Anastasia Saade

Ronan Garlantezec

Stéphane Jouneau

Christophe Paris

Keywords: Low-income, idiopathic pulmonary fibrosis, proxy variable, health, living conditions, mediators

We read with interest the recent manuscript « low income and outcome in idiopathic pulmonary fibrosis (IPF): an association to uncover » by Sesé and colleagues [START_REF] Sesé | Low income and outcome in idiopathic pulmonary fibrosis: An association to uncover[END_REF]. The authors compared the survival of 200 patients selected from the French national prospective cohort COFI according to their income, low versus high income (using a threshold of 18170 euros / year).

With an average follow-up of 33.8 ± 22.7 months, low-income was significantly associated with worse overall survival after adjustment in a multivariate model. Occupational exposure was not independently associated with survival. Exposure to fine particles was not included in the model probably for multi collinearity issue with occupational exposure.

The World Health Organization (WHO) defines social determinants of health as "the conditions, in which people are born, grow, work, live, and age, and the wider set of forces and systems shaping the conditions of daily life". Evaluation of health should address both social and behavioural causes [START_REF] Marteau | Changing behaviour: an essential component of tackling health inequalities[END_REF]. Living conditions are conditioned by how much people can afford for food, accommodation, transportation, healthcare… and is complex to measure. Individuals with restricted consumption budgets are confronted with deprivation, inability to acquire or consume certain goods, to reach a certain level of comfort or finally to honor certain compulsory Low-income families have altered life conditions including reduced access to healthcare (3).

Not only health access plays a role in health in general, environmental and occupational exposures have their own contribution.

Accepted manuscript / Final version

The relationship between economically disadvantaged households and health environmental conditions has been widely reported [START_REF] Hood | Dwelling disparities: how poor housing leads to poor health[END_REF][START_REF] Padilla | Air quality and social deprivation in four French metropolitan areas--a localized spatiotemporal environmental inequality analysis[END_REF][START_REF] Pasetto | Environmental Justice in Industrially Contaminated Sites. A Review of Scientific Evidence in the WHO European Region[END_REF]. Low-income households often bear a higher exposure to environmental pollution than high-income households. Air pollution is associated with increased death from lung cancer and cardiopulmonary disease and the development of fibrotic interstitial lung diseases [START_REF] Dockery | An association between air pollution and mortality in six U.S. cities[END_REF][START_REF] Harari | Fibrotic interstitial lung diseases and air pollution: a systematic literature review[END_REF]. In addition, poor housing leads to exposure to lead, molds, moisture, dust mites, and rodents, associated with the increased rate of chronic diseases such as asthma, obesity and diabetes [START_REF] Hood | Dwelling disparities: how poor housing leads to poor health[END_REF][START_REF] Ruiz | Disparities in Environmental Exposures to Endocrine-Disrupting Chemicals and Diabetes Risk in Vulnerable Populations[END_REF]. Over 50% of low-income housing had at least 3 exposure-related problems [START_REF] Adamkiewicz | Environmental conditions in low-income urban housing: clustering and associations with self-reported health[END_REF].

Furthermore, low-income results from low-wages occupations which are associated with higher occupational exposure risks [START_REF] St-Denis | Sociodemographic Determinants of Occupational Risks of Exposure to COVID-19 in Canada[END_REF]. Burgard and Lin clearly described the correlation between "bad Jobs" and "bad Health" [START_REF] Burgard | Bad Jobs, Bad Health? How Work and Working Conditions Contribute to Health Disparities[END_REF]. Interestingly, occupational risks were reported to be involved in the development of IPF. Second-hand smoke, breathable dust, asbestos, silica, wood dust, metal dust or fumes vapours, gases, dust or fumes were risk factors independently associated with IPF [START_REF] Abramson | Occupational and environmental risk factors for idiopathic pulmonary fibrosis in Australia: case-control study[END_REF][START_REF] Blanc | The Occupational Burden of Nonmalignant Respiratory Diseases. An Official American Accepted manuscript / Final version Thoracic Society and European Respiratory Society Statement[END_REF]. Air pollution plays a major role in both the natural history of IPF and lung function [START_REF] Johannson | Air Pollution Exposure Is Associated With Lower Lung Function, but Not Changes in Lung Function, in Patients With Idiopathic Pulmonary Fibrosis[END_REF][START_REF] Sesé | Role of atmospheric pollution on the natural history of idiopathic pulmonary fibrosis[END_REF]. The involvement of metal dust remains controversial [START_REF] Abramson | Occupational and environmental risk factors for idiopathic pulmonary fibrosis in Australia: case-control study[END_REF][START_REF] Koo | Occupational exposure and idiopathic pulmonary fibrosis: a multicentre case-control study in Korea[END_REF]. The predominant cause of death in patients with IPF is acute exacerbation of idiopathic pulmonary fibrosis that accounts for 40% mortality [START_REF] Natsuizaka | Epidemiologic survey of Japanese patients with idiopathic pulmonary fibrosis and investigation of ethnic differences[END_REF][START_REF] Collard | Acute exacerbations of idiopathic pulmonary fibrosis[END_REF][START_REF] Parambil | Histopathologic features and outcome of patients with acute exacerbation of idiopathic pulmonary fibrosis undergoing surgical lung biopsy[END_REF]. Exposure to fine particles and air pollution were observed as risk factors for acute exacerbation of IPF [START_REF] Tahara | Exposure to PM2.5 is a risk factor for acute exacerbation of surgically diagnosed idiopathic pulmonary fibrosis: a case-control study[END_REF][START_REF] Tomos | Long-term personal air pollution exposure and risk for acute exacerbation of idiopathic pulmonary fibrosis[END_REF][START_REF] Winterbottom | Exposure to Ambient Particulate Matter Is Associated With Accelerated Functional Decline in Idiopathic Pulmonary Fibrosis[END_REF].

In summary, low-income is a strong health determinant that often includes poor environmental and occupational conditions. In other words, low-income is a driver of several behavioural and social mediators that directly affect the outcome. Consequently, the question may be is lowincome a directly relevant variable associated with IPF survival? Or is low-income a proxy variable, with close correlation to living conditions? Consequently, the association between Accepted manuscript / Final version low-income and reduced survival in IPF should be interpreted with caution and warrants further studies to uncover unobservable or immeasurable variables linked to environment and occupational exposure in IPF in addition to path analysis to correctly interpret the association observed. As a result, assessment of both environmental and occupational exposures, by means of questionnaires, has to be systematically implemented.

  expenses. Insufficient outcome (what goes out) results from insufficient income (what comes in) but reflects a minor aspect of poverty and needs to be complemented by non-monetary indicators based on living conditions. In 2019, according to the French national institute of statistics and economic studies (INSEE), 11% to 13% of the French population was considered poor based on non-monetary indicators.


Nataliya Shcherbakova
email: nataliya.shcherbakova@ensiacet.fr

Ivonne Rodriguez -Donis

Jens Abildskov

Vincent Gerbaudª

Univolatility curves in ternary mixtures: geometry and numerical computation

Keywords: univolatility curves, differential continuation method, azeotropes bifurcation, double azeotropy

We propose a new non-iterative numerical algorithm allowing computation of all univolatility curves in homogeneous ternary mixtures independently of the presence of the azeotropes. The key point is the concept of generalized univolatility curves in the 3D state space, which allows the main computational part to be reduced to a simple integration of a system of ordinary differential equations.

Introduction

Accurate numerical computation is essential to synthesis and design of distillation processes. The change of the volatility order between two components i and j in a ternary diagram can be detected by tracing the associated univolatility curve (or αcurve) , , i.e., the set of points of equality between the distribution coefficients and [START_REF] Kiva | Azeotropic phase equilibrium diagrams: a survey[END_REF]. Univolatility curves divide composition space into different Korder regions, even for zeotropic mixtures. Their knowledge is essential to design extractive distillation process [START_REF] Gerbaud | Distillation: equipment and processes[END_REF].

Though the computation of univolatility curves starting at azeotropic points is straightforward, the detection of univolatility curves not associated with azeotropes is a more complicated and time-consuming process. A ternary diagram may contain up to 3 families of α-curves defined by their respective index. [START_REF] Zhvanetskii | Classification of K-order regions on the distillation line diagram for a ternary zeotropic system[END_REF], [START_REF] Reshetov | Diagrams of K-Ordered Regions with An Arbitrary Number of Unitary α-lines[END_REF] and [START_REF] Reshetov | Statistical analysis of the kinds of vapor-liquid equilibrium diagrams of three-component systems with binary and ternary azeotropes[END_REF] formulated the main principles of classification of the univolatility curves. The proposed nomenclature distinguishes the univolatility curves of type , connecting two points on the same binary side of the composition triangle from the curves of type , connecting two different binary sides of the composition triangle. Transitions between types , and , can occur as univolatility curves depend on pressure and temperature of VLE.

In this paper we present a new numerical algorithm for computing all univolatility curves in a ternary system independently of the presence of azeotropes. The method is based on the concept of a generalized univolatility curve in the 3D x -T state space, which can be computed by integrating a system of ordinary differential equations. The efficiency of the proposed algorithm is illustrated through the study of hexafluorobenzene -methyl propionate -benzene mixture exhibiting bi-binary and biternary azeotropy at 1 atm. We show the transformation of the ternary diagram with the variation of pressure, observing the formation of tangential azeotropy, bi-ternary azeotropy, saddle-node azeotropes, as well as singular configuration of univolatility curves not yet described in the literature.

Geometry of univolatility curves

Definitions and notations

Consider an open evaporation of a homogeneous ternary mixture kept at thermodynamic equilibrium at constant pressure. Let , , = 1,2,3 and T denote the mole fractions in the liquid and in the vapor phases and the temperature of the system. In the absence of chemical reactions a two-phase ternary mixture has three independent state variables. By selecting , and , the complete state space of the system reads = { , : ∈ , !, ∈ Ω, = 1,2}, where , are the minimum and maximum boiling temperatures of the mixture and Ω = { = , ∶ ∈ 0,1!, = 1,2} is the composition space. The distribution coefficients Ki describe the vaporliquid equilibrium (VLE) in terms of molar concentrations of the components in the vapor and liquid phases:

, , = , , , = 1,2,3. The following relation allows computing the temperature of the system: Φ , , = ∑ , ,

( ) -1 = 0. (1)
The ratio = / is the relative volatility of component i with respect to component j. If > 1, i is more volatile than j and vice versa. The sets of points in Ω satisfying = 1 are called the univolatility curve , (or -curve).

A ternary diagram may contain more than one univolatility curve of index " , .". Not all these curves are related to azeotropes. However, every azeotrope belongs to at least one univolatility curve. An intersection of two univolatility curves at some point implies that this point is either a tangential binary azeotrope or a ternary azeotrope. In both cases the third univolatility curve of complementary type passes through this point. A detailed analysis of the relation between the topology of -curves and the structure of the underlying residue curves maps (RCM) can be found in [START_REF] Kiva | Azeotropic phase equilibrium diagrams: a survey[END_REF].

The three dimensional geometry of univolatility curves

Geometrically, Eq. ( 1) defines a hypersurface in the state space M, referred to below as the boiling temperature surface W. Along with the boiling temperature surface, M contains three univolatility hypersurfaces defined by Ψ , = , -, = 0, , . = 1,2,3.

(2)

Their intersections with the surface W, are smooth curves in M called the generalized univolatility curves θ . The -curves are merely projections of the generalized univolatility curves on the , -plane.

Fig. 1 illustrates these geometrical concepts for the case of a zeotropic mixture acetone ( ) -ethyl acetate ( ) -benzene. Its ternary diagram is characterized by a unique univolatility curve, ,( , between ethyl acetate and benzene. This curve is a projection on Ω of the generalized univolatility curve 1 ( formed by the intersection of the surface W with the hypersurface Ψ ( , = 0.

Remark 1. The generalized univolatility curve 1 has isolated singularities at the points of common tangency between the surface W and the associated univolatility hyper-surface. In this case 3 = 3 = 3 ( , i.e., the curve 1 degenerates into a point.

A more detailed analysis [START_REF] Shcherbakova | A novel method for detecting and computing univolatility curves in ternary mixtures[END_REF] shows that a point is singular for the univolatility curves if and only if it is singular for the corresponding -curve in Ω. Such isolated singular points can be of elliptic or hyperbolic type. In the latter the resulting univolatility curve is formed by four branches meeting at the singular point. It is important to stress that singular points of univolatility curves are not necessarily related to the singular points of the underlying RCM. Section 3 and Figure 2 provide examples of these highly non-generic configurations.

Remark 2. The same geometrical argument can be directly applied to the computation of the unidistribution curves by setting Ψ , , = , , -1,

where K is any of the distribution coefficients.

Numerical computation of the univolatility curves

Eqs.

(3) provide an efficient tool for numerical computation of the univolatility curves using the standard Runge-Kutta schemes for the ODE integration. Initial points for such integration can be found following the idea of [START_REF] Kiva | Azeotropic phase equilibrium diagrams: a survey[END_REF]. They showed that the existence of the univolatility curves can be detected by analyzing the behavior of the binary distribution coefficients and distribution coefficient at infinite dilution along the binary sides of 9 (for more details see [START_REF] Shcherbakova | A novel method for detecting and computing univolatility curves in ternary mixtures[END_REF].

For example, consider a curve , starting from the 13 binary side, that is, from the -axis. The starting point of this curve is a projection of a point , : , 0, , :

in state space M. The whole curve , can be computed as the projection on 9 of the solution of the following initial value problem:

; ;< = ; 3 , , ‖3 , , ‖ , ; ;< = ; 3 , , ‖3 , , ‖ , ; ;< = ; 3 ( , , ‖3
, , ‖ , (4) 0 = , : , 0 = 0, 0 = , : , ; = < >? @3 A , : , 0, : BC.

(

) 5
Here s is the arc-length of the curve. The normalization of the vector field 3 avoids the possible stiffness of Eqs. (3). For the curves starting from other binary sides of Ω an appropriate modification of initial conditions and direction (eq. ( 5)) should be made. The general algorithm can be sketched:

1. Create a list of all possible starting points of the univolatility curves on each binary side of Ω [START_REF] Kiva | Azeotropic phase equilibrium diagrams: a survey[END_REF]. 2. Take a starting point from the list created in point 1 and solve the problem of type ( 4), ( 5) using any standard ODE solver. Continue numerical integration until the border of Ω is attained. Then both terminal points of the computed curve should be excluded from the list of starting points. 3. Go back to point 2 until the list of starting points is exhausted. This algorithm is implemented in Mathematica 9. It uses a standard solver for a pair of non-linear algebraic equations in 2D to find the starting points of integration, and a standard ODE solver in 3D for the rest of the computation. In particular, no specific iterative procedure is needed to compute the whole curve once the starting point is detected. Any implementation using one of the standard algorithmic languages is possible, though the use of a compatible library of automatic differentiation is suggested to in order to facilitate the writing of Eqs.(4) in explicit form.

Case study: hexafluorobenzene -methyl propionate -benzene.

We illustrate the efficiency of the proposed algorithm in Fig. 2, which shows the transformation of the topology of -curves under pressure variation for hexafluorobenzene ( ) -methyl propionate ( ) -benzene. The VLE parameters are computed according to Wilson model using the data from [START_REF] Myagkova | Physicochemical foundations of separation of biazeotropic mixtures[END_REF], the DIPPR equation and database were used for the vapor pressure computation.

At 1 atm (Fig. 2c) the binary mixture benzene -hexafluorobenzene exhibits two azeotropes D ( , D ( E . Each of these gives rise to a univolatility curve of ,( type. The binary azeotrope D ( (benzene-methyl propionate) is the origin of a ,( curve, the other curve of the same index, of ,( type, links two non-azeotropic points on the 12-side. The binary azeotrope D (hexafluorobenzene -methyl propionate) gives rise to the unique , type curve. Such a complex structure of -curves reflects a peculiar topology of the underlying RCM characterized by two ternary azeotropes: a saddle (D ( ) and a stable node (D ( E ). Lowering the pressure makes D ( E and D to merge into a tangential binary azeotrope D * at P≈0.85 atm (Fig. 2b). Three different univolatility curves meet at this point (the curve ,( follows the 12 edge). The tangential azeotrope D * disappears, as well as the curve ,( , at further pressure reduction (Fig. 2a). Other interesting phenomena occur at higher pressure. Indeed, raising the pressure forces the two curves ,( get closer, and at P ≈ 1.55 atm (Fig. 2d) they meet at a singular point, forming a cross-shape configuration as described in Remark 1. With the infinitesimal pressure increase, this singular configuration splits into a ,( curve connected to D ( and a new ,( curve linking two non-azeotropic points on the 13-side. Meanwhile the two branches of the ,( curve get closer as well as two ternary azeotropes. At P ≈ 4.39 atm (Fig. 2f) we again observe a particular configuration, this time with the ,( curve. This configuration splits into a ,( and ,( branch, the latter connecting two binary azeotropes, D ( and D ( E . A small pressure increase to P ≈ 4.40 atm (Fig. 2g) results in the fusion of two ternary azeotropes into a unique singular azeotrope D ( * of saddle-node type, which disappears with the further pressure increase (Fig. 2h).

Conclusions

The topology of RCMs and associated univolatility curves is non-trivial even in the case of zeotropic ternary mixtures. Our analysis shows that 2D representation in the composition space is not always sufficient to describe the true nature of the univolatility curves. A new method for the detection of univolatility curves is proposed in this paper. It uses the concept of a generalized univolatility curve in the 3D state space. The computational efficiency of the algorithm is illustrated by tracing the bifurcation in the univolatility curves topology with pressure variation in the case of hexafluorobenzenemethyl propionate -benzene. The method allows computing detailed phenomena like the formation of tangential azeotrope, bi-ternary azeotropy, saddle-node azeotropes, as well as singular configuration of univolatility curves not yet described in the literature.

Figure 1 .

 1 Figure 1. The univolatility curve ,( , the boiling temperature surface W and the univolatility hypersurface Ψ ( = 0 for acetone ( ) -ethyl acetate ( ) -benzene Consider a generalized univolatility curve 1 2 = 2 , 2 in M, solution to the pair of algebraic equations: Φ , = 0, Ψ , = 0. Let 3 = 3 , 3 , 3 ( denote a tangent vector to θ at some point. By construction, 3 is orthogonal both to the normal vector 4 5 = ∇Φ to the surface W and to the normal 4 = ∇Ψ to the univolatility hypersurface, as shown in Fig.1. Hence 3 = 4 5 × 4 , and: Theorem. The generalized univolatility curve 1 2 projecting on the univolatility curve = 1 is an integral curve of the vector field 3 = ∇Φ × ∇Ψ , i.e., it is a solution to the following system of ordinary differential equations in M: 8 = 3 , , , 8 = 3 , , , 8 = 3 ( , ,(3)

Figure 2 .

 2 Figure 2. RCM and univolatility curves of hexafluorobenzene ( ) -methyl propionate ( ) -benzene


Mai-Linh Doan

Gérard Gary

Rock pulverization at extreme strain rate near the San Andreas Fault

Pulverized rocks were found near the San Andreas Fault, as far as 400m away from the fault core. It is exceptional to observe such a dense fragmentation so far from the core. Here we show that extreme strain rate is necessary to intensely fragment intact rocks sampled near the fault. We performed laboratory experiments on natural rocks to understand this pulverization, using a Split Hopkinson Pressure Bars apparatus. When strain rate exceeds 150/s, samples break into numerous fragments, to a scale smaller than the initial grain size. To satisfy such strain rate, we propose that the pulverization damage is associated with supershear rupture.

In the northeast of Los Angeles, the Mojave segment of the San Andreas Fault displays unusual fault damage. The whole outcrop is made of rocks finely broken to a scale smaller than the initial grain size of about 1.5 mm. Such damage can be found as far as 400 m from the fault core 1 . This damage pattern affects mainly crystalline rocks.

These rocks are not extensively weathered, with only minor clay content [START_REF] Wilson | Particle size and energetics of gouge from earthquake rupture zones[END_REF][START_REF] Rockwell | Granulometric and mineralogical properties of pulverized rocks from Tejon Pass on the San Andreas Fault and from Tejon Ranch on the Garlock Fault, California[END_REF] . Another puzzling feature is the preservation of the initial isotropic grain pattern, indicating that these rocks experienced little strain [START_REF] Dor | Pulverized Rocks in the Mojave section of the San Andreas Fault Zone[END_REF] . Whereas grain comminution and gouge formation is common within the fault core, where much of the strain occurs, intense pulverization is very surprising so far from the fault core. In the damage zone around a fault, the strain is usually localized along fractures, leaving blocks relatively undamaged [START_REF] Caine | Fault zone architecture and permeability structure[END_REF][START_REF] Chester | Internal structure and weakening mechanisms of the San Andreas fault[END_REF] . Why does this not happen near the San Andreas Fault?

The objective of this paper is to investigate the role of dynamic loading on the pulverization of rocks near the San Andreas Fault. We performed laboratory experiments on rocks sampled near the San Andreas Fault. After presenting the experiments and their results, we discuss their implication on the physics of earthquakes.

Dynamic pulverization is a process where stress localization is inhibited, so that the whole medium can be finely fractured [START_REF] Grady | Dynamic rock fragmentation[END_REF] . Strain localization is the consequence of an unstable feedback: the largest pre-existing crack within the material is the most favourable for further fracture propagation; once extended, it becomes even more amenable to further propagation. Finally, this crack extends at the expense of the others.

However, at higher loading rate, this localization process may be inhibited. The favoured crack propagates at a finite rate, at least limited by the P-wave velocity of the medium, and cannot accommodate all the energy provided to the medium. Other fractures can propagate simultaneously and coalesce to produce numerous small fragments. The sample becomes eventually pulverized [START_REF] Grady | Dynamic rock fragmentation[END_REF] . Did this process apply to the damaged rocks bordering the core of the San Andreas Fault? The dynamic pulverization has been considered in fault cores [START_REF] Reches | Gouge formation by dynamic pulverization during earthquake rupture[END_REF] , but never in the damaged zone around a fault. To quantify its plausibility, we have submitted rocks sampled near the pulverized zone to high strain rate loading. These rocks were not originally pulverized as they laid further from the fault core (about 500 m away), but they remained more damaged than the host rock. This rock material therefore reproduces the state of damage just before pulverization: not yet pulverized, but still damaged.

The experiments were performed using a Split Hopkinson Pressure Bar (SHPB) apparatus [START_REF] Kolsky | Stress waves in Solids[END_REF] in the Laboratoire de Mécanique des Solides of the École Polytechnique, Palaiseau, France. Each sample was inserted between two bars impacted by an striker arriving at known velocity. The dimensions of the bars (3 meters long and 4 cm diameter) were so that the propagation of elastic waves in the bars was mainly onedimensional. Considering the analytical solution of the wave equation in the bars, forces and displacements applied to the bar extremities were retrieved. We checked that the forces were identical at the input and output bars, to verify the sample were homogenously loaded. We can then derive average stress, strain and strain rate in the sample [START_REF] Zhao | On the use of SHPB techniques to determine the dynamic behavior of materials in the range of small strains[END_REF] (Figure 3).

Even if the SHPB apparatus is relatively common in mechanical engineering, it is scarcely used in Earth Sciences [START_REF] Shan | Obtaining dynamic complete stress-strain curves for rock using the Split Hopkinson Pressure Bar technique[END_REF][START_REF] Shockey | Fragmentation of rock under dynamic load[END_REF] . In these experiments, we avoided two difficulties.

Firstly, the lack of confining stress was not a problem, as rock pulverization 100 m away from the fault zone has been so far observed at the ground surface. Secondly, there was no scaling issue, as the size of the tested samples (typically 3 cm long with a diameter of 2.5 cm) was similar to the size of the pulverized samples taken from the field. Moreover, the grain size of the rock studied here was more than ten times smaller than the sample size (Figure 1a).

Experimental results yielded three types of final states (figures 1 and 2): (1) an unbroken state, where insufficient loading did not allow to break the sample, (2) a simple fracturing state, where a sample was split by a few (at most three) longitudinal fractures -a common damage pattern for uniaxial loading at small strain rate -, and (3) a multiple fragmentation state, when the sample was ruined into multiple fragments, some with a size smaller than 1 mm. The experiments performed at a strain rate higher than 150/s provide samples finely broken, whereas those performed below 100/s gave samples broken into two or three fragments (figure 2). The grain size of fragments decreased with higher strain rate.

We will now discuss the pertinence of the laboratory results to explain the natural pulverization. As the tested samples were collected in the damage zone of the San Andreas Fault, their Young modulus was small (10±3 GPa, about one fifth of the tabulated value for granite [START_REF]CRC handbook of chemistry and physics: a ready-reference book of chemical and physical data[END_REF] ). The static strength of the material ranges between 50 and 90 MPa, about half the tabulated values for intact granite [START_REF]CRC handbook of chemistry and physics: a ready-reference book of chemical and physical data[END_REF] . These low values suggest that the initial samples exhibit a dense network of cracks. All these cracks compete with the most favourable crack and help preventing strain localization along a single fracture.

Consequently, we estimate that our threshold strain rate is a minimum value for the transition to fine fragmentation in the field.

Our transition does not only delimit different final damage patterns, but also corresponds to an increase in the apparent strength of the sample. These experimental results are in accordance with the statistical theory of Hild and Reynoual [START_REF] Hild | Single and multiple fragmentation of brittle geomaterials[END_REF] for the transition from single to multiple fracturing regimes. In both theory and experiments, rock strength starts increasing with strain rate, once in a pulverization regime.

Theoretically, there is an intrinsic increase of the material strength [START_REF] Li | About the dynamic strength enhancement of concrete-like materials in a split Hopkinson pressure bar test[END_REF] , as the propagation of several small fractures requires more energy than that of a single large fracture.

Experimentally, there is also a dynamic confinement phenomenon, when the sample does not instantaneously expand laterally at high strain rates. When computing the equivalent constraining stress with Forrestal formula [START_REF] Forrestal | The effect of radial inertia on brittle samples during the split Hopkinson pressure bar test[END_REF] , we retrieve dynamic confining pressures in the range of 2 to 10 MPa, corresponding to a burial depth of only 100 -500 m. This suggests that pulverization may also be found in the shallow subsurface.

We do not pretend to reproduce the exact state of pulverized rocks, especially its grain size distribution. The natural state is certainly the result of several earthquakes, each one damaging rocks by both compression and shear loading. Here, we focus only on the transition between localized fracturing and pervasive fragmentation.

We focused on dynamic pulverization to explain pulverization near faults. Other mechanisms may inhibit the strain localization process. The first one is the ductilefragile transition at high confining stress or high temperature [START_REF] Guéguen | Introduction to the Physics of Rocks[END_REF] . But this transition is too high (300 MPa, 350 °C16 ) for crystalline rocks at the ground surface. A second way to inhibit strain localization is to apply fast rotating stress. The sheared zones are then reworked over and over, so that the strain localization is annealed at the expense of the formation of preferential microstructure orientation [START_REF] Bystricky | High shear strain of olivine aggregate : rheological and seismic conséquences[END_REF] . This mechanism requires large strain and it is not the case for the pulverized rocks from the San Andreas Fault since they preserve their original facture.

High strain rate may be a necessary condition for pulverization, but it may not be a sufficient condition. Other phenomena can also modulate the onset of pulverization.

Under tensile loading, rock strength is reduced and this may explain the asymmetry of damage observed along faults [START_REF] Ben Zion | Properties and Implication of Dynamic Rupture along a Material Interface[END_REF] .

There are some issues in relating the laboratory experiments to the field observations. Our experiments were performed under a uniaxial loading. The transition strain rate depends on the speed of fracture propagation and on the interaction between cracks, controlled by the statistics of the initial crack population and the size of the stress shadow zones around each crack. These parameters do not strongly depend on the fracture mode [START_REF] Huang | A dynamic damage growth model for uniaxial compressive response of rock aggregates[END_REF] . The fracture speed propagation is close to the S wave velocity cs in all fracture modes. The shape of the stressed areas around a crack tip differs with fracture mode, but its size varies similarly, with stress decaying with distance to the fracture tip r as 1/r 1/2 . Hence, the transition from single to multiple fracturing is only weakly dependent on the fracture mode [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] . The strain rate condition obtained experimentally is a reasonable approximation to the natural case in that the sample is subject to both shear stress and normal stress.

Our results show that the transition zone is mainly related to the strain rate, rather than to the stress amplitude. Pulverized rocks are markers of events with high strain rate (>100/s). They also endured low strain amplitude, as their initial structure is preserved.

To determine in which conditions these two constraints are satisfied together, we computed the strain rate and the stress near a crack tip propagating at a constant speed.

Considering distances from the fault (r=100 m) small relative to the rupture size (more than 10 km), we calculate an asymptotic development on the distance to the rupture tip.

Details of the computation are given in supplementary material. The first term of the development [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] with a rupture speed below the S wave velocity (subshear rupture) yields a strain decaying as 1/r 1/2 , and a strain rate decaying as 1/r 3/2 . Figure 4 shows the shear strain rate of rocks located 100 m from the fault supposing that the maximum strain is 2%, a value we think representative of the maximum strain sustained by the pulverized rocks. The maximum strain rate stays two orders of magnitude below the experimental threshold strain rate to enable pulverization. Other terms of the Taylor expansion of strain [START_REF] Xu | Higher order asymptotic field at the moving crack tip[END_REF] give similar or lower strain rate for a strain smaller than 2%. Reaching the experimental threshold is therefore unlikely for subshear rupture in homogeneous material.

In the previous computation, we made two hypotheses: a homogeneous medium and a sub-shear rupture. For faults separating two different media, a sharp tensile pulse can be generated in the tensile part of the fault. If the Weertman pulse they describe induces pulverization, we expect to find pulverized rocks along bimaterial faults, with no sign of compression. However, pulverized sandstone along the San Jacinto Fault show compression features [START_REF] Dor | Damage characterization in sandstones along the Mojave section of the San Andreas Fault with a new method: implications for the depth and mechanism of rock pulverization[END_REF] . Moreover, pulverized rocks can be found on both sides of the San Andreas Fault near Lake Hughes area [START_REF] Dor | Pulverized Rocks in the Mojave section of the San Andreas Fault Zone[END_REF] . Pulverized rocks can be induced by several processes, and we propose a different mechanism for this high strain rate. Supershear rupture induces a shock wave, with a sharp front and with small decay with distance [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] . High strain rates of loading can be reached 100m far from the fault core as supershear rupture induces a shock wave [START_REF] Rosakis | Cracks faster than the shear wave speed[END_REF] and generates high frequency displacements [START_REF] Bizarri | Effects of supershear rupture speed on the highfrequency content of S waves investigated using spontaneous dynamic rupture models and isochrone theory[END_REF] . This is consistent with the discovery so far of pulverized rocks only near large strike-slip faults: San Andreas and San Jacinto Faults in California [START_REF] Dor | Pulverized Rocks in the Mojave section of the San Andreas Fault Zone[END_REF][START_REF] Rockwell | Granulometric and mineralogical properties of pulverized rocks from Tejon Pass on the San Andreas Fault and from Tejon Ranch on the Garlock Fault, California[END_REF][START_REF] Dor | Damage characterization in sandstones along the Mojave section of the San Andreas Fault with a new method: implications for the depth and mechanism of rock pulverization[END_REF] , Northern Anatolia Fault in Turkey [START_REF] Dor | Geological and geomorphologic asymmetry across the rupture zones of the 1943 and 1944 earthquakes on the North Anatolian Fault: possible signals for preferred earthquake propagation direction[END_REF] , Arima-Takatsuki Fault in Japan 27 , which are more amenable to supershear rupture. We predict that off-fault pulverization would be only found only along large strike-slip faults, and this provides a way to test our hypothesis in the future. This work gives constraints on the formation of pulverized rocks, which endured deformation rate larger than 100/s. As strong indicator of previous supershear rupture, moreover as an indicator independent of seismological data, pulverized rocks should be seriously considered for the risk assessment of such potentially damaging earthquakes. propagating with a constant below the Rayleigh velocity in a location 100 m away from the fault core. As the fracture tip passes near the monitoring location strain and strain rate varies. We normalized both the strain and strain rate so that the maximum strain amplitude is 2%, corresponding to the order of

Figure 1 .

 1 Figure 1. States of the samples after experiment. (a-left) At a low strain rate (here, 140/s), a granodiorite sample split into a few fragments when deformed in the Split Hopkinson Pressure Bar apparatus. (b-right) At a higher strain rate(here, 400/s), the sample was pulverized into numerous fragments with diameter smaller than the rock initial grain size. The ruler has centimetric marks.

Figure 2 .

 2 Figure 2. Summary of the 27 experiments performed, where the final state relative to the peak stress and the peak strain rate are represented. The transition from single crack fracture (blue circles, figure 1 left) to intense pulverization (red stars, figure 2 right) depends on strain rate. The threshold occurs between 100/s and 150/s. Some samples remain unbroken (green squares).

Figure 3 .

 3 Figure 3. Time-lapse snapshots of a sample being pulverized. The vertical bars lettered A, B, C and D denote the times of the above photographs, taken withfour independently triggered cameras. The stress and strain history measured in our experiment is reported in the lower graphs. The sample is broken during the first loading, before the subsequent loading associated to the waves reflected at the bar extremities. The sample breaks in an early stage but because of their inertia, the fragments fly away later. The sample is fragmented while in a pure uniaxial compression, and not during stress relaxation.

Figure 4 .

 4 Figure 4. Dilatational strain rate versus dilatational strain induced by a rupture

Acknowledgments. We thank Raymond Barre for is laboratory help, François Hild and Kamal Safa for their comments on fracturation regimes, Michel Bouchon and Pascal Favreau for discussions on supershear rupture, Roland Leau, Robert Guiguet and Jean-Pierre Gratier for sample preparation. We appreciated the support of the direction of the LGIT and the 3S-R laboratories for their support. We thank Emily Brodsky, François Renard, Yehuda Ben-Zion and Sébastien Boutareaud for reading previous drafts of this manuscript.

Competing Finantial Interest Statement. Both authors have no conflict interest to disclose. magnitude of the maximum strain experienced by the pulverized rocks near the San Andreas Fault. The curves were computed for several rupture velocities, all below the Rayleigh wave velocity (here c R ~0.8740301c S ). All curves achieve a maximum strain rate less than 0.25/s, three orders of magnitude below the 100/s pulverisation threshold we determined in the laboratory. Details of the computation are given in supplementary material.


Claude Pasquier

Laurent Théry

A distributed editing environment for XML documents

XML is based on two essential aspects: the modelization of data in a tree like structure and the separation between the information itself and the way it is displayed. XML structures are easily serializable. The separation between an abstract representation and one or several views on it allows the elaboration of specialized interfaces to visualize or modify data. A lot of developments were made to interact with XML data but the use of these applications over the Internet is just starting. This paper presents a prototype of a distributed editing environment over the Internet. The key point of our system is the way user interactions are handled. Selections and modifications made by a user are not directly reflected on the concrete view, they are serialized in XML and transmitted to a server which applies them to the document and broadcasts updates to the views. This organization has several advantages. XML documents coding selection and modification operations are usually smaller than the edited document and can be directly processed with a transformation engine which can adapt them to different representations. In addition, several selections or modifications can be combined into an unique XML document. This allows one to update multiple views with different frequencies and fits the requirement of an asynchronous communication mode like HTTP.

Introduction

XML is the central point of the far-reaching process of standardization that is going to alter the way information is handled. The deployment of XML related technologies will have a large impact on operations like structured editing, storage, information exchange, data transformation, querying, and rendering.

Two essential aspects of XML have been inherited from SGML [START_REF]Information processing -Text and office systems -Standard Generalized Markup Language (SGML)[END_REF]: the modelization of data in a tree like structure and the separation between information itself and the way it is displayed. An XML document is typically a serialization of a tree with every node and leaf tagged. It is a convenient user-readable and platform-independent representation, well-suited for transmission over a network. The separation between an abstract representation and one or several views on it allows the elaboration of specialized interfaces to visualize or modify data.

A lot of developments were made to interact with XML data but the use of these applications over the Internet is still under development. The World Wide Web Consortium proposes several standards to visualize XML documents (CSS [START_REF]CSS2 World Wide Web Consortium. Cascading Style Sheets, level 2[END_REF], XSL [START_REF]XSL World Wide Web Consortium. Extensible Stylesheet Language (XSL). W3C Working Draft[END_REF]) and to transform them (XSLT [START_REF]XSLT World Wide Web Consortium[END_REF]). However, these standards are mainly defined to display documents, not to interact with them. If we consider real e-commerce applications, displaying and modifying data will be needed. For example, a vendor may want to update via Internet (using a portable computer or a mobile phone) the database of its company with new information relative to a visited client.

On new XML browsers, some interaction with a displayed document can nevertheless be realized. Documents are internally modelized as DOM trees and can be manipulated with methods that access public DOM APIs. The problem with this solution is that only the concrete view of a document is edited, not the data itself. This paper presents a prototype of a distributed editing environment over the Internet. Its architecture is based on the software development environment Centaur [START_REF] Borras | CENTAUR: The system[END_REF] and ideas presented in [START_REF] Clément | A distributed architecture for programming environments[END_REF] and [START_REF] Déry | Distributed Architecture for Programming Environment[END_REF]. The key point of our system is the way user interactions are handled. Selections and modifications made by a user are not directly reflected on the concrete view. They are serialized in XML and transmitted to a server which applies them to the document and broadcasts updates to the views. Bidirectional correspondences between a source and a result tree are expressed in the declarative language XPPML. It is used to transform serialized user-interactions made on the source structure to equivalent operations applicable to the result structure.

Interacting with XML documents

Editing an XML document over an asynchronous, unstable and rather slow protocol like HTTP implies minimizing the amount of information exchanged. In particular when editing large documents, it is not a very good strategy to retransmit every time the modified document. User interactions are encoded as XML documents. Data transmitted on the network represent actions. The first set of actions declares external operations on the structure (selection, modification, redisplay, etc). The second set concerns updates to be applied on the structure (selection of a given subtree, deletion of a node, etc.). In our prototype, we have defined two actions, selection and modification, that are both based on an unambiguous way to identify a node on a tree. We take as an example the simple mathematical expression '2*(8/2+5)' which will be used throughout this paper. It is expressed by the following XML document called 'exp':

<?XML version='1.0'?> <exp> <mult> <int value='2'/> <plus>
 <int value='8'/> <int value='2'/>
 <int value='5'/> </plus> </mult> </exp>

Paths

One can identify a node in the tree with an expression specifying a path to this node. This can be expressed with the XPath [START_REF]XPath World Wide Web Consortium. XML Path Language. W3C Recommendation[END_REF] standard by starting from the document's root and specifying either the name of all the encountered child nodes or their relative positions. For example, the node '8/2+5' can be equally designated by the following XPath expressions:

/exp/mult/plus /*[1]/*[1]/*[2]
Note that identifying subtrees by a list of node names is ambiguous, since two different subtrees could have the same path. For this reason, we prefer the second solution. In order to transmit paths, we propose to represent a XPath location which uses relative positions by an ipath (IntegerPath) XML element. The location of the node '8/2+5' is thus expressed by:

<ipath> <move num='1'/> <move num='2'/> </ipath>
The starting point of the path is the root node (called documentElement in DOM) and not "the parent of the document element" as it is specified in XPath. Then, the empty element <ipath/> identifies the root of the document, <ipath> <move num='1'/></ipath> its first child.

Selection path

We use selection path to designate different elements of a document. These elements may belong to different selections. Selections are represented by symbolic names. A spath element is defined by the following DTD: Selection paths are general enough to be used for different purposes. On the client side it can be used to represent extension to an existing selection. On the server side it can be used to represent all the existing selections.

<!ELEMENT spath (select*, (move, spath)*)> <!ELEMENT select EMPTY> <!ATTLIST select

Modification path

A modification path memorizes modifications done on a document. The following DTD describes the structure of a mpath element:

<!ELEMENT mpath (element, (move, mpath)*)> <!ATTLIST mpath type (move|delete|insert|change) move> <!ELEMENT element (%targetElt;)> <!ELEMENT move EMPTY> <!ATTLIST move num NMTOKEN #REQUIRED>
A mpath has an attribute type which indicates the kind of modification to perform (deletion, insertion or replacement). By default, if no attribute is given, the type is move. Values that are changed or inserted are specified under the tag element.

As an example, one can evaluate the node '8/2' and replace it with its integer value. This is expressed by the following mpath element:

<mpath> <move num='1'/> <mpath> <move num='2'/> <mpath> <move num='1'/> <mpath type='change'> <element> <int value='4'/> </element> </mpath> </mpath> </mpath> </mpath>
A modification path can be applied to a document to perform the memorized operations. Note that this application can be delayed. Two mpath elements can be combined in a new modification path which amalgamates the successive operations. For example, the previous modification path and the one that corresponds to the deletion of the node identified by the selection 'selA' gives the following combined path:

<mpath> <move num='1'/> <mpath> <move num='2'/> <mpath> <move num='1'/> <mpath type='change'> <element> <int value='4'/> </element> </mpath> <move num='2'/> <mpath type='delete'/> </mpath> </mpath>
When applied to a document, this mpath will replace the element '8/2' with '4' and will remove the element '5'. The same mpath can also be applied to the current selection to compute a new selection path compatible with the updated structure of the document.

Modification paths are generic enough to represent any kind of modification. They represent a convenient way to interact with large documents over a network since only the new elements of the document need to be transmitted.

Communication with a user interface

Except for XML editors that let the user directly write the tags of XML documents, standard editing environments propose an interface between the user and the logical structure. Usually, selections and modifications are done on a concrete view of the data. User interactions are transformed into actions on the logical structure and the layout of the view is recomputed.

Our concepts of selection and modification paths can be applied to modelize the communication between a concrete view and an abstract representation. Representing actions by valid XML documents it possible to use standard tools to manipulate them. Let's take the example of an XML browser in which the displayed document is obtained by transforming an initial document. All modifications made on the source document can be reflected to the view by transmitting only a modification path.

Modification paths on the logical tree and the concrete tree can largely differ. However, one can be generated from the other by using a transformation process similar to the one used to get the initial document. This means that the transformation tool should be capable not only of processing the initial document but also of transforming selection and modification paths from one structure to another.

Standard transformation languages, like DSSSL [START_REF]DSSSL International Organization for Standardization[END_REF] or XSLT [START_REF]XSLT World Wide Web Consortium[END_REF] are wellsuited for processing XML documents. However, current implementations of these languages are based on a batch process that transforms the whole source document into a target one. Once this is done, no link is kept between the two structures. In addition, there is no available implementation capable of doing a reverse transformation (from a target structure back to the source one). Adding this dynamic capability to engines based on DSSSL or XSLT standards seems difficult because of the expressiveness of these languages. In our project, we have developed a transformation engine based on XPPML (Xml Pretty-Printing Meta Language) that is strictly less powerful than XSLT but satisfies all our dynamic requirements. It is a modified version of the transformation engine developed for the Aïoli system [START_REF] Théry | Presentation of Aïoli[END_REF].

XPPML

XPPML is an XML extension for the pretty-printing meta language PPML [START_REF]Centaur 1.2, chapter The PPML Manual[END_REF] defined in the Centaur system. An XPPML specification is a collection of unparsing rules associated with abstract syntax patterns. The concepts of XPPML are very close to those found in XSLT. Basically, it is a language for transforming XML documents into other representations. A transformation in the XPPML language is expressed as a well-formed XML document. It describes rules for converting a source tree into a result one by associating patterns with templates. The formatting machine generates a result structure by traversing the source tree and looking for a pattern that matches each node. When a match is found, the corresponding template is instantiated to create parts of the result tree. Features of XPPML include contextual formatting, conditional layout over external boolean functions and inclusion of user-defined external functions.

The formatting machine makes use of the XPPML rules to generate the path on the result tree corresponding to a given path on the source one but is also able to retrieve the position on a source structure corresponding to a position on a result one.

A typical XPPML document has the following structure:

<x:prettyPrinter ppName="..." langName="..." xmlns:x='http://www-sop.inria.fr/lemme/xppml/1.0'> <x:extension name="..."/> <x:import package="..."/> <x:rule> ... </x:rule> </x:prettyPrinter> An XPPML specification is fully identified by both the name of the pretty printer and the name of the language to which it applies. This allows one to refer and retrieve, for example, the standard pretty printer for the Java language or the pretty printer Y for the language Z. These two identifiers are stored in the attributes 'ppName' and 'langName' of the element 'prettyPrinter'.

Modularity of pretty-printing declarations is realized through one or several 'extension' elements where the names of other XPPML definitions to be included are specified. The 'import' element is used to declare the package where external Java functions should be searched.

The core of the XPPML declaration is defined by a list of 'rule' elements composed of a 'pattern' element and a layout. A pattern is defined by a 'template' element and zero or more additional constraints. The 'match' attribute in the 'template' element is a pattern that identifies the source node to which the rule applies. For example, the following XPPML specification defines the layout of our 'exp' document: XML namespaces are used to distinguish between XPPML constructs and elements corresponding to the output structure. This example uses Figue's syntax [START_REF] Conductier | Figue's Documentation[END_REF] for the layout but other output format like HTML or xsl:fo can equally be used. Figue is an incremental bi-directional layout engine that handles a limited set of combinators like 'h' to specify an horizontal layout, 'v' for a vertical one, 'atom' for a terminal box and some other specialized mathematical constructs.

The first rule matches a 'mult' node composed of exactly two children. Its layout part specifies that an horizontal box has to be created (use of Figue's horizontal combinator 'h') and filled with the concrete tree obtained by a recursive call of the formatting machine on the node's first child, a concrete leaf with the textual value '*' and the layout of the second child.

Contextual pattern are specified in the left-hand side of a rule, as in the second rule which matches a 'plus' node that have a 'mult' node as parent. It is also possible to define rules identified by a context name. When a contextual pretty-printing is required, a context name is added to the recursive call.

In the fourth rule, an external Java method called 'identitypp' is used to display the attribute 'value' of the node representing an integer. External functions can also be used with conditional layout constructs defined with the XPPML instructions 'if ' and 'case'.

Editing and rendering are handled by Figue. During the editing process, Figue communicates to the formatter the location of concrete subtrees which have been selected or modified. The formatter retrieves the corresponding abstract trees, performs the necessary modifications and returns back to the layout engine the selection or modification paths applicable on the concrete tree. With this information, Figue performs an incremental update on the view.

Aïoli can be used as a server which communicates with clients through HTTP requests. Documents are stored on the server side where updating and format-ting operations are performed. Information manipulated on the server side is represented by DOM objects. These are transformed to XML text format and are transmitted to the client through HTTP(S). Standard HTTP protocol is used; marshaling and unmarshaling are done with application methods.

In our current implementation, several clients can be connected to the same server. Figue is one of the possible clients but standard web browsers can also be used. New clients may join and watch the progress of existing editing sessions but problems specific to the implementation of a concurrent editing environment [START_REF] Israel | An Architecture for Multi-User Software Development Environments, Computing Systems[END_REF] have not yet been studied.

A typical editing session

If our example document is accessible at the url 'docURL' and our server is located on 'myserver', then a typical invocation of the transformation engine is done by the following HTTP request:

http://myserver&doc=docURL&ppml=std&type=figue
The arguments 'ppml' and 'type' correspond respectively to the name of the prettyprinter to use and the type of syntax used to codify the concrete tree. The client first declares a new selection called current:

clientMess #1: <setSelection type='Single' name='current'/>
The server answers with a reference to the new selection. This information is needed because if several clients are accessing the same document, the name of the selection may be already used. In that case, the server returns a selection with a different name: serverMess #1: <selection type="Single" name="current" /> Then the client requests for a copy of the document using the appropriate prettyprinting:

clientMess #2: <redraw />
The server sends the corresponding concrete tree using the Figue syntax: serverMess #2: <redraw><H><Atom Value="2"/><Atom Value="*"/> <H><Atom Value="("/><H><Atom Value="8"/> <Atom Value="/"/><Atom Value="2"/></H> <Atom Value="+"/><Atom Value="5"/> <Atom Value=")"/></H></H></redraw>

The client is now able to display the expression:

Let's suppose we want to edit the division. We first select the character '/' with the mouse:

In order to update a selection, the client needs to send a message to the server. It first computes the ipath corresponding to the selected expression in the result tree and sends a message asking for the current selection to be modified: clientMess #3: <updateSelection selName='current'><ipath> <move num="3"/><move num="2"/><move num="2"/> </ipath></updateSelection>

The server acknowledges this message:

serverMess #3: <done/>
The server translates the selection made on the result tree into a selection on the source tree. In our case, it is the tree
<int value='8'/> <int value='2'/>
 that has generated the character '/'. In order to get the value of the selection, the client requests the list of selection's changes that are memorized on the server side: clientMess #4: <commit type="select"/>

The server responds with a structure specifying that the current selection must be placed on the second child of the third node, i.e. the node representing '8/2'. serverMess #4: <commit type="select"><Change> <select name="current"/></Change><Path Rank="3"> <Path Rank="2"><ExtendSelection Name="current" /> </Path></Path></commit>

This information can be used to highlight the current selection in the editor:

Note that what we obtain is a structured selection: by selecting the single character '/' we get the whole division expression. Now that the division is selected, we can decide to evaluate it and replace it with 4:

clientMess #5: <change selName='current'><int value="4"/></change>

The modification is done on the abstract tree stored on the server side and an acknowledgement is sent.

serverMess #5: <done/>

To get the modification, the client just asks for the list of modifications that have been done on the abstract structure since the last commit request:

clientMess #6: <commit type="modif"/>

The server returns the modifications to be applied on the concrete tree:

serverMess #6: <commit type="modif"><mpath type="move"> <move num="3"/><mpath type="move"><move num="2"/> <mpath type="change"><element><Atom Value="4"/> </element></mpath></mpath></mpath></commit>

The client can then reflect these modifications on the editor:

The modification may have changed the values of some selections. So, the client asks for their new values: clientMess #7: <commit type="select"/>

The modification has only replaced a selected tree; nothing has changed: serverMess #7: <commit type="select"><Change> <select name="current"/></Change><Path Rank="3"> <Path Rank="2"><ExtendSelection Name="current"/> </Path></Path></commit>

The client can update the editor with the selection:

Conclusion

The solution described in this paper has several advantages. XML documents coding selection and modification operations are usually smaller than the edited document and they are directly processed by a transformation engine which adapts them to several representations. In addition, the possibility to group several selections or modifications into a single XML document allows us to update multiple views with different frequencies and fits the requirement of an asynchronous communication mode like HTTP. Several clients with different connection speeds can collaborate in a single editing session. For the moment, specific aspects concerning concurrent editing problems have not yet been tackled. We are aware that a lot of problems have to be resolved in order to provide a full featured collaborative environment. Still, we believe that what we have presented here can be used as a valuable basis for more elaborated protocols.

XPPML is less expressive than XSLT. Any XPPML specification can be easily translated into XSLT. XPPML has been designed so that a dynamic link between the source structure and the transformed one could easily be maintained. Doing this for XSLT seems more problematic. For example, a transformation rule in XSLT can have access to any node of the source document, even those localized outside the matched subtree. This means that a single modification in the source document can potentially affect the overall result.

Our system has been successfully tested with the distributed editing of large Java programs. User-interactions are quickly processed and reflected to the client. The unique problem of performance we had to face was only with the initial transmission of the formatted document which may be very large. We are studying possibilities to transmit only the part of the structure needed by the client (the subtree visible in the window for example) or to allow several clients to access different subtrees of a same document.

For the moment, selections and modifications can only be done on nodes of the tree. It is sufficient for editing highly structured structure with few unconstrained text fields, like the kind of documents we have presented here. However, for editing general XML documents, it is necessary to represent selections or modifications of parts of textual fields. The concept of range used in the specification of DOM level 2 [START_REF]DOM-Level-2 W3C (World Wide Web Consortium) DOM Level 2 W3C Recommendation[END_REF] which represents a selection by a pair node + offset will be implemented in the next version of our system.

In our organization, the amount of software on the side of the client is kept to a minimum. It is composed of a communication layer that sends and receives messages over the network and a layout engine. All the other components are concentrated in the server. It is then particularly adapted to situations where clients have sparse resources.

Acknowledgements

This work has been done in the framework of Dyade, the Bull-Inria Research Joint Venture.


Christelle Chauffeton

Gilles Wallez
email: gilles.wallez@sorbonne-universite.fr

Crystal-chemical investigation of the "cubic Al-Si spinel" issued from the thermal transformation of kaolinite and halloysite

Keywords:

published or not. The documents may come

Crystal-chemical investigation of the "cubic Al-Si spinel" issued from the thermal transformation of kaolinite and halloysite

Christelle Chauffeton, Gilles Wallez

Introduction

Kaolin (a general term for kaolinite (Al 2 Si 2 O 5 (OH) 4 )-based clays) has been used worldwide for millennia in the manufacturing of ceramics, in particular porcelain. Naturally aluminum-rich and low in impurities, it is appreciated for its refractory properties and its whiteness. At high temperature, kaolinite undergoes a series of irreversible chemical and structural transformations known through an abundant literature [START_REF] Brindley | Kaolinite-Mullite Reaction Series III, The High-Temperature Phases[END_REF][START_REF] Ortega | The Multistep Nature of the Kaolinite Dehydroxylation: Kinetics and Mechanism[END_REF][START_REF] Chakravorty | Kaolinite-Mullite Reaction Series: The Development and Significance of a Binary Aluminosilicate[END_REF][START_REF] De Aza | Time-resolved powder neutron diffraction study of the phase transformation sequence of kaolinite to mullite[END_REF][START_REF] Duncan | Kinetics and Mechanism of High-Temperature Reactions of Kaolinite Minerals[END_REF][START_REF] Gualtieri | Kinetic Study of the Kaolinite-Mullite Reaction Sequence. Part II: Mullite Formation[END_REF][START_REF] Sainz | XRD Microstructural Analysis of Mullites Obtained from Kaolinite-Alumina Mixtures[END_REF][START_REF] Lee | Phase Transformation Sequence from Kaolinite to Mullite Investigated by an Energy-Filtering Transmission Electron Microscope[END_REF][START_REF] Mackenzie | MAS NMR Evidence for the Presence of Silicon in the Alumina Spinel from Thermally Transformed Kaolinite[END_REF][START_REF] Okada | Characterization of Spinel Phase Formed in the Kaolin-Mullite Thermal Sequence[END_REF][START_REF] Sonuparlak | Spinel Phase Formation During the 980 °C Exothermic Reaction in the Kaolinite-to-Mullite Reaction Series[END_REF][START_REF] Johnson | Influence of impurities on high-temperature reactions of kaolinite[END_REF][START_REF] Brown | Outstanding Problems in the Kaolinite-Mullite Reaction Sequence Investigated by 29 Si and 27 Al Solid-State Nuclear Magnetic Resonance: High-Temperature Transformations of Metakaolinite[END_REF][START_REF] Liu | Mullite formation in kaolinite-α-alumina[END_REF]. At first, dehydroxylation between 400 and 650 °C yields metastable metakaolinite Al 2 Si 2 O 7 that decomposes at 947 °C [START_REF] Ptaček | The kinetics of Al-Si spinel phase crystallization from calcined kaolin[END_REF] into an amorphous silica-based phase and nanocrystalline needleshaped aluminum-rich primary mullite, along with a nanocrystalline "Al-Si spinel" phase that will be the main topic of this work. A similar transformation is observed for related 1:1 phyllosilicate clays like halloysite, imogolite, dickite and allophane [START_REF] Duncan | Kinetics and Mechanism of High-Temperature Reactions of Kaolinite Minerals[END_REF][START_REF] Okada | Characterization of Spinel Phase Formed in the Kaolin-Mullite Thermal Sequence[END_REF]. On further heating, secondary mullite grows by reaction between the spinel and the amorphous phase. This process goes together with a gradual substitution of silicon for aluminum in mullite and an increase of the crystals size. Although the growth of secondary mullite from pure kaolinite or a mixture with α-Al 2 O 3 [START_REF] Sainz | XRD Microstructural Analysis of Mullites Obtained from Kaolinite-Alumina Mixtures[END_REF][START_REF] Liu | Mullite formation in kaolinite-α-alumina[END_REF] starts around 1300 °C, this diffusion-based mechanism can be observed as low as 1000 °C in the presence of numerous metal oxides like Mg, Mn or Fe [START_REF] De Aza | Time-resolved powder neutron diffraction study of the phase transformation sequence of kaolinite to mullite[END_REF][START_REF] Sainz | XRD Microstructural Analysis of Mullites Obtained from Kaolinite-Alumina Mixtures[END_REF][START_REF] Johnson | Influence of impurities on high-temperature reactions of kaolinite[END_REF][START_REF] Shin | Mullitization from a Multicomponent Oxide System in the Temperature Range 1200°-1500[END_REF][START_REF] Yamuna | Phase-Pure Mullite from Kaolinite[END_REF][START_REF] Odaka | Cation dopant effect on phase transformation and microstructural evolution in M 2+ -substituted gamma-alumina powders[END_REF]. These "mullitizers", which are also of common occurrence in natural clays, have long been used in industrial ceramics manufacturing to catalyze the conversion into mullite [START_REF] Parmelee | Catalytic Mullitization of Kaolinite by Metallic Oxides[END_REF].

Conceivably, these irreversible thermal transformations which spread over hundreds of degrees could be monitored using X-ray diffraction (XRD) coupled to Rietveld analysis which offers an unparalleled potential in terms of crystal-chemical, microstructural and quantitative phases analysis (RQPA). However, though previous works endeavored exploring the thermal evolution of kaolinite through XRD, the phases quantification was relative [START_REF] De Aza | Time-resolved powder neutron diffraction study of the phase transformation sequence of kaolinite to mullite[END_REF] or based on calibration curves obtained by measuring the height or the surface of a single XRD peak on standards [START_REF] Chakravorty | Kaolinite-Mullite Reaction Series: The Development and Significance of a Binary Aluminosilicate[END_REF][START_REF] Shin | Mullitization from a Multicomponent Oxide System in the Temperature Range 1200°-1500[END_REF][START_REF] Chen | Kinetics of secondary mullite formation in kaolin-Al 2 O 3 ceramics[END_REF][START_REF] Okada | Characterization of Spinel Phase Formed in the Kaolin-Mullite Thermal Sequence[END_REF]. It is noteworthy that the Rietveld method was never fully implemented in the study of these materials because of several difficulties: the presence of a glassy phase, the low crystallinity of primary mullite and, last but not least, the still unknown crystal structure and chemical composition of the Al-Si spinel, a point which remains subject to debate more than sixty years after its discovery.

Actually, the so-called phase was first reported in 1959 by Brindley and Nakahira [START_REF] Brindley | Kaolinite-Mullite Reaction Series III, The High-Temperature Phases[END_REF] in the frame of the study of the kaolinite-mullite transformation. Diffuse XRD lines accounted for the presence of a nanocrystalline cubic phase, structurally related to spinel MgAl 2 O 4 . The irruption of this new phase in the intensively studied Al-Si-O system gave rise to divergent hypotheses concerning its formula -a point that still remains much debated. On the basis of the cubic symmetry and the cell parameter (a ≈ 7.90 Å), Brindley and Nakahira pointed out the obvious kinship with γ-Al 2 O 3 , that occurs by heating boehmite and alumina gels, but structural relationships with metakaolinite led them to conjecture an Al 2 O 3 :SiO 2 = 2:3 mol ratio [START_REF] Brindley | Kaolinite-Mullite Reaction Series III, The High-Temperature Phases[END_REF]. Using similar arguments, Low et al. [START_REF] Low | The structure and composition of Al-Si spinel[END_REF] concluded in a 2:1 proportion, whereas Chakraborthy et al. measured a 2:3 ratio by elemental analyses [START_REF] Chakraborthy | Reexamination of Kaolinite-Mullite Reaction Series[END_REF].

Nevertheless, many early authors kept hanging on to a pure-alumina model, like Percival et al. who reached this conclusion from IR spectroscopy [START_REF] Percival | Interpretation of the kaolinite-mullite reaction sequence from infrared absorption spectra[END_REF] and Léonard et al. from Radial Electron Density Distribution [START_REF] Léonard | Structural Analysis of the Transition Phase in the Kaolinite-Mullite Thermal Sequence[END_REF]. Brown et al. stated that "the spinel phase contains insufficient Si to be detected (…) and has a 27 Al NMR spectrum consistent with γ-Al 2 O 3 " [START_REF] Brown | Outstanding Problems in the Kaolinite-Mullite Reaction Sequence Investigated by 29 Si and 27 Al Solid-State Nuclear Magnetic Resonance: High-Temperature Transformations of Metakaolinite[END_REF]. This opinion would be tempered later on by some of these authors who detected a maximum of 3.9 w% SiO 2 (Al 2 O 3 :SiO 2 ≥ 14.5:1) by 29 Si NMR [START_REF] Mackenzie | MAS NMR Evidence for the Presence of Silicon in the Alumina Spinel from Thermally Transformed Kaolinite[END_REF], in contradiction with the maximum of 11.5 w% SiO 2 (4.5:1) found by Schneider et al. using the same method [START_REF] Schneider | Constitution of the γ-Alumina Phase in Chemically-Produced Mullite Precursors[END_REF].

Analyzing an alkali-leached heated kaolinite [START_REF] Okada | Characterization of Spinel Phase Formed in the Kaolin-Mullite Thermal Sequence[END_REF] and a xerogel [START_REF] Okada | Characterization of the spinel phase from SiO 2 -Al 2 O 3 xerogels and the formation process of mullite[END_REF], Okada et al. found out the same 6:1 composition, in fair agreement with Sonuparlak et al.'s result (< 5.3:1) using EDS in an analytical TEM [START_REF] Sonuparlak | Spinel Phase Formation During the 980 °C Exothermic Reaction in the Kaolinite-to-Mullite Reaction Series[END_REF]. From TEM micrographs, Lee et al. observed a topotactic relationship between metakaolinite and the spinel phase, and emitted the hypothesis that the Si content of the latter increased with temperature, as for mullite [START_REF] Lee | Phase Transformation Sequence from Kaolinite to Mullite Investigated by an Energy-Filtering Transmission Electron Microscope[END_REF]. Conceivably, the nanometric size of the crystallites and their dispersion in the silica-rich glassy phase play a role in this discrepancy.

So, despite most studies -in particular, the most recent ones -agree with the fact that silicon, if any, only occurs in a faint amount in the Al-Si spinel, many questions still remain unaddressed: the precise chemical formula and its possible variability, the crystal structure and the amount in the products of decomposition of the clays. Whereas all these questions are usually addressed through XRD, no such study has ever been reported in the literature concerning the title compound. Indeed, several intrinsic impediments must be taken into account prior to an in-deep crystallographic investigation:

1-the nanometric size of the crystallites results in very diffuse diffraction peaks, amongst which only few are intense enough to be measured; 2-some of these peaks are blurred by the intense and irregular background due to the amorphous silica-rich phase; 3-the XRD pattern of the compound exhibits misleading similarities (apparent cubic symmetry, cell parameters and probably crystal structure) with those of η-Al 2 O 3 (a = 7.906 Å [START_REF] Li | Structural models of eta-and gammaaluminas by X-ray Rietveld refinement[END_REF][START_REF] Shirasuka | The preparation of eta alumina and its structure, Yogyo Kyokaishi Shi[END_REF]) and γ-Al 2 O 3 (a = 7.938 Å [START_REF] Smrčok | Gamma-alumina: a single-crystal X-ray diffraction study[END_REF], (2V) ⅓ = 7.898 Å [START_REF] Li | Structural models of eta-and gammaaluminas by X-ray Rietveld refinement[END_REF] or 7.952 Å [START_REF] Paglia | Tetragonal structure model for boehmite-derived gamma-alumina[END_REF]); 4-silicon, if present in the compound, cannot be precisely distinguished from isoelectronic aluminum using X-ray; 5-probable cation/vacancy disorder -like in nanometric transition aluminas -may hamper the determination of the cation:anion ratio, hence of the composition.

Experimental

Starting materials and samples' preparation

The starting materials consisted in two minerals commonly used for the making of porcelain earthenware: a natural halloysite-rich clay originating from Les Eyzies, France (EZ in the following) and the kaolinite-rich ECC Standard Porcelain TM , Imerys Minerals Ltd., UK (SP). The compositions in elemental oxides measured by X-ray fluorescence with a Bruker S1 The clays were fired at various temperatures, with a particular attention paid to the domain of primary and secondary crystallization of mullite, between 950 and 1200 °C. The samples were put into an alumina crucible, introduced in the oven below 600 °C and heated at 300 °C.h -1 up to dwell temperature, then left during 1 h. The samples were ground into powders prior to XRD analysis.

X-ray powder diffraction and Rietveld analysis

The X-ray powder diffraction experiments were carried out in Bragg-Brentano configuration on a Panalytical Xpert Pro instrument with a Ge (111)-monochromated CuKα1 source and a PIXcel1D detector. The scans used for structure analyses were recorded over the 10-140 ° 2θ range, step 0.013 ° during 13 h. The Rietveld analyses were performed with the Fullprof suite [START_REF] Rodriguez-Carvajal | FULLPROF: A Program for Rietveld Refinement and Pattern Matching Analysis[END_REF]. Because of numerous faint and/or diffuse diffraction peaks, along with broad bumps due to the glassy phase, the background was fitted by interpolation between selected points without Bragg contribution of the spinel phase, at the same 2θ positions for all the samples. Various profile models were implemented depending on the nature of the crystalline phases.

The diffraction peaks of quartz, cristobalite and fluorite (used as an internal standard)

were fitted with the pseudo-Voigt function and the Caglioti FWHM polynomial. Quartz's structural data were refined to allow an optimal fit of this abundant phase, while those for cristobalite were taken from [START_REF] Dollase | Reinvestigation of the structure of low cristobalite[END_REF].

Concerning mullite, the Thompson-Cox-Hastings profile function [START_REF] Thompson | Rietveld refinement of Debye-Scherrer synchrotron X-ray data from Al 2 O 3[END_REF] was implemented along with a uniaxial Lorentzian contribution to peak broadening to define a hkl-dependent FWHM model that took into account both the small size of the crystallites (particularly in the domain of primary crystallization) and their well-known elongation following the c-axis.

Compared to the results obtained with the isotropic TCH function, this procedure allowed to reduce considerably the intensity residuals on the mullite peaks. Actually, the size effect can be considered to prevail according to previous Williamson-Hall [START_REF] Castelein | Shape, size and composition of mullite nanocrystals from a rapidly sintered kaolin[END_REF] and Warren-Averbach [START_REF] Sainz | XRD Microstructural Analysis of Mullites Obtained from Kaolinite-Alumina Mixtures[END_REF] analyses which concluded in the limited contribution of the strains to the broadening.

Mullite is a non-stoichiometric solid with general formula Al 4+2x Si 2-2x O 10-x , with x usually ranging from 0.2 to 0.9 (55 to 90 mol% Al 2 O 3 ) [START_REF] Schneider | Structure and properties of mullite -A review[END_REF][START_REF] Fischer | Formation of aluminum rich 9:1 mullite and its transformation to low alumina mullite upon heating[END_REF]. The site occupancy factors were refined in the secondary domain and kept fixed in the domain of primary crystallization where the diffracted intensities were not intense enough. In both cases, care was taken that the composition-sensitive a-cell edge and the Al:Si ratio remained in fair compliance with Fischer et al.'s relation [START_REF] Fischer | Formation of aluminum rich 9:1 mullite and its transformation to low alumina mullite upon heating[END_REF].

As the Bragg peaks of the spinel phase appear symmetric and regular-shaped, a cubic symmetry was assumed, with the same Fd-3m space group as the MgAl 2 O 4 archetype in compliance with the reflection conditions. However, due to the peaks' width, a slight lattice distortion like for γ-Al 2 O 3 [START_REF] Li | Structural models of eta-and gammaaluminas by X-ray Rietveld refinement[END_REF][START_REF] Paglia | Tetragonal structure model for boehmite-derived gamma-alumina[END_REF] cannot be ruled out. The peaks were fitted with a pseudo-Voigt function. As will be shown in the following, the cations are scattered over four, partlyoccupied sites. This disorder does not allow to refine the presence of silicon, if any. However, on the basis of the most recent research results and the analogy of the cell dimensions with those of transition aluminas, the present spinel phase can be thought of as pure alumina. The structure model was built from a fully occupied quasi close packing of oxygen atoms in 32e (x, x, x) position like in the spinel alumina models, then the cation array was determined by refining the occupancy of Al atoms in the five possible host sites (octahedral 16c (0, 0, 0) and 16d (1/2, 1/2, 1/2); tetrahedral 8a (3/8, 3/8, 3/8), 48f (x, 1/8, 1/8) and 8b (3/8, 3/8, 3/8)). After the 8b site was found to be empty, the total occupancy of the four other sites was constrained to comply with an Al:O = 2:3 atomic ratio.

An example of Rietveld plot can be seen on Fig. 1 (a comprehensive survey of the results is available in Table S1 (Supplementary Information)). As will be shown in the following, the growth of the secondary mullite coincides with the vanishing of the spinel phase. For this reason, the most consistent -and arguably, reliable -results have been obtained from samples which only exhibit low amounts of mullite. Likewise, some samples still containing minor unidentified impurities in the low temperature domain yielded somewhat divergent structural parameters.

Furthermore, despite our efforts, the structural model may be impacted by the difficulties in measuring the areas of the spinel's diffuse Lorentzian peaks and cannot pretend to reach the usual standards in terms of precision.

Results and Discussion

The crystal structure of the spinel phase

In the Fd-3m symmetry, the anion close packing provides two octahedral and three Concerning the present phase, the cation arrays refined for the diverse samples clearly exhibit strong similarities, in spite of discrepancies which often exceed the computed standard deviations (probably underestimated) but do not seem to be ascribable to the nature of the raw material, nor to the thermal treatment undergone (Fig. 2). Likewise, the Rietveld refinements did not reveal significant nor systematic variations of the cell parameter. Therefore, the following discussion will be based on a mean model derived from the data measured on samples containing high amounts of spinel, which consists in a cubic cell (a = 7.909(5) Å, similar to those of the pre-cited transition aluminas) with the atomic positions, site occupancy factors and Al-O bond lengths reported in Table 2. [START_REF] Smrčok | Gamma-alumina: a single-crystal X-ray diffraction study[END_REF] and -Al 2 O 3 [START_REF] Shirasuka | The preparation of eta alumina and its structure, Yogyo Kyokaishi Shi[END_REF] respectively: all these three forms feature the same occupied sites, but in -Al 2 O 3 , only 6 % of the cations (16d: 1.632, 16c: 0.056, 8a: 0.863, 48f: 0.114) are located off the 16d-8a regular positions vs. 39 % in -Al 2 O 3 (16d: 1.128, 16c: 0.540, 8a: 0.503, 48f: 0.498). As for the latter, the present compound (Table 2) exhibits a strongly (34 %) disordered cation array but any further comparison would be hazardous insofar as -Al 2 O 3 forms around 500 °C from bayerite or gelatineous pseudoboehmite and decomposes as low as 900 °C [37].

Microstructure

The size of the crystallites can be assessed from the FWHM of the 400 and 440 reflections, the only intense ones. The Williamson-Hall analyses performed on 5 SP and 8 EZ samples with high spinel content (see Fig. S2 in Supplementary Information section) led to an average L = 5.0(5) nm value. Here also, no significant variations due to the starting clay or the thermal treatment were observed. This size is of the same order as Rozita et al.'s HRTEM and XRD measurements on commercial -Al 2 O 3 powders (5-16 nm) [START_REF] Rozita | An investigation of commercial gamma-Al 2 O 3 nanoparticles[END_REF].

Application to the Rietveld Quantitative Phase Analysis

The structural data set of the spinel phase was used to measure the thermal evolution of the phases issued from the transformation of halloysite. The EZ clay was preferred to the SP because of its lesser content in minor oxides, which delays the growth of secondary mullite and extends therefore the domain of existence of the spinel phase. The refinements were performed on the EZ_980 to _1260 samples admixed with 10 w% CaF 2 for the calibration of the diffracted intensities. The spinel pattern was modelled with the same profile function for all the samples. Fig. 3 features the weight ratios of the spinel phase, mullite, cristobalite and the amorphous phase calculated by residual, considering that these phases sum to 100 % (quartz (5.3(6) w%) is not taken into account).

Fig. 3. Thermal evolution of the weight ratios of the spinel phase, mullite, cristobalite and amorphous phase issued from the EZ clay inferred from Rietveld quantitative phase analysis.

The domain of primary mullite crystallization (up to about 1140 °C) reveals a noteworthy phenomenon: the amount of spinel continues increasing all along, contrary to the common opinion that this phase directly stems from the collapse of metakaolinite. Indeed, it seems that a further nucleation occurs from the devitrification of the amorphous phase, but without growth according to the invariance of the crystallites' size, as expected from a mechanism occurring in an all-solid system. So, the spinel and the primary mullite develop independently until the secondary mullite begins growing, probably due to the increase of the diffusion process which allows silica and alumina to react with each other. The onset and the end of this step match fairly with those of the spinel decrease. For comparison, the amount of the spinel phase issued from the SP clay starts decreasing as low as 1100 and vanishes around 1180 °C (Table S1), probably because of the presence of about 10 w% of secondary oxides vs.

3 w% in EZ. Beyond this temperature, mullite growth, which cannot be fed anymore by the spinel phase as a source of aluminum, may still occur from a direct reaction within the residual amorphous phase. Besides, faint amounts of cristobalite are observed starting from 1200 °C with both clays.

Stability of the spinel phase

The spinel phase issued from the transformation of kaolinite and halloysite is known to persist up to temperatures far beyond the domain of stability of pure "transition" aluminas generally obtained by dehydroxylation of alumina gels or hydrated minerals, as gibbsite-χ-κ, boehmite-γ-δ-θ and bayerite-η-θ transformation routes all end with the collapse of the lowdensity spinels into the massive and stable α-form (corundum) between 1050 and 1100 °C [37]. The existence of these transition aluminas up to these high temperaturesan asset for their use as catalystsis generally ascribed to the high specific surface that results from their synthesis route.

However, studies carried out during the past decades on the substitution of various oxides in transition aluminas highlighted a stabilizing effect that can be ascribed to their higher solubility in a spinel-type structure than in α-Al 2 O 3 [START_REF] Okada | Effect of divalent cation additives on the gamma-Al 2 O 3 -to-alpha-Al 2 O 3 phase transition[END_REF][START_REF] Maglia | Energetics of aluminum vacancies and incorporation of foreign trivalent ions in γ-Al 2 O 3 : an atomistic simulation study[END_REF]. Both iso-and aliovalent substitutions are favored by the lacunary cation sublattice and the presence of tetrahedral host sites, whereas in α-Al 2 O 3 , only octahedral sites can be occupied. The case of silicon oxide is of particular interest for the present study because its presence in faint amounts in the "Al-Si spinel" cannot be excluded. Computational studies [START_REF] Nahif | Ab initio study of the effect of Si on the phase stability and electronic structure of γand α-Al 2 O 3[END_REF][START_REF] Jiang | Ab initio study of effects of substitutional additives on the phase stability of γ-alumina[END_REF][START_REF] Andersson | Ab initio calculations on the effects of additives on alumina phase stability[END_REF] all concluded that Si in low concentrations (0-5 at%) should take place into tetrahedral sites and build stronger bonds than Al, thus increasing the stiffness and the stability of the spinel framework. It is noteworthy however that the real solubility of Si in these phases remains unknown.

Conclusion

Although the composition of the so-called "Al-Si spinel" remains a pending issue, its crystal structure solved by the Rietveld method can be described as a new type of cubic transition alumina with a high cation/vacancy disorder. Despite approximate, the structural modelthe first one proposed more than sixty years after the discovery of the phase -allowed to measure the ratios of the phases issued from the transformation of an halloysite-rich clay, but the same process can be implemented for other minerals containing 1:1 phyllosilicates, in particular to assess the effect of minor oxides, known to act as mullitizers, or in experimental archaeology for the determination of the thermal treatment undergone by an earthenware artifact.

From a more fundamental point of view, it is worth noting that the spinel observed in the present work does not seem to "collapse" by itself, but rather reacts with the silica-rich glassy phase according to the simultaneous growth of secondary mullite. Arguably, this spinel form exhibits a higher thermal stability compared to all the transition aluminas, whether pure or doped with silicon. Although this study does not pretend to address this issue, further works could be intended in order to investigate the role played by some of the physicochemical peculiarities in its stability, that is, the crystal structure based on an original cation array, the very small size of the crystallites and their dispersion in the glassy matrix.

f f f f f f f - f f 1.05 SP 1180_1h 0.03 f f f f f f f - f f 5.
f f f f f f - f f 3.24 EZ 1220_1h 0.03 f f f f f f f - f f 7.37
* the occupancy of the "empty" 8b site was refined freely, whereas those of the other sites were constrained to comply with a global Al 2 O 3 formula.

"f" means fixed for insufficient intensity. Table S2. Comparison between the cations arrays in the averaged "Al-Si spinel" model and in some transition aluminas ((T) refers to a tetragonal cell distortion). [Saraswati] V. Saraswati, G.V.N. Rao, X-ray diffraction in gamma-alumina whiskers, J.

Cryst. Growth 83 (1987) 606.

Fig. 1 .

 1 Fig. 1. Rietveld plot for sample SP_1080_1h: i obs (red dots), i calc (black), i obs -i calc (blue), and contributions of the spinel phase (green) and mullite (purple). Sharp peaks correspond to quartz. Sample holder baseline subtracted.

  tetrahedral types of host sites for cations. The archetype spinel structure can be thought of as the interlocking of two similar sublattices, each made of [110] strips of equivalent octahedra bridged by two independent sets of tetrahedra. In an AB 2 O 4 ordered model, octahedrontetrahedron faces sharing only allows the occupation of one kind of octahedral (16d) and tetrahedral (8a) sites, but in transition aluminas, the cation deficiency (Al 2.667 O 4 in a spinel-type formulation) allows the simultaneous occupation of several octahedral and tetrahedral sites.

Fig. 2 .

 2 Fig. 2. Occupancy of the octahedral and tetrahedral sites of the spinel phase of selected samples, in number of Al atoms per Al 8/3 O 4 formula.

  Firing an Al 2 O 3 :SiO 2 = 95:5 w% gel in the frame of the earliest experiment-based study, Chakravorty & Ghos reported γ-Al 2 O 3 and nanometric "Al-Si spinel" at 1000 °C, which entirely transformed into α-Al 2 O 3 and mullite after 1 h at 1200 °C [44]. Mekasuwandumrong et al. investigated the effect of SiO 2 on solvothermal χ-Al 2 O 3 , but they only observed a retardation of 100 °C for the formation of α-Al 2 O 3[START_REF] Mekasuwandumrong | Impact of concentration and Si doping on the properties and phase transformation behavior of nanocrystalline alumina prepared via solvothermal synthesis[END_REF].

Fig. S1 .

 S1 Fig. S1. Occupancy scheme of the cation sites: octahedral (left) and tetrahedral (right).

Fig. S2 .

 S2 Fig. S2. Williamson-Hall plot based on the 400 and 440 reflections of the SP (red) and EZ (black). The lines are plotted from the mean values for each group of points.

Table 1 .

 1 Titan spectrometer are reported inTable1. From preliminary XRD-Rietveld analysis, the EZ clay contains 92 w% halloysite, vs. 84 w% kaolinite for the SP clay from the dealer's technical data chart. Elemental XRF assay of the Eyzies (EZ) and Standard Porcelain (SP) clays (oxygen not taken into account).

	element	Si	Al	S	K	Rb	Mg	Ca
	EZ	50.1(4)	45.7(5)	0.1	-	-	0.8(3)	0.1
	SP	45.2(4)	38.9(4)	0.2(1)	4.1(2)	0.2	2.7(5)	1.8(2)
	element	Sr	Ba	Ti	V	Cr	Mn	Fe
	EZ	-	0.3	0.1	0.1	0.1	0.1	1.1(2)
	SP	0.2	2.6(4)	0.1	-	-	0.1	3.9(2)

Table 2 .

 2 Averaged structural data for the spinel phase.

	atom site coordinates	occupancy atoms	/	B iso (Å 2 ) bond lengths (Å)
				Al 8/3 O 4
	O	32e (0.257(4), x, x)	1	4		0.8

This four-site array accounts for a highly disordered structure in agreement with the nucleation-type formation mechanism. A comparison can be made with Smrčok et al.'s and Shirasuka et al.'s models for -Al 2 O 3

Table S3 .

 S3 Calculated XRD diagram of the averaged spinel phase at λ CuKα1 = 1.5405 Å.

	3	3	3	1.522	60.81	8
	5	1	1	1.522	60.81	188
	4	4	0	1.398	66.86	1000
	5	3	1	1.337	70.37	6
	4	4	2	1.318	71.52	< 1
	6	2	0	1.251	76.05	6
	5	3	3	1.206	79.38	38
	6	2	2	1.192	80.49	< 1
	4	4	4	1.142	84.87	64
	7	1	1	1.107	88.14	< 1
	5	5	1	1.107	88.14	7
	6	4	2	1.057	93.58	11
	7	3	1	1.030	96.85	17
	5	5	3	1.030	96.85	44
	8	0	0	0.989	102.37	140
	7	3	3	0.966	105.73	2
	6	4	4	0.959	106.86	< 1
	6	6	0	0.932	111.47	3
	8	2	2	0.932	111.47	3
	5	5	5	0.913	115.02	20
	7	5	1	0.913	115.02	37
	6	6	2	0.907	116.22	1
	8	4	0	0.884	121.18	97
	7	5	3	0.868	125.07	< 1
	h 9	k 1	l 1	d (Å) 0.868	2θ (°) 125.07	I (‰) 4
	1 8	1 4	1 2	4.566 0.863	19.42 126.41	67 < 1
	2 6	2 6	0 4	2.796 0.843	31.98 132.03	58 2
	3 9	1 3	1 1	2.385 0.829	37.69 136.59	357 67
	2	2	2	2.283	39.44	101
	4	0	0	1.977	45.86	556
	3	3	1	1.814	50.24	3
	4	2	2	1.614	57.00	17

Table S4 .

 S4 Elemental oxides XRF assay of the Eyzies (EZ) and Standard Porcelain (SP) clays.

Acknowledgements

The authors wish to thank Dr Atika Chemmi, Head of the Applied Research Department of the Manufacture Nationale de Sèvres, France, for selecting and providing the clay samples and Dr Thibault Charpentier, Research Director at the CEA, France for his kind help in trying to solve the issue of the "Al-Si spinel" composition. CC is grateful to the Fondation Bettencourt-Schueller and the Cité de la Céramique Sèvres-Limoges, France for founding her work in the frame of her PhD thesis.

Supplementary information

The CIF file of the averaged crystal structure, details concerning the structure and lattice parameters of the spinel phase in the studied samples, a comparison with the known structural forms of transition aluminas, the Williamson-Hall plot and the calculated XRD pattern of the averaged crystal structure are provided as Supplementary Information.

Supplementary information section

Table S1. Main parameters concerning the spinel phase issued from the Rietveld analyses: spinel/mullite w/w ratio, oxygen atom coordinate x, site occupancy factors of aluminum in octahedral (16d and 16c) and tetrahedral (8a, 48f and 8b) sites, cell parameter a (Å), full width at half maximum of main 400 and 440 peaks (W 400 and W 440 , °) and R Bragg reliability factor (%). Shaded samples were not taken into account in the determination of the average spinel model due to too faint intensities or to the presence of impurity peaks.