P Y Hicher

H Wahyudi

D Tessier

Microstructural analysis of inherent and induced anisotropy in clay

Keywords: clay structure, anisotropy, triaxial tests, oedometer tests, scanning electron microscopy, transmission electron microscopy

The aim of this study was to relate the mechanical behaviour of saturated clays to their structural characteristics. Two clays were studied: a kaolinite and a bentonite. The evolution of the shape, the size, the concentration and the orientation of the elements which constitute the clay structure was examined by means of scanning and transmission electron microscopes. Particular care was taken to avoid as far as possible disturbing the micro-structure during observation. The results showed the existence of particle breakage and the creation of structural anisotropy during loading. We can conclude from this work that the mechanical behaviour to clayed materials is largely dependent on the changes which occur at the scale of the particles.

INTRODUCTION

Soils in general, and particularly saturated and non-saturated clays, are usually characterized by their &macroscopic' mechanical properties, that is, at the scale of the sample or at larger scales for in situ testing. Their macroscopic behaviour can be represented by their stress}strain relationships [1}4] and analysis of their properties at the scale of the crystal unit is usually done in the framework of physico-chemical or crystallographic studies which seek to determine a type of crystal, the ion exchange capacity, the number of exchangeable cations, the inter-crystal distance in relation to the type of water, etc. [5}12].

Since these two types of characterizing a!ect two completely di!erent scales of investigation, it is highly di$cult to relate the macroscopic properties to the di!erent heterogeneities of the studied systems, especially those which correspond to larger scales than the clay crystals themselves, i.e. particles and associations of particles, aggregates, "ssures, etc. The evolution of the microstructure during one-dimensional consolidation has been studied by Pusch [START_REF] Pusch | Microstructural changes in soft quick clay at failure[END_REF], Martin and Ladd [START_REF] Martin | Fabric of consolidated kaolinite[END_REF], Delage and Lefebvre [START_REF] Delage | Study of the structure of a sensitive Champlain clay and its evolution during consolidation[END_REF] and others. Several authors have also reported observations of microstructure and pore-size distribution in relation to the permeability of clayey materials [15}17].

The aim of this study is to provide some evidence of the connections between the evolution of the clay structures at di!erent scales of observation and the mechanical behaviour of clayey materials subjected to di!erent stress and strain loadings. We are particularly interested in connecting structural and mechanical anisotropy. For this purpose, di!erent methods of investigation have been used to allow us to describe the clay structure at scales between 1 mm and 10 A s along the one-dimensional and triaxial loadings.

In the following, terms referring to clay structure are de"ned as follows:

1. Clay layer: the elementary unit of structure comprising stacked units of tetrahedral and octahedral sheets forming the given clay mineral unit cell. 2. Crystal that consist of several layers having exact parallel orientation of a}b planes (MeH ring and Oberlin, 1971). 3. Clay particle formed by nearly parallel face-to-face grouping of crystals or stacking of elementary clay layer units into compound tactoid of quasi-crystal (Quirk and Aylmore, 1971). 4. Aggregates made up of particles that are not easily disaggregated and that contain voids or internal discontinuities between them.

TESTED MATERIALS AND METHODS

We studied two di!erent clays: a kaolinite and a bentonite, whose structural organizations are very di!erent. Table I summarizes their physical properties. A slurry was "rst prepared at a water content equal to two times the liquid limit. Figures 1 and2 present observations from scanning electron microscope (SEM) by the use of the low-temperature technique (CRYOSCAN) [START_REF] Tessier | Utilisation du microscope eH lectronique a`balayage dans l'eH tude des sols. Observation de sols humides a`di!eH rents pF[END_REF][START_REF] Delage | Use of cryoscan apparatus for observation of freeze-fractured planes of a sensitive QueH bec clay[END_REF]. The microstructures of the two clays appear very di!erent, in agreement with numerous studies on clay structures [START_REF] Collins | The form and function of microfabric features in a variety of natural soils[END_REF][START_REF] Tessier | Etude expeH rimentale de l'organisation des mateH riaux argileux[END_REF]. For the bentonite slurry, we can observe large voids limited by solid particles of large lateral extension and of strongly variable shapes. These aggregates represent roughly face-to-face associations of crystals made up of numerous layers with variable layer distances. The microstructure of the kaolinite slurry appears more compact, with voids of smaller sizes and aggregates of variable sizes, particularly in thickness. The crystals are roughly associated face to face, giving clay particles. Compared to the bentonite aggregates, they are more regularly shaped (platelets), #at and smaller (maximum length in the order of 10 m). The layer distance (7 A s ) corresponds to the kaolinite crystal. At the level of individual clay layers, a transmission electron microscope (TEM) can be used. The procedure followed in this study has already been presented by Tessier [START_REF] Tessier | Electron microscopy studies of clay microstructure[END_REF]. Water within the samples (section only 50 nm thick) was successively replaced by methanol, propylene oxide and "nally by Spurr's epoxy resin. The exchange occurred in a liquid medium so that any variation in the bulk volume of the sample could be prevented. After hardening and ultramicrotomy, the lattice images that correspond to each layer and interlayer distances could be observed under TEM. It was therefore possible to observe the organization of the crystals which constitute a clay particle. Each crystal is made up of a certain number of layers. The kaolinite particles were formed by thick and rigid aggregates of layers, while the bentonite particles appeared to be made of larger and more supple layers. These layers could be easily observed in the case of the bentonite (Figure 3).

The microstructure study was completed by the use of mercury intrusion porosimetry (MIP) to provide the pore size distribution inside the specimen at di!erent loading points. This technique has been reported and discussed by di!erent authors. Among them, Gillott [START_REF] Gillott | Importance of Specimen Preparation in Microscopy[END_REF], Delage and Lefebvre [START_REF] Delage | Study of the structure of a sensitive Champlain clay and its evolution during consolidation[END_REF], Gri$ths and Joshi [START_REF] Gri$ths | Change in pore size distribution due to consolidation of clays[END_REF] demonstrated that it is highly suitable for the study of clay microstructure, provided that the sample is dehydrated using a freeze-drying method. In these conditions the authors showed that the data from SEM and MIP were in good agreement; intra-aggregates spaces measured by SEM were of the same order of magnitude as those given by MIP. In conducting tests with the SEM or the TEM, we were obliged to extract small fragments from the samples whereupon it was always possible that the structures could be slightly disturbed. Furthermore, these fragments had to be treated either for the SEM Cryoscan technique or by exchanging the pore water with resin in the case of TEM. While interpreting these results on their own could prove to be a delicate matter, the use of complementary methods, such as MIP, could do much to support and to enhance these microscopic observations.

TESTING PROGRAMME

We performed two types of mechanical testing: a consolidated drained triaxial test and a onedimensional consolidation test.

One-dimensional and isotropic consolidation test results produced very di!erent compressibility coe$cients (C ! ) for the two clays (Figures 4 and5). For the kaolinite, C ! was constant and equal to 0.23 in the stress range 100}6000 kPa. For the bentonite C ! decreased with the stress increase from 1.5 for stress between 100 and 1000 kPa down to 0.23 for the maximum applied stress (6000 kPa), i.e. close to the value obtained for the kaolinite. In both the cases the isotropic and one-dimensional consolidation curves are parallel in the plane void ratio-mean e!ective stress. For a given mean e!ective stress the void ratio is higher in the case of isotropic consolidation. All these results will be discussed below in relation to the observations of the microstructure.

Drained triaxial tests were performed on normally consolidated and overconsolidated samples (OCR"10). Some results are presented in Figures 6}9. Normally consolidated clays were always contractant while overconsolidated samples were contractant at the beginning of loading and then dilatant up to failure. Under failure conditions, microstructural analyses of strain localization have been performed and presented previously [START_REF] Hicher | Microstructural analysis of strain localisation in clay[END_REF]. In a narrow zone along the failure surface, a strong reorientation and a breaking of the particles take place, creating maximum anisotropy. Large displacements and large rotations of particles are necessary to obtain this result.

The tests were stopped at di!erent values of axial strain. The samples were then cut into small parts and the structure was analysed by di!erent techniques. In each case, we examined two orientations of the cutting: in horizontal and in vertical planes. This procedure allowed us to follow the evolution of the clay structure during mechanical loading.

RESULTS OBTAINED ON KAOLINITE

The states which were analyzed by the SEM technique correspond to point O

) , O ) , O )
of the one-dimensional consolidation test, to point I ) of the isotropic consolidation test (Figure 4), to point A, B, C of the normally consolidated triaxial test (Figure 6), and to point D, E, F of the over consolidated triaxial test (Figure 7).

Isotropic and one-dimensional consolidation tests

The points of observations A, I ) (OCR"1) and D (OCR"10) correspond to di!erent isotropic consolidation stresses. In Figures 10 and11 we can see that the general orientation of clay particles is isotropic: there is no preferential organization in either the horizontal or vertical planes. The particles are plate shaped and form aggregates with a maximum diameter of about 10 m. The mean size of an individual particle is 1.1;0.8 m. Generally, the particles are associated edge-face although some face}face associations can also be observed. Point A (0.3 MPa) and D (0.1 MPa) correspond to almost identical void ratios (e"0.75). In the corresponding "gures, we can observe an identical density of particles, but the mean particles size is lower in the case of the overconsolidated sample, subjected to a higher maximum stress (1 MPa) which caused some aggregates to break. This phenomenon is found to be even more pronounced for the state represented by I ) with the consolidation pressure of 6 MPa. Here we can also see a strong densi"cation of the sample (e"0.5).

The examination of points O

) (Figures 4 and12) correspond to one-dimensional consolidations of 6.3 MPa. At this stress the organization is strongly anisotropic with a predominance of face}face associations. In this case the dimensions of the pores can become smaller rather than in the case of an isotropic organisation where the association is mainly side-face. As a consequence, for the same mean e!ective stress, the void ratio is smaller in the case of one-dimensional consolidation than in the case of isotropic consolidation (Figure 4).

Drained triaxial tests

The observation point B and C (Figure 6) correspond to drained triaxial compression of normally consolidated samples sheared up to axial stress of 9 and 19.5 per cent, respectively. We can observe in Figure 13 the state of the clay structure at point C. On a horizontal plane the particle orientation is isotropic, while on a vertical plane, we can observe at point C a reorientation of the particles forming an anisotropic structure. We can also observe a decrease of the numbers of large aggregates and more generally a decrease in the particle size. Figure 13 corresponds to a section selected inside the shear band which is visible on the photos. Several photos taken in di!erent zones along the shear band lead to the same conclusions. The orientation of the particles next to the failure surface is parallel to the direction of the shear band. In comparison with the others parts of the samples, the mean size of the particles in the zone of failure is smaller and the size of the particles is less variable. This means that strain localization leads to large displacements but also to large rotations of the particles in the shear band, producing two consequences: "rstly the creation of very large anisotropy of the structure in the central part of the band in a narrow zone along the failure surface and, secondly, the breaking of aggregates and even particles in this zone.

The observation point E and F (Figure 7) correspond to drained triaxial tests on overconsolidated samples (OCR"10) sheared up to axial strains of 6 and 17.4 per cent, respectively. The two points belong to the dilatant domain and have void ratios higher than the initial void ratio. The same conclusion as before can be drawn from the observations in Figure 14: on a vertical plane we can see a progressive reorientation of the particles forming an oriented structure. The initial state (point D) has already been characterized by a decrease in particle size, compared to the initial point A for normally consolidated samples. This phenomenon continues during triaxial loading and at point F, the big aggregates have disappeared almost completely. Therefore, rupturing of aggregates (as well as of smaller particles) does take place.

RESULTS OBTAINED ON BENTONITE

We have analysed samples of bentonite subjected to the same kinds of mechanical loading. The observations made during isotropic consolidation (point K, P, and I ) indicate a progressive increase in density due to a void ratio decrease. The particle size remains homogeneous and, contrary to the results obtained with kaolinite, we do not see particles concentrating to form big aggregates. Big voids still exist at point K (0.3 MPa) and P (1 MPa), but at point I (6 MPa), porosity is considerably lower and thus the voids are much smaller. The structure is isotropic and the arrangement of the particles is the same in both horizontal and vertical planes (Figure 15). One-dimensional consolidation tests (points O , O , O in Figure 5) are characterized by a strongly marked orientation of the particles in vertical planes. Due to the particularly long and #exible shape of bentonite particles, the possibility of reorientation is higher and hence the observations obtained at high consolidation stresses show a very high degree of the structural anisotropy in vertical planes. We observe also a decrease in particle size due to breakage, deformation and disintegration of the stacks of #akes as in the case of kaolinite (Figure 16).

Triaxial tests on normally consolidated samples (points L M, N) and overconsolidated samples (points Q and R) also reoriented particles in vertical planes, but these phenomena are less pronounced and are visible mainly at large strain (points M, N, R), accompanied by a decrease in particle size (Figure 17).

QUANTITATIVE ANALYSIS OF CLAY STRUCTURE EVOLUTION

Evolution of shape, size and concentration of clay particles

The photos obtained by the SEM technique permitted us to focus our study on quantitative measuring of the evolution of clay structure elements, mainly of aggregates and single particles. However, this was possible only for the kaolinite because the shape of the bentonite particle was too complicated for our kind of analysis. Our analysis was based on the following three parameters:

1. I , the shape index, is the value of the ratio between the length (A) and the width (B) of the individual particle element or individual aggregate element. It is sometimes also called aspect ratio. 2. S , the mean surface, is measured by considering the element as an ellipse: S " AB/2.

D

, the particle concentration, represents the number of individual particles by surface unity expressed in m.

Shape evolution

The histograms of the shape index (Figure 18) allow us to conclude that the increase of the axial strain during loading creates an increase of the mean value of the shape index. The minimum value of I is 1 for round (or square) particles and an increase of I signi"es that the shape becomes more angular. The value of I is in general larger on a vertical plane than on a horizontal plane. This result was obtained from both triaxial and one-dimensional consolidation tests. However, the evolution of I on a horizontal plane was found to be less important in the case of one-dimensional tests. This is due to the fact that the particles rotate during loading and end up in a #at position on a horizontal plane; this phenomenon becomes pronounced during onedimensional testing (Figures 20 and21).

Size evolution

The histograms representating S show at "rst a high degree of scattering of the values because of the initial grain distribution and the sedimentation process which creates aggregates of di!erent sizes (Figure 19). The evolution of the histograms corresponding to di!erent loading steps shows a decrease in size during loading, accompanied by a decrease in scattering. This indicates that there is a breakage of the aggregates and that the biggest ones split into smaller elements at "rst, leading to a more homogeneous size distribution. At large deformations, the big aggregates disappear almost completely. These observations can be made for both triaxial and onedimensional tests (Figures 20 and21) and have been also reported by various investigators [START_REF] Pusch | Microstructural changes in soft quick clay at failure[END_REF].

If we relate the evolution of the shape to that of the size, we can conclude that the breakage of the aggregates into smaller particles a!ects mainly the width of the aggregates. Delamination occurs along the plane direction of the individual particles constituting the aggregates. During the sedimentation process, the particles agglomerate to create an aggregate. Their surfaces of contact, however, remain the weakness zones within the aggregates which, upon being subjected to shear forces, will be the "rst to break.

Evolution of particle concentration

Particle concentration is not only connected to the density of particle assembly, which is equal to the density of the sample, but is also connected to the shape and the size of the particles. If these two parameters remain equal, we can then assume that the evolution of particle concentration is the same as that of sample density. However, as seen before, a signi"cant change in shape and size takes place during loading. In general, an increase of axial strain in all tests created an increase of the mean value of the particle concentration which could be interpreted as being chie#y a consequence of particle breakage (Figures 20 and21).

Evolution of particle orientation

In order to study particle orientation, we tried to construct rose diagrams from photos obtained with the SEM. In connection to each photo, the plane was divided into a given number of quadrants (each of them equal to 153). We then counted the number of particles from which the long axis (length A) was oriented within a given quadrant. By dividing this number by the total number of particles, we obtained a percentage of particles for a given orientation, which allowed us to drawn a radius of the same orientation starting from the centre with a length proportional to this percentage. With this construction, we could see that, for any given isotropic structure, all the radii would have the same length (equal in our case to "0.083) and they will de"ne a circle. Each rose diagram was constructed by superposing the results obtained from ten photos at the same mechanical state. For example, in Figures 22 and23, we could see that isotropic consolidation in kaolinite and bentonite led to rose diagrams of circles in both horizontal and vertical planes. The presence of some peaks at di!erent orientations corresponds to the rupture of big aggregates which occurred as a consequence of delamination along particle planes while the individualized particles stayed parallel to each other. We could see that the diagrams corresponding to large consolidation stresses were more regular due to more systematic ruptures of the aggregates of di!erent sizes.

The evolution of the rose diagrams along triaxial loading show an increasing di!erence between horizontal and vertical planes, with the existence of a privileged direction in the latter case that corresponds to the direction of loading. For example, in the case of kaolinite, an axial strain of 20 per cent at the end of the test created a reorientation of the particles in the direction perpendicular to the vertical axis which groups together 19}21 per cent of the particles (as opposed to 8.3 per cent for an isotropic structure). The same order of magnitude was obtained for the bentonite tested under the same conditions. The most radical orientation was obtained with one-dimensional loading. For a vertical stress of 800 kPa ( "13.75 per cent), the percentage of particles oriented perpendicular to the vertical axis was I "25 per cent and reached 30 per cent when the vertical stress was equal to 6.3 MPa. The results obtained with the bentonite was even more remarkable: for a vertical stress of 1 MPa, the same percentage was equal to 31 percent and went up to 33 percent for a vertical stress equal to 6.3 MPa (Figure 23).

We obtained therefore a very strong structural anisotropy due to large reorientation of particles in the direction perpendicular to the direction of loading. Some analysis from X-ray di!raction con"rmed the existence of induced anisotropy during triaxial loading. Figure 24 presents Debye}Scherres diagrams on a kaolinite samples in two di!erent states: initial slurry and normally consolidated sample subjected to a triaxial test up to 17 percent of axial strain. The "rst ring starting from the centre corresponds to the one of the kaolinite (d"7.15 A s ). In the case of the slurry, it has a homogeneous intensity indicating an isotropic structure, while for the other we can observe a variation in intensity which corresponds to a very de"nite anisotropy in the clay structure.

EVOLUTION OF THE PORES

Mercury intrusion porosimetry was applied to di!erent samples of bentonite and kaolinite submitted to various loading history. Contrary to the SEM which characterizes the organization of the solid phase, we are investigating here the pores and their evolution during mechanical loading. The results can be accounted for by two curves:

1. The cumulative pore volume as a function of pore diameter.

2. The derivative of the previous curve as a function of pore diameter.

Some results obtained on the bentonite are presented in Figure 25. Two major groups of pores emerged after an isotropic or one-dimensional consolidation at small stresses: one centres around 8 m and the other around 10 nm. The "rst group corresponds to the inter-aggregate pores while the second is related to the inter-particulate pores. The number of big pores corresponding to inter aggregate pores progressively decreases when consolidation stress increases, while the dimensions of small pores, corresponding to interparticulate pores, become progressively smaller down to 4 nm for high consolidation stress (6.3 MPa for the vertical stress during one-dimensional consolidation). At these high stresses, the big pores nearly disappear. This phenomenon corresponds to signi"cant change in void ratio at the scale of the sample and in the mechanical properties at a macroscopic level. The compressibility Index C ! decreases from 1.51 for small stresses to 0.23 for high stresses, due to the disappearing of the big pores which leads to an increase in the sti!ness of the clay structure.

The results of the mercury porosimetry show that during mechanical loading the change in the pore geometry is noticeable mainly because of the progressive disappearing of the big pore family (around 8 m). When, by using the SEM technique, we examined the solid phase, we could conclude that signi"cant breakage of the biggest aggregates occurred under the same mechanical conditions. These two phenomena are completely connected and the simultaneous use of these two techniques o!er us complementary ways of reaching the same conclusions regarding the behaviour of the clay at the scale of particles.

MICRO-ORGANIZATION AT THE LEVEL OF LAYERS

Observations at the scale of the crystals using TEM lead to the same conclusions as the ones obtained from the SEM. In particular we could follow the increase of the anisotropy of the structure during one-dimensional and triaxial tests in vertical planes, while the arrangement of the particles remained isotropic in a horizontal plane.

The number of layers which constitutes the tactoids can vary during the loading history of the samples. In general, the measured number of layers was somewhere between 50 and 80. Within the tactoids, internal discontinuities as large as 10 nm in size could be observed, especially for small consolidation stresses. These micropores play an important role during the shearing or the consolidation process. Two major phenomena were observed. At large strains, there is a decrease in the volume of the micropores and the layers can becomes perfectly parallel, creating a stronger rigidity of individual particles. The micropores, and especially the bigger ones, can form a zone of weakness for the particles: at larger stresses or strains, a delamination of the particles, initiated at the level of these discontinuities inside the particles, could be observed (Figure 26). This evolution at the scale of the tactoids is thus very similar to the one observed at the scales of the aggregates and it contributes to the evolution of the clay structure: development of anisotropy and changes in the dimension of the constitutive elements, aggregates and particles.

INFLUENCE OF STRUCTURAL ANISOTROPY ON MECHANICAL PROPERTIES OF CLAY

In order to quantify the anisotropy of the structure, we created a parameter I , called the orientation index, which was de"ned as follows: if Q is the zone of maximum orientation intensity Q(per cent) in a rose diagram, we would consider the neighbouring zones P and R and zone L directly perpendicular to Q and its neighbouring zones K and M; I will be the ratio between (K (per cent)#L (per cent)#M (per cent)) and (P (per cent)#Q (per cent)#R (per cent)). I can vary from 1 for an isotropic structure to 0 for a perfectly anisotropic structure. Figure 27 shows an example of I evolution during triaxial and one-dimensional loading in horizontal and vertical planes. We notice that on a vertical plane, a strong decrease of I occurs when the axial strain increases. This is more signi"cant in the case of one-dimensional loading because it expresses a progressive increase of the structure's anisotropy. We can also see a slight decrease of I on a horizontal plane. This corresponds to what was said earlier about the rupture of aggregates into parallel particles. In that particular case, however, the decrease of I is small and tapers o! quickly.

In order to analyse the e!ects of the structure anisotropy on mechanical properties, we performed two one-dimensional consolidation tests on kaolinite specimen, one up to "150 kPa and the other up to 2.25 MPa. In the "rst case, I "0.67 and, in the second, I "0.23. Two samples were prepared, one in a vertical direction and another in a horizontal direction and they were subjected to a triaxial loading starting from a mean e!ective stress equal to the one existing at the end of each one-dimensional consolidation (K "0.54) in order to avoid a signi"cant change in the initial clay structure. The results are presented in Figure 28. We can see that for I "0.67 the horizontal and vertical samples are similar. In the case of a strong anisotropy (I "0.23), however, the specimen subjected to a triaxial loading in the direction of the previous loading which created anisotropy (vertical sample), has larger sti!ness and larger strength than the one previously loaded in a perpendicular direction (horizontal sample). For example, the tangent modulus is 1.5 higher in the "rst case. The maximum strength is less a!ected, which indicates the in#uence of an induced anisotropy which has superimposed itself on the initial one. This phenomenon can also be observed on the evolution of the secant modulus which decreases more rapidly in the case of the vertical sample. Besides these di!erences due to the initial anisotropy, we also observe a di!erence in the stress}strain relations obtained at low and high consolidation stresses. At high stresses, the clay samples appeared to be more ductile and the maximum strength (expressed by the ratios / , or the friction angle) was lower. This is probably due to the e!ect of particle rupture during shearing. The same e!ect of grain crushing was obtained from the stress}strain relationship for other granular materials such as sands, gravels or glass balls. This phenomenon was widely studied in clay by Naskos [START_REF] Naskos | Quelques aspects du comportement meH canique de l'argile satureH e, consolideH e sous fortes pressions[END_REF].

Some three-dimensional triaxial tests were also performed on samples of normally consolidated kaolinite [START_REF] Hicher | Induced anisotropy in normally consolidated clay[END_REF]. Initial anisotropy was created on isotropically consolidated samples by an axisymmetric drained compression up to a major principal strain of 5 per cent. The corresponding structure of the clay samples is similar to the one associated with point B in Figure 6 and is represented by the rose diagram in Figure 21. Figure 29 presents the results of "ve threedimensional tests.

We have two tests at b"0 (b"( ! )/( ! )), one with in the same direction as in the previous loading, one with perpendicular to that direction; two tests at b"1 with parallel and perpendicular to the direction of previous loading; one test at b"0.5 with in the direction of previous loading, N represents the angle between the initial loading and the following stress path in the octahedral plane.

Strength development is much faster when is applied in the same direction as for the initial compression ( N "03, 303, 603). The deviatoric strain isovalue lines express this change in initial sti!ness with the orientation of major principal stress (Figure 30). In comparison with the ones obtained from isotropic samples, they present a translation along the axis of initial loading which suggests a kinematic hardening created by plastic deformations due to particle reorientation during the axisymmetric compression.

The in#uence of initial anisotropy is erased by subsequent large plastic deformations. This can be observed in the octahedral plane where the deviatoric strain isovalue curves becomes identical for isotropic and anisotropic samples. As a consequence, the maximum shear strength and therefore the failure criterion appear in this case to be independent of the initial anisotropy.

CONCLUSION

In attempting to relate the mechanical behaviour to the structural characteristics of clay samples, we have used a variety of experimental techniques in order to describe these relationships at di!erent scales ranging from the scale of layers, the nanometic level, to the scale of aggregates, the micrometic level. These techniques were based on quantitative and qualitative physical methods, such as scanning electron microscopy in an initially hydrated state. Particular care was taken to avoid as for as possible disturbing the micro-structure during observation.

Micro-organization of clay particles

Progressive ruptures of the clay particles took place during mechanical loading. The big aggregates (composed of several individual particles) were "rst a!ected and so they progressively split into smaller elements. The individual particles could also be a!ected; that is, rupturing occurred by delamination in a plane that was parallel to the layers. In the case of the bentonite micropores, inter-crystallite and/or discontinuities were found within a tactoid which created a zone of weakness that facilitated the delamination process. This continuous breaking during loading reduced the size of the clay elements which subsequently also reduced the size of the pores. Results from our mercury porosimetry con"rmed these changes in the pore size.

Along with this important size change, we observed a structural re-organization during onedimensional and triaxial testing. This phenomenon was predominant in the case of one-dimensional loading, especially in the bentonite samples, because these particles were larger and more #exible than in the kaolinite samples. This led, for high consolidation stresses, to a very strong anisotropy in the clay structure. The use of rose diagrams from the photos obtained by the SEM allowed us to quantify the anisotropy by determining the di!erent orientations of the particles in horizontal and vertical planes.

Relationship between micro-organization and mechanical properties

Mechanical properties of clays are much in#uenced by changes which occur in the microstructure. An increase in density leads to an increase of mechanical properties (sti!ness and maximum strength) which can be considered proportional to the maximum consolidation stress as long as the breakage of the particles remains minimal. For high consolidation stresses, this is no longer the case and an in#uence of grain breakage can be observed on the evolution of the stress}strain relationship which shows an increase in ductility accompanied by a decrease in the maximum strength ratio (or in the friction angle).

The anisotropy of the arrangement of the particles created an anisotropy of their mechanical behaviour. Conventional triaxial tests on one-dimensionally consolidated samples show an important change in sti!ness between &vertical' samples (where loading occurs in the direction of consolidation) and &horizontal' ones. The rigidity of the particle assembly is stronger when the load is applied in a direction perpendicular to the main direction of particle orientation, rather than in the same direction. The observed sti!ness of &horizontal' clay samples can easily be two times lower than that of the &vertical' samples. For the former it can also be observed that and are di!erent. Three-dimensional tests performed on anisotropic samples (submitted to a previous axisymetric loading) have generally shown that sti!ness decreases signi"cantly when the direction of the initial loading passes from the direction of the principal major stress to that of the intermediate stress and "nally to the minor stress.

It therefore appears that mechanical loading has an impact at various levels on the structural organization of a clay and that the changes produced should be taken into account in order to understand better the mechanical behaviour of clay samples along stress and strain paths.

APPENDIX

Figure 1 .

 1 Figure 1. Observation of kaolinite slurry by SEM.

Figure 2 .

 2 Figure 2. Observation of bentonite slurry by SEM.

Figure 3 .

 3 Figure 3. Observation of bentonite at particle scale by TEM.

Figure 4 .

 4 Figure 4. Isotropic and one-dimensional consolidation of kaolinite specimen.

Figure 5 .

 5 Figure 5. Isotropic and one-dimensional consolidation of bentonite specimen.

Figure 6 .

 6 Figure 6. Drained triaxial test on normally consolidated kaolinite.

Figure 7 .

 7 Figure 7. Drained triaxial test on over consolidated kaolinite.

Figure 8 .

 8 Figure 8. Drained triaxial test on normally consolidated bentonite.

Figure 9 .

 9 Figure 9. Drained triaxial test on over consolidated bentonite.

Figure 10 .

 10 Figure 10. Observation of a kaolinite specimen isotropically consolidated at 300 kPa (point A).

Figure 11 .

 11 Figure 11. Observation of a kaolinite specimen isotropically consolidated at 1 MPa: (a) horizontal plane, (b) vertical plane.

Figure 12 .

 12 Figure 12. Observation of a kaolinite specimen: one-dimensional consolidation of 6.3 MPa (point O ) ): (a) horizontal plane, (b) vertical plane.

Figure 13 .

 13 Figure 13. Observation of a kaolinite specimen submitted to a drained triaxial test up to "19.5 per cent (point C): (a) horizontal plane, (b) vertical plane.

Figure 14 .

 14 Figure 14. Observation of an over consolidated kaolinite specimen submitted to a drained triaxial test up to "17.4 per cent (point F): (a) horizontal plane, (b) vertical plane.

Figure 15 .

 15 Figure 15. Observation of a bentonite specimen isotropically consolidated at 6 MPa (point I ).

Figure 16 .

 16 Figure 16. Observation of a bentonite specimen: one-dimensional consolidation at 1 MPa (point O ): (a) horizontal plane, (b) vertical plane.

Figure 17 .

 17 Figure 17. Observation of a bentonite specimen submitted to a drained triaxial test up to "17 per cent (point M).

Figure 18 .

 18 Figure 18. Shape index histograms of kaoline specimen for drained triaxial tests.

Figure 19 .

 19 Figure 19. Mean surface histograms of kaolinite specimen for drained triaxial tests.

Figure 20 .

 20 Figure 20. Evolution of shape index, mean surface and particle concentration as a function of the major principal strain during drained triaxial tests.

Figure 21 .

 21 Figure 21. Evolution of shape index, mean surface and particle concentration as a function of the major principal strain during one-dimensional consolidation tests.

Figure 22 .

 22 Figure 22. Rose diagrams of kaolinite particle orientation after drained triaxial tests.

Figure 23 .

 23 Figure 23. Rose diagrams of bentonite particle orientation after one-dimensional consolidation tests.

Figure 24 .

 24 Figure 24. X-ray di!raction tests on kaoline specimen: (a) initial slurry, (b) triaxial test up to "17 per cent.

Figure 25 .

 25 Figure 25. Mercury porosimetry results for bentonite specimen after one-dimensional loading: cumulative pore volume as a function of pore diameter.

Figure 26 .

 26 Figure 26. TEM observation of a bentonite particle.

Figure 27 .

 27 Figure 27. Evolution of bentonite particle orientations as a function of major principal strain.

Figure 28 .

 28 Figure 28. In#uence of the structure anisotropy on the stress-strain relationship of kaolinite specimen during drained triaxial tests.

Figure 29 .

 29 Figure 29. 3D tests on anisotropic kaolinite specimen.

Figure 30 .

 30 Figure 30. In#uence of initial anisotropy on the 3D behaviour of kaolinite. Isodeformation lines in the octahedral plane for proportional 3D loading.

Table I .

 I Physical properties of studied clays

	Properties			Kaolinite	Bentonite
	Mineralogy			Kaolinite	Smectite Mg
				few illite, quartz	few feldspar, illite,
					quartz
	Speci"c surface of the clay powder (m/g)	10.80	288.38
	%(80 m			9 6	7 9
	%(2 m			4 7	6 7
	Liquid limit = * Plastic limit = . Activity	(%) (%)		40 20 0.42	100 46 0.80
	Volumetric weight of the solid 1 Swelling index C 1	(t/m)	2.65 0.049	2.70 0.128


N Saiyouri

P Y Hicher

D Tessier

Microstructural approach and transfer water modelling in highly compacted unsaturated swelling clays

Keywords: X-ray, swelling compacted clay, potential, di!usion, microstructure, water transfer

A compacted smectite clay was studied in order to understand the swelling phenomenon. The in#uence of the sample initial state on the swelling mechanisms was demonstrated. A complete physical}chemical and mineralogical characterization of the clay was achieved using X-ray by re#exion and thermal methods. The X-ray di!raction by transmission method allowed to describe the various levels of the hydrated clay organization and the microstructure evolution. The pore distribution and the water partition are shown. Other important parameters: water retention, inter-layer distances, particle size, number of layers per particle were examined. This information gives the ability to follow the swelling which occurred in two steps, and to calculate the swelling potential versus the water content, or the void ratio.

In application to this study, a water transfer in unsaturated clay was studied with the laws of continuous medium mechanics. A "nite di!erence program was used to examine two cases: prevented and free swelling in an oedometer. In order to take into account the material deformation, the swelling potential was introduced in the di!usion coe$cients. The simulation results of the water transfer obtained on this basis, were in good agreement with the experiments.

INTRODUCTION

The aim of this study was to understand and to model the swelling mechanisms in highly compacted clays, and to integrate these mechanisms into a water transfer calculation in a clay matrix. The practical application concerns, in particular, the construction of barriers in deep nuclear waste repositories.

The work focused on the modelling of clay behaviour based on a microstructural approach of the swelling mechanism. The studied clay is a smectite, highly compacted at small water content, in order to develop very high swelling pressure when hydrated at constant volume in a nuclear waste repository site.

The swelling phenomenon was experimentally studied by an important number of authors. } Several modelling works exist also on this subject. } The conclusion of all these works is that the macroscopic swelling for sodium or calcium smectites occurs in two microscopic steps: the "rst one is the water insertion between the elementary clay layers. Afterwards, spaces of larger sizes are "lled. The relative importance of the two mechanisms is di!erent when the clay is calcium or sodium. However, the studied clays were usually prepared from gels or powders, i.e. at a very high void ratio. It is therefore important to see if the initial state of the clay specimen will a!ect the swelling mechanisms.

The second aspect of our work is to introduce these mechanisms in a modelling of water transfer in compacted clay specimens.

Generally, the transfer phenomena are described by partial derivative equations (linear or non-linear), with physical parameters in the continuum of the porous medium. The models are established from the associated physical processes which are mass conservation and power balance-sheet, or Darcy's law for water #ow and Fick's law for gas #ow. They allow the prediction of space and temporal evolution of hydraulic or chemical or also thermal state in a porous medium. To solve them, one needs the knowledge of initial and boundary conditions, or some macroscopic factors like hydraulic or thermal conductivity or "nally di!usion or dispersion coe$cients.

To model the transfers in rigid or deformable unsaturated soils, numerical methods using "nite element or "nite di!erence program are now mainly used. Philip used an iterative method for horizontal and vertical in"ltration. He introduced the Boltzmann transformation and then used a "nite di!erence program to solve the resulting equations. Other works on #uid transfers in porous media with hydromechanical coupling are also presented in the literature. } In cases of deformable media, the transfer equations have to be coupled with the mechanical model describing the stress}strain relationship for a given medium. Di!erent approaches have been developed to take into account the mechanical behaviour of unsaturated soils. Several authors showed that the e!ective stress principle (Terzaghi) is not correct for unsaturated soils. Some others, Matyas et al. and Fredlund et al., proposed a description of soil behaviour which uses the constitutive surface area or state surface notion. Gens, proposed an elastoplastic modelling for unsaturated soils based on an approach which considers an independent role of the stress and the negative pore water pressure.

The main aspect of our work was to study the water transfer introducing the swelling in#uence on the #ow. This question dealt with attention to both experiment validation of the clay hydration, and the microscopic study of the clay especially by the X-ray di!raction method. The numerical code used for the #ow simulation was deliberately simpli"ed in order to enable the simple introduction of the underlying mechanisms shown by the microstructural observations. A monodirectional #ow was assumed which allowed to simulate a test in a one-dimensional condition, and to validate our approach on two typical cases: free and prevented swelling.

MATERIAL AND METHODS

The studied clay is called FoCa7 (Fourges-Cahaignes clay) from the Paris Basin. A complete physical}chemical characterization of this material was achieved, which is a smectite}kaolinite clay. The speci"c surface area was measured using the monoethyl ether ethylene glycol method (EGME) which was developed by Heilmen et al. The value found was 453)66 m/g. Usually in the literature, for a very high content of smectite clay (up to 90%), the average value of the speci"c surface area is about 800 m/g. This di!erence is explained by the natural state of the material which was used in our study, while higher values are obtained on puri"ed samples. The di!erent impurities were determined by thermal analysis and X-ray methods as described later. Nevertheless, it is su$cient to remember that the impurities reduce considerably the speci"c surface area of a clay. The analysis of the nature and quantity of the exchangeable cations (counterions) was performed, and the cationic exchange capacity (CEC) was measured in order to characterize the physical properties of the clay. The results obtained by the Soil Analysis Laboratory at Arras-France are shown in Table I.

These values con"rm that the FoCa7 clay is mainly a calcium smectite with a CEC average value comparable to those given in the literature.

The total proportion of cations and the number of exchangeable cations in the FoCa7 clay have also been determined. The results showed that the total calcium quantity present in this material is important. This is compatible with the previous analysis on the ratio of exchangeable calcium. The results also showed that FoCa7 contains a substantial fraction of iron (5)44 g/100 g of clay). This is an additional factor which decreases the swelling. Another conclusion is that the total potassium presence indicates the existence of illite in the clay.

In order to determine the mineralogical components of this material, we performed a powder analysis by X-ray apparatus in re#exion which indicated that the mineralogical components are essentially kaolinite and smectite. Because of the natural state of the clay, we noted also the presence of several impurities (quartz, gibbsite and goethite). The smectite fraction present in the sample is very important, around 80%, while only traces of the other components, except the kaolinite. A thermal method allowed to quantify the kaolinite present in the sample (about&7%).

Swelling experiments were performed on initially highly compacted clays. These samples were prepared from a powder which was one dimensionally compacted up to 60 MPa in the oedometer cell used for the hydration experiments. Their initial water content was&10%, which is the value corresponding to equilibrium with the atmospheric relative humidity of the experimental room (60% of relative humidity). Knowing the water content and the clay mass, the dry density was obtained by prescribing the total volume of the sample.

A set of experiments was carried out on FoCa7 samples in order to understand the swelling phenomena at the microscopic scale by the X-ray di!raction method. Especially, it is important to know and to explain the location of water between the layers and the water "lling spaces in larger pores. Thus, the samples were hydrated from the initial state, to high water content values, by decreasing the imposed suction within every sample.

The methods used were the pF cells, the Richard's cell apparatus ((1 MPa) and the relative humidity in dessicators. These methods are very well described in Reference 9. The principle for each method is brie#y described herein. The "rst and the second methods are based on pneumatic pressure which is "xed in the cell. The samples are in contact across a porous stone to de-ionized water which hydrates them according to the cell pressure value. They remain in the cell until the equilibrium between this pressure value and their suction is reached. On average it was obtained after 4 or 5 days. The pF tubes correspond to 0)1 at 100 KPa, while the Richard's cell goes up to 2 MPa. In the dessicators, the pressure is imposed to the sample by saturated salt solutions. These salt solutions are chosen to impose a particular relative humidity. The suction values reached by this method are very important (about 100 MPa). The greater imposed pressure value was 107 MPa. The equilibrium was obtained in more than 1 month.

Establishment of moisture proxles~water content and apparent volume

After equilibrium at each suction level by the three methods described previously, the water content and volume were measured. The water content was determined after heating the sample in an oven at 1053C for 48 h. For apparent volume measurements, we used a method developed by Tessier and Berrier: the volume was determined from the Archimed's pressure exerted on the samples submerged in kerosene (AFNOR standard).

Principle of the X-ray method

The hydrated samples were also analysed by the X-ray di!raction method in order to evaluate the swelling potential. This X-ray di!raction by transmission is performed on hydrated samples which were prepared at di!erent values of suction, and had their water content and void ratio measured.

The diagrams were obtained by transmission according to Pons. This method allowed information on the distance evolution between the layers d G , on the particle size distribution or on the number of layers per particle M. It also gives the probabilities p G , to have those values of di!erent quoted parameters.

The interpretation of the diagrams is based on the comparison between theoretical and experimental curves. Figure 1 presents an example at the suction value of 1 MPa. The parameters are obtained by a "tting of the theoretical curve compared to the experimental one.

The theoretical curve is obtained from the di!used intensity expression by the system which is crossed by the X-ray beam.

According to Pons et al., the di!used intensity by a smectite can be written as

I(s)" 1 2 "F (s)" s (M) G(s, M, p G , d G )
where s is the di!usion vector, F(s) is the structure factor in the direction perpendicular to the layering, (M) is the weight distribution in particle thickness, G(M) is the modulation function determined by Mering, and G(M) can be written as

G(s, M, p G , d G )"1#2 Re +\ L M!n M Q+ (s, p G , d G )
where

Q (s, p G , d G
) is a function which takes into account the interference between two adjacent layers.

In

Q(s)" P G p G exp (2 isd G ) with P G p G "1
In the particular case of montmorillonite, the structure factor is known. Thus, the layer number per particle M, and the interlayer distances d G with corresponding probabilities p G can be simultaneously found.

RESULTS AND DISCUSSION

Water retention measurement. Water content and volume change curves

The void ratio and the volumetric and mass water content values become very large when the imposed suction values decrease. The water content and the void ratio versus the suction values are "rst presented in Figures 2 and3, respectively. The water content value reached at the lower imposed suction value (1 kPa) in the FoCa7 clay is about 68%. This is interpreted as due to the nature of smectite sheets present in the samples which can absorb large quantities of water and promote similar hydration and swelling. However, due to the presence of a more important number of divalent cations (Ca) in FoCa7 than of monovalent ones (Na), this amount of water content is small if compared to the water content reached for sodium montmorillonite, for example, which can be about 200% for low suction values.

In order to more clearly present the results concerning the water retention by the clay, the relation between the degree of saturation and the suction pressure can be used. The grain dry weight used is 2)67 g/cm for FoCa7. Figure 4 indicates that the clay is not saturated at the beginning of the cycle, but it becomes saturated at the end.

X-ray results and microstructures

The X-ray analysis gives the distances between the clay layers. These distances correspond to the number of water layers adsorbed between the clay layers. They can be equal to 10, 12)6, 15)6, 18)6 or 21)6 A s which corresponds to 0, 1, 2, 3, or 4 water layers. Figure 5 gives an example of adsorption of water between the clay layers.

In the following, we present the average layer distances and the clay particle sizes. The number of layers per clay particle decreases as the water content increases in the clay. For the lower value of the water content, a particle is composed of 100 layers (Figure 6). But when the water content increases, the number of layers per particle decreases until reaching only 10 layers Figure 7. Average distance between layers of clay versus suction Figure 6. Number of layers per particle and water content of the clay versus suction by particle. The particle size decreases in a ratio of 10 between the initial state and the very hydrated one.

Figure 7 presents the inter-layer distances for FoCa7, calculated using the Pons model. From the &dry' state (10% water content), we can see that a displacement from 15)6 to 18)6 A s , was observed from high-to-low values of suction.

Swelling mechanism

Clay particle size change. It is usually reported in the literature that the clay swelling mechanism is mainly due to a change in layer distances. Our work shows that the size of particles is also a very important parameter to understand the clay behaviour, and especially the swelling. As shown in Figure 8, the convention we have chosen is that an aggregate is a set of particles or crystals, with the particles themselves composed of layers.

It appears from our experiments that the swelling mechanism of the clay begins by an adsorption of water between individual clay layers which is accompanied by a decrease of the particle size, or in other words by a subdivision of the clay particles.

Layer distance evolution. In Figure 7, FoCa7 clay presents only one water layer at about 50 MPa. We observe that the clay layer distance increases with the water content from 1 to 3 water layers from low to higher water content. When another water layer is adsorbed, the layer distance goes from 12)6 to 15)6 A s . This transition arises at a suction of about 7 MPa. The same behaviour had been observed by Delvaux for another smectitic clay: a sodium montmorillonite. We know, however, after Delvaux et al., and also after experiments performed and not shown in this paper, that another transition appears at about 60 kPa for a sodium clay and leads to a layer spacing of 21)6 A s (four water layers) and never more. We then conclude that the crystal swelling is limited to four water layers, while according to the study at low suctions the layer distance can reach 100 A s . Our interpretation is that the initial strongly compacted state of the clay limited the crystal swelling to four water layers.

Limited crystal swelling. The hydration is accompanied by the insertion of water layers between the clay layers, which causes a division of the initial particles (see Figure 9). The resulting smaller particles are separated by distances which increase considerably up to some water layers. The Figure 9. Swelling mechanism interpretation conclusion that it exists in spaces other than interlayer ones, comes from the calculation of water ratio in the interlayers compared to the total water in the clay. This question will be discussed in the following paragraph.

External water. The Pons model allows us to estimate the interlayer sizes in the hydrated clay by the probabilities p G to have one to four water layers in the interlayer spaces. The product of the speci"c surface area which is the surface that water can reach, by the size of water molecule gives the elementary volume of water. This later can be multiplied by the water layer number and the probability to have it in order to obtain the total water in each porosity scale. The results are the water volumes <1, <2, <3 and <4 which correspond to the water quantity in the spaces containing, respectively, one, two, three or four water layers. The di!erence between the measured total water content and the water in inter-layer spaces, i.e. the <1, <2, <3, and <4, constitutes the external water "lling interparticles pores. These water volumes are presented in Figure 10. The external water is also present in the same graph which shows that the external water appears very important compared to the di!erent <1, 2 , <4 when the suction values are small (several KPa). However, when the water content decreases, the external water decreases too, and the proportion of water in the volumes <1, 2 , <4 increases. It means that when the clay approaches a dry state, water is only present between the clay layers and there is nearly no external water. But when wet states are reached, the external water becomes important and its proportion compared to the interlayer water is higher.

It can be also deduced that when the suction is important, the volumes <1 and <2 are prevailing and when the suction decreases, the proportion of the volumes <3 and <4 increases. This means that in a very hydrated clay the probability to have clay layers separated by one or two water layers is very weak compared to the one to have three or four water layer. Moreover, the external water is simultaneously more important than the interlayer water. Swelling Modelling. After having established the existence of di!erent kinds of water corresponding to di!erent pore sizes, we can say that the swelling phenomenon in our clay occurs in two steps: "rst the insertion of adsorbed water between the clay layers, which is accompanied by the division of initial particles (in other words, the surface area is changing). Afterwards the particles are separated by distances which are greater than 21)6 A s . We can try to calculate those distances. This can be based on the water quantities already assessed in the previous paragraph (i.e. the external water), the number of layers and the speci"c surface area. The curve in Figure 11 presents the calculated distances versus the swelling pressure which represents the imposed pressure. We can therefore deduce the relationship between the external water (or interparticle distance) and the corresponding swelling pressure.

It can be noticed that a linear relationship between the distances and the log of the swelling pressure is obtained. We can therefore propose a model taking into account the overall amount of water as a function of the swelling pressure. In this model the interlayer distances can be plotted horizontally (Figure 12). If we try to achieve the same representation for FoCa7 clay, we take the results of Figure 11, and we add the values of interlayer distances versus the swelling pressure. This leads to Figure 13. We can say that the water between the clay layers gives a swelling less important than the interparticle water because of the repulsion with high distances between particles. Those distances are compatible with the existence of a di!use double layer around the particle surface.

APPLICATION TO THE MODELLING OF WATER TRANSFER

Having proposed a model of swelling at a microscale, we will try now to introduce it at a macroscale. A part of the di$culty to understand the unsaturated soils behaviour arises because of the phenomenon associated with volume change. Some speci"c assumptions concerning the description of the soil potential must be considered "rst.

(w"water) is the total potential in the system and it can be composed of the sum of several terms: S is the speci"c entropy, is the applied pressure potential, corresponding to the external applied pressure, is the matricial potential, corresponding to the suction pressure in the soil, is the pneumatic potential, corresponding to the air pressure and is the gravitational potential corresponding to the gravitation e!ect.

d "!(S /g) d¹#d #d #d #d
In condition of isothermy (dt"0) and air at atmospheric pressure, we can write " # #

In the case of swelling, the applied potential is the swelling potential, i.e. we can write " # # is the swelling potential. Swelling clays develop high suction and swelling potential values, the gravitational potential can, thus, be neglected compared to the two other terms.

Two cases are presented: prevented and free swelling.

In the prevented swelling case (no deformed matrix), no volume change occurs. The single contribution is that of the suction potential, i.e. "

When the swelling is free and in order to take into account the swelling deformation, we introduced the potential of swelling in the total water potential of the sample, instead of the suction potential alone in the case of no deformed matrix. The swelling potential was the one measured experimentally, more precisely, deduced from the swelling pressure presented in the previous paragraphs. So, the swelling potential was added to the suction potential to take into account the total potential in the sample. This represents the originality of our approach concerning the water transfer study in swelling soils.

" # If D ( ) is the Darcy's di!usivity (or di!usion) coe$cient, it can be related to the potential derivative as follows (Figure 14):

D ( )"k * *
where K is the permeability to water coe$cient (called K in the following). is the concentration or water content.

In the non-deformed matrix case, the di!usion term can be written as

D "D D "!K * * D
is the matricial di!usion (obtained from the suction). In the free swelling case the di!usion coe$cient can be written as

D "D #D
where D "!K* /* D is the swelling di!usion (obtained from swelling potential).

In order to test the validity of our approach, we considered a simple case of one-dimensional swelling that we had experimental results at our disposal, on the same clay FoCa7. In the tests performed by Imbert the clay sample was 51 mm high, its initial water content was 10% and the water content at saturation was 52%. The test consisted to humidify the sample from the bottom and to follow the saturation front evolution in an oedometric cell, by the gamma spectroscopy method. The pro"les of water content and dry density of the clay were obtained every 3 mm from the bottom of the sample, during all the tests (Figure 15). Two cases were studied: free and prevented volume change. The equations controlling the water #ow have been studied by several authors. } A "nite di!erence method was used in order to solve these equations in a one-dimensional case. The aim of this section is to demonstrate that using the equations discussed before, we are able to model the water #ow with some speci"c boundary conditions according to each case. The results of calculation will be presented and compared to experimental results in the cases of no-volume change and free swelling for a highly compacted specimen of FoCa7 clay.

The hypotheses are the following:

(a) #ows are one-dimensional and deformations are small. (b) E!ect of gravity is neglected. (c) Darcy's law can be applied.

The #ow in porous media is governed by two equations: mass conservation for all phases (i), with concentration G and Darcy's law for #uid #ow behaviour. The combination of these two equations gives the transfer equation. In the one-dimensional case, for the aqueous phase only, the obtained standard di!usion equation with non-linear coe$cients which is called Richard's equation can be written:

* *t " * *x D( ) * *x
where is the total potential in the sample. The following boundary and initial conditions are considered: at t"0, " ; for all x when t'0, " at x"0 being the value of at saturation. To solve Richard's equation, a "nite di!erence program was used. An implicit scheme was tested on FoCa7 clay.

In the following, we use the notation:

(i x, k t)" I G
where x and t are, respectively, space and time steps.

As no humidity exchange is permitted with the outer medium of the sample, we can take another boundary condition which is no #ow at the head of the sample.

In the calculations, the di!erence between the case without swelling and the free swelling one is only in the form of the potential and then of the di!usion as discussed before.

To achieve the calculations, we had to introduce the values of several parameters. The initial and saturation water content were taken directly from the experiments. The permeability of the clay was experimentally measured by Imbert using another specimen with the same characteristics. The suction potential was deduced from the water retention curve, which was also measured by Imbert independently by using saturated salt solutions to impose the suction values in the sample. The di!usion coe$cient was calculated from those values using the relation:

D "!K * *
Having determined these coe$cients, the calculation of water transfer in the non-deformed case could be performed. The calculation of free swelling case needed also the swelling potential term in order to evaluate the di!usion factors. It was derived from the experimental curves presented above giving the relation between water content and swelling pressure (Figure 11). The function obtained between the swelling di!usion and the water content has roughly an exponential form. Thus, the swelling potential was added to the suction potential to take into account the total potential in the sample. The total di!usion which was deduced from the total potential is presented in Figure 14. These calculations gave the moisture pro"les versus time at every height in the samples, which could be compared to experimental results obtained by gamma spectroscopy. The results of calculation for the non-deformed matrix are presented in Figures 16 and17, respectively, for the distances 3 and 13 mm from the bottom of the sample, as well as the experimental values.

The graphs indicate that the calculation results are acceptable compared to the experimental measures. The complete saturation of the specimen occurs after 3 months for numerical results, in agreement with experimental results.

The results presented in Figures 18 and19 correspond to the comparison between our calculation with the conditions of free swelling, and the experimental data given in Reference 28 about water content pro"les at two distances 3 and 51 mm from the bottom, 51 mm is the top of the clay sample for the same FoCa7 clay. These "gures show a satisfactory agreement between calculations and experiments. In particular, the necessary time to achieve a complete saturation of the specimens is well predicted by the calculation.

We used the same model for undeformable matrix and for free swelling. The di!erence was only in the di!usion factors evaluation and so in the water potential considered in each case. The comparison between calculated and experiment results is satisfactory for the two cases. For the intermediate cases the actual formulation needs to be completed by a given stress}strain relationship of the clay particles, allowing us to compute the relation between the swelling pressure and the volume change of the clay specimen at a microscopic level. This point can be a logical following of this work.

CONCLUSIONS

The experimental study of FoCa7 clay allowed us a better understanding of microscopic aspects of swelling. The X-ray method gave us the ability to follow the microstructure evolution of the clay samples during hydration. The conclusion about swelling is that it occurs in two stages with We demonstrated that the initial microstructure, i.e. before hydration, plays a key role in the mechanisms of the water adsorption. Our specimens were highly compacted (one dimensional compression up to 60 MPa). After swelling, when we measured the distances between clay layers by X-ray method we did not observe any distances exceeding 21 A s , which represents four water layers, while distances about 100 A s were measured in the case of hydration of clay samples which were gels or powders (suspension after hydration). So we can say that the initial compaction of the samples provoked a very close structure which resisted the water insertion and lead to the division into smaller particles.

Using the results of the experimental analysis, we proposed a modelling of water transfer in unsaturated compacted clays. The results of calculation in the case of one directional hydration compared well to experimental results in conditions of non-deformed matrix and free swelling.

Figure 1 .

 1 Figure 1. Modelling of the experimental curve obtained by X-ray di!raction on FoCa7 sample at a suction of 1 MPa

Figure 2 .Figure 3 .Figure 5 .

 235 Figure 2. Water content at equilibrium

Figure 4 .

 4 Figure 4. Degree of saturation versus suction pressure

Figure 8 .

 8 Figure 8. Notion of clay particle, cristal and aggregate structure

Figure 10 .

 10 Figure 10. Water between layers and external water

Figure 11 .Figure 12 .

 1112 Figure 11. No inter-layer distances versus swelling pressure

Figure 13 .

 13 Figure 13. Inter-particular and inter-layer separation versus the swelling pressure for FoCa7

Figure 14 .

 14 Figure 14. Di!usivity function due to the total potential

Figure 15 .

 15 Figure 15. Experimental principle of oedometric hydration of the clay samples

Figure 16 .

 16 Figure 16. Prevented swelling moisture pro"les versus time at 3 mm from the bottom; experiment and calculation

Figure 17 .

 17 Figure 17. Prevented swelling moisture pro"les versus time at 13 mm from the bottom; experiment and calculation

Figure 18 .Figure 19 .

 1819 Figure 18. Free swelling moisture pro"les versus time at 3 mm from the bottom; experiment and calculation

Table I

 I

	. Exchangeable cations and cationic
	exchange capacity for FoCa7
	CEC	Na	Ca
	(meq/100 g	(meq/100 g	(meq/100 g
	of clay)	of clay)	of clay)
	69)1	2 )64	62)90

  our case, i.e. where the transition between two layers is done by r translations d

	with respective probabilities p	, p	, 2 , p P	, the Q (s, p G	, d G	, d ) function takes the following form: , 2 , d P

ACKNOWLEDGEMENTS

This work was supported by ANDRA (National Agency of nuclear waste management) under contracts.


Raphaël Butez
email: butez@ceremade.dauphine.fr

The largest root of random Kac polynomials is heavy tailed

We prove that the largest and smallest root in modulus of random Kac polynomials have a non-universal behavior. They do not converge towards the edge of the support of the limiting distribution of the zeros. This non-universality is surprising as the large deviation principle for the empirical measure is universal. This is in sharp contrast with random matrix theory where the large deviation principle is non-universal but the fluctuations of the largest eigenvalue are universal. We show that the modulus of the largest zero is heavy tailed, with a number of finite moments bounded from above by the behavior at the origin of the distribution of the coefficients. We also prove that the random process of the roots of modulus smaller than one converges towards a limit point process. Finally, in the case of complex Gaussian coefficients, we use the work of Peres and Virág [PV05] to obtain explicit formulas for the limiting objects.

Introduction

Consider a random polynomial of the form:

P n (z) = n k=0 a k z k = a n n k=1 (z -z (n) k )
where a 0 , . . . , a n are i.i.d. random variables and z (n) 1 , . . . , z (n) n are the complex zeros of P n . These polynomials are often called Kac polynomials. The zeros of these polynomials are known to concentrate on the unit circle of C as their degree tends to infinity under some moment condition on the coefficients. This universal behavior has been studied by many authors since the work of Sparo and Shur [ŠŠ62] and we refer to the book [START_REF] Bharucha | Random Polynomials: Probability and Mathematical Statistics: a Series of Monographs and Textbooks[END_REF] for more precise information on the history of the topic. The most precise result about this convergence was given by Ibragimov and Zaporozhets in [START_REF] Ibragimov | On distribution of zeros of random polynomials in complex plane[END_REF] where they prove that for any bounded and continuous function f and any ε > 0

P 1 n n k=1 f (z (n) k ) - 1 2π 2π 0 f (e iθ )dθ > ε ---→ n→∞ 0
if and only if E(log(1 + |a 0 |)) < ∞ and P(a 0 = 0) < 1.

This result means that a proportion going to one of the zeros cluster uniformly on the unit circle. It does not prevent a negligible part of the zeros to be real or to be away from the unit circle. In this situation, it is natural to ask if max k |z k | converge towards 1 as n goes to infinity. In this note we prove that the behavior of the extremal zeros of P n is not universal and that the random variable max |z (n) k | is usually a heavy tailed random variable. We give an upper bound on the number of finite moments depending on the cumulative distribution function of the coefficients at 0.

To study the zeros of P n for large n, we may want to see P n as the partial sum of a random entire series. If we assume that the random variable |a 0 | is non-deterministic and satisfies E(log(1 + |a 0 |)) < ∞, then the entire series P ∞ (z) = ∞ k=0 a k z k has almost surely a radius of convergence equal to 1 and P ∞ is a random non-constant holomorphic function on the unit disk. Hence P ∞ has a countable set of zeros {z (∞) k } which are counted with multiplicity. As P ∞ is almost surely a non-constant analytic function, its zeros are isolated and have no accumulation point inside the open unit disk. This ensures that there is a finite number of zeros inside any compact set in the disk.

Theorem 1.1 (Main result). Assume that the random variable a 0 satisfies E(log(1

+ |a 0 |)) < ∞, that |a 0 | is not deterministic and that P(a 0 = 0) = 0. Let n ∈ N ∪ {∞} and x (n) 1 = min k |z (n) k | and if n < ∞, x (n) n = max k |z (n) k |.

The random variable x (n)

n has the same distribution as 1/x

(n) 1 .

2. There exists three constants C 1 > 0 , r > 0 and A > 0 depending only on the distribution of |a 0 | such that

∀ 0 < t < C 1 , P x (n) 1 ≤ t ≥ P |a 0 | ≤ rt 2 A.
(1)

3. If there exists k ≥ 0, a > 0 and δ > 0 such that

∀ t < δ, P(|a 0 | ≤ t) ≥ at k then E((x (n) n ) k ) = ∞.

Almost surely, the point process

χ n = z (n) k | |z (n) k | < 1 converges weakly in the space of Radon measures towards χ ∞ = z (∞) k
. More precisely, for any continuous and compactly supported function f defined on the open unit disk D(0, 1) we have

k f (z (n) k ) a.s ---→ n→∞ k f (z (∞) k ).

The random variable x

(n) 1 converges almost surely towards x (∞) 1 and x (n) n converges in distribution towards x (∞) := 1/x (∞) 1 .
The condition P(a 0 = 0) = 0 ensures that the degree of P n is n. Hence the zero set {z (n) k } is well defined. If we remove this condition, the theorem is still true if we take the convention that for a polynomial of degree k < n, z

(n) k+1 = • • • = z (n) n = ∞.
In the case of real Gaussian coefficients, Majumdar and Schehr proved a similar result to (3) for the largest real root in [START_REF] Schehr | Real roots of random polynomials and zero crossing properties of diffusion equation[END_REF]. They use a Taylor expansion of the first intensity function of the real zeros at 0 to prove that the density of x (n) n decays at infinity like 1/t 2 . Notice that the first three points of Theorem 1.1 are valid for any fixed n. The heavy tail behavior of the largest root is not asymptotic. The points (4) and (5) imply that the heavy tail phenomena do not vanish at infinity, namely that x (∞) does not have more finite moments that x (∞) n . The point (4) is a deterministic statement, and is a direct application of Hurwitz's theorem in complex analysis. It will allow us obtain Corollary 1.2 which gives the limit of the point process χ n for complex Gaussian coefficients.

What does this theorem say for some classical distribution of the coefficients? If the distribution of |a 0 | is absolutely continuous with respect to the Lebesgue measure on R + with density g, then the point (3) can be linked to the density g at zero:

1. if g is continuous at 0 and g(0) > 0 then the largest root in modulus has infinite mean;

2. if g(t) ∼ αt at zero, then x

(n)
n has an infinite second moment.

These two situations cover most of the classical examples of random variables. Real Gaussian random variables, exponential random variables, Cauchy random variables and many others are covered by the first part of the remark. Radial complex random variables such as complex Gaussian random variables are covered by the second point. This is a consequence of the polar change of coordinates which adds a factor 2πr to the density. This phenomenon is illustrated in Figure 1: when the coefficients are complex, the density of x

(n) 1 vanishes at zero. If the distribution of the a k 's is supported in an annulus bounded away from zero, then all the roots lie in an annulus. This can be seen as a consequence of the Gershgorin circle theorem of localization of the eigenvalues of matrices, applied to the companion matrix of the polynomial P n . In this setting, we also know the empirical measure of P n converges deterministically towards the uniform measure on the unit circle [START_REF] Hughes | The zeros of random polynomials cluster uniformly near the unit circle[END_REF].

This theorem can be surprising if we compare it to similar results in random matrix theory. There is a strong analogy of results and techniques between random polynomials and random matrices. For Ginibre random matrices or Kac random polynomials, the empirical measures converge towards a deterministic measure and the explicit distribution of the eigenvalues (or zeros) can be computed when the coefficients are Gaussian (real or complex). Large deviation principles for the empirical measures were obtained with the same speed and very similar rate functions ([HP00], [START_REF] Ben | Large deviations from the circular law[END_REF] for Ginibre, [ZZ10] and [START_REF] Butez | Large deviations for the empirical measure of random polynomials: revisit of the zeitouni-zelditch theorem[END_REF] for Kac polynomials). For random matrices, the large deviation principle for the empirical measure is known not to be universal [START_REF] Bordenave | A large deviation principle for wigner matrices without gaussian tails[END_REF] and to depend on the tail of the coefficients of the matrix. For random polynomials, the large deviation principle is universal [START_REF] Butez | Universal large deviations for kac polynomials[END_REF]. For random matrices, the fluctuations of the largest eigenvalue at the edge of the limiting distribution have been studied by many authors since [START_REF] Tracy | Level-spacing distributions and the airy kernel[END_REF] and have proven to be universal.

When the a k 's are i.i.d. standard complex Gaussian random variables, the zeros of P n form a Coulomb gas in C n (see [START_REF] Zeitouni | Large deviations of empirical measures of zeros of random polynomials[END_REF]) with density of the form (z

(n) 1 , . . . , z (n) n ) ∼ 1 Z n exp   i =j log |z (n) i -z (n) j | -(n + 1) log 1 2π 2π 0 n k=1 |e iθ -z (n) k | 2 dθ   .
This is similar to the eigenvalues of complex Ginibre random matrices where the density of the eigenvalues on C n is of the form (see [START_REF] Ginibre | Statistical ensembles of complex, quaternion, and real matrices[END_REF])

(λ 1 , . . . , λ n ) ∼ 1 Z n exp   i =j log |λ i -λ j | -n n k=1 |λ i | 2 2   .
They have in common the same interaction between the particles, but the confining term is different. Why does the largest particle have such a different behavior for polynomials and matrices? If we look at the behavior of the confining term in each variable for random polynomials, we see that it grows at infinity like log(|z|) while the confining term for Ginibre is V (z) = |z| 2 /2. We believe that it is a general fact for Coulomb gases: when the confining term is of order log |z| at infinity, the largest particle has a heavy tail and when the confining term is stronger than logarithm, the largest particle should converge towards the edge of the limiting distribution. The potential energy of the largest particle is approximativelylog |z -w|dµ ∞ (w) + V (z), where µ ∞ is the limiting distribution of the particles. Hence, if V grows faster than logarithm, the cost of having a particle far from the support of µ ∞ grows at infinity. On the other hand, if V and log |z -w|dµ ∞ (w) are of same order, the cost of having a particle far from the support of µ ∞ is finite. To our knowledge, this phenomenon is not treated in the literature. The heuristic above only relies on "energetic" considerations and has not been proved so far. The same phenomenon should appear for real Gaussian coefficients, as the distribution of the zeros of P n [START_REF] Butez | Large deviations for the empirical measure of random polynomials: revisit of the zeitouni-zelditch theorem[END_REF] is also very similar to the distribution of the eigenvalues of the real Ginibre ensemble [Ede97]: both form a mixture of Coulomb gases, each gas having a fixed number of particles on the real line.

If we compare the hypotheses of Theorem 1.1 with the one of the main theorem from [BZ17], we see that both rely on the behavior of the distribution of the coefficients at zero. Within the universality class of random Kac polynomials with Gaussian coefficients, the number of finite moments for the largest root is constant.

Complex Gaussian coefficients.

Theorem 1.1 states that x (n) 1 and x (n) n converge in distribution but do not give any precise information on the limit. In the case of complex Gaussian coefficient, we can give more precise results on this limiting distribution as the zeros of P ∞ have been studied in the case of Gaussian Analytic Functions (GAF) by Peres and Virág in [PV05]

P ∞ (z) = k∈N a k z k .
The corollary below is just a combination of Theorem 1.1 along with [HKPV09, Theorem 5.1.1 and Corollary 5.1.7].

Corollary 1.2 (Gaussian case). Let (a k ) k∈N be a sequence i.i.d. standard complex Gaussian random variables.

The point process z

(n) k | |z (n) k | < 1

converges towards the determinantal point process in the open unit disk D(0, 1) with the Bergman Kernel

∀z, w ∈ D(0, 1), K(z, w) = 1 π(1 -z w) 2 .
As a consequence, the set {|z 2. The smallest root in modulus of P ∞ , z 1 , has a rotationally invariant distribution and its modulus,

x (∞)
1 , has a cumulative distribution function defined on (0, 1) given by:

F x (∞) 1 (t) = 1 - ∞ k=1 (1 -t 2k ).
This theorem is no more than the combination of the work of Peres and Virág [START_REF] Peres | Zeros of the iid gaussian power series: a conformally invariant determinantal process[END_REF] with Theorem 1.1. This allows us to compute the exact limit distribution of the smallest modulus of the zeros of P n . Notice that the cumulative distribution function

F x (∞)
1 is of order t 2 around zeros, which implies that 1/x (∞) 1

has an infinite variance. Figure 2 for complex Gaussian coefficients and density of x

(∞)
1 .

Proofs of the results

We start with a lemma that will be essential in the proof of Theorem 1.1.

Lemma 2.1 (Lemma 4.1 in [START_REF] Kabluchko | Asymptotic distribution of complex zeros of random analytic functions[END_REF]). Let (a k ) k∈N be i.i.d. random variables. Fix ε > 0. Then

sup k∈N |a k | e εk < ∞ a.s. ⇔ E(log(1 + a 0 )) < ∞.
Proof of the lemma. For every non negative random variable X we have:

∞ k=1 P(X ≥ k) ≤ E(X) ≤ ∞ k=0 P(X ≥ k).
Those inequalities come from the relation: E(X) = R + P(X ≥ x)dx. Now we use this inequality to the non negative variable

X = 1 ε log(1 + |a 0 |).
We deduce that

∞ k=1 P( |a k | e εk > 1) < ∞
and so we have, thanks to the Borel-Cantelli lemma lim sup

|a k | e εk ≤ 1, which implies sup k∈N |a k |
e εk < ∞. The reverse implication relies on a similar reasoning, and will not be used in the proof of the theorem.

Proof of Theorem 1.1. Proof of 1. Let P n (z) = a 0 + a 1 z + • • • + a n z n and Q n (z) = z n P n (1/z) = a n = a n-1 z + • • • = a 0 z n .
As the a k are i.i.d. random variables, the distribution of the random polynomials P n and

Q n are the same. If {z (n) k } is the set of zeros of P n then the set of zeros of Q n is {1/z (n) k }. This implies that x (n) n and 1/x (n) 1 have the same distribution. Proof of 2. Fix n ∈ N ∪ {∞}. The random variable M := sup k≥2 |a k |/e k is
almost surely finite. This is obvious for n < ∞ and this is a consequence of lemma 2.1 for n = ∞. There exists K such that P(M < K) > 0. Let us define C 2 = P(M < K) > 0. The key idea of this proof is to use Rouché's theorem [START_REF] Lang | Complex analysis[END_REF]p. 181] to show that P n and P 1 (z) = a 0 + a 1 z have the same number of roots in a neighborhood of the origin. Rouché's theorem is the following: if γ is a closed path holomogous to 0 in some open set U such that γ has an interior and f and g are two analytic functions on U such that for any z ∈ γ

|f (z) -g(z)| < |f (z)|
then f and g have the same number of zeros in the interior of γ. To bound from below the probability that x (n) 1 is smaller than t, we compare it to the modulus of the root of P 1 .

P(x

(n) 1 ≤ t) ≥ P x (n) 1 ≤ 2 |a 0 | |a 1 |
|z|=2|a 0 |/|a 1 | |P n (z) -P 1 (z)| < inf |z|=2|a 0 |/|a 1 | |P 1 (z)| and 2 |a 0 | |a 1 | ≤ t .
We notice that the triangle inequality implies that inf

|z|=2|a 0 |/|a 1 | |P 1 (z)| = inf |z|=2|a 0 |/|a 1 | |a 0 + a 1 z| ≥ inf |z|=2|a 0 |/|a 1 | |a 1 ||z| -|a 0 | = |a 0 | and that sup |z|=2|a 0 |/|a 1 | |P n (z) -P 1 (z)| ≤ k≥2 |a k | 2|a 0 | a 1 k ≤ M k≥2 e k 2|a 0 | a 1 k ≤ M 4e 2 |a 0 | 2 /|a 1 | 2 1 -2e|a 0 |/|a 1 | .
Let r be a constant such that C 3 = P(|a 1 | > r) > 0, then we obtain

P(x (n) 1 ≤ t) ≥ P M 4e 2 |a 0 | 2 /|a 1 | 2 1 -2e|a 0 |/|a 1 | < |a 0 | and 2 |a 0 | |a 1 | ≤ t ≥ P K 4e 2 |a 0 | 2 /|a 1 | 2 1 -2e|a 0 |/|a 1 | < r|a 0 | |a 1 | and |a 1 | > r and 2 |a 0 | |a 1 | ≤ t and M < K
As t < 1/(A + 1) then A t 2 1-t < t, we obtain that the event

2 |a 0 | |a 1 | ≤ t and |a 1 | > r contains the event K 4e 2 |a 0 | 2 /|a 1 | 2 1 -2e|a 0 |/|a 1 | < r |a 0 | |a 1 | if t ≤ e -1 K/r + 1 .
For t ≤ e -1 K/r+1 = C 1 , we get, using the independence of the a k 's

P(x (n) 1 ≤ t) ≥ P 2 |a 0 | |a 1 | ≤ t and |a 1 | > r and M < K ≥ P |a 0 | < rt 2 P (|a 1 | > r) P (M < K) ≥ C 2 C 3 P |a 0 | < rt 2 .
Proof of 3. Assume that there exists k ≥ 0, a > 0 and δ > 0 such that

∀ t < δ, P(|a 0 | < t) ≥ at k .
Let X be a non-negative random variable. Then the Fubini theorem implies that

1 k + 1 E(X k+1 ) = ∞ 0 t k P(X ≥ t)dt.
Using this along with the point (1) we get

1 k E((x (n) n ) k ) = ∞ 0 t k-1 P(x (n) n ≥ t)dt = ∞ 0 t k-1 P(x (n) 1 ≤ 1/t)dt ≥ A ∞ 1/C 1 t k-1 P(|a 0 | ≤ 2 rt )dt ≥ A 2 k r k ∞ 1/C 1 at k-1 t -k dt. This implies that E((x (n) n ) k ) = ∞.
Proof of 4. Thanks to Lemma 2.1, we obtain that the radius of convergence of the random entire function P ∞ is almost surely 1. This implies that, almost surely, P n converges uniformly towards P ∞ on any closed disk D(0, ρ) with radius ρ < 1. In this setting, the almost sure convergence of the zeros of P n inside D(0, ρ) is exactly Hurwitz's theorem [START_REF] Conway | Functions of one complex variable[END_REF]p. 152] in complex analysis. Hurwitz's theorem is a consequence of Rouché's theorem, which is a consequence of the argument principle. We give a proof of the convergence of the point processes using directly Rouché's theorem. Let (Ω, F, P) be a probability space on which the a k 's are defined. Let N be a negligible set such that for any ω ∈ Ω \ N , P ∞ is a non-constant entire series with radius of convergence one.

Fix ω ∈ Ω \ N and let z (∞) be a zero of P ∞ , with multiplicity β. As the zeros of P ∞ are isolated, for any ε small enough, P ∞ has no other zero than z (∞) in the closed disk D(z (∞) , ε). Thanks to Rouché's theorem, we know that if

sup |z-z (∞) |=ε |P n (z) -P ∞ (z)| < inf |z-z (∞) |=ε |P ∞ (z)| (2)
then P n and P ∞ have the same number of zeros inside D(z (∞) , ε). The inequality (2) is automatically satisfied for n large enough, as we fixed ε such that P ∞ does not have a zero on the boundary of the disk D(z (∞) , ε).

Here we proved that for any zero of multiplicity β of P ∞ , for any ε > 0 sufficiently small, one can find β zeros of P n at a distance at most ε of z (∞) . This implies that any fixed finite number of zeros of P n converges almost surely towards zeros of P ∞ .

Proof of 5. The fact that x

Comments.

Notice that the proof of points (1), (2) and (3) of Theorem 1.1 for finite n does not use the assumption E(log(1 + |a 0 |)) < ∞. These three points are always valid. This assumption is only needed to make sure that, almost surely, the a k 's do not grow faster than e k . An alternative proof of Theorem 1.1 uses Jensen's formula [Lan13, p. 341] for analytic functions. We chose to use an approach based on Rouché's Theorem as it also implies the convergence of the point process of the small roots.

k

  | and min k |z

Figure 1 :

 1 Figure 1: Histogram of the x (500) 1 for exponential coefficients with mean 1 (left) and dµ = 1/2πe -|z| dz on C (right).

k

  |} k≥1 has the same law as the set {U 1/2k k } k≥1 where (U k ) k≥1 is a sequence of i.i.d. uniform random variables on [0, 1].

  is an illustration of the point (2) of Corollary 1.2. The histogram of x (500) 1 is very similar to the graph of the density of x ∞ 1 .

Figure 2 :

 2 Figure 2: Histogram of x (500) 1