Karin El Rifaii

Henricus H Wensink

Thomas Bizien

Jean-Christophe P Gabriel

Laurent Michot

Patrick Davidson

Destabilization of the nematic phase of clay nanosheet suspensions by polymer adsorption

Keywords: Clay, beidellite, polymer, polyethylene oxide, dextran, nematic, liquid-crystal, adsorption

Complex aqueous mixtures comprised of swelling clays and hydrosoluble polymers naturally occur in soils and play a major role in pedogenesis. They are also very often used for formulating oil-well drilling fluids, paints, and personal-care products. The suspensions of some natural clays, thanks to their large nanoparticle aspect ratio, spontaneously form nematic liquid-crystalline phases where the particles align parallel to each other, which affects their flow properties. We observed that adding small amounts of hydrosoluble polymers to these clay suspensions destabilizes the nematic phase with respect to the isotropic (disordered) phase. The polymers that we used (polyethylene oxide and dextran) were too small to adopt particlebridging conformations and small-angle X-ray scattering experiments showed that the structure of the nematic phase is not altered by polymer doping. However, the adsorption isotherm shows that the macromolecules adsorb onto the clay nanosheets, effectively coating them with a polymer layer. Our extension of Onsager's theory for polymer-coated platelets properly captures the experimental phase diagram and shows how the nematic phase destabilization can be due to the polymer adsorbing more on the platelet faces than at the rim. Because the flow properties of the nematic phase are very different from those of the isotropic phase, the presence or absence of the former phase is an important factor to be determined and considered to explain the rheological behavior of these complex systems.

INTRODUCTION

Thanks to their low cost and wide availability, clay minerals find numerous industrial applications in very diverse areas such as cosmetics, paints, paper coatings, composites, and oildrilling fluids. Hydrosoluble polymers, like polyethylene oxide (PEO), are often added to aqueous clay formulations either to fine-tune their rheological properties for specific applications [START_REF] Haraguchi | Nanocomposite Hydrogels: A Unique Organic-Inorganic Network Structure with Extraordinary Mechanical, Optical, and Swelling/De-Swelling Properties[END_REF] or, more recently, to produce nanocomposite materials for pharmaceutical applications. [START_REF] Schexnailder | Nanocomposite Polymer Hydrogels[END_REF] Moreover, the interactions of dissolved organic matter with clays were also reported as playing an important role in soils where they control to a large extent water and nutrient retention capacity, and are also of prime importance in water treatment operations. [START_REF] Kalbitz | Controls on the Dynamics of Dissolved Organic Matter in Soils: A Review[END_REF][START_REF] Arias-Estevez | The Mobility and Degradation of Pesticides in Soils and the Pollution of Groundwater Resources[END_REF] Consequently, many fundamental studies have explored the structure and rheology of claypolymer aqueous mixtures over the last decades. Although several articles report on the investigation by neutron scattering of the adsorption of PEO on vermiculite, [START_REF] Smalley | Bridging Flocculation in Vermiculite-PEO Mixtures[END_REF][START_REF] Swenson | Structure of Bridging Polymers[END_REF][START_REF] Swenson | Mechanism and Strength of Polymer Bridging Flocculation[END_REF][START_REF] Swenson | Interlayer Structure of a Clay-Polymer-Salt-Water System[END_REF] the vast majority of these studies focused on the laponite -PEO system, probably because laponite is a synthetic clay mineral that is well characterized and easy to purchase. [START_REF] Lal | Interaction of Polymer with Clays[END_REF][START_REF] Nelson | Dynamic Light Scattering Studies of Poly(Ethylene Oxide) Adsorbed on Laponite: Layer Conformation and Its Effect on Particle Stability[END_REF][START_REF] Nelson | A Small-Angle Neutron Scattering Study of Adsorbed Poly(Ethylene Oxide) on Laponite[END_REF][START_REF] Mongondry | Influence of Pyrophosphate or Polyethylene Oxide on the Aggregation and Gelation of Aqueous Laponite Dispersions[END_REF][START_REF] Loizou | Large Scale Structures in Nanocomposite Hydrogels[END_REF][START_REF] Baghdadi | Rheology and Gelation Kinetics in Laponite Dispersions Containing Poly(Ethylene Oxide)[END_REF][START_REF] De Lisi | Adsorption of Triblock Copolymers and Their Homopolymers at Laponite Clay/Solution Interface. Role Played by the Copolymer Nature[END_REF][START_REF] Baghdadi | Evidence for Re-Entrant Behavior in Laponite-PEO Systems[END_REF][START_REF] De Lisi | Aqueous Laponite Clay Dispersions in the Presence of Poly(Ethylene Oxide) or Poly(Propylene Oxide) Oligomers and Their Triblock Copolymers[END_REF][START_REF] Schexnailder | Heterogeneity in Nanocomposite Hydrogels from Poly(Ethylene Oxide) Cross-Linked with Silicate Nanoparticles[END_REF][START_REF] Takeda | Rheo-SANS Studies on Shear Thickening in Clay-Poly(Ethylene Oxide) Mixed Solutions[END_REF][START_REF] Fall | Shear Thickening of Laponite Suspensions with Poly(Ethylene Oxide)[END_REF][START_REF] Lorthioir | Segmental Motions of Poly(Ethylene Glycol) Chains Adsorbed on Laponite Platelets in Clay-Based Hydrogels: A-NMR Investigation[END_REF][START_REF] Zulian | Dual Aging Behaviour in a Clay-Polymer Dispersion[END_REF][START_REF] Zulian | Influence of an Adsorbing Polymer on the Aging Dynamics of Laponite Clay Suspensions[END_REF] These articles described very puzzling rheological properties, such as sample rheo-thinning, shear-thickening, aging and rejuvenation, depending on the PEO concentration and molecular weight. A notable example is that of the "shake-gels" which are laponite/PEO fluid systems that become stiff enough, when strongly sheared, to support their own weight, yet turn fluid again after several hours. [START_REF] Pozzo | Reversible Shear Gelation of Polymer-Clay Dispersions[END_REF][START_REF] Zebrowski | Shake-Gels: Shear-Induced Gelation of Laponite-PEO Mixtures[END_REF] Most of these original rheological properties were explained by the steric barrier against gelation due to the adsorption of PEO and, in the case of long enough PEO, by the presence of "bridging" conformations that give rise to a hybrid particle/polymer network. However, laponite is singular by its small particle size (D  25 nm) compared to natural clays (D  200 -500 nm) and its behavior may not be quite representative of that of swelling clay minerals. For example, aqueous suspensions of some natural clays, like beidellite and nontronite, spontaneously form a nematic liquid-crystal phase at low weight fraction (a few wt%) whereas laponite suspensions undergo a sol/gel transition that prevents reaching the nematic phase at complete thermodynamic equilibrium. [START_REF] Gabriel | Observation of Nematic Liquid-Crystal Textures in Aqueous Gels of Smectite Clays[END_REF][START_REF] Michot | Liquid-Crystalline Aqueous Clay Suspensions[END_REF][START_REF] Paineau | Liquid-Crystalline Nematic Phase in Aqueous Suspensions of a Disk-Shaped Natural Beidellite Clay[END_REF][START_REF] Paineau | Liquid-Crystalline Properties of Aqueous Suspensions of Natural Clay Nanosheets[END_REF][START_REF] Lemaire | The Measurement by SAXS of the Nematic Order Parameter of Laponite Gels[END_REF] One reason for this difference in behavior is probably the huge aspect ratio of natural clay nanosheets (D/t > 300, where t = 0.65 nm is the nanosheet thickness) compared to that of laponite (D/t < 40) since the onset of liquid-crystalline ordering generally takes place at lower concentrations for increasing particle diameter.

In this article, we report on the effect of PEO addition on the isotropic/nematic (i.e. disordered/ordered) phase transition of beidellite clay aqueous suspensions. Since this transition is of first-order, its most salient feature is the coexistence, in some clay concentration range, of two phases in the same sample: a nematic birefringent phase at the bottom and an isotropic phase at the top. Surprisingly, we observed that adding PEO to beidellite suspensions completely destabilizes the nematic phase with respect to the isotropic one. Since the adsorption of PEO on some other clays is well documented in the literature, [START_REF] Mongondry | Influence of Pyrophosphate or Polyethylene Oxide on the Aggregation and Gelation of Aqueous Laponite Dispersions[END_REF][START_REF] De Lisi | Adsorption of Triblock Copolymers and Their Homopolymers at Laponite Clay/Solution Interface. Role Played by the Copolymer Nature[END_REF][START_REF] Yuang | Determination of the Surface Area of Smectite in Water by Ethylene Oxide Chain Adsorption[END_REF][START_REF] Rossi | Influence of Non-Ionic Polymers on the Rheological Behaviour of Na+-Montmorillonite Clay Suspensions. Part II. Homopolymer Ethyleneoxide and Polypropylene Oxidepolyethylene Oxide ABA Copolymers[END_REF] we developed a theoretical model, based on the Onsager theory, that considers "dressed" clay nanosheets, coated with polymer, and we thus show that this unexpected experimental observation may be the sign that PEO adsorbs more onto the beidellite nanosheet faces than at its edge.

MATERIALS AND METHODS

Materials:

The beidellite clay preparation procedure was already described in detail elsewhere [START_REF] Paineau | Liquid-Crystalline Nematic Phase in Aqueous Suspensions of a Disk-Shaped Natural Beidellite Clay[END_REF] and we therefore only describe it here briefly. Beidellite from Idaho was purchased from the Source Clays Repository at Purdue University, West Lafayette, Indiana, USA. 200 g of the as-received sample were ground in an agate mortar, dispersed in 5 liters of a 1M NaCl solution and stirred for a few hours. Due to the high ionic strength in this suspension, once stirring was stopped, flocculated clay particles settled at the bottom of the vessel. The supernatant was then discarded and replaced by fresh 1M NaCl solution. This operation was repeated three times, which ensures that all clay layers are exchanged with Na + counterions. The sample was then redispersed in 10 liters of deionized water, placed in various dialysis tubes (Visking 14000Da) and dialyzed against at least 25 liters of deionized water. The water was changed every 24 to 48 hours and the conductivity of the bath was checked. Dialysis was stopped when the conductivity was as low as 20 µS/m. The suspension was then poured into Imhoff settling cones and, after 24 hours of sedimentation, the material located at the bottom of the cones was discarded. The final suspension was then subjected to successive centrifugation steps at 4600, 16000, and 34000 g (average values in the rotor). After each centrifugation step, the bottom fraction was collected and dispersed in deionized water to yield a free flowing suspension, whereas the supernatant was centrifuged at higher speed. Three size fractions were thus obtained. The one used in the present work is the last one; it has the smallest nanosheets, with an average equivalent particle diameter of D = 200 nm, as determined by transmission electron microscopy (TEM); the particle thickness is t = 0.65 nm. Moreover, TEM experiments showed that the beidellite particles have a rather irregular shape and a polydispersity distribution of about 30% (Figure SI 1). Moreover, assuming that the clay sheets are fully delaminated in suspension, the specific surface area, SSA = 770 m 2 .g -1 , can be estimated from the particle dimensions and density (2.7). The weight fraction of the beidellite suspension initial batch was determined by its weight loss upon drying to be 1.6 wt%. The zeta potential of the clay particles in suspension is -45 mV and the particle charge density is 0.1 C.m -2 . The pH of the stock clay suspension is 5.8, which is probably due to the dissolution of atmospheric CO 2 . These aqueous suspensions of beidellite clay show very good colloidal stability over many years.

Most polymer / clay mixtures were prepared with PEO of molar mass M W = 100 kDa, which corresponds to an average polymerization degree of 2300, purchased from Sigma-Aldrich.

Additional experiments were also performed with PEO of M W = 10 kDa molar mass, as well as with dextran (a natural polysaccharide also purchased from Sigma-Aldrich) of molar mass M W = 100 kDa. The polydispersity distributions of these polymers were not determined by the provider. The concentrations of the initial pure polymer solutions were defined by precisely weighing the constituents.

Sample preparation:

Colloidal mixtures of beidellite nanosheets and PEO were prepared by directly mixing the suspensions of nanosheets with solutions of PEO in deionized (18 M.cm) water. The composition of each mixture was defined by the weight fractions, C beid , and C PEO , of the beidellite nanosheets, and PEO, respectively, to explore 2-dimensional phase diagrams. For this purpose, the proper volumes of nanosheet suspensions and PEO solutions were precisely measured with a calibrated micropipette and dilution lines, at constant C beid , or C PEO , were prepared. The density of beidellite is 2.7, so that the clay volume fraction,  beid , in the mixture is simply given by C beid /2.7. The density of PEO is 1.1 and the volume fraction of PEO,  PEO , in the mixture is given by C PEO /1.1. The ionic strength of the mixtures is essentially set by the NaCl molarity (10 -4 M) used in the last dialysis of the clay suspensions. All mixture samples had the same 1 mL volume and were poured into 2 mL glass vials equipped with tight Teflon-lined caps.

All samples were stored in the dark.

Visual observation of the samples:

Series of samples, at constant C beid or C PEO , were examined with the naked eye, either in natural light or between crossed polarizers to assess the number of coexisting phases and to check for their birefringence. Photographs of the samples were recorded with an Olympus XZ-1 camera, at normal incidence. The samples were examined at different times after preparation, over a year, but most of the results shown here were obtained after about two months of sample aging.

Small-angle X-ray scattering experiments:

A series of small-angle X-ray scattering (SAXS) measurements of the beidellite / PEO system was made at the Swing SAXS beamline of the SOLEIL synchrotron facility at Saint-Aubin, France. The X-ray energy was 12 keV ( = 0.1033 nm), the sample-to-detector distance was 6.226 m, so that the scattering vector modulus, q (q = (4sin)/ where 2 is the scattering angle) ranged roughly between 10 -2 and 1 nm -1 . The beam size was 375 × 75 µm 2 at the sample level, the scattering was recorded by an Eiger-4M detector, and the exposure time was typically 0.5 s. The usual SAXS data reduction procedures were made, using the Swing data reduction software (Foxtrot 3.4.9), and the data was displayed either as 2-dimensional SAXS patterns or as plots of the scattered intensity versus scattering vector modulus, I(q), obtained by azimuthal averaging of the SAXS patterns. The samples were filled into cylindrical Lindemann glass capillaries of 1.0 ± 0.1 mm diameter (Glas-Technik & Konstruktion, Germany) and sealed with silicone glue. Each capillary tube was scanned along its vertical axis by recording the SAXS patterns every few millimetres.

Adsorption isotherm:

The adsorption isotherm of PEO on beidellite was obtained using the following protocol. A starting PEO solution at 20 g/L was diluted to obtain a series of 9 solutions with concentrations ranging from 0 to 20 g/L. The viscosity of each of these solutions was then measured using an Anton Paar automatic Microviscosimeter AMVn, which yields a calibration curve relating viscosity with PEO concentration (Figure SI 2). 2 mL of each of these solutions were then mixed with 0.5 mL of a beidellite suspension of known weight fraction (typically around 15 g/L). After four hours, the suspensions were centrifuged at 105000 g to separate the liquid from the solid.

The viscosity of the supernatant was measured and the content in PEO was deduced on the basis of the calibration curve. The adsorbed amount was then measured by the rest method, i.e.

𝑄 𝑎 = (𝑐 𝑖 -𝑐 𝑒 ) * 𝑉

𝑀

where c i and c e are the initial and equilibrium concentrations in PEO, respectively, V the suspension volume and M the mass of solid initially dispersed in the suspension. Moreover, several data points of the adsorption curve were checked by additional thermogravimetric analyses of the supernatant.

EXPERIMENTAL RESULTS AND DISCUSSION

The addition of PEO to the suspensions of beidellite clay nanosheets did not result in any obvious effect such as flocculation or coacervation. In fact, detecting the effect of PEO doping on the samples required their observation between crossed polarizers. Under these conditions, the isotropic phase looks completely dark whereas the nematic phase, owing to its birefringence, looks bright. Consequently, samples belonging to the biphasic domain of the phase diagram, between the nematic and isotropic phases, show the isotropic/nematic phase coexistence in polarized-light, with the bright nematic phase residing at the bottom of the vial whereas the dark isotropic phase is found at the top. The isotropic phase has a concentration of clay nanosheets which is large enough to show strong flow-birefringence and Tyndall scattering. Indeed, we also checked, by measuring the weight loss upon drying, that the clay concentration in the isotropic phase is not much lower than that in the nematic phase, as expected from previous work. [START_REF] Paineau | Liquid-Crystalline Nematic Phase in Aqueous Suspensions of a Disk-Shaped Natural Beidellite Clay[END_REF] The observation of four series of samples of constant beidellite concentration and increasing PEO content, placed between crossed polarizers, reveals that the nematic phase is strongly destabilized with respect to the isotropic one (Figure 1). For example, in the series with C beid = 1.28 wt% (Figure 1b), the first sample, with no PEO, is almost completely nematic whereas in the last sample, with C PEO = 1 wt%, the nematic phase has almost completely vanished. In a similar way, in the series with C beid = 1.12 wt% (Figure 1c), the first sample is biphasic with almost equal proportions of nematic and isotropic phases whereas the sample, with C PEO = 0.5 wt%, has no nematic phase left. Our synchrotron SAXS study of the beidellite / PEO system confirmed this phase identification (Figure 2) as the SAXS patterns were indeed those expected for the isotropic and nematic phases. The patterns of the nematic phase were anisotropic (Figure 2a) when the nematic part of the sample was aligned by the shear flow that occurred when the capillaries were filled.

The anisotropy of the SAXS patterns demonstrates the long-range orientational order of the phase and, combined with the absence of any sharp scattering features, proves its nematic nature. [START_REF] Paineau | Liquid-Crystalline Properties of Aqueous Suspensions of Natural Clay Nanosheets[END_REF] In contrast, the SAXS patterns of the disordered phase (Figure 2b) only showed a strong isotropic halo and the corresponding azimuthally-averaged scattered intensity (see Materials and Methods section), I(q), was a featureless, monotonously decreasing, function of the scattering vector modulus q.

In the nematic phase, I(q) shows marked scattering peaks. Moreover, in first approximation, since the particle form factor decays as q -2 in this q-range (called "intermediate" range), the quantity obtained by multiplying I(q) by q 2 (Kratky representation) is proportional to the structure factor, S(q), of the phase (Figure 2c,d). [START_REF] Feigin | Structure Analysis by Small-Angle X-Ray and Neutron Scattering[END_REF] S(q) in the nematic phase has strong oscillations that classically arise from some liquid-like positional order of the nanosheets. In this system of negatively charged nanosheets that mostly interact through strong electrostatic repulsions, the average distance, d, between nanosheets along the nematic director is related to the position, q 0 , of the first broad scattering peak by d  2π/q 0 . We therefore simply deduce the distance d from the peak position. Although q 0 varies with the clay weight fraction, its dependence on PEO doping (Figure SI 3) did not display, within experimental accuracy, the typical large increase associated with the appearance of bridging conformations of the polymer. [START_REF] Swenson | Mechanism and Strength of Polymer Bridging Flocculation[END_REF] Note that this average distance between nanosheets is rather large, typically around 100 nm, due to the very low (< 2 wt%) nanosheet weight fraction of the system. Moreover, we did not observe any systematic influence of PEO addition on the higher-order peaks, either in terms of peak intensity or width. The green lines show the boundaries of the isotropic/nematic coexistence region.

Our investigation of PEO adsorption onto beidellite nanosheets shows that this polymer has large affinity for the clay, as illustrated by the adsorption isotherm (Figure 4). Indeed, an initial sharp adsorption stage is clearly observed, followed by a saturation plateau at a value of  0.7 g adsorbed PEO per g of clay. This result corresponds to 0.9 mg/m 2 , a value quite comparable to those previously reported (0.6 -0.9 mg/m 2 , depending on the polymer molecular weight) for the adsorption of PEO on laponite. [START_REF] Mongondry | Influence of Pyrophosphate or Polyethylene Oxide on the Aggregation and Gelation of Aqueous Laponite Dispersions[END_REF] The series of samples shown in Figure 1 all belong to the regime of initial steep adsorption rise in Figure 4. Our adsorption measurements, in line with previous reports on laponite, [START_REF] Lal | Interaction of Polymer with Clays[END_REF][START_REF] Nelson | A Small-Angle Neutron Scattering Study of Adsorbed Poly(Ethylene Oxide) on Laponite[END_REF][START_REF] Mongondry | Influence of Pyrophosphate or Polyethylene Oxide on the Aggregation and Gelation of Aqueous Laponite Dispersions[END_REF] means that the PEO also adsorbs on beidellite, with a very similar adsorption isotherm. Lal et al and De Lisi et al who used neutron scattering techniques reported that both short and long PEO chains adsorbed on laponite form a thin layer of only a few nanometers thickness. 9, [START_REF] De Lisi | Aqueous Laponite Clay Dispersions in the Presence of Poly(Ethylene Oxide) or Poly(Propylene Oxide) Oligomers and Their Triblock Copolymers[END_REF] Burchill et al reached a similar conclusion for short PEO chains adsorbed on Na-montmorillonite. [START_REF] Burchill | Smectite Polymer Interactions in Aqueous Systems[END_REF] Then, the classical depletion interactions [START_REF] Lekkerkerker | Colloids and the Depletion Interaction[END_REF] are probably irrelevant for this hybrid system, at least for C PEO < 0.5 wt% where most of the destabilization of the nematic phase takes place. Moreover, the principal signature of depletion interactions on the phase diagram is to widen the biphasic gap, not to tilt it towards higher particle concentration. The values found in the literature [START_REF] Kuhl | Modulation of Interaction Forces between Bilayers Exposing Short-Chained Ethylene Oxide Headgroups[END_REF][START_REF] Devanand | Asymptotic Behavior and Long-Range Interactions in Aqueous Solutions of Poly(Ethylene Oxide)[END_REF] for the gyration radius of free PEO in water at room temperature are somewhat scattered but a value of R g  25 nm seems reasonable for PEO with 100 kDa molar mass. In principle, the partially adsorbed PEO coils should show a succession of loops, adsorbed trains, and dangling tails, or they might even "bridge" neighboring cofacial nanosheets. [START_REF] Fleer | Polymers at Interfaces[END_REF] The value of 0.7 g adsorbed PEO per g of clay corresponds to about 230 PEO coils per beidellite nanosheet (115 on each side), which is roughly six times the full particle coverage if the PEO coils adsorbed in random-coil conformation, which rules out this possibility.

Moreover, bridging conformations seem quite improbable because the average distance between nanosheets is at least 80 nm for all samples, and therefore much larger than the PEO end-to-end distance, resulting in an elastic energy penalty for stretching  k B T(d/R g ) 2 > 10 k B T. [START_REF] Liu | Reversible Control of Spacing in Charged Lamellar Membrane Hydrogels by Hydrophobically Mediated Tethering with Symmetric and Asymmetric Double-End-Anchored Poly(Ethylene Glycol)s[END_REF] This is consistent with the absence of any sign of gelation or polymer bridging flocculation in our samples. Therefore, in the following theoretical model (see next section), we consider that the PEO macromolecules adsorb as a succession of trains, loops, and tails, giving rise to a polymer coating layer on the beidellite nanosheets.

This assumption is further confirmed by our observation that the nematic phase of the beidellite suspensions is also destabilized (see Figure SI 4-6) by adding roughly the same amount of PEO of M w 10 kDa as with PEO of M w 100 kDa, which means that the total PEO length is not a relevant parameter (at least in the range that we explored). Note also that PEO does not bear any electric charge at neutral pH and therefore should not alter, at such low concentrations, the electrostatic repulsions between nanosheets.

We also observed the destabilization of the nematic phase of beidellite clay suspensions using dextran instead of PEO (See Figure SI 7). Although both polymers are hydrosoluble, the molecular structure of dextran is quite different from that of PEO because the former is a polysaccharide. Moreover, similar experiments done with nematic suspensions of nontronite clay [START_REF] Michot | Liquid-Crystalline Aqueous Clay Suspensions[END_REF] showed the same effect (See Figure SI 8). Therefore, the disappearance of the nematic phase upon addition of adsorbing polymer seems to be a rather general phenomenon, possibly amenable to a generic theoretical description.

THEORY OF ORDERING TRANSITIONS IN FLUIDS OF POLYMER-COATED NANOSHEETS

SECOND-VIRIAL THEORY FOR POLYMER-COATED PLATELETS:

Inspired by a previously reported model for polymer-coated rods [START_REF] Shundyak | Theory of the Isotropic-Nematic Transition in Dispersions of Compressible Rods[END_REF] we consider a hard cylindrical platelet of thickness L and diameter D enveloped by a uniform layer of adsorbed polymer of thickness . Near-field electrostatic interactions between the platelets, which we shall not consider in further detail, may be incorporated through an effective platelet thickness that is larger than the bare hard-core value. While multi-particle correlations unmistakably play a role in the thermodynamics of discotic fluids, [START_REF] Duncan | Theory and Computer Simulation for the Cubatic Phase of Cut Spheres[END_REF][START_REF] Masters | Virial Expansions[END_REF] we restrict ourselves to considering interactions between particle pairs alone, in the supposition that such a second-virial approximation provides valuable qualitative insight into the ordering properties of nanosheets. The second virial coefficient between two such platelets at random orientation indicated by the enclosed angle between their normals  reads:

𝐵 2 (𝛾) = - 1 2 ∫ 𝑑𝑟[exp(-𝛽𝑈(𝑟; 𝛾)) -1]
(1) with  -1 = k B T the thermal energy in terms of Boltzmann's constant k B and temperature T, U the pair potential and r the distance between the centres-of-mass of the platelets. A simple approximation is to assume the pair interaction to be infinitely repulsive when the hard-cores overlap, but of finite repulsion  > 0 when the polymer layer of one particle overlaps with either the layer or hard-core of the other particle. Since the repulsive potential is constant within the cylindrical envelope of the disc, the second-virial coefficient is simply proportional to the excluded volume v ex (details can be found in Reference [START_REF] Vroege | Phase-Transitions in Lyotropic Colloidal and Polymer Liquid-Crystals[END_REF] ), so that we write for the coated discs:

𝐵 2 (𝛾) = - 1 2 [(𝑒 -𝛽𝜀 -1)𝑣 𝑒𝑥 (𝐿 𝛿 , 𝐷 𝛿 ) -𝑒 -𝛽𝜀 𝑣 𝑒𝑥 (𝐿, 𝐷)] (2)
with L  = L+2 the total thickness of the platelet (hard-core plus polymer coating) and, similarly, D  = D + 2 for the diameter. Here,  is a measure for the amount of edge adsorption compared to face adsorption, so that  = 0 refers to no edge adsorption,  = 1 to equal adsorption at the face and edge, while  > 1 corresponds to preferential adsorption at the plate edge. The key quantity featuring in B 2 is the orientation-dependent excluded volume, defined as: [START_REF] Onsager | The Effects of Shape on the Interaction of Colloidal Particles[END_REF] 𝑣 𝑒𝑥 (𝐿, 𝐷) = with E() an elliptic integral of the second kind. Unlike rods, [START_REF] Shundyak | Theory of the Isotropic-Nematic Transition in Dispersions of Compressible Rods[END_REF] the excluded volume of platelets, at least up to leading order for flat particles, does not depend on the thickness L which only enters through higher-order correction terms that depend quite subtly on the inter-particle orientation. This will likely impact the phase diagram in ways that are qualitatively different from that of polymer-coated rods. [START_REF] Shundyak | Theory of the Isotropic-Nematic Transition in Dispersions of Compressible Rods[END_REF]

ISOTROPIC-NEMATIC COEXISTENCE:

The total Helmholtz free energy F of a platelet fluid composed of N particles reflects a balance between orientational entropy and pair interactions encoded in the second-virial coefficient: [START_REF] Vroege | Phase-Transitions in Lyotropic Colloidal and Polymer Liquid-Crystals[END_REF][START_REF] Onsager | The Effects of Shape on the Interaction of Colloidal Particles[END_REF] 𝛽𝐹 𝑁 ~ln 𝜌 -1 + 〈ln 4𝜋𝑓〉 𝑓 + 𝜌〈〈𝐵 2 (𝛾)〉〉 𝑓 (4) where the brackets denote an average with respect to the orientational distribution function f which is a priori unknown. The chemical potential and pressure follow from the free energy using standard thermodynamic derivatives and read:

𝛽𝜇~ln 𝜌 + 〈ln 4𝜋𝑓〉 𝑓 + 2𝜌〈〈𝐵 2 〉〉 𝑓 𝛽𝑃~𝜌 + 𝜌 2 〈〈𝐵 2 〉〉 𝑓 (5)
A simple Gaussian approximation for f enables us to render the free energy in algebraic form. [START_REF] Vroege | Phase-Transitions in Lyotropic Colloidal and Polymer Liquid-Crystals[END_REF] Indeed, up to leading order we find for strongly aligned nematics and for isotropic order, respectively: [START_REF] Wensink | Isotropic-Nematic Density Inversion in a Binary Mixture of Thin and Thick Hard Platelets[END_REF] 〈ln 4𝜋𝑓〉 𝑓 ~ ln𝛼 -1 (𝑁) 〈ln 4𝜋𝑓〉 𝑓 ~ 0 (𝐼) (6) with  a measure for the nematic order. Similarly, we find for the angular contributions in

B 2 〈〈|sin 𝛾|〉〉 𝑓 ~ √ 𝜋 𝛼 (𝑁) 〈〈|sin 𝛾|〉〉 𝑓 ~ 𝜋 4 (𝐼) (7)
and:

〈〈𝐸(𝛾)〉〉 𝑓 ~ 𝜋 2 (𝑁) 〈〈𝐸(𝛾)〉〉 𝑓 ~𝜋2 8 (𝐼) (8)
and finally:

〈〈|cos 𝛾|〉〉 𝑓 ~ 1 (𝑁) 〈〈|cos 𝛾|〉〉 𝑓 ~1 2 (𝐼) (9)
Minimizing the free energy for  we find a simple quadratic relationship between the degree of nematic order and the plate concentration  = N/V:

𝛼~𝜋 4 𝜌 2 𝛼 0 2 (10)
with

𝛼 0 ~1 2 [(1 -𝑒 -𝛽𝜀 ) ( 𝜋 2 𝐷 𝛿 3 + 2𝐿 𝛿 2 𝐷 𝛿 ) + 𝑒 -𝛽𝜀 ( 𝜋 2 𝐷 3 + 2𝐿 2 𝐷)] (11)
In what follows we will use c = ( 2 /16)D 3 as a dimensionless measure for the plate concentration which yields the well-known asymptotic values for the coexistence concentrations, c I = 3.45 and c N = 5.12 for strongly anisotropic particles in the limit of zero thickness L/D → 0. [START_REF] Vroege | Phase-Transitions in Lyotropic Colloidal and Polymer Liquid-Crystals[END_REF] Next, we explore the effect of an increasing thickness  of adsorbed polymer, induced by an increasing amount of polymer added to the system, onto the isotropic-nematic phase diagram.

For simplicity, we will set the hard-core aspect-ratio to zero (L/D = 0) and we further assume that increasing the amount of added polymer does not markedly change the soft repulsion  ~ 1/k B T between overlapping polymer coatings. Figure 5 showcases the subtle effect of edge adsorption on the phase behavior; while strong edge adsorption ( = 1) favors nematic order, a marked weakening or absence of edge adsorption ( = 0) clearly stabilizes isotropic order, as observed in the experiment. One could also envisage a scenario where adsorption produces a coating of fixed size  which becomes increasingly dense in polymer, thus leading to enhanced repulsion  as more polymers adsorb. We find that keeping  = 0.1 and increasing  produces scenarios that are qualitatively similar to the ones in Figure 5.

We reiterate that the model presented here does not claim to give quantitative predictions for the experimental systems studied. In fact, comparing the experimental and theoretical phase diagrams, we find that the experimental concentrations are typically a factor 4 lower than the theoretical ones (even in the absence of a polymer layer). We attribute this discrepancy to the intricate effects of far-field electrostatic forces acting between pairs, triplets, etc. of discs, [START_REF] Jabbari-Farouji | Interplay of Anisotropy in Shape and Interactions in Charged Platelet Suspensions[END_REF] which to this day remain largely elusive and therefore cannot straightforwardly be incorporated into the theoretical model.

We further mention that the current predictions could be qualitatively affected by diameter and thickness dispersity of the discs as present in the experimental situation. In principle, this could be scrutinized via a straightforward extension of the present second-virial model by allowing the disc dimensions to follow a continuous distribution (e.g. of the log-normal form) with a prescribed standard deviation. [START_REF] Wensink | Phase Equilibria in Systems of Hard Disks with Thickness Polydispersity[END_REF]

EFFECT OF LAYER COMPRESSION

Reference [START_REF] Shundyak | Theory of the Isotropic-Nematic Transition in Dispersions of Compressible Rods[END_REF] provides a simple route to addressing the effect of polymer compression on the fluid phase behavior. The central assumption is that compression of the polymer layers due to crowding of neighboring platelets induces an entropic penalty but at the same time reduces the excluded volume of the platelets. A compression factor 0 < 𝛾 𝑐 < 1 expresses the degree of compression. The free energy penalty associated with this reads:

𝛽𝐹 𝑁 ~𝑘 2 (𝛾 𝑐 2 + 𝛾 𝑐 -2 -2) (12)
which reduces to zero for the uncompressed case (𝛾 𝑐 = 1) as it should. Further, k > 0 is a rigidity constant reflecting the "softness" or elasticity of the polymer layer. It should be roughly proportional to the amount of polymer adsorbed onto the platelet. Intuitively, the local polymer concentration within the adsorbed layer should also determine the layer-to-layer repulsion strength , so we expect these two energy scales to be intimately linked. The effective dimensions of the coated platelet become:

𝐿 𝛿 = 𝐿 + 2𝛾 𝑐 𝛿 𝐷 𝛿 = 𝐷 + 2𝛾 𝑐 𝛿 (13)
The trade-off between the entropic penalty due to overlapping polymers and reduction of excluded volume results in an equilibrium compression which lies somewhere between full compression (𝛾 𝑐 = 0) and no compression at all (𝛾 𝑐 = 1). Minimization of the total free energy, including Eq. ( 12) with respect to 𝛾 𝑐 yields:

𝑘(𝛾 𝑐 -𝛾 𝑐 -3 ) + 𝜌 𝜕〈〈𝐵 2 (𝛾)〉〉 𝜕𝛾 𝑐 = 0 (14)
This condition, which cannot be resolved analytically, gives the equilibrium value for the compression factor for a given particle concentration and rigidity constant k. Note that the compression will generally be different for the isotropic and nematic phase. Results are shown in Figure 6. Even though the degree of compression is very small (𝛾 𝑐 =0.995 at most for both phases) the change in phase behavior is quite dramatic, with a stable reentrant feature emerging at low polymer addition in the absence of edge adsorption ( = 0). The model for compressible plates demonstrates that even the tiniest amount of polymer compression (about 5 %) leads to significant deviations in the isotropic-nematic phase behavior compared to fully incompressible discs. Despite the dilute nature of the systems considered in the experiment, direct disc-disc collisions must be frequent enough to generate nematic order.

Each collision event is affected by the compressibility of the adsorbed polymer layers, even for extremely rigid polymer layers (say with k >>100).

CONCLUSIONS

As expected for systems of highly anisotropic plate-like nanoparticles, beidellite clay aqueous suspensions spontaneously show nematic order. However, doping these suspensions with small amounts of adsorbing hydrosoluble polymer, with gyration radius much smaller than the average distance between particles, destabilizes their nematic order. Our SAXS experiments showed that polymer doping does not affect the structure of the nematic phase but only weakens its thermodynamic stability. According to our theoretical model, this phenomenon can be explained by the selective adsorption of the polymer on the faces of the nanosheets rather than at their edge. Previous neutron scattering and NMR studies of PEO adsorbed on laponite had inferred that a large number of polymer chains were "wrapped" around the laponite rim. [START_REF] Nelson | A Small-Angle Neutron Scattering Study of Adsorbed Poly(Ethylene Oxide) on Laponite[END_REF] However, in that study, the gyration radius of the PEO was about equal to the particle size whereas it is ten times smaller than the particle size in the present case. Moreover, neutron scattering techniques are less sensitive to the loops and dangling chains which are not adsorbed on the particles but nevertheless take part of the soft repulsion between the particles. Our model, based on Onsager's theory, shows that this additional soft repulsion may indeed bring about the destabilization of the nematic phase if polymer adsorption is stronger on the faces than on the edge of the clay nanosheet. This explanation is in good agreement with previous reports showing that PEO chains adsorb on the clay faces through complexation with the interlayer charge compensating cations. [START_REF] Ruiz-Hitzky | Polymer-Salt Intercalation Complexes in Layer Silicates[END_REF][START_REF] Montarges | Intercalation of Al-13-Polyethyleneoxide Complexes into Montmorillonite Clay[END_REF][START_REF] Aranda | Ethylene Oxide)-Silicate Intercalation Materials[END_REF] The destabilization of the nematic phase through polymer adsorption seems a fairly general phenomenon since we observed it for a wide range of PEO molecular weight, provided that there are no bridging conformations, not only with PEO but also with dextran, and also with nematic suspensions of nontronite clay. Furthermore, we found that the molar mass of the polymer is not a much relevant parameter, in the range that we explored, as the destabilization of the nematic phase seemed independent of the molar mass. From the point of view of applications, this work is relevant to most situations involving flow of clay / polymer systems because nematic liquid crystals have hydrodynamic properties (anisotropic viscosity, orientation-dependent shear-thinning …) very different from those of usual isotropic liquids. [START_REF] Gennes (de) | The Physics of Liquid Crystals[END_REF] This means that the alignment state of these systems should be carefully monitored in such situations to help interpret unexpected rheological behaviors.

ASSOCIATED CONTENT Supporting Information.

Figure 1 .

 1 Figure 1. Photographs of four series of sample tubes observed between crossed polarizers. For

Figure 2 .

 2 Figure 2. SAXS patterns of a) the nematic part and b) the isotropic part of the same biphasic

Figure 3 .

 3 Figure 3. Experimental phase diagram of the PEO (100 kDa) / beidellite colloidal suspensions.

36

 36

Figure 4 .

 4 Figure 4. Adsorption isotherm of PEO (100 kDa) on beidellite, showing the adsorbed mass of

Figure 5 .

 5 Figure 5. Isotropic-nematic phase behavior for polymer-coated platelets at different degrees of

Figure 6 .

 6 Figure 6. I-N phase diagram of compressible platelets with k = 100. For the no-edge adsorption

Figure SI 1 :

 1 Figure SI 1: Transmission electron microscopy image of the beidellite nanosheets.

Figure SI 2 :

 2 Figure SI 2: Calibration graphs of the PEO concentration versus the suspension viscosity for

Figure SI 3 :

 3 Figure SI 3: Graph showing the average distance, d, between nanosheets in the nematic phase

Figure SI 4 :

 4 Figure SI 4: Photographs of series of sample tubes showing that the nematic phase of beidellite

Figure SI 5 :

 5 Figure SI 5: Phase diagram of the (10 kDa) PEO / beidellite colloidal suspensions.

Figure SI 6 :

 6 Figure SI 6: Adsorption isotherm of PEO (10 kDa) on beidellite.

Figure SI 7 :

 7 Figure SI 7: Photographs of series of sample tubes showing that the nematic phase is

Figure SI 8 :

 8 Figure SI 8: Photographs of series of sample tubes showing that the nematic phase of

Table of

 of Contents graphic:


Alice Le Brigant

Nicolas Guigui

Sana Rebbah

Stéphane Puechmorel

Classifying histograms of medical data using information geometry of beta distributions

Keywords: Information geometry, histogram analysis, classification, clustering, medical imaging

come

INTRODUCTION

The differential geometric approach to probability theory and statistics has met increasing interest in the past years, from the theoretical point of view as well as in applications. In this approach, probability distributions are seen as elements of a differentiable manifold, on which a metric structure is defined through the choice of a Riemannian metric. Two very important ones are the Wasserstein metric, central in optimal transport, and the Fisher information metric (also called Fisher-Rao metric), essential in information geometry. Unlike optimal transport, information geometry is foremost concerned with parametric families of probability distributions, and defines a Riemannian structure on the parameter space using the Fisher information matrix, see [START_REF] Fisher | On the mathematical foundations of theoretical statistics[END_REF]. The Fisher information is the Hessian of the Kullback-Leibler divergence, a popular measure of dissimilarity between probability distributions that does not verify the properties of a distance. In parameter estimation, it can be interpreted as the quantity of information contained in the model about unknown parameters. [START_REF] Rao | Information and the accuracy attainable in the estimation of statistical parameters[END_REF] showed that it could be used to locally define a scalar product on the space of parameters, interpreted as a Riemannian metric.

An important feature of this metric is that it is invariant under any diffeomorphic change of parameterization. Indeed, it is induced by a more general Riemannian metric on the infinite-dimensional space of smooth, non paramet-Sponsor and financial support acknowledgment goes here. Paper titles should be written in uppercase and lowercase letters, not all uppercase.

ric probability densities, and therefore it is invariant to coordinate change; see e.g. [START_REF] Friedrich | Die Fisher-information und symplektische strukturen[END_REF]. It can be seen as a natural choice of metric as it is the only Riemannian metric that is invariant with respect to transformation by a sufficient statistic, or a diffeomorphic transformation of the support in the non-parametric case [START_REF] Cencov | Statistical decision rules and optimal inference[END_REF][START_REF] Bauer | Uniqueness of the Fisher-Rao metric on the space of smooth densities[END_REF]. Arguably the most famous example of Fisher information geometry of a statistical model is that of the univariate Gaussian model, which is hyperbolic. The geometries of other parametric families such as the multivariate Gaussian model [START_REF] Atkinson | Rao's distance measure[END_REF][START_REF] Skovgaard | A Riemannian geometry of the multivariate normal model[END_REF], the family of gamma distributions [START_REF] Arwini | Information geometry: near randomness and near independence[END_REF][START_REF] Rebbah | The geometry of the generalized gamma manifold and an application to medical imaging[END_REF], or more generally location-scale models [START_REF] Said | Warped Riemannian metrics for location-scale models[END_REF], among others, have also received a lot of attention.

In this work, we focus on beta distributions, a family of probability measures on [0, 1] used to model random variables defined on a compact interval in a wide variety of applications, e.g. in Bayesian inference as conjugate prior for several discrete probability distributions [START_REF] O'neill | Bayesian inference for partially observed stochastic epidemics[END_REF], or to model percentages and proportions in genomic studies [START_REF] Yang | Beta approximation of ratio distribution and its application to next generation sequencing read counts[END_REF]. Up to our knowledge, the information geometry of beta distributions has not yet received much attention. In Section 2, we give new results and properties for this geometry by deriving the metric matrix, and the geodesic equations. We compute the sectional curvature and show it has negative sign everywhere. This result is of particular interest to ensure the existence of Fréchet means. By numerically solving the geodesic equation, we may compute geodesic distances and means between probability distributions.

In Section 3, we exemplify the use of this geometry in the analysis of histograms extracted from segmented medical images. Traditional statistical methods are challenged in medical data analysis as data-sets often have very few samples (∼ 10 -100) whereas observations are very high dimensional (∼ 10 4 -10 6 ) and missing data are ubiquitous. Moreover anatomical measurements are often obtained through complex processing pipelines that result in various numbers of features (e.g. mesh representations of organs' surface may have different numbers of vertices), resulting in the lack of a common representation space for the data.

To bypass these issues, it is thus helpful to consider histograms across the whole region of interest and to represent them in the space of beta distributions. The obtained representation has only two dimensions which is useful for visualization and interpretation of the results. We demonstrates that classification in this space achieves similar accuracy in discriminating patients from healthy controls as in the high-dimensional data space.

FISHER GEOMETRY OF BETA DISTRIBUTIONS

The Fisher information metric

Let Θ be an open set of R d and let µ be a measure defined on a measurable space (E, T ). A parameterized family of distributions on the parameter space Θ is a set of probability measures absolutely continuous with respect to µ, that is:

P Θ = {p(•; θ)µ, θ ∈ Θ}. When the mapping θ ∈ Θ → p(x, θ) is differentiable for µ- almost all
x ∈ E and the model satisfies certain regularity conditions, the Fisher information matrix at θ is defined to be:

I(θ) = [E∂ i l(x, θ)∂ j l(x, θ)] 1≤i,j≤d ,
where l(x, θ) = log p(x, θ). As an open subset of R d , Θ is a differentiable manifold with T θ Θ R d . The Fisher information defines a Riemannian metric on Θ called the Fisher information metric (u, v) ∈ T θ Θ → g θ (u, v) = u t I(θ)v, where u t denotes the transpose of the vector u. By a slight abuse of language, we will talk of the geometry of P Θ , implicitly referring to the metric structure induced on P Θ by (Θ, g).

Here we consider the parametric family of beta distributions, a family of probability measures on [0, 1] with density with respect to the Lebesgue measure parameterized by two positive scalars x, y > 0

p(t; x, y) = Γ(x + y) Γ(x)Γ(y) t x-1 (1 -t) y-1 , t ∈ [0, 1].
We consider the Riemannian manifold composed of the parameter space Θ = R * + × R * + and the Fisher information metric g, and by extension denote by beta manifold the pair (B, g), where B is the family of beta distributions B = {B(x, y) = p(•; x, y)dt, x > 0, y > 0}. Here dt denotes the Lebesgue measure on [0, 1]. The beta family is a natural exponential one with log-partition function ϕ(x, y) = ln Γ(x + y) -ln Γ(x) -ln Γ(y).

(1) Fig. 1. Geodesic balls in the beta manifold.

and so the Fisher information admits a simple expression: I(x, y) = -Hess ϕ(x, y). Straightforward computations then yield

I(x, y) = ψ (x) -ψ (x + y) -ψ (x + y) -ψ (x + y) ψ (y) -ψ (x + y) (2)
where ψ denotes the digamma function, i.e.

ψ(x) = d dx ln Γ(x). (3)
The matrix form of the Fisher metric g is given by ( 2), and its infinitesimal length element is

ds 2 = ψ (x)dx 2 + ψ (y)dy 2 -ψ (x + y)(dx + dy) 2 .

Geodesics and geodesic distance

The distance between two beta distributions is defined as the geodesic distance associated to the Fisher metric in the parameter space

d F (B(x, y), B(x , y )) = inf γ 1 0 g( γ(t), γ(t))dt, (4)
where the infimum is taken over all paths γ : [0, 1] → Θ such that γ(0) = (x, y) and γ(1) = (x , y ). To compute this distance in practice, one needs to solve the geodesic equation.

Proposition 1. The geodesics t → (x(t), y(t)) of the beta parameter space Θ are solutions of

ẍ + a(x, y) ẋ2 + b(x, y) ẋ ẏ + c(x, y) ẏ2 = 0, ÿ + a(y, x) ẏ2 + b(y, x) ẋ ẏ + c(y, x) ẋ2 = 0, (5)
where

a(x, y) = ψ (x)ψ (y) -ψ (x)ψ (x + y) -ψ (y)ψ (x + y) 2d(x, y) , b(x, y) = - ψ (x + y)ψ (y) d(x, y) , c(x, y) = ψ (y)ψ (x + y) -ψ (y)ψ (x + y) 2d(x, y) , d(x, y) = ψ (x)ψ (y) -ψ (x + y)(ψ (x) + ψ (y)).
No closed form for the geodesics is known, but they can be computed numerically by solving (5), see Section. 2.4. Nonetheless we can notice that, due to the symmetry of the metric with respect to parameters x and y, the line of equation x = y is a geodesic, where the parameterization is fixed by the unique geodesic equation given by (5).

Negative curvature and Fréchet mean

Extension of basic statistical objects such as the mean, the median or the variance to the setting of Riemannian manifolds is now well known; see e.g. [START_REF] Pennec | Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements[END_REF].

A popular choice to define the notion of mean in a Riemannian manifold is the Fréchet mean, also called intrinsic mean, which is defined as the minimizer of the sum of the squared distances to the points that we want to average. In our setting, the intrinsic mean of a set of {B i , i = 1, . . . , n} of beta distributions is given by

B = argmin B∈B n i=1 d F (B, B i ) 2 .
The Fréchet mean is in general not unique and refers to a set. Uniqueness holds however when the curvature of the Riemannian manifold is negative [START_REF] Karcher | Riemannian center of mass and mollifier smoothing[END_REF], which is the case here.

Theorem 1. The beta manifold (B, d F ) has negative sectional curvature.

The proof of this theorem, given in the appendix, relies on certain regularity conditions of the polygamma functions, and in particuler on the sub-additivity of the ratio ψ /ψ which was proven recently by [START_REF] Yang | Some properties of the divided difference of psi and polygamma functions[END_REF].

The space of beta distributions being moreover simply connected and complete as shown in Le [START_REF] Brigant | Fisher-rao geometry of dirichlet distributions[END_REF], it is a Hadamard manifold. Therefore any set of beta distributions admits a unique Fréchet mean for the Fisher geometry. This property of the Fisher geometry is an important argument in favor of its use for supervised and unsupervised classification of histograms. Indeed, the popular K-means algorithm, also called Lloyd's algorithm [START_REF] Lloyd | Least squares quantization in PCM[END_REF], which seeks to minimize intra-cluster variance, relies on the computation of the mean of the clusters.

It can be easily extended to non-linear manifolds by replacing the Euclidean distance and mean by the Riemannian distance and Fréchet mean.

Implementation

Geodesics (see Fig. 1) are computed through the exponential and logarithm maps of the Riemannian manifold, which can be implemented by solving respectively the initial and boundary value problems associated to the geodesic equation ( 5). This allows to approximate the distance, by averaging the norm of the velocity of the discretized geodesic between two points. The Fréchet mean can be found in practice using a gradient descent algorithm commonly referred to as the Karcher flow. It simply consists in applying the following update on the current estimate B of the mean at each iteration:

B ← exp B τ n n i=1 log B (B i )
for some step size τ > 0. Here exp and log denote the Riemannian exponential and logarithm maps. Our implementation is available in the open-source Python package geomstats http://geomstats.ai.

APPLICATIONS TO MEDICAL DATA ANALYSIS

In this section with exemplify the theoretical results of the previous section on histograms computed from medical data in two different contexts.

Data-sets

Cardiac Shape Deformations Firstly, we use a data-base of 204 shapes of right ventricles (RV) of the heart extracted from 3d-echocardiographic sequences [START_REF] Moceri | Three-dimensional right-ventricular regional deformation and survival in pulmonary hypertension[END_REF]. 3d-meshes were extracted from the sequences with semiautomatic software (TomTec 4D RV-Function 2.0) and post-processed to compute markers of deformation during the cardiac cycle. We focus on the systole, that is the period during which the RV contracts to eject blood through the pulmonary valve. As a marker of deformation, we use the Area Strain (AS), that represents the local stretching of the RV surface. This feature is of growing interest among clinicians to assess cardiac function. It was shown to be a predictor of survival in pulmonary hypertension [START_REF] Moceri | Three-dimensional right-ventricular regional deformation and survival in pulmonary hypertension[END_REF].

For each subject, the RV at time t is represented by a 3d-mesh of 822 vertices and 1587 triangular cells. Note that all vertices and cells correspond to one-another across subjects, but this may not be the case in most applications. This allows us to compare statistics computed in the original data space and in the space of beta distributions. We compute the relative area change of each cell k between the end of diastole (t 0 ) and the end of systole (t 1 ):

AS k = a t1 k -a t0 k a t0 k , (6)
where a t k is the area of the triangle k at time t for a given mesh. This results in a distribution of values greater than -1. Due to the tissue's nature, the values should in fact be bounded by constants p < q. However, the acquisition noise causes some values to lie outside [p, q] although physically infeasible. We thus normalize the data to lie in the interval [0, 1] by applying the transformation x → min(max(x, p), q) -p q -p .

Finally, we map each patient's histogram to the space of beta distributions by estimating the parameters x, y by maximum likelihood (ML). Fig. 2 represents the data for a given patient with the estimated probability density function and Fig. 3 the obtained representation of the population's histograms in the manifold of beta distributions.

Cortical Thickness maps The measure of brain atrophy is a crucial tool in the analysis of neurodegenerative diseases. In particular, Cortical Thickness (CTh) measured by structural Medical Resonance Imaging (MRI) has been proposed as a biomarker for the early diagnosis of Alzheimer's disease [START_REF] Pini | Brain atrophy in Alzheimer's disease and aging[END_REF][START_REF] Busovaca | Is the Alzheimer's disease cortical thickness signature a biological marker for memory?[END_REF] and for the prediction of conversion from Mild Cognitive Impairment to Alzheimer's Disease [START_REF] Wei | Prediction of conversion from Mild Cognitive Impairment to Alzheimer's disease using MRI and structural network features[END_REF][START_REF] Sun | Detection of conversion from Mild Cognitive Impairment to Alzheimer's disease using longitudinal brain MRI[END_REF]. In many of these studies, central tendency measures such as the mean or the median are used to represent the biomarker, either in the whole cortex, in regions of interest or in the voxels, leading to an important loss of information. More recent ones used histogram analysis [START_REF] Giulietti | Whole brain white matter histogram analysis of diffusion tensor imaging data detects microstructural damage in mild cognitive impairment and Alzheimer's disease patients[END_REF][START_REF] Ruiz | Alzheimer's disease computeraided diagnosis: Histogram-based analysis of regional MRI volumes for feature selection and classification[END_REF], and in [START_REF] Rebbah | The geometry of the generalized gamma manifold and an application to medical imaging[END_REF], the authors use the Fisher geometry of generalized gamma distributions to perform this analysis. However, cortical thickness is a bounded quantity, and therefore beta distributions seem to be more natural candidates to model their distribution. Moreover, the negative curvature of the beta geometry makes it a suitable candidate in clustering procedures based on computation of cluster means, such as the K-mean algorithm used here.

The data used in this application were extracted from MR scans selected from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database 1 . Indeed, the initial subjects were not age-and sex-matched and our procedure consisted in randomly selecting subjects. In addition, some of the subjects were excluded because of a low diagnosis reliability (according to ADNI criteria) and others because CTh was from 3D T1-weighted MR images using the Matlab toolbox CorThiZon [START_REF] Querbes | Early diagnosis of Alzheimer's disease using cortical thickness: impact of cognitive reserve[END_REF]. There is no unique definition of CTh. In [START_REF] Macdonald | Automated 3-D extraction of inner and outer surfaces of cerebral cortex from MRI[END_REF] it is obtained as the distance between uniquely associated pairs of points on bounding surfaces. Uncoupled methods do not rely on a given set of associated landmarks but create them using techniques coming from shape matching [START_REF] Tagare | Deformable 2-D template matching using orthogonal curves[END_REF]. Finally, a quite different category of algorithms makes use of an elliptic partial differential equation to obtain a field of deformation between the boundaries [START_REF] Yezzi | An Eulerian PDE approach for computing tissue thickness[END_REF]. The CorThiZon toolbox, developed at Inserm, implements a Laplace equation based method that falls within this last category.

For each subject, the data consist in measurements of CTh along the whole cortical ribbon. The overall procedure is the one described in [START_REF] Querbes | Early diagnosis of Alzheimer's disease using cortical thickness: impact of cognitive reserve[END_REF], with a normalized voxel size of 1mm along all directions. Due to the variability of the size and shape of the brain among the population of study, we obtain samples of unequal length. This results in the lack of a common representation space for the data, a problem that is solved by the histogram-based approach that we propose here. Indeed, just as for the AS data, we compute for each patient the histogram of CTh measurements, from which we estimate the parameters of a beta distribution by ML. Histograms are normalized beforehand with respect to the maximal CTh value among the population.

Methodology

In both cases, the subjects are divided in two classes: diseased and controls. In order to assess whether the proposed representation and geometry are relevant, we perform supervised and unsupervised classification in the Riemannian manifold of beta distributions. We compare the results both when using the Riemannian metric and the Euclidean metric on the parameter space (R + * ) 2 . We chose to use the K-nearest-neighbor (KNN) and K-means algorithms as they only require computation of distances on non-linear manifolds. Both supervised (SKM) and unsupervised (UKM) versions of K-means are compared. Finally, in the case of the AS data, as we have cell-to-cell correspondence, we can run the algorithms in the original data space R1587 to assess the loss of information when considering only two-parameter distributions instead of the whole histograms. We also compare our method with a standard dimensionality reduction performed by principal component analysis (PCA). This cannot be done for the CTh data since the mapping between values and brain regions differs across patients. All models are trained and tested in a 5-fold cross-validation fashion, and we assess the classification accuracy in all cases. We used the scikitlearn package [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF] to perform these experiments.

Results

Supervised classification Tables 1 and2 show the mean classification accuracy over 5-fold cross validation for each setting: (1) the space of parameters of the beta distributions fitted to the data-point histograms and, when applicable, (2) the Euclidean space of the original data, and (3) the space spanned by the first two principal components obtained by PCA. The number of neighbors for the KNN algorithm is chosen to be K = 7. This choice is compared to other values through 5-fold cross validation, see Fig. 5 and6. These results show that in the space of beta parameters, the Fisher information geometry performs significantly better than Euclidean geometry. Moreover, this performance is comparable to that of classification in the original data space, despite the considerable reduction of dimension (from 1587 to 2). This is illustrated in Fig. 5, where both methods perform alternatively better as the number of neighbors changes.

In comparison, the two-dimensional representation space provided by PCA performs significantly worst on average. This can be explained by the fact that the performance of dimensionality reduction of PCA depends on the choice of training set, and certain train/test splittings lead to poor performance. This is illustrated in Fig. 7: in one of the cross-validation splittings, the test set for each class in the PCA representation is closer to the center of the other class computed from the training test. This has the effect of lowering the average accuracy over the whole cross-validation process. This problem does not occur in the two-dimensional beta representation, as the mapping from one data-point to a pair of parameters (x, y) does not depend on the other points present in the data-set. Unsupervised classification The clustering performance of the K-means algorithm is shown for the same settings in Tables 3 and4. These results confirm that the Fisher information geometry in the space of beta parameters is more suited to compare histograms than Euclidean geometry. Unlike in the supervised setting, the two-dimensional representation given by PCA performs well, since in this case the whole data-set was used to perform the dimensionality reduction.

CONCLUSION

In this work, we have studied the geometry of the beta manifold and shown in particular that it was negatively curved. We have illustrated its use as a non linear representation space for histograms of medical data, which presents the advantage of being low-dimensional and can easily be visualized. This common representation is also particularly helpful when the number of measurements vary from one subject to another. The examples studied here tend to show that the Fisher information metric is a more adapted choice of metric on this space than the Euclidean one to compare and classify histograms. Finally, the dimensionality reduction performed in the process does not result in a significant loss in performance. Moreover, unlike a standard method like PCA, the quality of the dimensionality reduction does not rely on the choice and size of the data-set: each data-point is mapped to an element of the beta manifold independently of the others. This makes the beta representation particularly interesting for small data-sets. Comparison with other geometries, such as that of the gamma and generalized gamma manifolds, and exploration of other supervised and unsupervised classification algorithms will be the object of future work.

Fig. 2 .

 2 Fig. 2. Example of the normalization and ML representation of a subject's AS distributions in the space of beta distributions

1

  Fig. 4. CTh data represented in the parameter space of beta distributions. of unsuccessful CTh measurement due to poor image quality. The resulting population is composed of 143 subjects: 71 healthy controls subjects and 72 Alzheimer's disease patients.

Fig. 5 .

 5 Fig. 5. Mean accuracy of KNN classification on the AS data over 5-fold cross validation. Vertical segments indicate standard deviation for the original Euclidean and the beta Riemannian representations.

Fig. 7 .

 7 Fig. 7. Supervised K-means on the AS strain data in beta and PCA representations.

Table 1 .

 1 Mean (and standard deviation) of the classification accuracy for the AS data on 5fold cross-validation

		Original	PCA	Beta
		Euclidean	Euclidean	Euclidean	Riemannian
	KNN	0.81 (0.05)	0.67 (0.32) 0.77 (0.09)	0.83 (0.05)
	SKM	0.85 (0.03) 0.72 (0.28) 0.66 (0.06)	0.81 (0.06)

Table 2 .

 2 Mean (and standard deviation) of the classification accuracy for the CTh data on 5fold cross-validation

		Beta
		Euclidean	Riemannian
	KNN 0.77 (0.05)	0.79 (0.04)
	SKM	0.66 (0.10)	0.80 (0.05)

Table 3 .

 3 Clustering accuracy for the AS data

		Beta		Original	PCA
		Euclidean Riemannian Euclidean Euclidean
	UKM	0.60	0.80	0.84	0.84

Table 4 .

 4 Clustering accuracy for the CTh data

		Beta
		Euclidean Riemannian
	UKM	0.61	0.82

ACKNOWLEDGEMENTS

The authors would like to warmly thank Pamela Moceri (CHU Nice) and Nicolas Duchateau (Université Lyon 1) for acquiring and curating the heart data. N. Guigui has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement G-Statistics No 786854). This research has been conducted within the FP2M federation (CNRS FR 2036)

where the Γ k ij 's denote the Christoffel symbols of the second kind. These can be obtained from the Christoffel symbols of the first kind Γ ijk and the coefficients g ij of the inverse of the metric matrix Γ k ij = Γ ijl g kl . Here we have used the Einstein summation convention. Since the Fisher metric is a Hessian metric (see [START_REF] Totaro | The curvature of a Hessian metric[END_REF]), the Christoffel symbols of the first kind can be obtained as

where ϕ is the log-partition function (1) and lower indices denote partial derivatives. Straightforward computation yields the desired equations.

Appendix B. PROOF OF THEOREM 1

The sectional curvature of a Hessian metric is given by

). Factorizing the numerator by ψ (x)ψ (y)ψ (x + y) yields

where F = ψ /ψ . Since ψ is negative, the first factor is negative. Moreover, it has been shown in (Yang, 2017, Corollary 4) that the function F is sub-additive. This means that the second factor is positive, yielding the desired result.