Qiong Wang

Prof Yu-Jun Cui
email: yujun.cui@enpc.fr

Anh Minh Tang

Pierre Delage

Behrouz Gatmiri

Wei-Min Ye

Long-term effect of water chemistry on the swelling pressure of a bentonite-based material

Keywords: Radioactive waste disposal, bentonite based materials, pore water chemistry, time dependence, swelling pressure

Compacted bentonite-based materials have been proposed as possible sealing andbackfill materials in geological repositories for the high-level radioactive waste disposal in several countries.During the long time lifespan of a repository, as the chemical composition of porewater can change, the swelling and sealing capacity of the material mayalso change.From a point of view of storage safety assessment, it is important to evaluate thispossible change of swelling capacity. In this study, a mixture of bentonite and crushedCallovo-Oxfordian claystone was investigated. The long-term effect of pore waterchemistry on the swelling pressure was studied at constant-volume conditions for 700days. Distilled water and synthetic water having the chemical composition similar tothat of in-situ pore water were used for hydration. The results obtained in an initial period of100 hours revealed no significant influence of the water composition on the swellingpressure evolution, and the maximum swelling pressures observed were close to4.30 MPa for a dry density of 1.70 Mg/m 3 . Over a longer time period, on the contrary, the swelling pressure eventually decreased for allsamples, especially for the sample saturated with synthetic water. In addition,comparison of a one-step soaking test with a multi-step soaking test showed nowetting procedure effect on the long term swelling behaviour. All the results areanalysed by considering the physico-chemical interaction between the minerals ofclaystone, minerals of bentonite and different fluids involved.

INTRODUCTION

Compacted bentonite-based materials have been proposed as possible sealing/backfillmaterials in deep geological repositories for high-level radioactive wastes (HLW) inseveral countries. Due to their favourable swelling characteristics, these materials areexpected to fill up all voids left in the system after construction and emplacement of the waste packages. An important long-term safety function of bentonite-based barriers is to ensurea relatively impermeable zone around the high-level radioactive waste thereby limiting groundwater flow and waste package degradation rates and, ultimately, waste leaching rates.

This roletermed as "sealing" [START_REF] Komine | Simplified evaluation for swelling characteristics of bentonites[END_REF][START_REF] Komine | New equations for swelling characteristics of bentonite-based buffer materials[END_REF] is essential to ensurethe overall safety of the geological disposal: the disposed waste is isolated from thebiosphere over long time. For this purpose, the swelling property becomes a key factorin the design and fabrication of the sealing/backfill materials, and needs to be studiedin depth.

The swelling properties (i.e. swelling pressure or swelling potential) ofbentonite-based materials have been widely investigated [START_REF] Pusch | Mineral-water interactions and their influence on the physical behavior of highly compacted Na bentonite[END_REF]Komine andOgata, 1994[START_REF] Mata | Hydraulic behaviour of bentonite based mixtures in engineered barriers: The Backfill and Plug Test at the A¨ spo¨ HRL[END_REF], 2004a;[START_REF] Delage | The relationship between suction and swelling properties in a heavily compacted unsaturated clay[END_REF][START_REF] Agus | Swelling pressures and wetting-drying curves of a highly compacted bentonite-sand mixture[END_REF][START_REF] Komine | Swelling characteristics of bentonites in artificial seawater[END_REF]. They depend strongly on the initial state such as dry density and water content [START_REF] Komine | Experimental study on swelling characteristics of compacted bentonite[END_REF][START_REF] Villar | Influence of dry density and water content on the swelling of a compacted bentonite[END_REF]. After being installed in the field,especially in some geological host formations relatively rich in salts (e.g. saltformation, claystone, granite formations, etc.), site water of certain salinity caninteract with bentonite [START_REF] Herbert | Swelling pressures of MX-80 bentonite in solutions of different ionic strength[END_REF]; as a result, the hydromechanicalbehaviourof the bentonite-based materials and in turn the performance of the barrier maybe notably affected. Moreover, the water composition can change over time.

Frequently, bentonite/sand mixtures are considered for the reason of good control ofswelling pressure, enhanced thermal conductivity as compared to pure bentonite and a better mechanical resistance. Forinstance, a 70/30 bentonite-sand mixture was proposed as buffer material in Japanwhile a 50/50 bentonite/sand mixture was suggested in the Canadian repositoryconcept [START_REF] Dixon | A study of the compaction properties of pot ential clay-sand buffer mixtures for use in nuclear fuel waste disposal[END_REF]. In France, Callovo-Oxfordian (COx) claystone has been considered as apossible geological host-rock for high-level radioactive waste disposal, and themixture of bentonite and crushed Callovo-Oxfordian (COx) claystone is proposed as apossible sealing/backfill material for environmental and economic reasons (Andra,2005;Tang et al., 2011a;Tang et al., 2011b;[START_REF] Wang | Experimental study on the swelling behaviour of bentonite/claystone mixture[END_REF]. If physico-chemicalinteractions essentially take place only between groundwater and bentonite in the case ofsand/bentonite mixture, it is not the case for the bentonite/claystone because of the additional interactions between these two materials and the water. Due to the low permeability of thebentonite-based materials, these interactions might develop overlongtime and the resulting effects will not be immediate. Hence, particular attention should be paid to their long-term behaviour.

This study focuses on the swelling pressure of the compacted mixture of MX80bentonite and crushed COx claystone. The long term effect of water chemistry on theswelling pressure and microstructure were investigated after 700 days in contact with water.Emphasis was put on the physico-chemical interaction between the minerals ofclaystone, minerals of bentonite and different fluids involved. The wetting procedureeffect on the long term swelling behaviour was also investigated by comparing a one-step soaking test with multistep soaking test.

MATERIALS AND METHODS

Materials

The bentonite used in this study is a commercial MX80 Na-bentonite, which containslarge quantities of montmorillonite (75-90%). Silt is the dominant remaining fraction,which mainly consists of quartz and feldspars as well as micas, sulphides, and oxides [START_REF] Pusch | Mineral-water interactions and their influence on the physical behavior of highly compacted Na bentonite[END_REF]. Table 1 details the main mineralogical components of MX80 bentoniteobtained by several authors.

The total cation exchange capacity (CEC) is between0.78 and 0.85 meq/g (Table 2), the cation population at exchange positions is mainlycomposed of Na + (60.0-67.0 meq/100g), other population being much lower: Ca 2+ (5.0-8.0 meq/100g), Mg 2+ (3.0-4.0meq/100g) and K + (0.2-1.3 meq/100g).The grain size distribution (Fig. 1) determined by hydrometer (AFNOR NF P94-057)shows that the fraction of clay-size particles (< 2 μm) is 84%. The bentonite tested has anaverage specific gravity of 2.76, a liquid limit of 520%, and a plastic limit of 42%.

Callovo-Oxfordian (COx) claystone was taken at 490-m depth from the UndergroundResearch Laboratory (URL) at Bure in eastern France. It contains 40-45% clayminerals (mainly interstratified minerals of illite-smectite), 20-30% carbonates(mainly calcite) and 20-30% quartz and feldspar [START_REF] Hoteit | Geological Radioactive Waste Disposal Project in France: Conceptual Model of a Deep Geological Formation and Underground Research Laboratory in Meuse/Haute-Marne Site[END_REF]Lebon andGhoreychi, 2000;[START_REF] Zhang | Experimental study of the hydro-mechanical behaviour of the Callovo-Oxfordian argillite[END_REF]. The in-situ water content is 2.8-8.7 %; the bulkdensity is 2.32-2.61

Mg/m 3 and the specific gravity is 2.70 [START_REF] Hoteit | Geological Radioactive Waste Disposal Project in France: Conceptual Model of a Deep Geological Formation and Underground Research Laboratory in Meuse/Haute-Marne Site[END_REF]Tanget al., 2011a). The excavated claystone was air-dried and crushed into a powder. Fig. 1 depicts thegrain size distribution of the crushed powder obtained by dry sieving; it contains 18%of fine grains (0.08 mm). The grain size distribution determined by hydrometerconfirms that the content of clay-size particles (< 2 μm) is 40%.

Both distilled water and synthetic water were used for hydration. The chemicalcomposition of the synthetic water (see Table 3) is similar to that of the groundwaterat Bure site. The pH values are between 7.00 and 7.28; the concentration of Na+ istwice that of Ca 2+ and Mg 2+ and four times that of K + [START_REF] Gaucher | Modelling the porewater chemistry of the Callovian-Oxfordian formation at a regional scale[END_REF]Marty etal., 2010).

Sample preparation

In this study,all tests were performed on samples of compacted bentonite/claystonemixture with a bentonite content of 70 % in dry mass. Bentonite and claystonepowders, with the initial water contents of 11.8 % and 2.64 % respectively, were firstcarefully mixed prior to compaction. Samples were statically compacted to a drydensity of 1.70 Mg/m 3 in a metallic cell (70 mm in internal diameter) at a controlledrate of 0.05 mm/min. After compaction, the sample height is about 10 mm. The total suction measured in these specimens using ahygrometer was s = 90 MPa. The specimensare then carefully introducedinto constantvolume cells (having the same diameter as the compaction cell) forthe swelling pressure experiment.

Experimental methods and programme

The constant-volume cell used in this study is presented in Fig. Four swelling pressure tests were performed using the constant-volume cell (Table 4).In tests LT01 and LT02 (LT stands for long term), the samples were saturated withsynthetic water and distilled water, respectively. The duration of these testscarried out to study the long-term effect of water chemistry on the swellingpressurewas 700 days. In test LT03, three decreasing suctions (57 MPa, 38 MPa and 12.6 MPa) were firstapplied in steps using vapour equilibrium technique, prior to distilled water floodingin the last stage. This test also lasted 700 days. This test was performed to investigate the effect of the wetting procedure on the long termswelling behaviourby comparisonwith test LT02. Test ST (ST for short term) was performed by directly injectingsynthetic water as in test LT01, but with a shorter duration of 100 hours.

After certain hydration time (see Table 4), the specimen was taken out of the cell for the analysis of pore-size distributionby Mercury Intrusion Porosimetry (MIP). water. This indicates that the wettingprocedure did not affect the shortterm swelling behaviour for this material.

EXPERIMENTAL RESULTS

In Fig. 5, all results from the three tests LT01, LT02 and LT03 during 700 days arepresented.

It appears that over this long period, the swelling pressure decreased somewhat for allsamples, especially for the sample saturated with synthetic water (LT01). The finalswelling pressure was 3.95 MPa (corresponding to a decrease of 9%), and 4.19 MPa(decrease of 3%) for samples saturated with synthetic water (LT01) and distilled water(LT02 and LT03), respectively.

The pore size distribution curve for all the samples taken at the end of the tests arepresented in Fig. 6. It is observed in Fig. 6a that the final values of intruded mercuryvoid ratio (em= intruded mercury volume over soil solid volume) are between 0.4 and0.5, much lower than the global void ratio (e = 0.61). This shows that a large amount ofporosity isinaccessible to mercury. Comparison between the curves obtained after different durations clearly shows that the amount of accessible porosity after 700 days is larger in both cases of distilled water (LT02 and LT03) and synthetic water (LT01) than after 100 hours with synthetic water. In addition, for samples hydrated for 700 days, more quantity of accessible porosity is observed in the case of synthetic water (LT01).

The incremental pore volume dem/dlog(d) (Fig. 6b) refers to the volume of mercurythat intrudes into the pores between pressure increments. For all samples a typicalbimodal porosity can be observed; thus, two pore groups can be defined: a group ofmicro-pores having a mean size of 0.02 μm and a group of macro-pores having amean size of 20 μm.

After 700-day saturation (LT01, LT02 and LT03), themacro-pores and micro-pores quantity increased when comparing to the short-termtest (ST). Change in macro-pores is more significant especially for the samplesaturated with synthetic water (LT01). For tests LT02 and LT03 which were hydratedwith distilled water, very similar curves were observed suggesting no effect of thewetting procedure on the pore-size distributions. As far as changes in pores size are concerned,it can be observed that over time the group of micro-pores had a size slightlydecreased. On the contrary, the group of macro-pores had a size slightlyincreased.

Changes for test LT01 weremore marked.

INTERPRETATION AND DISCUSSION

Various studies showed that the swelling capacity of bentonite is lower in saline water;the higher the dissolved salt concentration, the lower the swelling pressure. This phenomenonbecomes less pronounced in case of high dry density [START_REF] Studds | The effects of salt solutions on the properties of bentonite-sand mixtures[END_REF]Karland etal., 2005;[START_REF] Suzuki | In situ observations of the swelling of bentonite aggregates in NaCl solution[END_REF]Karland et al., 2006;[START_REF] Castellanos | Chemical impact on the hydromechanical behaviour of high-density febex bentonite[END_REF]Katsumi etal., 2008;Siddique et al., 2011). In general, with a high void ratio (low dry density)and/or high salt concentrations, the influence of pore water chemistry on bentonitebehaviour is significant and can be qualitatively explained by the diffuse double layer(DDL) theory (Karland, 1997;[START_REF] Mata | Hydraulic behaviour of bentonite based mixtures in engineered barriers: The Backfill and Plug Test at the A¨ spo¨ HRL[END_REF][START_REF] Castellanos | Chemical impact on the hydromechanical behaviour of high-density febex bentonite[END_REF]. For densebentonite, since there is little water present between adjacent clay platelets, there canbe little or no development of a DDL associated with the clay particles [START_REF] Dixon | Porewater salinity and the development of swelling pressure in bentonite-based buffer and backfill materials[END_REF][START_REF] Pusch | Microstructure of Smectite Clays and Engineering Performance[END_REF]; thus it becomes difficult even impossible to describe thebentonite behaviour by the DDL theory.

In this regard, [START_REF] Karnland | Bentonite swelling pressure in NaCl solutions-Experimentally determined data and model calculations[END_REF] studied the effect of salinity on theswelling pressure of MX-80 bentonite, and noted that the effect of salinity of thesaturating fluid was relatively lower with a higher density. For MX80 bentonite, [START_REF] Dixon | Porewater salinity and the development of swelling pressure in bentonite-based buffer and backfill materials[END_REF] also concluded that at an effective bentonite dry density (bentonite drydensity in the bentonite/sand mixture) higher than 1.22 Mg/m 3 , changes in waterchemistry do not significantly affect the swelling behaviour due to the small numberof water molecule layers between clay sheets.

Confirmation was made by [START_REF] Castellanos | Chemical impact on the hydromechanical behaviour of high-density febex bentonite[END_REF] on the FEBEX bentonite: an increase in salt concentration reduced theswelling pressure, but this change was much less marked for high density. It can bededuced that for the studied bentonite/claystone mixture, the high density (1.70Mg/m 3 ) and the low salinity of the synthetic water (see Table 3) representative of the present in situ conditions resulted together in thenegligible effect of water chemistry on the swelling pressure in short term.

Basically, the swelling pressure of bentonite is mainly related to the crystallineswelling and double layer swelling. Crystalline swelling is caused by the hydration ofexchangeable cations (K + , Na + , Ca 2+ , Mg 2+ ) between mineral layers that have astructure with one alumina octahedral sheet sandwiched between two silica tetrahedralsheets. It is a process which involves the adsorption of a maximum number of hydratesdepending on the nature of cations.

After three to four water monolayers, i.e. atinterlayer separation distances ≥1 nm, the surface hydration becomes less significant andthe electrical double-layer repulsion becomes the main swelling mechanism (Bradburyand Baeyens, 2003;[START_REF] Suzuki | In situ observations of the swelling of bentonite aggregates in NaCl solution[END_REF]. For clay at high density, the low amount ofwater taken up is to all intents and purposes pseudo-crystalline interlayer water, andinsufficient to form the DDL [START_REF] Pusch | Microstructure of Smectite Clays and Engineering Performance[END_REF]. Therefore, the swellingpressure is mainly governed by the crystalline swelling (interaction between the layersurfaces and water) and the double layer repulsion makes only a limitedcontribution. In thiscase, the exchangeable cations are the key elements that control the clay-waterinteraction [START_REF] Abdullah | On the physicochemical aspects of compacted clay compressibility[END_REF].

As far as the pore water is concerned, it may significantly affect the chemicalcompositions of clays through the exchangeable cations [START_REF] Yukselen-Aksoy | Seawater effect on consistency limits and compressibility characteristics of clays[END_REF]. Thecation exchange reaction is mainly controlled by the exchange capacity of the clayminerals [START_REF] Mata | Hydraulic behaviour of bentonite based mixtures in engineered barriers: The Backfill and Plug Test at the A¨ spo¨ HRL[END_REF]; the ease with which a cation of one type can replace a cationof another type depends mainly on the valence, relative abundance of the differentcation types, and the cation size. Other factorsbeing equal, the higher the valence ofthe cation, the higher the replacing power; for cations of the same valence, thereplacing power increases with the size of the cation [START_REF] Laine | Long-Term Stability of Bentonite A Literature Review[END_REF].

Atypical replaceability order is: Na + <K + <Mg 2+ <Ca 2+ [START_REF] Mitchell | Fundamentals of Soil Behavior[END_REF][START_REF] Pusch | Experimental study of the effect of high porewater salinity on the physical properties of a natural smectitic clay[END_REF][START_REF] Mata | Hydraulic behaviour of bentonite based mixtures in engineered barriers: The Backfill and Plug Test at the A¨ spo¨ HRL[END_REF]. According to this order, for the Na-bentonite as in this study, ittends to change from sodium (Na -) to other types (i.e. K -, Mg -or Ca -) depending onthe cations present in the permeating water.

One of the most common cation exchange reactions in the Na-bentonite is the oneinvolving sodium and calcium [START_REF] Mata | Hydraulic behaviour of bentonite based mixtures in engineered barriers: The Backfill and Plug Test at the A¨ spo¨ HRL[END_REF]. [START_REF] Mata | Hydraulic behaviour of bentonite based mixtures in engineered barriers: The Backfill and Plug Test at the A¨ spo¨ HRL[END_REF] investigated the evolutionof Ca 2+ and Na + in a specimen of sodium MX80 bentonite hydrated with saline water(NaCl and CaCl, 50/50 by mass) and distilled water. Analysis of the collected porefluid showed an increaseof sodium concentration and a decrease of calciumconcentration as compared to the injected water. This suggeststhat the sodium bentonitewas transformed to a calcium bentonite. [START_REF] Montes-H | Sorption kinetic of water vapour of MX80 bentonite submitted to different physical-chemical and mechanical conditions[END_REF][START_REF] Montes-H | ESEM observations of compacted bentonite submitted to hydration/dehydration conditions[END_REF] investigatedthe chemical transformation of Na/Ca-MX80 bentonite (Na0.18Ca0.10) in arepresentative pore water (pH = 7.3) from Bure site (-180 m), the Na/Ca-to Camontmorillonite conversion was identified as the main chemical transformation. At aconstant temperature, this transformation is affected by the groundwater composition(pH and cation concentrations) and solid/liquid ratio as well as the reaction time [START_REF] Mitchell | Fundamentals of Soil Behavior[END_REF][START_REF] Herbert | Long-term behaviourof the Wyoming bentonite MX-80 in high saline solutions[END_REF][START_REF] Fernández | Geochemical behaviour of a bentonite barrier in the laboratory after up to 8 years of heating and hydration[END_REF]Laine and Karttunen,2010). For the short term swelling pressure test in this study, it was actually the lowcation concentration (low salinity of synthetic water mentioned above) and highsolid/liquid ratio (i.e. high density) that led to the negligible effect of water chemistryon the swelling pressure. Nevertheless, after a long time period of reactions, the effectbecame evident because the presence of Ca 2+ in Bure site water (see Table 3) enabledpart of Na-montmorillonite to change to Ca-montmorillonite which has a much lowerexpansibility. This process resulted in a decrease of swelling pressure overtime asshown in Fig. 5.

The transformation from sodium to calcium montmorillonite is critically dependent onthe amount of available calcium (Ca 2+ ) in the pore water (Muurinen and Lehikoinen,1999;[START_REF] Fernández | Analysis of the porewater chemical composition of a Spanish compacted bentonite used in anengineered barrier[END_REF][START_REF] Fernández | Geochemical behaviour of a bentonite barrier in the laboratory after up to 8 years of heating and hydration[END_REF]. Forthe sample saturatedwith distilled water, the pore water chemistry evolved over time to reachequilibrium with the claystone minerals (the claystone contains 20-30% carbonates, mainlycalcite).Consequently, it was as if diluted Bure site water infiltrated to the MX80bentonite, leading to the degradationof swelling pressure. However, with a much lower cationconcentration (only 30% claystone in the mixture), the decrease of swellingpressure was less significant than in the casewith synthetic water.

For the sample first wetted by three steps of suction (LT03), after the sample wasflooded with distilled water, the maximum swelling pressure was found close to thatobtained by waterflooding the sample directly (LT02), indicting anegligible effectof wetting procedure on the swelling pressure (Figs. 3 and4). This phenomenoncan be explained bythe mechanism identified by [START_REF] Cui | Microstructure changes of a confined swelling soil due to suction controlled hydration Unsaturated soils[END_REF]: when hydrating bydecreasing suction (57-38-12.6 MPa) under confined conditions, the macro-pores wereprogressively cloggedby the invasion of exfoliated clay particles, whereas the micro-pores remained almost un-affected. Themicro-pores started to change only when the water saturation is approached. Thissuggests that step-wetting by suction control and direct flooding with water may leadto similar microstructures after saturation, thus similar maximum swelling pressures.After 700-day hydration with distilled water, not only similar swelling pressures (Fig. 5), but also similar pore-size distributions were observed for tests L02 and L03 (Fig. 6),which enhanced this conclusion. In addition, the repeatability of swelling pressure testand the MIP test was verified.

It was noted that the chemical effect on the HM behaviour of clayey soils is due tochanges at different structural levels and the interaction between them [START_REF] Mata | Hydraulic behaviour of bentonite based mixtures in engineered barriers: The Backfill and Plug Test at the A¨ spo¨ HRL[END_REF].From the pore-size distribution observation in this study, it can be seen that the amount ofaccessible porosity increased after a long time period for both distilled (LT02 andLT03) and synthetic water (LT01), with more increase for synthetic water (Fig. 6). Ifwe relate these phenomena to the swelling pressure, it is clear that the sample withlower swelling pressure had a higher intruded mercury void ratio in both micro-andmacro-pores. For samples with higher swelling capacity, more interlayer hydrationoccurred, leading to a constriction of accessible pores.

When referring to theincremental pore volume, the same tendency can be identified, for the sample withhigher swelling pressure, both the macro-pores and micro-pores decreased.

However,most of changes in microstructure occurred in macro-pores family. The sameobservation can be made in terms of pore size changes. This is in agreement with theobservation made by [START_REF] Mata | Hydraulic behaviour of bentonite based mixtures in engineered barriers: The Backfill and Plug Test at the A¨ spo¨ HRL[END_REF] who conducted MIP test on the compacted sodiumbentonite MX80 and sand mixture (70/30) (dry density of 1.37 and 1.67

Mg/m 3 )saturated with distilled water and saline water, and found that the saline water effectswere significant on the macro-pores, the micro-pores remaining almost unaffected.

CONCLUSION

Long term swelling pressure tests and MIP tests on a bentonite/claystone mixturewere performed. The effects of water chemistry as well as the hydration procedure onthe swelling pressure were investigated in both short andlong terms. The results areanalysed by considering the physico-chemical interaction between the minerals ofclaystone, the minerals of bentonite and the different pore water chemistries involved.

There was no obvious effect of water chemistry on the swelling pressure in short term(100 hours) due to the high dry density of the bentonite-based material and the low salinity of the synthetic water. However, aftera long period of 700 days, the swelling pressure decreased for all samples, especiallyfor the sample saturated with synthetic water.

Different hydration procedures led to comparable long term swelling behaviour and similarpore-size distributions.

The effect of the chemical composition of the pore water on the swelling pressure can be interpreted in terms of changes inmicrostructure. The sample with higher swelling pressure showed less macro andmicro-pores associated with a higher interlayer hydration.

From a practical point of view, the specification and the design of the sealing/backfillmaterial are often made based on the swelling properties measured in short term.However, a nonnegligible decrease (9%) of swelling pressure was observed after 700days even with a water of low salinity. This expected reduction of swelling pressure has thus to be taken into account when designing a repository to ensure that the bentonite-based barrier will also meet the requirements on which its long term performance rests.Considering the possible pore water chemical composition changes over the very long term lifespan of the repository is also recommended to avoiddeterioration of the swelling capacity of these materials and thus of the self-sealing capacity of the system.

List of Tables

List of Figures

  2. It consists ofthreeparts: (1) the bottom part containing a porous stone and a drainage system; (2) themiddle cell (70 mm inner diameter, 10 mm height) used to prevent radial swelling,with two air outlets; (3) the top part incorporating a total pressure sensor (workingpressure of 5 MPa) to monitor the swelling pressure. The sample was wetted byconnecting the water inlet to a water reservoir or a suction control system usingvapour equilibrium technique at an ambient temperature of 201 o C. More details can be found in Tang et al. (2011b), Wanget al. (2012).

Fig. 3

 3 Fig.3 presents the results from tests LT01, LT02, and ST during the first 100 hours afterthe

Figure 1 .

 1 Figure 1. Grain size distribution curves of MX80 bentonite and crushed COx claystone Figure 2. Constant-volume cell for swelling pressure measurement Figure 3. Evolution of swelling pressure for tests LT01, LT02 and ST during the first 100 h Figure 4. Evolution of swelling pressure for samples wetted with decreasing suction Figure 5. Evolution of swelling pressure for tests LT01, LT02, and LT03 for 700 days; (a) scale from 0 to 5 MPa, (b) scale from 3.6 to 4.6 MPa Figure 6. Pore size distribution curves; (a) cumulative curves, (b) derived curves

Table 1 .

 1 Mineral composition of the MX80 bentontie (%)

Table 2 .

 2 Exchangeable cations of MX80 bentonite (meq/100g)

Table 3 .

 3 Chemical composition of the synthetic water

Table 4 .

 4 Test programme

Acknowledgements

The support from the PHC Cai Yuanpei project (24077QE) and that of the ChinaScholarship Council (CSC) are greatly acknowledged.


Pablo Torres
email: ptorres@fceia.unr.edu.ar

Mario Valencia-Pabon

Stable Kneser Graphs are almost all not weakly Hom-Idempotent *

Keywords: Cartesian product of graphs, Stable Kneser graphs, Cayley graphs, Homidempotent graphs

A graph G is said to be hom-idempotent if there is an homomorphism from G 2 to G, and weakly hom-idempotent if for some n ≥ 1 there is a homomorphism from G n+1 to G n . Larose et al. [Eur. J. Comb. 19:867-881, 1998] proved that Kneser graphs KG(n, k) are not weakly hom-idempotent for n ≥ 2k + 1, k ≥ 2. We show that 2-stable Kneser graphs KG(n, k) 2-stab are not weakly hom-idempotent, for n ≥ 2k + 2, k ≥ 2. Moreover, for s, k ≥ 2, we prove that s-stable Kneser graphs KG(ks+1, k) s-stab are circulant graphs and so hom-idempotent graphs.

Introduction

Let [n] denote the set {1, . . . , n}. For positive integers n ≥ 2k, the Kneser graph KG(n, k) has as vertices the k-subsets of [n] and two vertices are connected by an edge if they have empty intersection. In a famous paper, Lovász [START_REF] Lovász | Kneser's conjecture, chromatic number and homotopy[END_REF] showed that its chromatic number χ(K(n, k)) is equal to n -2k + 2. After this result, Schrijver [START_REF] Schrijver | Vertex-critical subgraphs of Kneser graphs[END_REF] proved that the chromatic number remains the same when we consider the subgraph KG(n, k) 2-stab of KG(n, k) obtained by restricting the vertex set to the k-subsets that are 2-stable, that is, that do not contain two consecutive elements of [n] (where 1 and n are considered also to be consecutive). Schrijver [START_REF] Schrijver | Vertex-critical subgraphs of Kneser graphs[END_REF] also proved that the 2-stable Kneser graphs are vertex critical (or χ-critical), i.e. the chromatic number of any proper subgraph of KG(n, k) 2-stab is strictly less than n -2k + 2; for this reason, the 2-stable Kneser graphs are also known as the Schrijver graphs. After these general advances, a lot of work has been done concerning properties of Kneser graphs and stable Kneser graphs (see [START_REF] Frankl | On the chromatic number of the general Kneser graph[END_REF][START_REF] Frankl | Extremal problems concerning Kneser graphs[END_REF][START_REF] Larose | On normal Cayley graphs and Hom-idempotent graphs[END_REF][START_REF] Braun | Symmetries of the stable Kneser graphs[END_REF][START_REF] Meunier | The chromatic number of almost stable Kneser hypergraphs[END_REF] and references therein). For example, it is well known that for n ≥ 2k + 1 the automorphism group of the Kneser graph KG(n, k) is the symmetric group induced by the permutation action on [n]; see [START_REF] Godsil | Algebraic graph theory[END_REF] for a textbook account. More recently, Braun [START_REF] Braun | Symmetries of the stable Kneser graphs[END_REF] showed that the automorphism group of the 2-stable Kneser graphs KG(n, k) 2-stab is the dihedral group of order 2n.

The cartesian product G H of two graphs G and H has vertex set V (G) × V (H), two vertices being joined by an edge whenever they have one coordinate equal and the other adjacent. This product is commutative and associative up to isomorphism.

An homomorphism from a graph G into a graph H, denoted by G → H, is an edge-preserving map from V (G) to V (H). If H is a subgraph of G and φ : G → H has the property that φ(u) = u for every vertex u of H, then φ is called a retraction and H is called a retract of G. If φ : G → H is a bijection and φ -1 is also a homomorphism from H to G, then φ is an isomorphism and we write G H. In particular, if G = H then φ is an automorphism if and only if it is bijective. Two graphs G and H are homomorphically equivalent, denoted by G ↔ H, if G → H and H → G. A graph G is called a core if it has no proper retracts, i.e., any homomorphism φ : G → G is an automorphism of G. It is well known that any finite graph G is homomorphically equivalent to at least one core G • , as can be seen by selecting G • as a retract of G with a minimum number of vertices. In this way, G • is uniquely determined up to isomorphism, and it makes sense to think of it as the core of G. It is widely known that Kneser graphs are core. Moreover, it is not difficult to deduce that any χ-critical graph is a core. Therefore, any 2-stable Kneser graph is also a core, because they are χ-critical graphs [START_REF] Schrijver | Vertex-critical subgraphs of Kneser graphs[END_REF].

Let A be a group and S a subset of A that is closed under inverses and does not contain the identity. The Cayley gragh Cay(A, S) is the graph whose vertex set is A, two vertices u, v being joined by an edge if u -1 v ∈ S. Cayley graphs of cyclic groups are often called circulants.

A graph G is said vertex-transitive if its automorphism group AUT(G) acts transitively on its vertex-set. It's well known that Cayley graphs and Kneser graphs are vertex-transitive. However, 2-stable Kneser graphs are not vertex-transitive in general. For example, no automorphism in KG(6, 2) 2-stab sends {1, 3} to {1, 4}, since AUT(KG(6, 2) 2-stab ) is the dihedral group of order 12 acting on the set {1, 2, . . . , 6}.

We write G n for the n-fold cartesian product of a graph G. A graph G is said hom-idempotent if there is a homomorphism from G 2 to G, and weakly hom-idempotent if for some n ≥ 1 there is a homomorphism from G n+1 to G n . Larose et al. [START_REF] Larose | On normal Cayley graphs and Hom-idempotent graphs[END_REF] showed that the Kneser graphs are not weakly hom-idempotent. However, the technique used by Larose et al. [START_REF] Larose | On normal Cayley graphs and Hom-idempotent graphs[END_REF] cannot be extended directly to the 2-stable Kneser graphs.

A subset S ⊆ [n] is s-stable if any two of its elements are at least "at distance s apart" on the n-cycle, that is, if s ≤ |i -j| ≤ n -s for distinct i, j ∈ S. For s, k ≥ 2 and n ≥ ks, the s-stable Kneser graph KG(n, k) s-stab is the subgraph of KG(n, k) obtained by restricting the vertex set of KG(n, k) to the s-stable k-subsets of [n].

In this paper, we show that almost all 2-stable Kneser graphs are not weakly hom-idempotent. Moreover, for s, k ≥ 2, we show that s-stable Kneser graphs KG(ks + 1, k) s-stab are circulant graphs and so hom-idempotent graphs. In the sequel, we will use the term modulo [n] to denote arithmetic operations on the set [n] where n represents the 0.

2-stable Kneser graphs

As we have mentioned in the previous section, Braun [START_REF] Braun | Symmetries of the stable Kneser graphs[END_REF] showed that the automorphism group of the 2-stable Kneser graph KG(n, k) 2-stab is the dihedral group D 2n of order 2n. We denote the elements of D 2n as follows (arithmetic operations are taken modulo [n]):

• Rotations: Let σ 0 be the identity permutation on [n] and, for 1

≤ i ≤ n -1, let σ i = σ i-1 • σ 1 ,
where σ 1 is the circular permutation (1, 2, . . . , n -1, n).

• Reflexions:

-Case n odd. For 1 ≤ i ≤ n, let ρ i be the permutation formed by the product of the transpositions

(i + 1, i -1)(i + 2, i -2) . . . (i + n-1 2 , i -n-1 2 )
, where i is a fix point. -Case n even. For 1 ≤ i ≤ n 2 , we have two types of reflexions: let ρ i be the permutation formed by the product of the transpositions (i+1, i-1)(i+2, i-2) . . . (i+ n 2 -1, i-n 2 +1), where i and i + n 2 are fix points; and let δ i be the permutation formed by the product of transpositions (i, i -1)(i + 1, i -2) . . .

(i + n 2 -1, i -n 2 ) without fix point. An automorphism φ of a graph G is called a shift of G if {u, φ(u)} ∈ E(G) for each u ∈ V (G).
In other words, a shift of G maps every vertex to one of its neighbors [START_REF] Larose | On normal Cayley graphs and Hom-idempotent graphs[END_REF].

Lemma 1. Let n ≥ 2k + 2. Then, the only two shifts of the 2-stable Kneser graph KG(n, k) 2-stab are the rotations σ 1 and σ n-1 .

Proof. It is very easy to deduce that the circular permutations σ 1 = (1, 2, 3, . . . , n -1, n) and σ n-1 = (σ 1 ) -1 = (n, n -1, n -2, . . . , 2, 1) are both shifts of the graph KG(n, k) 2-stab . In order to prove that they are the only two shifts of KG(n, k) 2-stab , we will proceed by cases. The arithmetic operations are taken modulo [n].

• Rotations. Clearly, the identity permutation σ 0 is not a shift. Now, we claim that for each i ∈ {2, 3, . . . , n -2}, there exists a vertex v i in KG(n, k) 2-stab such that {1, i + 1} ⊆ v i . In fact, vertex v i can be computed as follows:

-

If 2 ≤ i ≤ 2k -1 then, set v i = {1, 1 + 2.1, 1 + 2.2, . . . , 1 + 2.(k -1)} if i is even, otherwise set v i = {1, 2 + 2.1, 2 + 2.2, . . . , 2 + 2.(k -1)}.
-

If 2k ≤ i ≤ n -2 then, set v i = {1, 1 + 2.1, 1 + 2.2, . . . , 1 + 2.(k -2), i + 1}.
Now, for each 2 ≤ i ≤ n -2, we know that σ i (1) = 1 + i and therefore, 1 + i ∈ σ i (v i ) which implies that {v i , σ i (v i )} is not an edge of KG(n, k) 2-stab . Thus, for 2 ≤ i ≤ n -2, σ i is not a shift of KG(n, k) 2-stab .

• Reflexions. We consider two cases:

-Case n odd. For each 1 ≤ i ≤ n, let v i be a vertex in KG(n, k) 2-stab such that i ∈ v i . Trivially, such vertex v i always exists. Now, we know that i is a fix point under the permutation ρ i and thus, i ∈ ρ i (v i ) which implies that {v i , ρ i (v i )} is not an edge of KG(n, k) 2-stab . Thus, for 1 ≤ i ≤ n, ρ i is not a shift of KG(n, k) 2-stab .

-Case n even. Analogous to the previous case, we can show that ρ i is not a shift of

KG(n, k) 2-stab , for 1 ≤ i ≤ n 2 . Now, for each 1 ≤ i ≤ n 2 , let v i = {i + 1, i + 1 + 2.1, i + 1 + 2.2, . . . , i + 1 + 2.(k -2), i -2}. Clearly, v i is a 2-stable set, since i + 1 + 2.(k -2) and i -2 are not consecutive integers modulo [n]. So, v i is a vertex of KG(n, k) 2-stab such that {i + 1, i -2} ⊆ v i . However, {i + 1, i -2} ⊆ δ i (v i ) which implies that {v i , δ i (v i )} is not an edge of KG(n, k) 2-stab . Thus, for 1 ≤ i ≤ n 2 , δ i is not a shift of KG(n, k) 2-stab .
Larose et al. [START_REF] Larose | On normal Cayley graphs and Hom-idempotent graphs[END_REF] showed the following useful results:

Proposition 1 (Proposition 2.3 in [7]). A graph G is hom-idempotent if and only if G ↔ Cay(Aut(G • ), S G • ),
l i (v) = ks + 1. Lemma 3. Let s, k ≥ 2.
The number of vertices of the graph KG(ks+1, k) s-stab is equal to ks+1.

Proof. Again, let C be the cycle on ks + 1 points labeled by integers 1, 2, . . . , ks + 1 in the clockwise direction. From Lemma 2, we have that each vertex of KG(ks + 1, k) s-stab is uniquely determined by a clockwise circular interval of length s + 1 in C. Trivially there exist ks + 1 distinct clockwise circular intervals of length s + 1 in C and the lemma holds.

Proposition 3. Let s, k ≥ 2. Then, G(ks + 1, k) KG(ks + 1, k) s-stab .
Proof. Let C be a cycle on ks + 1 points. We assume that the vertices of G(ks + 1, k) are disposed over C in clockwise increasing order from 0 to ks. By Lemmas 2 and 3, we are able to define the application φ : G(ks + 1, k) → KG(ks + 1, k) s-stab as follows: let u be a vertex of G(ks + 1, k) such that u = jk + i, where 0 ≤ j ≤ s -1 and 0 ≤ i ≤ k -1. Then, φ(u) = {u 1 , . . . , u k } where,

u r = j + 1 + (r -1)s, if 1 ≤ r ≤ k -i j + 2 + (r -1)s, if k -i + 1 ≤ r ≤ k.
Finally, define φ(ks) = {s + 1, 2s + 1, . . . , ks + 1}. It is not difficult to prove that φ is a bijective function. It remains to show that φ is indeed a graph isomorphism. Let u, v be two vertices in C(ks + 1, k). In the sequel, we assume that v > u. In fact, if u > v we can always swap u and v. Let u = jk + i, where 0 ≤ j ≤ s -1 and 0

≤ i ≤ k -1. Let t = v -u, where 1 ≤ t ≤ k -1. By construction, φ(v) \ {v p : k -i -t + 1 ≤ p ≤ k -i} ⊂ φ(u),
-u = k or v -u = k(s -1) + 1 then, φ(u) ∩ φ(v) = ∅. Therefore, let t = v -u,
where k + 1 ≤ t ≤ k(s -1) and assume that there exists r ∈ [k] such that u r ∈ φ(u) ∩ φ(v). We consider the following cases:

• Case 1 ≤ r ≤ k -i. As u = jk + i then, v = jk + i + xk + y, where 0 ≤ j ≤ s -1, 0 ≤ i ≤ k -1, 1 ≤ x ≤ s -1, and 0 ≤ i + y ≤ k -1. If u r = v r , with 1 ≤ r ≤ k -(i + y)
then, u r = j + 1 + (r -1)s = j + x + 1 + (r -1)s = v r and thus x = (r -r )s. As x ≥ 1 then, r -r > 0 and thus x ≥ s which is a contradiction. If k -(i + y) + 1 ≤ r ≤ k then, j + 1 + (r -1)s = j + x + 2 + (r -1)s which implies that x = (r -r )s -1. However, r ≤ k -i and r ≥ k -i -y + 1 and so, x ≤ (y -1)s -1. Moreover, as x ≥ 1 then, y ≥ 2 and thus x = s -1, but it implies that t ≥ k(s -1) + 2 which is a contradiction.

• Case k -i + 1 ≤ r ≤ k. As the previous case, v = (j + x)k + (i + y), with 1 ≤ x ≤ s -1 and 0 ≤ i + y ≤ k -1. If u r = j + 2 + (r -1)s = j + x + 1 + (r -1)s = v r , with 1 ≤ r ≤ k -(i + y) then, x = (r -r )s + 1 which implies that x ≥ (y + 1)s + 1 ≥ s, that is a contradiction. If k -(i + y) + 1 ≤ r ≤ k then, j + 2 + (r -
1)s = j + x + 2 + (r -1)s. Then, x = (r -r )s and as x ≥ 1 then r -r > 0, but it implies that x ≥ s which is again a contradiction. In fact, it is not hard to see that D 2n is a subgroup of AUT(C

(r) n ). Let α ∈ AUT(C (r) n ) and let N [i] denote the closed neighborhood of a vertex i in C (r)
n . Assume that α sends two consecutive vertices i, j in the cycle to non consecutive vertices i , j in the cycle, respectively. Observe that

|N [i] ∩ N [j]| = 2r, but |N [i ] ∩ N [j ]| ≤ 2r -1 since there exists at least two vertices in N [i ] \ N [j ].
Therefore, any automorphism of C (r) n sends consecutive vertices to consecutive vertices. From this fact it is easy to see that any automorphism of C Note that all Cayley graphs on abelian groups are normal, and thus hom-idempotents. In particular, the circulant graphs are Cayley graphs on cyclic groups (i.e., cycles, powers of cycles, complements of powers of cycles, complete graphs, etc). Therefore, by Proposition 3 and Lemma 4, we have the following result. Theorem 3. Let s, k ≥ 2. Then, KG(ks + 1, k) s-stab is hom-idempotent.

  where the arithmetic operations are taken modulo [k]. As t ≤ k -1 then, we must have that φ(u) ∩ φ(v) = ∅. Analogously, let k(s-1)+2 ≤ v-u ≤ ks and let t = ks+1-(v-u). Then, φ(u)\{u p : k-i-t+1 ≤ p ≤ k-i} ⊂ φ(v), where the arithmetic operations are taken modulo [k]. Again, as t ≤ k -1 then, we must have that φ(u) ∩ φ(v) = ∅. Notice that, by the previous argument, if v

nRemark 1 .

 1 ) is also D 2n . For r ≥ 1 and n ≥ 2r + 3 the automorphism group of C (r) n is the dihedral group D 2n .

n

  belongs to D 2n . Therefore, by Proposition 3, the automorphism group of the graph KG(ks + 1, k) s-stab is the dihedral group D 2(ks+1) . However, the problem to compute the automorphism group of the graph KG(n, k) s-stab for k ≥ 2, s > 2, and n > ks + 1 is still open.Another direct consequence of Proposition 3 is that χ(KG(ks + 1, k) s-stab ) = s + 1. This fact has been found also by Meunier (see Proposition 1 in[START_REF] Meunier | The chromatic number of almost stable Kneser hypergraphs[END_REF]). Let Cay(A, S) be a Cayley graph. If a -1 Sa = S for all a ∈ A, then Cay(A, S) is called a normal Cayley graph.Lemma 4 ([5]). Any normal Cayley graph is hom-idempotent.

  Lemma 2. Let s, k ≥ 2 and let v = {v 1 , v 2 , . . . , v k } be a vertex of KG(ks + 1, k) s-stab . Then, l i (v) ∈ {s, s + 1} for all i ∈ [k]. Moreover, there exists exactly one i ∈ [k] such that l i (v) = s + 1.Proof. By definition, l i (v) ≥ s for any i ∈ [k]. The result follows from the fact that k i=1

where Aut(G • ) denotes the automorphism group of the core G • of G and S G • denotes the set of all shifts of G • .

  Therefore, vertices u, v in C(ks + 1, k) are adjacent if and only if vertices φ(u), φ(v) in KG(ks + 1, k) s-stab are adjacent.Notice that if n ≥ 2r + 3 the automorphism group of the r th power of a n-cycle, C

	(r) n , is the
	dihedral group D 2n (see Remark 1) and thus, AUT(C

* Partially supported by MathAmSud Project 13MATH-07 (Argentina-Brazil-Chile-France) and by the French ANR JCJC CombPhysMat2Tens grant.

Theorem 1 (Theorem 5.1 in [START_REF] Larose | On normal Cayley graphs and Hom-idempotent graphs[END_REF]). Let G be a χ-critical graph. Then G is weakly hom-idempotent if and only if it is hom-idempotent. Proposition 2. Let n ≥ 2k + 2 and let G denotes the graph KG(n, k) 2-stab . Then, G → Cay(Aut(G), S G ), where S G are the shifts of G.

Proof. We know that the automorphism group of the graph KG(n, k) 2-stab is the dihedral group D 2n on [n]. Moreover, by Lemma 1, we known that the only two shifts of KG(n, k) 2-stab are the circular permutations σ 1 = (1, 2, . . . , n -1, n) and its inverse permutation σ n-1 = (σ 1 ) -1 = (n, n -1, . . . , 2, 1). Therefore, the Cayley graph Cay(D 2n , {σ 1 , (σ 1 ) -1 }) is a 2-regular graph, that is, each vertex (i.e. each automorphism in D 2n ) has exactly two neighbors. In fact, notice that the vertex identity σ 0 has as neighbors the vertices σ 1 and σ n-1 , and as it's well known, any Cayley graph is vertex-transitive, and thus, any vertex in Cay(D 2n , {σ 1 , (σ 1 ) -1 }) has exactly two neighbors. So, Cay(D 2n , {σ 1 , (σ

As mentioned in the previous section, we know that any 2-stable Kneser graph is a core. Therefore, by Propositions 1 and 2, and by Theorem 1, we have the following result.

Theorem 2. For any n ≥ 2k + 2, the 2-stable Kneser graphs KG(n, k) 2-stab are not weakly hom-idempotent. The pth power of a graph G, that we denoted by G (p) , is the graph having the same vertex set as G and where two vertices are adjacent in G (p) if the distance between them in G is at most equal to p, where the distance of two vertices in a graph G is the number of edges on the shortest path connecting them.

Let n ≥ 2k be positive integers. The Cayley graphs Cay(Z n , {k, k + 1, . . . , n -k}), that we denoted by G(n, k), are known as circular graphs [START_REF] Vince | Star chromatic number[END_REF][START_REF] Hahn | Graph homomorphisms: structure and symmetry[END_REF], where Z n denote the cyclic group of order n. It is well known that the Kneser graph KG(n, k) contains an induced subgraph isomorphic to G(n, k). In fact, let C(n, k) be the subgraph of KG(n, k) obtained by restricting the vertex set of KG(n, k) to the shifts modulo [n] of the k-subset {1, 2, . . . , k}, that is, {1, 2, . . . , k}, {2, 3, . . . , k + 1}, . . . , {n, 1, 2, . . . , k -1}. Define φ : G(n, k) → C(n, k) by putting φ(u) = {u + 1, u + 2, . . . , u + k} where the arithmetic operations are taken modulo [n]. Clearly, φ is a graph isomorphism. Notice also that the graph G(n, k) is isomorphic to the graph C (k-1) n , i.e. the complement graph of the (k-1)th power of a cycle C n . Vince [START_REF] Vince | Star chromatic number[END_REF] has shown that χ(G(n, k)) = n k .

In the remaining of this section, we will always assume w.l.o.g. that any vertex

If C is the cycle on ks + 1 points labeled by integers 1, 2, . . . , ks + 1 in the clockwise direction and v = {v 1 , v 2 , . . . , v k } is a vertex of the s-stable Kneser graph KG(ks + 1, k) s-stab , then l i (v) gives the distance in the clockwise direction between v i and v i+1 in C.