19)
κ κ κ κ - + - = , for κa > 1, ( 21
)
where σ is the electrical conductivity (in S m -1 ), Σ the surface conductance of the electrical double layer, the subscripts "p", "s", "b" corresponding, respectively, to the particle "interior", the particle surface and the surrounding medium (the bulk electrolyte). The surface conductance expresses the excess of electrical conductivity at the solid surface compared to that of the bulk electrolyte. K s corresponds to the well-known "Dukhin number", Du (see [18,[START_REF] Dukhin | Dielectric phenomena and the double layer in disperse systems and polyelectrolytes[END_REF] and [START_REF] Crespy | [END_REF] for more details). According to Eqs. (15) to (17) and by replacing K s with Du, we obtain:

[ ]ζ κ η ε μ ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + - - + = 1 ) ( Du 2 2 Du 2 1 2 1 3 2 a f K K p p e . ( 22
)
In the absence of surface conductance and in the case of an insulating particle, Henry's theory leads to von Smoluchowski's equation for large κa values (f(κa) = 1.5) and Hückel's equation for κa << 1 (thick double layer, f(κa) = 1). For κa >> 1, f(κa) = 1.5, and Eq. ( 22) reduces to:

ζ η ε μ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + - - + = Du 2 2 Du 2 1 1 3 2 p p e K K . (23)
In the case of insulating particles, K p = 0, and Eq. ( 23) corresponds to O'Brien's formula [43] within the limit of a DC applied electric field and disregarding the inertial term in his theory. O'Brien [43] developed a complete picture of the frequency dependent dielectric response of a dilute suspension of spheres with thin double layers. Note that all of the equations presented here, except Smoluchowski's equation, consider a Debye-Hückel ionic atmosphere, i.e. that electric potential in the diffuse layer follows a Debye-Hückel distribution. Consequently, the analytical equations that we use to estimate the zeta potential from the electrophoretic mobility are valid for low zeta potentials (⎜ζ ⎜≤ 25.7 mV in the case of 1:1 electrolyte at T = 298 K), but they can still be applied for zeta potentials of greater amplitude [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF].

In addition, Eqs. ( 11) to (23) do not consider spheroidal particles, particle volume fraction, polydispersity of the sample, i.e. agglomerates of different sizes, and diffuse layer overlapping. Several electrokinetic models make it possible to estimate surface conductance for spheroidal particles (for example, [44]), consider polydispersivity of the sample and nanoparticles agglomeration [45], and diffuse layer overlapping [46].

Moreover, Mangelsdorf and White's numerical model, which takes into account particle size effects and the adsorption of ions and their mobility in the inner part of the EDL [21], can be more accurate than the analytical solutions we use, especially at low ionic strength and at pH values far from the PZC where the amplitude of the surface electric potential is high.

However, the electrical conductivity of the particle "interior" σ p can be determined using the so-called differential self-consistent model [47,48]. This model considers small contiguity between the particles and allows the determination of the electrical conductivity of the agglomerate corresponding to the final concentration of inclusions (here the elementary nanoparticles of conductivity σ e ) by addition of infinitesimal portions of inclusions:

( ) ∫ ∫ - - = - + φ σ σ σ σ σ σ σ σ 1 0 1 3 2 Ω dΩ d p b e e , ( 24
)
where φ is the intra-aggregate porosity, Ω = v/(v+V) is the volume fraction of the elementary nanoparticles in the agglomerate with v and V the total volume (in m -3 ) of elementary particles and of water, respectively. By integrating Eq. ( 24), we obtain:

φ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - D p b e b e p σ σ σ σ σ σ , (25)
where D = 1/3 in the case of spherical elementary particles. Eq. ( 25) has been generalized to non-spherical particles, but it is not very practical because it is an equation of the form ) , , , , (

p e b p σ σ σ D f σ φ =
. Revil [48] found an analytical solution of Eq. ( 25) by considering D = 1/2 for disk-shaped particles:

( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + - + - - + = FΘ Θ Θ Θ FΘ F b p 4 1 1 1 2 1 2 σ σ , ( 26
) b e s b e a Θ σ σ σ Σ = = 2 , (27) 2
- = φ F . ( 28
)
According to Revil [48], D = 1/2 corresponds to a particle's shape usually found in most porous media. In addition, following the approach of [45], the electrophoretic mobility due to the polydispersity of the sample, e μ , i.e. to agglomerates possessing different sizes, can be calculated by:

∑ ∑ = = Δ Δ = N i i i i N i i i i i e e a a a f a a a f 1 3 1 3 ) ( ) ( μ μ , ( 29
)
where N is the number of different radii, f(a i ) represents the discretized version of the Particle Size Distribution (PSD), Δa i the radius intervals, and i e μ the electrophoretic mobility of agglomerates with radius a i ± Δa i /2. According to Eq. ( 29), the electrophoretic mobility of the polydisperse system is the volume average of the mobilities of particles with different sizes in the distribution.

Henry [START_REF] Henry | [END_REF] and O'Brien [43] considered that only the counterions in the diffuse layer are responsible for the surface conductivity. They did not consider the influence of the Stern layer on particle surface conductivity [18][19][20][21][22][23]. According to these authors, the zeta potential of (electrically) charged suspensions like polystyrene lattices, determined from electrophoresis, are lower in amplitude than the zeta potential values estimated from electrical conductivity measurements. In the case of electrophoresis, these authors assume that the lateral movement of ions at the Stern layer in response to the applied electric field can explain such discrepancy. The presence of mobile counterions at the Stern layer and in the diffuse layer lowers the amplitude of zeta potentials inferred from mobility measurements and raises those from conductivity measurements, compared to the zeta potentials that correspond to the intrinsic particle charge.

The "anomalous" surface conductance linked to the Stern layer, i.e. to ions below the shear plane, can be determined for oxide minerals using electrical conductivity measurements [25,27,[START_REF] Crespy | [END_REF]49] or a surface complexation model like a Triple Layer Model (TLM, [15,50]) where the associated equilibrium sorption constants are calibrated by titration experiments. Surface complexation models calculate the excess of counterions at the interface and thereby make it possible to estimate the (specific) surface conductance of the particle [50,51].

Determination of the surface conductance

The diffuse layer surface conductance

The surface conductance of the Electrical Double Layer (EDL) is the enhanced conductivity due to the presence of a double layer at the particle surface ( [START_REF] Henry | [END_REF][49][50][51]; see Fig. 4). We consider here the surface conductance due only to the presence of the diffuse layer at the particle surface. In the following chapter, we will consider the influence of the Stern layer on the surface conductance. According to Revil and Glover [51], the surface conductance Σ s (in S) can be described by the individual specific surface conductance of each ion in the diffuse layer,

s i Σ , ∑ = i s i i s ez Σ Σ . ( 30
)
The total surface conductance s Σ is made up of an electro-migration, e s Σ , and an electro-osmotic surface conductance, os s Σ ,

os s e s s Σ Σ Σ + = . ( 31
)
The electro-migration surface conductance is due to the excess of Ohmic conductivity in the EDL [15],

[ ]χ σ χ σ χ d D b e s ∫ - = 0 ) ( Σ , ( 32
)
where χ represents the surface/solution distance in m and χ D is the total thickness of the diffuse layer (usually, χ D = 2κ -1 ). The parameters σ(χ) and σ b are the electrical conductivity at the solid/solution interface and in the bulk pore water defined, respectively, by:

∑ = i s i s i i c Fz ) ( ) ( χ β χ σ , ( 33
)
∑ = i b i b i i b c Fz β σ , ( 34
)
where b s i , β is the ion mobility (in m 2 s -1 V -1 ) at the particle surface and in the bulk pore water. The electrical conductivity of the solution due to electro-migration is given by an equivalent Ohm's law determined using the generalized Nernst-Planck equations and the electrochemical potential equations for the ion species [14,[START_REF] De Groot | Non-equilibrium thermodynamics[END_REF] (electro-osmosis is neglected). The local electrical conductivity σ(χ) is given by analogy with the free electrolyte conductivity ( [51]). By incorporating Eqs. (33) and (34) in Eq. ( 32), we obtain:

[ ] ∫ ∑ - = D d c c Fz i b i b i s i s i i e s χ χ β χ β 0 ) ( Σ . (35)
According to Eq. ( 35), by considering the same ion mobility at the interface (diffuse layer here) and in the bulk pore water, i.e.

[ ] ∑ ∫ - = i b i s i i i e s d c c z F D χ χ β χ 0 ) ( Σ , ( 36
) ∑ ∫ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ - ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ± - = i i b i i i e s d RT F z c z F D χ χ ψ β χ 0 1 ) ( exp Σ , (37)
( )

κχ d - = exp ) ( ψ χ ψ , ( 38
)
where ψ d is the electric potential at the head end of the diffuse layer. The term ± corresponds to the sign of the electrical charge associated with the species i ("+" for cations and "-" for anions).

It is relatively easy to find an analytical solution for the ion contribution to the surface conductance using Eqs. ( 30), ( 37)-( 38) and linearizing the exponential function of the Boltzmann distribution ( [51] and Eqs. ( 191) to (194) of [54]):

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ± - ≈ ∑ - 1 2 exp 2 Σ 1 RT F z c z F d i b i i i i e s ψ β κ , ( 39
)
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ± - ≈ - 1 2 exp Na 2 Σ 1 RT F z c d i b i i e i ψ β κ , ( 40
)
where Na is the Avogadro Number (6.022×10 23 sites mol -1 ) and by considering a diffuse layer having a thickness of

1 2 - = κ χ D .
At the surface of the titanium dioxide particle (for conditions other than the IEP and for low ionic strength, typically ≤ 0.01 M for oxide minerals, [10]), counter-ions are predominant in the diffuse layer. When an electric field is applied, it results in a solvent convection and, consequently, a surplus conductivity called electro-osmotic conductivity. The electro-osmotic contribution to the surface conductance is described by [51]:

∫ = D d os os s χ χ χ β χ ρ 0 ) ( ) ( Σ , (41)
∑ = i os i i os s ez Σ Σ , ( 42
)
( ) ∑ ± = i s i i c z F ) ( ) 1 ( χ χ ρ , ( 43)
[ ]

d os ψ χ ψ η ε χ β - = ) ( ) ( , ( 44
)
where β os is the electro-osmotic mobility (in m 2 s -1 V -1 ) and ρ the volume charge density in the diffuse layer (in C m -3 ). According to Bikerman [55]:

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ± - ≈ ∑ - 1 2 exp Na 4 Σ 1 RT F z c RT d i i b i os s ψ κ η ε . ( 45
)
The ion contribution of the electro-osmotic surface conductance is determined using Eqs. ( 42) and ( 45):

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ± - ≈ - 1 2 exp Na 4 Σ 1 RT F z c ez RT d i b i i os i ψ κ η ε . ( 46
)
By combining Eqs. ( 40) and ( 46), the total ion contribution to the surface conductance is given by:

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ± - = - 1 2 exp Na 2 Σ 1 RT F z c B d i b i i s i ψ κ , ( 47
) i i i ez RT B η ε β 2 + = . (48)
In Eqs. ( 47) and ( 48), we have assumed that the ion mobilities are the same in the bulk electrolyte and in the diffuse layer. These equations correspond those of Bikerman [55] for characterizing the ion contribution to the surface conductance of the particle (due to the applied electric field). In his approach, the mobile counterions and coions in the diffuse layer are only responsible for the surface conductance. Moreover, according to Eq. ( 48), the ionic electro-osmotic contribution to the surface conductance must not be neglected. For example, for a NaCl solution and at T = 298 K, the Na + and Cl -ion mobility values in the bulk electrolyte are 5.17×10 -8 and 7.89×10 -8 m 2 s -1 V -1 , respectively [15]. The second term in Eq. ( 48), for

z i = 1, ε = 78.3×8.85×10 -12 F m -1 , η = 0.895×10 -3 Pa s, F = 96485 C mol -1 , R = 8.314 J mol -1 K -1 , and T = 298 K, is ∼ 3.98×10 -
For a binary symmetric electrolyte, z + = z -= z, and the surface conductance is determined using Eqs. ( 30) and ( 47):

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = - 1 2 cosh 4 Σ 1 RT Fz B c Fz d b s ψ κ , (49)
assuming that

B B B = = + + ) ( ) ( , (50)
and

b b b c c c = = - + ) ( ) ( . (51)
Eq. ( 49) assumes that both anions and cations have the same mobility. By using Eqs. ( 19), ( 34), (49), and the Nernst-Einstein relationship for ion diffusivity,

Fz RT D i i β = , (52)
we obtain Bikerman's equation [25,49,55]:

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 2 2 1 1 2 cosh 2 Du Fz RT D RT Fz a d η ε ψ κ . ( 53
)
In Eq. ( 53), the influence of pH on surface conductivity is not taken into account.

Taking into consideration the presence of H + and OH -ions, we obtain the following equation for surface conductance, according to Eqs. ( 30), (47), and for a binary symmetric background electrolyte like NaCl or KCl [50]:

( ) ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - × + = + - + + - 1 2 exp 10 1000 2 Σ H pH ) ( ) ( 1 RT Fz B B c Fz d b s ψ κ ( ) ⎭ ⎬ ⎫ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × + + - - - - 1 2 exp 10 1000 OH p pH ) ( ) ( RT Fz B B c d K b f ψ , (54)
where pK f is the negative log of the dissociation constant of water (14 at T = 298K).

The Stern layer and total surface conductance

For some highly-charged minerals like clays, the contribution of the compact Stern layer to surface conductivity must not be neglected [START_REF] Leroy | [END_REF]. However, it has not been extensively studied for titanium dioxide nanoparticles. According to several authors (e.g. [18][19][20][21][22][23]), both the diffuse and Stern layers contribute to the specific surface conductance:

Stern diffuse Σ Σ Σ s s s + = . ( 55
)
The Stern layer contribution can be described by [50,[START_REF] Leroy | [END_REF]:

∑ = i i i i s e z St St Stern Γ Σ β , (56)
where the superscript "St" corresponds to the Stern layer, and St Γ i is the surface site density of adsorbed counterions at the Stern layer (in sites m -2 ). Eq. ( 56) describes the lateral movement of adsorbed cations and anions [56,57], and consequently assumes that the ion species are not immobile in the Stern layer [16]. The ion mobility values at the Stern layer are still relatively unknown. For clay minerals and quartz, Revil and Glover [50] and Revil et al. [58] considered that the ion mobilities of several cations like Na + , K + , Ca 2+ are at least one order of magnitude smaller than those in the bulk electrolyte. For silica minerals, Leroy et al. [15] have hypothesized that the surface ion mobilities are similar to those of the bulk. Molecular dynamics simulations at the solid/water interface must be of particular importance to characterize the ion and water mobilities in the compact layer. For example, in their work concerning the characterization of the surface properties of rutile, Predota et al. [32] considered a surface water mobility that is 10% of the bulk ion mobility at 3.7 Å from the surface.

According to Eqs. ( 54) to (56), the total surface conductance of a particle can be described by:

( ) ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - × + = + - + + - 1 2 exp 10 1000 2 Σ H pH ) ( ) ( 1 RT Fz B B c Fz d b s ψ κ ( ) ∑ + ⎭ ⎬ ⎫ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × + + - - - - i i i i d K b e z RT Fz B B c f St St OH p pH ) ( ) ( Γ 1 2 exp 10 1000 β ψ . ( 57
)
As seen by Eq. ( 57), it is easy to estimate surface conductance if the electrical potential at the head end of the diffuse layer ψ d and the surface sites Below, we use this approach to determine the zeta potential of TiO 2 nanoparticles using the particle size and the electrophoretic mobility data of Foissy [START_REF] Foissy | Analysis of adsorption of ions and polyacrylic acids in aqueous dispersions of titane dioxide[END_REF].

Comparison with experimental data

Modeling strategy

Firstly, the function f(κa) is determined according to Eqs. [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF], (20) and (21). Secondly, titanium dioxide surface complexation models estimate the surface conductance of the double layer Σ s (Eq. ( 57)). The subsequent calculation of the electrical conductivity of the bulk electrolyte based on its ion composition, σ b , (Eq. ( 34)) and the radius of an agglomerate made up of elementary particles, a, make it possible to determine the Dukhin number, Du (Eq. 19). Thirdly, the parameter corresponding to the electrical conductivity of the agglomerate's interior, K p , is calculated thanks to Eqs. ( 26) to [START_REF] Parkhurst | User's guide to PHREEQC (Version 2) -A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF] for a given value of the intra-aggregate porosity φ and elementary particle's radius a e .

Finally, the zeta potential is estimated from the electrophoretic measurements using the calculated values of f(κa), Du, K p , and Eq. ( 22) for the case of Henry's theory (see Fig. 5). By comparison, the von Smoluchowski equation (Eq. ( 11)) is also used to convert electrophoretic mobilities to zeta potential values. The surface complexation calculations were done with PHREEQC v2.17 [START_REF] Parkhurst | User's guide to PHREEQC (Version 2) -A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF] and the conversion calculations were done using a MATLAB routine. Raw electrophoretic measurements are necessary for these calculations. Unfortunately, only pre-processed zeta potentials are available in most publications. Foissy [START_REF] Foissy | Analysis of adsorption of ions and polyacrylic acids in aqueous dispersions of titane dioxide[END_REF] provides electrophoretic measurements for titanium dioxide P25 for a wide range of electrolyte concentrations but in a NaCl electrolyte background. According to the comparative modeling of Bourikas et al. [9], NO 3 -and Cl -behave similarly at the TiO 2 surface. This could be verified by satisfactorily reproducing the surface charge data that Ridley et al. [2] obtained on the anatase ST-01 using the same model as the ESM listed in Table 1 and the same parameters for Cl -as for NO 3 -(not shown).

Input data of the electrokinetic model

Foissy [START_REF] Foissy | Analysis of adsorption of ions and polyacrylic acids in aqueous dispersions of titane dioxide[END_REF] did titration and electrophoretic measurements on TiO 2 P25 (Degussa, Germany). The crystalline form is anatase (95 %). The density measured by helium pycnometry after drying is 3.76 g cm -3

. The estimated specific surface area (using N 2 adsorption and immersion calorimetry) is 54 ± 3 m 2 g -1 . The size of an elementary particle is approximately 30 nm, but P25 nanoparticles can form aggregates measuring several micrometers [1]. Foissy [START_REF] Foissy | Analysis of adsorption of ions and polyacrylic acids in aqueous dispersions of titane dioxide[END_REF] measured electrophoretic mobilities for pH values comprised between 4 and 11, and salinities (NaCl) of 10 -4 , 10 -3 , and 10 -2 M.

Furthermore, he used X-ray measurements to obtain the particle size distribution as a function of pH at 10 -3 M.

At 10 -3 M, we use a normal distribution in agreement with the particle size distribution measured by Foissy [START_REF] Foissy | Analysis of adsorption of ions and polyacrylic acids in aqueous dispersions of titane dioxide[END_REF] and calculate the Debye length to estimate the function f(κa).

For other salinities, we also use a normal distribution for the particle size. Eqs. ( 12), ( 48) and ( 57) make it possible to calculate the specific surface conductance using the values of ψ 2 (ψ 2 = ψ d here), St Na Γ , St Cl Γ determined by the ESM surface complexation model (Fig. 6). The Na + , Cl -, H + and OH -ion mobility values in the bulk and diffuse layer (β i ) are 5.17 [15] for silica minerals, to estimate surface conductance. Surface conductance increases significantly with salinity and with the difference between pH and IEP (6.5 here) (Fig. 6). Note that we restrict the investigated pH range at 10 -4 and 10 -3 M to have a constant ionic strength. According to Eq. ( 19), the Dukhin number (Du) is determined using (1) a normal distribution for the particle size a (in agreement at 10 -3 M with the particle size distribution measured by Foissy [START_REF] Foissy | Analysis of adsorption of ions and polyacrylic acids in aqueous dispersions of titane dioxide[END_REF]), (2) the estimated surface conductance Σ s , and (3) the electrical conductivity of the bulk electrolyte σ b . The fitting parameters are the intraaggregate porosity φ and the PSD at 10 -4 and 10 -2 M.

Zeta potentials from electrophoretic data

According to the particle size measurements of Foissy at 10 -3 M [3] (Fig. 7), titanium dioxide nanoparticles are highly agglomerated at pH values close to the IEP, leading to 20-µm agglomerates. At extreme pH values (pH = 4, pH =10) agglomerates reach sizes comprised between one and three micrometers. Thus, κa >>1 and increases significantly at pH values close to the IEP. As a result, the Dukhin number is low in amplitude except for basic pH where it increases strongly (Fig. 7). Zeta potentials inferred from electrophoresis using our model are in very good agreement with the ESM predictions assuming ψ 2 = ψ d (Fig. 7). We use φ = 0.4, which is in accordance with the intraaggregate porosity value measured recently by Xu et al. [59] for P25 (φ = 0.38) from the N 2 desorption isotherm using the cylindrical pore model (BJH method). The zeta potential calculations using the Smoluchowski equation are also presented by comparison. Their amplitudes are not in agreement with the ESM predictions because the Smoluchowski equation neglects surface conductivity. PSD found at other salinities seem to be physically realistic (Fig. 8). Size of agglomerated nanoparticles is pH-and salinity-dependent, and increases significantly close to the IEP and at high ionic strength. The Dukhin number also diminishs significantly in these physico-chemical conditions. In addition, for 10 -4 and 10 -2 M, zeta potentials calculated from electrophoresis using our modeling approach are also in very good agreement with the ESM predictions (Fig. 9), except for pH values close to the IEP. We use φ = 0.4.

Our modeling results suggest that the shear plane may be located close to the OHP, in contradiction with the hypothesis of a stagnant diffuse layer having a salinity-dependant thickness at the TiO 2 water interface [1,9].

Conclusions

We have developed an extended Stern layer model to characterize the electrochemical properties of titanium dioxide in a 1:1 electrolyte (NaCl). This model significantly lowers the amplitude of electrical potential at the OHP compared to that of other recent surface complexation models [1,9] without altering the quality of the titration data predictions.

Henry's model [START_REF] Henry | [END_REF] is used to convert electrophoretic mobility measurements to zeta potential values taking into account agglomerate's size and surface conductance.

Electrical conductivity inside the agglomerate is calculated using the differential selfconsistent model of Sen et al. [47]. The theory of particle's surface conductance due to the diffuse and Stern layers is described in depth.

By combining the excess of electrical charge calculated at the compact Stern layer and in the diffuse layer with Henry's equations [START_REF] Henry | [END_REF], we show that the shear plane may be located close to the OHP, contradicting the assumption of the presence of a stagnant diffuse layer at the TiO 2 /water interface as mentioned by Bourikas et al. [9] and Panagiotou et al. [1].

In the future, we will use a numerical model to convert electrophoretic and electrical conductivity measurements to zeta potential values under arbitrary conditions including high zeta potential amplitudes, particle volume fraction, polydispersity of the sample, diffuse layer overlapping, and surface conductance. In addition, we will compare our modeling results to a much more refined set of experiments (spherical and monodisperse/polydisperse particles) including electrical conductivity measurements. 1).

Fig. 3. TiO 2 surface properties predicted by different surface complexation models at three ionic strengths in KNO 3 . Lines depict the TPM reference model results while symbols depict the results of alternative models (Table 1).

Fig. 1 .

 1 Fig. 1. Schematic drawing of the basic Stern model of Bourikas et al. [9], our basic

Fig. 2 .

 2 Fig. 2. TiO 2 surface charge and potential at the head end of the diffuse layer predicted

Fig. 3 .

 3 Fig. 3. TiO 2 surface properties predicted by different surface complexation models at

  used the Stokes equation and calculated the electrical (using Poisson equation) and viscous forces on an element of volume of the liquid to express the electrophoretic mobility as a function of the zeta potential. The von Smoluchowski equation linearly relates the electrophoretic mobility μ e (in m 2 s -1 V - 1 ) to the electrical potential at the shear plane (ζ, in V):

Fig. 4 .

 4 Fig. 4. Schematic representation of the electro-chemical properties of a suspension of

Γ

  i are known. These parameters can be estimated by a complete surface complexation model of the TiO 2 /water interface. The ion mobilities of the counterions in the compact layer t S i β are the only unknown.

Fig. 5 .

 5 Fig. 5. Modeling strategy for determining the zeta potential from electrophoresis.

  , 7.89, 36.20, and 20.49×10 -8 m 2 s -1 V -1 , respectively (from PHREEQC's phreeqd.dat database). The ion mobilities at the Stern layer remain unknown. We used i i β β = St , according to the surface conductivity model of Leroy et al.

Fig. 6 .

 6 Fig. 6. The electrical potential at the 2-plane, the surface site densities of adsorbed

Fig. 7 .Fig. 8 .

 78 Fig. 7. Input data of the model of Henry [40] and zeta potentials versus pH at 10 -3 M

Fig. 9 .

 9 Fig. 9. Zeta potentials versus pH at 10 -4 , 10 -3 , and 10 -2 M from electrophoresis. The lines are the ESM predictions assuming ψ 2 = ψ d .

Figure captions Fig. 1 .

 captions1 Figure captions

Fig. 2 .

 2 Fig. 2. TiO 2 surface charge and potential at the head end of the diffuse layer predicted

Fig. 4 .

 4 Fig. 4. Schematic representation of the electro-chemical properties of a suspension of

Fig. 5 .

 5 Fig. 5. Modeling strategy for determining the zeta potential from electrophoresis.

Fig. 6 .

 6 Fig. 6. The electrical potential at the 2-plane, the surface site densities of adsorbed

Fig. 7 .

 7 Fig. 7. Input data of the model of Henry [40] and zeta potentials versus pH at 10 -3 M

Fig. 8 .

 8 Fig.8. Input data of the model of Henry[START_REF] Henry | [END_REF] versus pH at 10 -4 , 10 -3 , and 10 -2 M.

Fig. 9 .

 9 Fig. 9. Zeta potentials versus pH at 10 -4 , 10 -3 , and 10 -2 M from electrophoresis. The lines are the ESM predictions assuming ψ 2 = ψ d .

Table

  ).

	Panagiotou et al. [1] reported zeta potential values far below the potential values of the
	2-plane (in absolute value). They interpreted this to be a consequence of a shift of the
	shear plane (where the zeta potential is located) from this 2-plane. While calculating the
	distance d from the 2-plane to the shear plane, they found a log-log linear relationship
	between the ionic strength I (in M) and d (in nm):
	log	d	=	5115 . 0 -	log	I	-	7229 . 0	.	(7)
	2.2. Alternative models
	While the data of Panagiotou et al. [1] could be very well reproduced by their TPM
	model, it could also be reproduced with the simpler Basic Stern Model (BSM). The
	BSM parameters (Table

Table 1 .

 1 Surface and interface parameters for the Panagiotou et al. model[1] and alternative models.

	Parameters Original	Reference	BSM	ESM	Bidentate
		Panagiotou et	model (TPM)
		al. model
	log K 1	7.8		7.6	7.6	7.55	7.55
	log K 2	4.6		4.2	4.2	4.15	4.15
	log K Na (K)	-1.7 (-1.1)	-1.7 (-1.1)	-1.7 (-1.7)	-1.4 (-1.1)	-1
	log K NO3	-2.3		-2.3	-99 *	-1.5	-1.5
	Δz 0 Na	-		-	-	-	0.55
	Δz 1 Na (K)	0.7 (0)	0.7 (0)	n.a.	1 (0.7)	0.45 (0.7)
	Δz 2 Na (K)	0.3 (1)	0.3 (1)	n.a.	0 (0.3)	0 (0.3)
	Δz 1 NO3	-0.7		-0.7	n.a.	-1	-1
	Δz 2 NO3	-0.3		-0.3	n.a.	0	0
	C 1 (F m -2 )	3.2		3.2	2.5	2.5	2.5
	C 2 (F m -2 )	4.2		4.2	∞	1	1
	* In the BSM, NO 3	-anions act as indifferent anions

m 2 s -1 V -1 , which is of the same order of magnitude as the first term.

Acknowledgments

This study was done within the framework of the NANOSEP Project (ANR-08-ECOT-009). The authors are grateful to French National Research Agency for financial support.


Aurelie Peyrefitte

Michel Diament

Guillaume Martelet

Sylvain Bonvalot

Using GOCE satellite gravity data for the exploration of the African lithosphere

Using GOCE satellite gravity data for the exploration of the African lithosphere

A.Peyrefitte, M.Diament, G.Martelet, S.Bonvalot

The GOCE (Gravity field and steady-state Ocean Circulation Explorer) satellite mission, launched by the European Space Agency (ESA) in 2009, aims to map the Earth's gravity with unprecedented detail (1°x1° at global scale). It provides a new class of gravity observations (gradiometric measurements) that can be used to investigate the lithospheric structures and processes from regional to global scales. The African continent is characterized by ancient cratons as well as more recent orogenic belts and large basins which, in place, are affected by recent to actual intraplate volcanism, rifting or hotspots. This complex geology attests for past and on-going tectonic and geodynamic processes which affect the African plate at the lithospheric scale. Such patterns are well known to provide a large variety of density contrasts, so gravity data gathered from ground or airborne surveys are extensively used since several decades to investigate the crustal or lithospheric structures of the African plate. We present here the first gravity maps (Fig. 1) computed from GOCE datasets shortly delivered and discuss their use for understanding the main structural patterns of the African lithosphere and the geodynamic processes.

Comparison of the model deduced from the 2 first months GOCE data with EGM2008 model based on ground surveys and satellite data shows significant differences over Africa mainly related to the heterogeneous ground data distribution (see Fig. 2).

Our aim is to investigate the crustal gravity signature of the entire African plate in order to 1better discriminate between mantle and crustal gravity signatures and 2-bring constraints on the crust itself.

For this purpose, we realized a first 3D forward model in which the African crust is discretized in spherical prisms, by jointly using two global seismological models: CRUST 2.0 and the global digital map of sediment thickness. Our model consists of three sedimentary and three crystalline crust layers and has a spatial resolution of 1°x1°. We computed the gravity (Fig. 3) and tensor component effects of our model using a dedicated software based on a discretization into spherical prisms developed by Uieda et al.

We substracted this gravity effect to a Bouguer map that we derived from the first GOCEbased free air anomalies.

Short wavelength residuals attest for shortcomings in the input crustal seismological model. Long wavelength residuals highlight mantellic gravity components which reflect deep thermal/compositional variations, such as the effect of the African Superplume.

Then, we will take full advantage of the availability of the gravity gradients by incorporating them in the inversion process to improve the crustal density structure of our model.

Figure 1 :

 1 Figure 1: Free air and Bouguer anomalies computed from the first GOCE gravity model (mGal)

Figure 2 :Figure 3 :

 23 Figure 2: Difference between EGM2008 and model deduced from first GOCE data (mGal) Figure 3: Gravity response of the 3D forward model computed in spherical geometry (mGal)


P Henrik Alfredsson

Nils Tillmark

Takahiro Tsukahara

Instabilities and transition in three-dimensional flows with rotation SPANWISE ROTATION EFFECTS ON SHEAR FLOWS

Introduction

Spanwise rotation on shear flows may give rise to strong instabilities due to the influence of the Coriolis force (f Coriolis = 2U × Ω). Here we define spanwise rotation as rotation along the z-axis, whereas the undisturbed flow U is in the x-direction and the shear is in the y-direction. In plane Poiseuille flow one half of the channel is destabilized whereas the other is stabilized. In plane Couette flow on the other hand the full flow field is either stabilized or destabilized depending on the direction of rotation (see figure 1). Another interesting complication arises if the flow in addition is affected by a centrifugal force which would occur if a curved channel flow is subjected to spanwise rotation. In certain limits these flows can also be described by the Taylor-Couette system. Both rotation and centrifugal effects may have strong influence in many technical applications such as rotating machinery (e.g. compressors, turbines, pumps and fans) but also in the natural sciences such as planetary flows and astrophysical situations.

In this presentation we will illustrate the strong influence on rotation in several laboratory shear flows, both stabilizing and destabilizing effects as well as in combination with centrifugal forces. The presentation is based on results obtained in our laboratory (see e.g. [START_REF] Tsukahara | Flow regimes in a plane Couette flow with system rotation[END_REF]) but also results from other experiments, theory and DNS.

Results from linear stability analysis

Rotating shear flows can be described by two parameters, one is the Reynolds number Re = U h ν and the other describes the rotation effect as for instance

Ro = 2Ω z h U or Ω = 2Ω z h 2 ν
The introduction of two different non-dimensional numbers characterizing the effects of rotation is due to the fact that the linear stability problem is preferably solved using Ro since the problem then has certain symmetry properties, whereas for an experimental study Ω is preferable since then Re can be changed by changing U without affecting the rotation parameter, i.e. Ω. Assuming that the primary instability occurs in the form of counterrotating vortices we can write a neutral disturbance in the form v = v(y) exp (iβz/2h) and the stability equation reduces to and this gives the lowest critical Re as 20.7 at Ro = 0.5. As can be seen instability only exists in the range 0 < Ro < 1, otherwise the flow is stable.

∇ 6 v -Re 2 Ro(U -Ro)β 2 v = 0 Plane Poiseuille flow! Couette flow! Ω z x y z U (y) U (y) Ω z

Experimental results

The experimental apparatus used for the rotating Couette flow studies are shown in figure 2. The Re -Ω space has been mapped in great detail and up to 17 different flow regimes have been observed. One example is shown in figure 3. The details of this mapping and specific flow cases will be given in the presentation.

Figure 1 :

 1 Figure 1: Schematics of spanwise rotating plane Poiseuille and Couette flows.

Figure 2 :

 2 Figure 2: Schematic of the rotating Couette flow channel.

Figure 3 :

 3 Figure 3: Flow visualization of rotating Couette flow at Re = 250 and Ω = 21 from ref. [1].


Svebor Karaman
email: svebor.karaman@labri.fr

Jenny Benois-Pineau

Rémi Mégret
email: remi.megret@ims-bordeaux.fr

Nested Graph Words for Object Recognition

Keywords: I.4.7, I.4.8 [Image Processing and Computer Vision]: Scene Analysis -Object recognition, Feature Measurement -Feature representation Algorithms, Measurement, Design, Experimentation Nested features, Bag-of-Visual-Words, Graph Words, Delaunay triangulation, Context Dependent Kernel

In this paper, we propose a new, scalable approach for the task of object based image search or object recognition. Despite the very large literature existing on the scalability issues in CBIR in the sense of retrieval approaches, the scalability of media and scalability of features remain an issue. In our work we tackle the problem of scalability and structural organization of features. The proposed features are nested local graphs built upon sets of SURF feature points with Delaunay triangulation. A Bag-of-Visual-Words (BoVW) framework is applied on these graphs, giving birth to a Bag-of-Graph-Words representation. The nested nature of the descriptors consists in scaling from trivial Delaunay graphs -isolated feature points -by increasing the number of nodes layer by layer up to graphs with maximal number of nodes. For each layer of graphs its proper visual dictionary is built. The experiments conducted on the SIVAL data set reveal that the graph features at different layers exhibit complementary performances on the same content. The nested approach, the combination of all existing layers, yields significant improvement of the object recognition performance compared to single level approaches.

INTRODUCTION

Visual object retrieval in images and videos is one of the most active field of research in the community. One important aspect of the most popular techniques addressing this task are the use of local features, SIFT (Scale Invariant Feature Transform) of Lowe [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] or SURF (Speed-Up Robust Features) of Bay [START_REF] Bay | Surf:Speeded up robust features[END_REF] key points for instance. SIFT and SURF key points are robust and discriminative local features. SIFT points are detected on local minima/maxima of a Difference of Gaussians (DoG) image computed at different scales. The SIFT key point feature is a orientation histogram in a close spatial neighborhood of the key point. SURF is based on sums of approximated Haar wavelet responses and use integral images in order to speed-up key points extraction.

In the trending approach of Bag-of-Visual-Words [START_REF] Sivic | Video google: a text retrieval approach to object matching in videos[END_REF], the features are quantized in visual dictionaries by clustering and images are depicted as a distribution of the visual words within them. The Bag-of-Visual-Words approach is an adaptation of the text retrieval approach Bag-of-Words (BoW) to images. The BoVW operates on local key points when BoW operates on words, the semantic power of a word is much higher than one's of a local key point. A visual word is also much more ambiguous than a text word. Moreover, the BoVW approach discard all spatial information about the relations between key points. Having a similar neighborhood of key points in two images indicates stronger similarity of content than sparse isolated key points.

To overcome this limitation of the BoVW, some approaches have been presented in the past few years. The spatial pyramid matching proposed in [START_REF] Grauman | The pyramid match kernel: Discriminative classification with sets of image features[END_REF] and used in [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF] and [START_REF] Bosch | Representing shape with a spatial pyramid kernel[END_REF] for its application to object or scene recognition enables to compare area generated from arbitrary splitting instead of the whole image. In [START_REF] Albatal | Visual Phrases for automatic images annotation[END_REF] an approach called "Visual Phrases" is used to group visual words according to their proximity in image plane. The visual phrases are represented by a histogram containing the distribution of the visual words in the phrase. In these works, the common idea is to build local signature according to a visual dictionaries from an arbitrary splitting for the spatial pyramid matching or on a set built by a proximity criterion for visual phrases. Our approach is introducing the local topological information within the visual features.

In this paper we propose a spatial embedding of features with local Delaunay graphs. The motivation for building such graphs comes from the invariance of Delaunay triangulation with regard to affine transformations of image plane: rotation, translation and scale. Hence, with invariant key point features such as SURF the global invariance of graphs is maintained. We plunge these graph features into a Bag-of-Visual-Words framework, building visual dictionaries by clustering graphs. Then the state-of-the-art visual signatures are used for object retrieval. Increasing the number of nodes in graphs yields a layered approach where each layer induce a stronger spatial embedding within graph features. We call this approach "nested" .It combines visual signatures of all graphs from trivial graphs which are isolated SURF points to the largest graphs we work with.

The paper is organized as follows, in section 2 we discuss the process of building these graphs and introduce their nested construction. In section 3, we introduce the dissimilarity metric used to compare graphs and built visual dictionaries by clustering. The latter are presented in section 4. Experiments with these new features are presented in section 5. Conclusions and perspectives are given in section 6.

GRAPH FEATURE CONSTRUCTION

Let us consider a graph G=(X,E) with X a set of nodes corresponding to some feature points x k,k=1,.,K , in image plane and E={e kl } ,k=1,.,K,l=1,.,K , where e kl =(x k ,x l ), a set of edges connecting these points. We call such a graph a "graph feature". We will build these features upon sets of neighboring feature points in image plane. Hence we propose a spatial embedding of local features with graphs. To build such graphs t be addressed : i) the choice of feature points sets design of connectivity as edges E.

To define the feature points sets X upon which we are looking for a set of feature points that we call the "seeds" Around them other feature points will be selected to build each graph feature. One could consider all detected seeds, however that would lead to a strong overlapping of graphs and would induce high redundancy in the set of graph features created. Therefore, the selected seeds have to points which are more likely to be detected in the same object. This selection can be done using a criterion on the "size" of the key point or on the octave in which it has been detected. This would mean giving priority to SURF points detected in higher scale or lower scale which has no real significant impact on their repeatability. The best criterion for this task is the response of the SURF key points approximated determinant of the Hessian. detected where local maxima of the approximated determinant are reached [START_REF] Bay | Surf:Speeded up robust features[END_REF]. SURF points with higher response are then more likely to be more repeatable. considered for building the graphs will be the SURF points with highest responses. Considering a fixed number can define the set of seeds S:

{ } Seeds N s s S ,..., 1 =
Given S, our aim is to add partial structural information of the object while keeping the discriminative power of SURF key points. We will therefore define graphs over the seeds and their neighboring SURF points. For the edges we use the Delaunay triangulation which is invariant with regard to affine transformations of image plane preserving angles: translation, rotation and zoom. regarding the future extensions of this work to video, the choice of Delaunay triangulation is also profitable for its good properties in tracking of structures [START_REF] Mahboubi | Joint tracking of polygonal and triangulated meshes of objects in moving sequences with time varying content[END_REF]. The set of all vertices used for building the graph G i is X Gi , the union of the seed and its neighborhood

ܺ ீ ൌ ൛ݔ ଵ ீ , … , ݔ ீ ൟ ൌ ܲ ∪
A Delaunay triangulation is computed on the points of building triangles according to the Delaunay constraint. An edge e ij =(x i Gi ,x j Gi ) is defined between two vertices of the graph edge of a triangle connects these two vertices. nce we propose a spatial embedding of local features with graphs. To build such graphs two questions have to be addressed : i) the choice of feature points sets X and ii) the upon which graphs will be built of feature points that we call the "seeds". other feature points will be selected to build each detected SURF points as strong overlapping of graphs redundancy in the set of graph features have to form a set of SURF points which are more likely to be detected in various instances of n can be done using a criterion on the octave in which it has been priority to SURF points detected in higher scale or lower scale which has no real

The best criterion for this task is the response of the SURF key points, which is the approximated determinant of the Hessian. SURF points are local maxima of the approximated Hessian SURF points with higher response are then more likely to be more repeatable. Hence, the seeds for building the graphs will be the SURF points with number of seeds N Seeds , we , our aim is to add partial structural information of the object while keeping the discriminative power of SURF key

We will therefore define graphs over the seeds and their spatial nearest SURF gives the set of neighbors P i : Hence the set of nodes for each graph upon a seed point is built.

we use the Delaunay triangulation which is invariant with regard to affine transformations of image plane ion, rotation and zoom. Furthermore, ding the future extensions of this work to video, the choice of Delaunay triangulation is also profitable for its good properties in f all vertices used for building , the union of the seed and its neighborhood:

∪ ሼݏ ሽ
A Delaunay triangulation is computed on the points of X Gi , building triangles according to the Delaunay constraint. An edge is defined between two vertices of the graph G i if an edge of a triangle connects these two vertices. Introducing a layered approach, where each layer add structural information we can define graphs of increasing size while moving from one layer to the upper one. To avoid a large number of layers, the number of nodes added at each layer should induce a significant change of structural information.

Delaunay triangulation, at least two points have to be added to the seed at the second layer. Adding one more node triangles instead of just one, resulting in pattern. Therefore, the number of the upper one is fixed to three. one containing only one SURF point, the seed, and the containing a graph built upon the seed and see examples in Figure 1. One layer will always contain of all the lower layers, hence we call illustrate it in Figure 2.

GRAPH COMPARISON

In order to integrate these new graph features in a Bag Words framework a dissimilarity measure and a clustering method have to be defined. In this section measure.

When defining such a measure features only or only topology should be much more interesting to t Introducing a layered approach, where each layer adds more structural information we can define graphs of increasing size one layer to the upper one. To avoid a large number of layers, the number of nodes added at each layer should of structural information. To built at least two points have to be added to the ayer. Adding one more node may yield three resulting in a more complete local pattern. Therefore, the number of nodes added from one layer to the upper one is fixed to three. We define four layers, the bottom aining only one SURF point, the seed, and the top one containing a graph built upon the seed and its 9 nearest neighbors, One layer will always contain the points of all the lower layers, hence we call this approach "nested" and : The nested approach. Bottom to top: SURF seed depicted as the white node, 3 neighbors graph where are in black, 6 neighbors graph and 9 neighbors

GRAPH COMPARISON

In order to integrate these new graph features in a Bag-of-Visualframework a dissimilarity measure and a clustering method section, we define the dissimilarity measure, the (dis)similarity of node or only topology could be considered. However, it should be much more interesting to take into account both SURF descriptors -node features-and graph topology. To achieve this we will investigate the use of the Context Dependent Kernel (CDK) presented in [START_REF] Sahbi | Robust matching and recognition using context-dependent kernels[END_REF]. The definition of the CDK relies on two matrices: D which contains the distances between node features, and T which contains the topology of the graphs being compared.

Considering two graphs A and B with respective number of nodes m and n, let us denote C the union of the two graphs:

The feature correspondence square matrix D of size (m+n)x(m+n) contains the "entrywise" L 2 -norm (i.e., the sum of the square values of vector coefficients) of the difference between SURF features:

The topology square matrix T of size (m+n)x(m+n) defines the connectivity between two vertices x i C and x j C . In this work we define a crisp connectivity as we set T ij to one if an edge connects the vertices x i C and x j C and 0 otherwise. Hence, only sub matrices where both lines and columns in I A or I B are not entirely null. More precisely, we can define sub matrices T AA and T BB corresponding to the topology of each graph A and B respectively, while sub matrices T AB and T BA are entirely null, vertices of graphs A and B are not connected.

The CDK denoted K is computed by an iterative process consisting of the propagation of the similarity in the description space according to the topology matrix.

Where exp represents the coefficient-wise exponential and ||M|| 1 =Σ ij |M ij | represents the L1 matrix norm. Similarly to the definition of sub matrices in topology matrix T we can define sub matrices in the kernel matrix K. The sub matrix K AB (t) represents the strength of the inter-graph links between graphs A and B once the topology has been taken into account. We can therefore define the dissimilarity measure that will be used for clustering:

This dissimilarity measure will be applied separately on each layer. However, for the bottom layer, since there is no topology to take into account for isolated points we will use directly the "entrywise" L 2 -norm of the difference between SURF features as an approximation. This approximation is valid if we consider high enough values for β, see Appendix for details.

VISUAL DICTIONNARIES

The state-of-the-art approach for computing the visual dictionary of a set of features is the use of the K-means clustering algorithm with a large number of clusters, often several thousands. The code-word is either the center of a cluster or a non-parametric representation like a K-Nearest Neighbors (K-NN) voting approach.

Both of these approaches are not suitable for the graph-features as using the K-means clustering algorithm implies iteratively moving the cluster centers with interpolation whereas defining a mean graph is a difficult task; and a fast K-NN needs an indexing structure which is not available in our graph feature space since it is not a vector space. Therefore, we present in the following section the selected method which is a two pass agglomerative hierarchical clustering. The model of a cluster is chosen as its median instead of the mean.

Clustering method

In order to quantize a very large database, it can be interesting to use a two pass clustering approach as proposed in [START_REF] Gosselin | Combining visual dictionary, kernel-based similarity and learning strategy for image category retrieval[END_REF], as it enables gain in terms of computational cost. Here, the first pass of the agglomerative hierarchical clustering will be run on all the features extracted from training images of one object. The second pass is fulfilled on clusters generated by the first pass on all objects of the database. To represent a cluster, we use the following definition of the median:

∑ = - ∈ = m i G i v V G 1

argmin median

With V -a cluster and v i -members of a cluster, G the candidate median and ‖ ‖ is a distance or dissimilarity measure in our case.

For the first pass, the dissimilarities between all the features, of the same layer, extracted on all the images of an object are computed. For the second pass, only the dissimilarities between all the medians of all object clusters are computed. Each layer being processed independently, we obtain a visual dictionary per layers of graphs with 1, 3,…, N max nodes.

Visual signatures

The usual representation of an image in a BoVW approach is to compute a histogram of all the visual words of the dictionary within the image. We use this representation without rejection, a feature is always assigned to the closest word in the dictionary.

Once the visual signatures of images have been computed, one can define the distance between two images as the distance between their visual signatures. In preliminary experiments we have compared results when using Hamming distance, Euclidean distance and L 1 distance for this task. The L 1 distance giving better results, final results are presented using this measure only.

{ } [ ] [ ] 1 , 0 ) , ( 2 ) , ( ) , ( ) , ( 1 , 0 K = B) s(A, , ) ( ij ∈ - + = ∈ ∑ ∈ ∈ B A s B B s A A s B A G F I j I i t ρ 2 ) ( C j C i ij ij ij x x d where d D - = = [ ] [ ]    = + + ∈ ∈ = = ∈ ∈ = ∪ = - B B m i C i A A i C i I n m m i for B x x I m i for A x x with B A C .. 1 .. 1    ∃ = = otherwise x x e if T where T T C j C i ij ij ij 0 ) , ( 1 ) ( ) exp( ) ( ) ( ) ( , ) exp( ) exp( ) 1 ( 1 ) 1 ( ) 1 ( ) ( 1 ) 0
( T TK D K G K G K G K D D K t t t t - - - + - = = - - = β α β β β

EXPERIMENTS

The SIVAL (Spatially Independent, Variable Area, and Lighting) data set [START_REF]SIVAL[END_REF] includes 25 objects, each of them being present in 60 images taken in 10 various environment and different poses. This data set is quite challenging as the objects are depicted in various lighting conditions and poses, see a snippet of the data set in Figure 3. It has also been chosen as the future of this works is the recognition of objects of the daily living that may appear in different places of a house, like a hover that may be moved in all the rooms in one's house. We separate learning and testing images by a random selection of half of the images for each object. We only take into account the content of a bounding box of each object as the aim of this paper is only object recognition and not yet localization. SURF key points are extracted within the bounding box, the numbers of seeds for the graphs building process is fixed to 300. The second layer corresponds to graphs built upon the seeds and their 3 nearest neighbors, the third layer with the 6 nearest neighbors and the fourth and last layer with the 9 nearest neighbors. For the CDK, α is set to 0.0001, β to 0.1 (ensuring K is a proper kernel) and the number of iterations is fixed to 2, H. Sahbi [START_REF] Sahbi | Robust matching and recognition using context-dependent kernels[END_REF] has shown that the convergence of the CDK is fast. The pass clustering compute 500 clusters for each object. The final dictionary size varies in the range 50-5000. Each layer will yield its own dictionary.

The performance is evaluated by the Mean Average Precision (MAP) measure. For each test images, all images in the learning set are ranked from the closest (in terms of L 1 distance between visual signatures) to the furthest. The average precision is evaluated for each test image of an object, and the MAP is the mean of these values for all the images of an object in the test set.

SURF vs Graphs

First of all, it is interesting to analyze if the graph words approach where each layer is taken into consideration separately obtains similar performances compared to the classical BoVW approach using only SURF features. This is depicted in Figure 4. Here we can observe a really similar behavior between isolated SURF features (dotted lines) compared to a single layer of graph words (dashed lines) even if the approach using 9 nearest neighbor graphs seems to slightly outperform the others. The decrease in performance on the right side of the curve can be imputed to the two pass agglomerative clustering. Indeed as we selected 500 clusters for each object in the first pass, the second pass has only 12500 points to build 5000 clusters in our experimental set-up. This will conduct to an over segmentation of the dictionary where similar features are not grouped in the same clusters. The decrease on the left side is due to a too small dictionary size where not enough words are discriminative between objects. This similar average performance hides however some very important differences in the performance of each feature on some specific objects. To illustrate this we select four objects categories where graphs features and SURF give different performances in Figure 5 and Figure 6. For the objects "apple" and "spritecan", the isolated SURF features outperform the graph approach, see Figure 5, whereas for the "wd40can" and "feltflowerrug" objects the graphs features perform better, see Figure 6. This unequal discriminative power of each layer leads naturally to the use of the combination of the different layers in a single visual signature.

The nested approach

The combination of graphs and SURF features is done by the concatenation of the signatures of each layer. The three curves in solid lines in Figure 4 correspond to the nested approach using only the two bottom layers (SURF + 3 nearest neighbors graphs) in red, the three bottom layers (SURF + 3 nearest neighbors graphs + 6 nearest neighbors) in green and all the layers in blue. The improvement in the average MAP is really significant, and each addition of layer improves the results.

The detailed results presented in Figure 5 and Figure 6, where the nested approach results are the solid curves, show that the combination of the visual signatures computed on each layer separately performs better or at least as well as the best isolated feature.

CONCLUSION AND PERSPECTIVES

In this paper, we have presented new graph features built upon SURF points as nodes and expressing spatial relations between local key points. The nested approach of using growing neighborhoods in several layers enables to capture the most discriminative visual information for different types of objects. Using growing spatial neighborhood clearly improves the results while each layer taken separately yields similar results.

The future of this work is the application of the method to the recognition of daily living objects in videos. The approach could be enhanced by refining some steps of the graphs construction and comparison. For instance, the selection of seeds could be performed by an adaptive method and the topology matrix be defined with a soft connectivity. In order to be efficient when processing a large amount of images, i.e. in videos, a graph embedding procedure could be applied to use an indexing structure that would speed up the recognition process.

ACKNOWLEDGMENTS

This work is partly supported by a grant from the ANR (Agence Nationale de la Recherche) with reference ANR-09-BLAN-0165-02, within the IMMED project.

APPENDIX

Let A and B be two SURF points. The application of the same dissimilarity measure as the one used for the graphs cannot be iterate since no topology can be defined for an isolated point. The dissimilarity measure can only be estimated on K (0) .

1 =

 1 Finding the k spatial nearest SURF neighbors of each seed s i gives the set of neighbors Hence the set of nodes for each graph upon a seed point is built.

9 Figure 1 :

 91 Figure 1: SURF and graph features on a crop object ajaxorange from SIVAL database

Figure 2 :

 2 Figure 2: The nested approach. Bottom to top: SURF seed depicted as the white node, 3 neighbors graph where neighbours are in black, 6 neighbors graph.

(b) 3 -

 3 nearest neighbor graphs nearest neighbor graphs (d) 9-nearest neighbor graphs SURF and graph features on a cropped image of the object ajaxorange from SIVAL database.

Figure 3 :

 3 Figure 3: The SIVAL data set.

Figure 4 :

 4 Figure 4: Average MAP on the whole SIVAL data set. Isolated SURF features are the dotted curves, single layer Graphs

Figure 5 :

 5 Figure 5: Detailed MAP for a selection of objects where isolated SURF features (dotted curves) outperforms graphs (dashed curves). The nested approach is depicted as solid curves. For better readability only performances of graphs built with 9 nearest neighbors and corresponding SURF features are presented.

Figure 6 :

 6 Figure 6: Detailed MAP for a selection of objects where graphs (dashed curves) outperforms isolated SURF features (dotted curves). The nested approach is depicted as solid curves. For better readability only performances of graphs built with 9 nearest neighbors and corresponding SURF features are presented here.

  Let us denote d the "entrywise" L 2 -norm of the difference between SURF features of A and B. If the parameter β of the CDK is set to be high enough with regards to the values of d, the dissimilarity ρ defined in section 3 can be approximated by d.


David Fabre

Jérôme Mougel

Laurent Lacaze

Instabilities and transition in three-dimensional flows with rotation SLOSHING MODES AND SINGULAR INERTIAL MODES IN A CYLINDRICAL TANK ROTATING AROUND ITS AXIS

Introduction

The problem of sloshing of a liquid in a partially filled tank is relevant to many fields in engineering. The particular case where the tank is rotating is relevant, for instance, to the fuel tanks of satellites and spatial launchers. It is known that the fluid motion occurring in these tanks can have a destabilizing effect on the motion of the vehicle, so a prediction of the properties of these motions is highly desirable. The problem is also relevant to the tides in a closed lake or internal basin.

Here we consider the case of a partially filled cylindrical tank rotating around this axis. It is well known that in this case, a steady solution corresponds to a solid-body rotation of the liquid with a parabolic free surface. Previous investigations [START_REF] Miles | Free surface oscillations in a rotating tank[END_REF][START_REF] Miles | Surface oscillations of a rotating liquid[END_REF] mostly studied the modes and stability of small oscillations in the inviscid case. A striking result is that the nature of the problem and of its solutions sharply varies with the ratio of the relative frequency of the eigenmode (i.e. with respect to the rotating frame) to twice the rotation rate. When this parameter is larger than 2 or smaller than -2, the problem is elliptic and admits well-defined eigenmodes which do not penetrate deeply into the fluids. On the other hand, when this parameter is within the interval [-2, 2], the problem is hyperbolic, and nothing guaranties the existence of regular eigenmodes. Regular eigenmodes were actually identified in this range in particular cases : they have a different structure, more akin to the tidal waves first investigated in the shallow-water theory of Lord Kelvin. The works dealing with the viscous problem [START_REF] Bauer | Axisymmetric natural damped frequencies of a viscous liquid in a circular cylindrical container-an alternative semi-analytical solution[END_REF] seem to have been limited to the case of axisymmetric waves.

In this work we investigate the problem with three methods : a shallow-water theory, a semi-analytical method where the solutions are expanded in series of special functions, and a finite-difference solution of the viscous problem. The results are compared with previous works. A striking result is the discovery, in the hyperbolic case (as defined above), of a new set of viscous eigenmodes with a pecular structure, akin to the singular inertial modes found in other geometries such as spherical shells [START_REF] Rieutord | Inertial waves in a rotating spherical shell[END_REF] or the atmosphere of rotating stars [START_REF] Dintrans | Oscillations of a rotating star: a non-perturbative theory[END_REF].

Base flow, Stability equations and resolution methods

We consider a cylindrical recipient partially filled with water and rotating around its axis. We note R the radius of the container, Ω the rotation rate and g the acceleration of gravity. It is a classical exercise to demonstrate that the free surface takes, at equilibrium, a parabolic shape defined by

z = h(r) ≡ h 0 + Ω 2 r 2 2g - Ω 2 R 2 4g , (1)
where h 0 is the average depth of the water ; more precisely the depth that the same volume of water would fill in the recipient if the rotation was stopped. We investigate the linear stability properties of this flow by considering eigenmodes with the form û(r, z)e imθ e σt , where r, θ, z are cylindrical coordinates, m is an azimuthal wavenumber and σ = σ r + iσ i a complex frequency. These eigenmodes have to be computed as nontrivial solutions of an eigenvalue problem which can be written in the following symbolic form :

Aû = σBû. (2)
Three methods of resolutions are used to tackle this problem. The first method is a shallow-water theory which allows to reduce the problem to a single, second-order equation which turns out to have analytical solution in terms of hyperbolic functions. This method is by nature limited to the case where the aspect ratio χ = h 0 /R of the tank is small (but viscosity may be included with an ad-hoc modeling).

The second method, which is also limited to the inviscid case, consists of finding semi-analytical solutions expanded over a basis of functions with separable variables, with the following form (for the pressure component): Where J m is a Bessel function, k j , ξ j are vertical and radial "wavenumbers" which have to be taken among a discrete set of characteristic values, and X j are amplitude coefficients to be solved for. Such an expansion automatically satisfies the boundary conditions at the bottom and outer boundary. Truncation of this series to a finite N and imposition of the free-surface condition at N points leads system of equations for the N amplitudes X j , which admits nontrivial solutions for a discrete set of eigenvalues.

p(r, z) = Σ ∞ j=1 X j cosh(k j z) cosh(k j h 0 ) J m (ξ j r), (3
The third method consists of discretizing the set of partial differential equations formed by the eigenvalue problem with a finite element methods. We do this by means of the software FreeFem++. This method allows to consider the general, viscous problem.

Results

Before considering the general case, we consider three situations where a comparison between our three methods of resolution (and with previous work) is possible. The first particular case corresponds to weak rotation. In this case, the eigenmodes belong to two distinct sets. The first ones are "pure sloshing modes" identical to those existing in a non-rotating container. The second ones are "pure inertial modes" which are identical to those existing in a cylindrical container with a flat, rigid surface. The second particular case corresponds to weak aspect ratio h 0 /R and allows use of the shallow-water approximation. The third particular case corresponds to large aspect ratio h 0 /R. In this case inviscid solutions can be found in closed form [START_REF] Miles | Free surface oscillations in a rotating tank[END_REF] We then consider the general, viscous case. In addition to the modes already identified in the litterature, we evidence a new set of eigenmode solutions with a singular structure, akin to the singuliar modes existing in spherical shells [START_REF] Rieutord | Inertial waves in a rotating spherical shell[END_REF] or the atmosphere of rotating stars [START_REF] Dintrans | Oscillations of a rotating star: a non-perturbative theory[END_REF]. The structure of these modes is concentrated along "rays" forming a closed pattern in the geometry. Figure 1 shows sample specimens of such modes.

)Figure 1 :

 1 Figure 1: Exemples of singular eigenmodes in a tank with a parabolic surface


Arturo Héctor

Gonz Ález Araya

Instabilities and transition in three-dimensional flows with rotation TRANSITION CHARACTERISTICS OF A SWIRLING ANNULAR FLOW

1 Introduction and scope of the work:

The subject of this paper is transition of a swirling annular flow induced by the Tollmien-Schlichting mechanism which is characterised by waves propagating with a certain speed in the shear flow. One of the outstanding characteristics of this flow is the helical shape of its streamlines. Flows with such streamlines are very difficult to successfully simulate with CFD, as the Reynolds stress tensor has a strong anisotropy [START_REF] Rocklage-Marliani | Three-dimensional laser-Doppler velocimeter measurements in swirling turbulent pipe flow[END_REF]. The object of this work is to try to gain insight into the physical features both in qualitative and quantitative terms through studying the bifurcation characteristics of this flow on the verge of transition. The focus of our attention is on the shape of the Reynolds stress tensor, for which the relevant properties of the bifurcating solution of the nonlinear equation of motion for disturbances to the basic flow are examined. A set of (nonlinear) differential equations governing the dynamics of disturbances in a swirling flow is derived for the flow in the annular gap between concentric circular cylinders. The analysis is undertaken for the swirling flow when an axial pressure gradient (parameterized by Re, Reynolds number), a rotation of the outer cylinder (parameterized by S a , rotation number) and the axial translation of the inner cylinder (parameterized by T w , translation number) act simultaneously to maintain the swirling flow.

Outline of the method of Stewartson and Stuart to gain an understanding of the influence of flow parameters in this problem

The starting point of our work is a linear stability analysis for the flow described above. To this end it is necessary to derive the nonlinear equations of motion for the flow in which the pressure is eliminated as an unknown. The system of equations derived is referred to as the extended Orr-Sommerfeld and Squire equations which together with the continuity equation lead to the following equation system which is written in matrix notation Dv = N :

D =   D OSr D OSϕ D OSx D Sqr D Sqϕ D Sqx D Cor D Coϕ D Cox .   , v =   v r v ϕ v x   , N =   N OS N Sq N Co   (1)
Here D OS , D Sq , D Co , v and N represent the operator in the classical Orr-Sommerfeld equation, Squire equation, continuity equation, velocity disturbance and nonlinear terms in the radial, azimuthal, and streamwise directions respectively. When the velocity disturbance is considered small the nonlinear terms can be neglected and the equation system above becomes the classical eigenvalue problem with which it is possible to determine the neutral stability surface of the flow. The diagrams above demonstrate the stabilizing effect of both the rotation of the outer cylinder from Sa = 0.01 to Sa = 0.3 and the axial translation of inner cylinder from T w = 0.001 to T w = 0.05. The figures of projections of these neutrally stable curves were evaluated numerically through MATLABprograms. Following the steps developed by Stewartson and Stuart for a weakly nonlinear theory, elementary waves centered around the neutral stability curve which were determined in the linear stability theory are used as part of a wave packet which for large times is described as superposition of several elementary waves moving with the group velocity and whose amplitude varies slowly. The slow variation of the amplitude is designated here as B.

The wave packet has the following form:

v = A N (y) exp[iΘ N ]exp[ω N i t] + c.c.. (2)
Θ N (r, ϕ, x) = (λ N x x + n N ϕ ϕ + ω N t) (3)
Here the frecuency ω N is permitted to be complex, the wavenumber λ N x is real whereas n ϕ may assume only real integer values. The subscript N denotes values on the surface of neutral stability and i the imaginary part.

The above form of the wave involves the identification of a small amplitude parameter A with the temporal growth rate ω N i of the wave packet.

When the departure of the parameters of the flow (Re, Sa and T w) is small relative to those of the neutral stability surface, the growth rate ω N i can be approximated by its Taylor expansion through the flow parameters [START_REF] Huerre | Hydrodynamic Instabilities of Open Flows[END_REF].

A = ω N i = d Re |Re -Re N | + d Sa |S a -S aN | + d Tw |T w -T wN | (4)
With the help of this parameter it is possible to define new scales in the problem solving the nonlinear problem with the method of multiple scales. An asymptotic expansion is introduced for the velocity perturbation v as well as the extension of the partial derivatives of the equations in the new scales [START_REF] Stewartson | A nonlinear instability theory for a wave system in plane Poiseuille flow[END_REF].

v 1 2 A v 1 + A v 2 + 3 2 A v 3 + o( 3 2 A ) (5)
By substitution of (5) in (1) yields a new system of equations which is sorted according to the order of the approximations. In the compact matrix-operator notation these may be written as follows:

O(

1 2
A ) :

Dv 1 = (RHS) 1 = 0. O( A ) : Dv 2 = (RHS) 2 O( 3 2
A ) :

Dv 3 = (RHS) 3 (6)
The solution for v 1 is sought in a product form as follows:

v 1r = B r (New scales)A N r (y) exp[iΘ N ] + c.c v 1ϕ = B ϕ (New scales)A N ϕ (y) exp[iΘ N ] + c.c v 1x = B x (New scales)A N x (y) exp[iΘ N ] + c.c (7)
Formulating the solvability condition for the equations to the orders O( A ) and O(

A ) in (6) , which involves determining the solution of adjoint problem of original eigenvalue problem, one of the three equations necessary for the evolution equation of the amplitude B can be obtained. The remaining equations for B can be obtained through the continuity equation and determining the direction of flow of kinetic energy in the fluctuation motion, matching the ratio of the partial derivatives of the kinetic energy to new scales with the ratio of the partial derivatives of ω N i to the wave numbers of the problem. Besides it is expected that the form of the Reynolds stress tensor in the wall-normal direction is given by components of the tensor product of (A N r ; A N ϕ ; A N x ), with itself.

Figure 1 :

 1 Figure 1: Neutral surfaces for a swirling annular flow.