C S Tang

A M Tang

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

P Delage

C Schroeder

B Shi

A Study of the Hydro-Mechanical Behaviour of Compacted Crushed Argillite

Keywords: Compacted crushed argillite, radioactive waste disposal, swelling pressure, microstructure, grain size effect, hydro-mechanical coupling

The argillite extracted from Bure site (France) is proposed, after being crushed and compacted, as a possible sealing and backfill material in the geological high-level radioactive waste disposal. In this study, the effects of the grain size distribution and the microstructure on the hydro-mechanical behaviour of the compacted crushed argillite have been investigated. The volume change properties were investigated by running one-dimensional compression tests under constant water content (2.4-2.8%) with loading-unloading cycles. Under various vertical stresses, water flooding tests were carried out in constant volume condition. Depending on the vertical stress level, either swelling or collapse behavior was observed in the sense that vertical stress increased or decreased upon flooding respectively. A clear effect of grain size distribution has been also identified: finer samples exhibit stiffer compression behaviour and higher swelling potential. To provide a microstructure insight into the macroscopic behaviour feature observed, both mercury intrusion porosimetry (MIP) and scanning electron microscopy (SEM) observations were performed, evidencing that: (i) at the same dry density, the size of inter-aggregates pores is larger for the coarser crushed material; (ii) mechanical compression only reduces the inter-aggregate porosity in the stress range considered; (iii) the micro-mechanisms governing the flooding under constant-volume condition include the swelling of the clay particles, the increase of the intra-aggregate pores and the collapse of the inter-aggregates pores. The results show a strong effect of the grain size distribution on the hydro-mechanical behaviour and thus the close link between the microstructure and the hydro-mechanical behaviour.

INTRODUCTION

Deep argillaceous formations are considered as potential host rock for radioactive waste disposal in many countries such as Belgium, Germany, France, Japan and Switzerland. In order to ensure the safety of the whole storage system, it is of prime importance to well understand the involved clay properties on different time scales: during the construction, during the exploitation and during the long term storage life of the waste [START_REF] Landais | Clays in natural & engineering barriers for radioactive waste confinement[END_REF]. In addition to the coupled thermo-hydro-mechanical behaviour of the host rocks, the engineering properties of backfilling/sealing materials are also an important issue to be investigated because these materials are usually used to avoid preferential pathways for water, gas and radionuclide migration, and they correspond therefore to an important part of the high-level radioactive waste underground repository safety system. In the past decades, bentonites have been widely studied as possible backfilling and sealing materials because of its low permeability, adequate self-sealing potential and high retention capacity [START_REF] Lloret | Mechnical behaviour of heavily compacted bentonite under high suction changes[END_REF][START_REF] Tang | Controlling suction by the vapour equilibrium technique at different temperatures and its application in determining the water retention properties of MX80 clay[END_REF]Hoffman et al., 2007). In France, ANDRA (the French Radioactive Waste Management Agency) has been constructing an Underground Research Laboratory (URL) at Bure site (eastern France, at 445-490 m depth) in order to investigate the suitability of the Callovo-Oxfordian (COx) argillaceous formation for the disposal of high-level radioactive waste [START_REF] Lebon | Knowledge of the three French underground laboratory sites[END_REF]. Unlike in the currently well-accepted design using compacted bentonite, the excavated COx argillite has been proposed at this URL as a possible sealing/backfilling material to isolate the waste canisters from the concrete plug and the access galleries. The main advantages of this design are: (i) the recycling of the excavated COx argillite as sealing/backfilling material reduces the negative impact on the environment; (ii) replacing commercial bentonite by excavated COx argillite reduces the cost; (iii) problems related to the mineralogical or physico-chemical incompatibility between the host rock and the engineered barrier can be avoided [START_REF]Référentiel des matériaux d'un stockage de déchets à haute activité et à vie longue -Tome 4: Les matériaux à base d'argilites excavées et remaniées[END_REF]. In addition, a series of tests have been carried out to investigate the geotechnical, hydraulic, mechanical and compaction behavior of the crushed COx argillite. Based on the obtained results, [START_REF]Référentiel des matériaux d'un stockage de déchets à haute activité et à vie longue -Tome 4: Les matériaux à base d'argilites excavées et remaniées[END_REF] concluded that the COx argillite is suitable to be used as a sealing/backfilling material.

It is expected that after installation of engineered barriers, water infiltration from the host rock takes place and the compacted argillite will expand and develop swelling pressure under nearly confined conditions. This swelling pressure should be high enough to ensure the sealing effectiveness. On the other hand, the swelling properties of the compacted argillite must be compatible with the in-situ stress state, i.e. the swelling pressure should not excess the in-situ minor principal stress (about 7 MPa at Bure URL) so as to ensure the mechanical stability of the system.

The present work aims at investigating the hydro-mechanical properties of compacted crushed COx argillite. Two crushed materials with different grain size distributions (GSD) were considered, that were obtained by crushing the excavated COx argillite with two different procedures. The compressibility of the two crushed materials was investigated by running one-dimensional compression tests at constant water content (2.4 -2.8%) with loading-unloading cycles. After compaction, water flooding was carried out under constant volume conditions. The effects on the hydro-mechanical behavior of the axial stresses ap-plied and the grain size distributions were identified. Finally, the soil microstructure changes before and after flooding were investigated based on the observations at mercury intrusion porosimetry (MIP) and scanning electron microscopy (SEM).

SOIL STUDIED

The soil studied was the Callovo-Oxfordian (COx) argillite taken during the construction by excavation of the ANDRA URL of Bure. It contains 40 -45% clay minerals with predominance of illite-smectite interstratified minerals, 30% carbonates and 25 -30% quartz and feldspar. The in-situ water content is 2.8 -8.7 %; the bulk density is 2.32 -2.61 Mg/m 3 and the specific gravity is 2.70 [START_REF] Fouché | Fabric control on strain and rupture of heterogeneous shale samples by using a non-conventional mechanical test[END_REF]. After [START_REF] Zhang | Experimental study on the thermo-hydromechanical behaviour of indurated clays[END_REF], the swelling pressure of COx argillite reaches 10 MPa.

The COx argillite was crushed following two different procedures, leading to two soils: (i) soil A was the coarser one with the maximum grain size of 8 mm; its air-dried water content was 2.8 %; (2) soil B was the finer one with the maximum grain size of 0.25 mm; its air-dried water content was 2.4 %. To obtain soil A, the excavated COx argillite taken from the underground laboratory was crushed using an industrial crusher to reach its maximum grain size of 8 mm and then air-dried. For soil B, a small quantity of soil A was then ground using a laboratory crusher to obtain a finer powder (maximum grain size of 0.25 mm). The particle size distributions of the two soils determined by dry sieving are presented in Fig. 1. Soil A contains 18% fine grains (< 0.08 mm) while soil B contains 70 % fine grains. In addition, the grain size distribution was also determined by dry-sieving method after washing [START_REF] Afnor | Soil investigation and testing -Granulometric analysis -Dry sieving method after washing[END_REF] for elements larger than 80 µm and by hydrometer method [START_REF] References Afnor | Soil investigation and testing -Granulometric analysis -Hydrometer method[END_REF] for elements smaller than 80 µm. Interestingly, the two curves are similar and show a large fraction of fine grains. More than 95 % grains are smaller than 0.08 mm. This observation constitutes the first illustration of the water content-dependent nature of the compacted crushed argillite: the argillite grains disaggregate when the water content is increasing, giving rise to smaller grains.

EXPERIMENTAL METHODS

Experimental set-up

Fig. 2 presents the experimental set-up used to study the compaction behaviour and the hydro-mechanical coupling. An oedometer cell (50-mm inner diameter) was placed on the top plate of the displacement shaft, and can moves upwards/downwards together with the displacement shaft. A force transducer was used to monitor the vertical stress σ v while a displacement transducer was used to monitor the vertical strain ε v . A volume/pressure controller was connected to the bottom base of the cell for the purpose of water flooding. A porous disc was placed between the specimen and the piston allowing water/air flow from the soil sample through the gap between the piston and the oedometer ring. All data were recorded by a data logging system. Note that the deformation of soil during the test was measured directly by the displacement transducer which was fixed on the piston (Fig. 2). The deformation of the piston and the oedometer cell during compaction was found to be negligible. Therefore, the measured displacement by the transducer was directly used to calculate the axial strain of soil specimen.

Experiment procedure

During the compaction, the crushed material was firstly placed inside the oedometer cell. After installation of the piston, the press was manually run to statically compress the soil to an initial dry density ρ d of 1.5 Mg/m 3 , the corresponding initial height of the sample being 2.67 cm. The compaction was then started at a constant rate of 0.05 mm/min. At the end of compaction, the axial stress was unloaded to a given value, the specimen was then flooded under constant-volume condition. The constant-volume condition was ensured by fixing the displacement shaft and, if necessary, by manually adjusting it according to the records of the displacement transducer. During flooding, the full saturation of the specimen was checked by using an absorbing paper to check whether free water flows out from the gap between the piston and oedometer ring. Table 1 summarizes the test program.

Description of the test program

For the tests E01, E02, E03, and E04, when ρ d reached 1.7 Mg/m 3 , the soil was unloaded until σ v was equal to 0.5 MPa and then reloaded at the same rate. This unloading/reloading cycle was also repeated at ρ d equal to 1.8 and 1.9 Mg/m 3 . When ρ d reached 2.0 Mg/m 3 , the soil was unloaded to σ v of 7 or 0.5 MPa prior to flooding under constant-volume conditions by injecting distilled water at a vertical stress of 100 kPa. Note that in practice, before the backfilling material is placed in the repository, it is first compacted to a target density. The unloading steps performed in this investigation aims to understand the rebound behavior of the compacted backfilling material. At the end of each loading step, about 1 hour was given to reach apparent equilibrium. The axial strain change was checked by the displacement transducer. The corresponding dry density and void ratio were therefore calculated according to the recorded displacement values during loading/unloading procedures.

For the tests E05 and E06, the soil sample was compacted directly to ρ d = 2.0 Mg/m 3 without any loading/unloading cycles, and was flooded following the same procedures that were applied to the tests E01-E04.

For the tests E07 and E08, a procedure similar to that of swell-consolidation test described by [START_REF] Basma | Laboratory assessment of swelling pressure of expansive soils[END_REF] was applied: the initially compacted sample was wetted and allowed to fully swell under a low sitting load. After that a standard consolidation test was conducted by applying incremental loads. The load required to revert the specimen to its initial void ration (or height) is defined as the swelling pressure. The high-pressure load frames described by [START_REF] Marcial | On the high stress compression of bentonites[END_REF] was used in these two tests. The soil was first compacted to ρ d = 2.0 Mg/m 3 using the equipment described in Fig. 2. After unloading, the oedometer cell with the compacted specimen was removed to the load frame allowing application of constant σ v and monitoring the soil vertical strain. A low σ v of 0.1 MPa was firstly applied and then water flooding was carried out. After stabilization of the soil strain, σ v was increased in steps to 0.2, 0.4, 0.9, 1.7, 3.5, 6.9, 13.8, 27.7 and 55.3 MPa, each loading step having been maintained for 24 h. The same steps were adopted for unloading. Note that this duration is recommended by the French Standard [START_REF] Afnor | Geotechnical investigating and testing, Laboratory testing of soils, Part 5: Incremental loading odometer test[END_REF] for strain stabilization.

Microstructural observation

Soil properties at the microscopic level are directly related to its macroscopic engineering behaviour and physical properties. Basically, measurements and observations at the microstructural level involving clay units and their aggregations and pore size distribution characteristics are very important, since they help better understand the behaviour of the soil under various hydro-mechanical stress state conditions.

In order to analyse the soil microstructure changes during these hydro-mechanical loadings, MIP tests were performed on soil compacted at ρ d equal to 1.8 and 2.0 Mg/m 3 following the static compaction procedure (including the unloading steps) described previously.

These tests were also performed on soil compacted at ρ d equal to 2.0 Mg/m 3 and flooded under constant volume conditions, where the initial applied vertical stress σ v was equal to 7 or 0.5 MPa. For the purpose of comparison, an additional test was equally performed on intact COx argillite at an initial water content w = 7.0 %. In addition, SEM tests were performed on soil samples having ρ d equal to 2.0 Mg/m 3 before and after flooding under an initial vertical stress σ v equal to 7 or 0.5 MPa at constant volume conditions.

For each MIP and SEM test, two parallel soil samples were prepared by repeating the hydro-mechanical loading in the oedometer cell, and then were rapidly frozen using liquid nitrogen previously vacuum-cooled below its boiling temperature (-196 °C) in order to avoid any boiling during immersion of the sample. Freezing was also accelerated by using small sticks of soil (1-2 g in weight). Finally, the frozen samples were subjected to sublimation in a true vacuum (10 -2 mm Hg) so as to eliminate the water without the formation of air-water capillary menisci, which cause volume shrinkage on drying.

During MIP test, the sample is placed in a dilatometer that is put under vacuum. Mercury intrusion was then performed by progressively increasing mercury pressure up to 200 MPa. Due to the pressure increase, mercury progressively penetrates into the voids of sample, starting, at lower pressures, with the largest pores and ending up at highest pressures by penetration into the smallest pores that can be reached. The volume of the pores penetrated during the increase in pressure is continuously monitored by a system that follows the decrease of the mercury level in the dilatometer. In this case, blank tests are needed to correct errors associated with the system compressibility and temperature effects (see [START_REF] Delage | Ageing effects in a compacted bentonite:a microstructure approach[END_REF] for more detail). Prior to the SEM test, the freeze-dried samples were coated with aurum to increase the electric conductivity.

EXPERIMENTAL RESULTS

Compression behaviour

The variations of ρ d with σ v during the compression of the crushed materials are shown in Fig. 3. The compression curves are similar for each soil: curves E01, E02 and E05 for the coarser soil A and curves E03, E04 and E06 for the finer soil B. A significant difference is however observed between the two soils. The pre-compression stress that corresponds to the pressure required to obtain the initial ρ d of 1.5 Mg/m 3 is lower than 0.4 MPa for soil A while that of soil B is close to 2 MPa. σ v required to reach ρ d equal to 2.0 Mg/m 3 for soil A is 27 MPa in test E01, 24 MPa in test E02 and 37 MPa in test E05, significantly lower than that of soil B: 47 MPa in test E03, 39 MPa in test E04 and 47 MPa in test E06.

In Fig. 4, the results of these tests are presented in the plane of void ratio (e) versus logσ v . During loading, when σ v exceeded the pre-compression stress, e decreased drastically and a clear linear relationship can be observed between e and logσ v . The slopede/dlogσ v corresponds to the compression index at the initial water content, C cwi . For soil A, C cwi is equal to 0.238 for E01, 0.240 for E02 and 0.229 for E05. Larger values were obtained for soil B: C cwi is equal to 0.316 for E03, 0.317 for E04 and 0.384 for E06.

In a similar fashion, the swelling index at the initial water content, C swi , was determined that corresponds to the slope -de/dlogσ v on the unloading paths. This operation was done for different ρ d values: 1.7, 1.8, 1.9 and 2.0 Mg/m 3 . In Fig. 5, the variations of C swi versus ρ d are plotted for tests E01, E02, E03 and E04. The results show that C swi increases almost linearly with ρ d . The difference between the two soils is not clear. Compared to C cwi , C swi seems to be of one order of magnitude smaller.

Hydro-mechanical behaviour

After compaction to ρ d equal to 2.0 Mg/m 3 , the soil was unloaded to σ v equal to 0.5 or 7 MPa, and the final dry density was in the range of 1.98-1.99 Mg/m 3 . The displacement shaft was then blocked and distilled water was subsequently injected at a pressure of 100 kPa from the bottom of the oedometer cell. Fig. 6 presents the changes of σ v and e versus time (t) during water flooding. Note that even though the "constant-volume" conditions were imposed by fixing the displacement shaft, slight changes in soil volume were recorded by the displacement transducer because of the deformability of the system (connection between the load cell and the piston, the soil specimen, the base of the oedometer cell, etc., see Fig. 2).

For test E01, flooding the soil A decreased the vertical stress σ v from 7.4 MPa to 5.3 MPa after 120 h (Fig. 6a). A slight decrease of e was equally observed, from 0.364 to 0.360, which corresponds to a volumetric strain of 0.3%. In tests E03 and E06, soil B was equally flooded at high σ v (7.6 MPa for E03, Fig. 6b; and 8.0 MPa for E06, Fig. 6c). The results obtained from the two tests were similar: σ v decreased, then increased and finally decreased to reach a stabilization value of 7.2 MPa; the changes of e were in the same fashion as σ v changes. The final volumetric strain was small, less than 0.15%. Compared to soil A, soil B had a higher final σ v after water flooding.

For tests E02, E04, and E05, the soil specimens were flooded under lower σ v . In the test E02, due to the failure of volume control, the results were not presented. A similar procedure was then applied in test E05, the results were presented in Fig. 6e. It can be seen that flooding soil A resulted in a quick increase of σ v from the initial value of 0.69 MPa to an equilibrium value of 4.23 MPa after 20 h. A slight variation of e was also observed and the corresponding volumetric strain is lower than 0.15%. In test E04, water flooding was performed on soil B. The results show a quick increase of σ v in the beginning (Fig. 6d); e increased from 0.369 to 0.371 after 6 h. A manual adjustment was done at this moment to bring the void ratio e back to its initial value. After 50 h, e was equal to 0.367 and σ v was equal to 5.08 MPa. Compared to soil A flooded at low stress, soil B showed a higher final stress: 5.08 MPa against 4.23 MPa (E05).

The results of oedometer tests E07 (soil A) and E08 (soil B) are presented in Fig. 7. The soil was firstly compacted at ρ d equal to 2.0 Mg/m 3 following the same procedure of tests E05 and E06. After compaction, the cell was put in an oedometer frame to perform swellconsolidation tests. Fig. 7a presents changes in void ratio e versus time. An initial σ v of 0.1 MPa was first applied prior to water flooding. Swelling was observed in the two tests: e increased from 0.350 to 0.442 for test E07 (soil A, after 100 h) and to 0.503 for test E08 (soil B, after 300 h). When the swelling stabilized, subsequent loading/unloading steps were applied, each step having been maintained for 24 h (this duration appeared to be sufficient for the stabilisation). In Fig. 7b, the values of e at equilibrium are plotted versus the logarithm of σ v . The compressibility parameters were determined from these curves: for soil A, the compression index C c and the swelling index C s are respectively equal to 0.172 and 0.082; for soil B, C c and C s are respectively equal to 0.164 and 0.075. After [START_REF] Basma | Laboratory assessment of swelling pressure of expansive soils[END_REF], this test can be also used to determine the soil swelling pressure that is considered as the vertical stress required to bring the void ratio back to its initial value. As shown in Fig. 7b, a swelling pressure of 2.5 MPa was obtained for soil A (E07), and 5.0 MPa for soil B (E08).

Mercury intrusion porosimetry

Before presenting the results from the mercury intrusion porosimetry investigations, the different clay microstructural levels and the corresponding terminologies considered in this study are first introduced. The description is based on that given by [START_REF] Delage | Ageing effects in a compacted bentonite:a microstructure approach[END_REF], where various clay microstructural levels were summarized in detail. In a compacted unsaturated soil, clay particles (made up of stacked clay layers) are aggregated together to form a typical aggregate structure (also termed as double structure). In this structure, various levels of porosity can be indentified: (i) planar interlayer spaces between the clay layers inside the clay particles, interpreted as inter-layer or intra-particle pores;

(ii) small voids between clay particles inside the aggregates, interpreted as inter-particle or intra-aggregate pores;

(iii) large voids between the aggregates made up of clay particles, interpreted as interaggregate pores.

Fig. 8 presents the results of MIP tests obtained on soil A and on the intact COx argillite . The intruded mercury void ratio (e m ) was defined as the ratio of mercury intrusion volume V m to soil solid volume V s :

s m m V V e =
(1)

The pore entrance diameter (d) was determined from the intrusion pressure p by assuming a cylindrical pore shape:

p d θ σ cos 4 = (2)
where σ = 0.484 N/m is the mercury-solid interfacial tension and θ = 141.3° is the mercury-solid contact angle (see [START_REF] Delage | Ageing effects in a compacted bentonite:a microstructure approach[END_REF] for more detail). It can be noted from the e m -logd plot (Fig. 8a) that the final values of e m are slightly lower than the values of e. This can be mainly explained by the limited range that the MIP technique can cover, i.e. from 0.008 to 15 µm (corresponding to the range of p from 200 MPa to 0.1 MPa); the value of e m does not include the pores larger than 15 µm (which may exist in the sample compacted at ρ d equals 1.8 Mg/m 3 , e equals 0.50) and the pores smaller than 0.008 µm.

The pore size distribution curves, de m /dlog(d) versus logd, allow the analysis of the soil microstructure. For the intact argillite sample, only a population of intra-aggregate pores having a diameter d close to 0.05 µm was observed (Fig. 8f). On the contrary, for the samples of soil A, two populations of pores were observed: intra-aggregate pores that are smaller than 0.1 µm and inter-aggregates pores lager than 0.1 µm. For the soil sample compacted at ρ d equal to 1.8 Mg/m 3 (e equal to 0.50), the population of inter-aggregates pores was close to 3 µm while that of intra-aggregate pores was close to 0.03 µm (Fig. 8b).

For the sample compacted at ρ d equal to 2.0 Mg/m 3 (e equal to 0.35), the population of inter-aggregates pores moved leftwards to 1.5 µm while that of intra-aggregate pores did not change (Fig. 8c). Subsequent flooding under constant-volume conditions (Fig. 8d,e) decreased the modal size of inter-aggregates pores to 0.3 -0.4 µm, the changes in intraaggregate pores size having been not significant. Fig. 8d shows that flooding under high stresses (initial σ v equal to 7 MPa) created larger intra-aggregate pores whose size reached 0.05 µm.

The results obtained from soil B and from the intact COx argillite are presented in Fig. 9. As observed previously on soil A, the values of e m for soil B were slightly lower than the values of e. The pore size distribution curve, de m /dlog(d) versus logd, of soil B compacted at ρ d equal to 1.8 Mg/m 3 (e equal to 0.50) shows a clear population of inter-aggregates pores of 0.7 µm (Fig. 9b). The population of intra-aggregate pores observed previously on soil A was no longer observed on soil B. Fig. 9c shows that compaction at ρ d equal to 2.0 g/m 3 (e equal to 0.35) decreased the size of the population of inter-aggregates pores to 0.4 µm. Subsequent flooding under constant-volume conditions at high stresses (initial σ v equal to 7 MPa, Fig. 9d) slightly decreased the size of the inter-aggregates pores population to 0.3 µm. At low stresses (initial σ v equals 0.5 MPa, Fig. 9e), this decrease of the population size of inter-aggregates pores when flooding was found to be not significant.

Scanning electron microscope

SEM tests were performed on the soil samples compacted at ρ d equal to 2.0 Mg/m 3 before and after flooding at 0.5 and 7 MPa vertical stresses. The pictures were taken on two scales and presented in Fig. 10 (125 x 100 µm) and Fig. 11 (25 x 20 µm), respectively.

Fig. 10a shows that the compacted sample of soil A was constituted of several clay aggregates. The size of the aggregates varies from about 1 µm to larger than 10 µm. Note that the aggregates larger than 100 µm could not be observed with this magnification (the size of picture taken is 125×100 µm). Inter-aggregates pores larger than 2 µm were obvious in this figure. The aggregate circled in Fig. 10a is enlarged in Fig. 11a, showing that it is made up of clay particles of about 1 µm in diameter.

The microstructure of soil A changed completely after flooding under constant-volume conditions (Fig. 10c,e): clay aggregates and large inter-aggregate pores were no longer clearly observed; only small inter-aggregate pores (smaller than 1 µm) were observed in Fig. 10e. The enlargement of these soil samples (Fig. 11c,e) shows that the clay particles were relatively more separate than before.

In case of soil B (Fig. 10b), several clay aggregates smaller than 10 µm were observed. The sizes of inter-aggregates pores were not apparent at this magnification (size of picture: 125×100 µm). At a higher magnification (Fig. 11b, size of picture 25×20 µm), several inter-aggregates pores smaller than 1 µm can be observed. The structure of clay aggregates formed by clay particles is also clear in Fig. 11b. After flooding, Fig. 10d and Fig. 10f show that the clay particles of soil B are more separate than before. Inter-aggregates pores smaller than 1 µm can be observed at a higher magnification (Fig. 11f, size of picture 25×20 µm).

DISCUSSION

Two crushed materials, A and B, crushed from the same COx argillite were studied in the present work. The grain size distribution curves determined by dry sieving show that soil A was coarser than soil B (Fig. 1). The results presented in Fig. 3 evidence that the compression behaviour of the crushed material was strongly affected by the grain size distribution.

For instance, the vertical stress σ v required to reach the initial dry density ρ d of 1.5 Mg/m 3 was higher for the finer soil: σ v = 2 MPa for soil B against σ v = 0.4 MPa for soil A. Furthermore, the vertical stress σ v required to reach the final dry density ρ d of 2.0 Mg/m 3 was also higher for the finer soil: σ v = 39 -47 MPa for soil B against σ v = 24 -37 MPa for soil (2008), showing that the dry density of a finer soil at standard Proctor optimum is lower. Moreover, Tien et al. (2004) studied the static compaction properties of various bentonite-sand mixtures and equally found that the compaction stress required to reach to same dry density was higher with higher bentonite weight ratios. This effect of grain size distribution on the compaction behaviour can be explained by the interactions between the soil grains at contact points in terms of friction and cohesion: the number of contact points being larger for the finer soil, a higher energy is needed to reach a considered density.

The results shown in Fig. 4 evidence that the compression index at initial water content C cwi of the finer soil (soil B) is larger than that of soil A. Similar effect of the soil structure on C cwi was equally observed by Smith et al. (1997) and Imhoff et al. (2004). In the present work, at low stresses, the finer soil B had higher void ratio; for instance, at σ v of 2 MPa, e is equal to 0.8 for soil B and 0.6 for soil A. This difference is closely related to the original structures of the two soils: assuming that the clay grains kept similar structure to that of intact COx argillite whose dry density is estimated at 2.25 Mg/m 3 , as soil B was crushed to smaller grain size (<0.025 µm) than soil A, when the same mass of soil was put in oedometer cell for compaction, the finer soil B must be looser with a larger initial void ratio, thus a higher compressibility. Under loading, the effect of the soil initial structure is progressively suppressed and the two compression curves tend to merge. Note that Tang et al. (2009) in-vestigated the compression behaviour of soil A in the same manner, and found that the compression index decreased when the sample was saturated.

Unloading/reloading cycles applied at various ρ d showed that the swelling index at constant water content (C swi ) increased with ρ d . As this parameter depends mainly on the compressibility of soil aggregates [START_REF] Tang | Thermo-mechanical behaviour of a compacted swelling clay[END_REF], this increase of C swi can be explained by the creation of micro-cracks by compression stresses on the soil aggregates which results in a decrease in rigidity [START_REF] Wright | Altération et rupture d'échantillon d'argilite lors de sollicitations thermomécaniques en conditions humides contrôlées[END_REF]. That explains equally why C swi was almost independent of the grain size distribution.

It is commonly accepted that hydration initially occurred in the hydrophilic zone of the clay, i.e. on the mineral surface and around the exchangeable cations in the interlayer spaces inside the clay aggregates. After [START_REF] Saiyouri | Microstructural approach and transfer water modeling in highly compacted unsaturated swelling clays[END_REF], the clay particles with a number of stacked clay layers are divided, when hydrated, into thinner ones that are made up of a smaller number of stacked layers. This is in agreement with the results presented in Fig. 1, which clearly show the decrease of the size of clay aggregates after soaking with water. The pictures obtained from SEM tests (Fig. 10 and Fig. 11) confirm that flooding separated the clay particles from the clay aggregates.

It has been observed that flooding compacted soils at constant-volume conditions induced a decrease of σ v when the initial σ v was high (7 MPa) and an increase of σ v when the initial σ v was low (0.5 MPa). The changes in swelling pressure of compacted expansive soils upon wetting under constant-volume conditions were also studied by [START_REF] Komine | Experimental study on swelling characteristics of sandbentonite mixture for nuclear waste disposal[END_REF] and [START_REF] Lloret | Mechnical behaviour of heavily compacted bentonite under high suction changes[END_REF], and similar observations were made. The observed phenomenon can be explained based on the double-structure model described by [START_REF] Gens | A framework for the behaviour of unsaturated expansive clays[END_REF], Alonso et al. (1999) and Sanchez et al. (2005), Tang and Cui (2009). On the one hand, wetting increases the interlayer distance (between clay particles) resulting in swelling of clay aggregates. On the other hand, this decreases the mechanical resistance of the structure. As a consequence, swelling of the clay aggregates induces an increase of confining stresses when the soil is wetted under constant-volume conditions at low initial stresses. On the contrary, when the soil is wetted under constant-volume conditions but at high initial stresses, wetting induces collapse of inter-aggregates pores and thus decrease of the confining stresses. The flooding-induced collapse of inter-aggregates pores was also evidenced by the MIP tests: the results in Fig. 8 and Fig. 9 show a decrease in size of the inter-aggregates pores population after flooding. The SEM pictures in Figs. 10 and 11 also illustrate these mechanisms. Microstructural investigations show that the initial vertical stress has a negligible effect on the microstructure of the soil after flooding. Actually, flooding the sample at low initial stresses (under constant-volume conditions) increases the vertical stress on one hand and decreases the yield stress (mechanical weakening) on the other hand. The collapse of interaggregates pores takes place when the vertical stress reaches the yield stress. As a result, event though flooding the sample under 0.5 or 7 MPa led to different responses in terms of vertical stresses, the consequences from the microstructural point of view are the same: collapse of inter-aggregates pores. [START_REF] Imbert | Hydro-mechanical response of a bentonite pellets/powder mixture upon infiltration[END_REF] also performed flooding tests in the oedometer cell under constant-volume conditions in order to monitor the swelling pressure of a bentonite pellets/powder mixture. Tests showed a common swelling pressure development pattern: an initial peak followed by a decrease of swelling pressure that eventually increased again. Maugis and Imbert (2007) performed a numerical analysis of these tests by using of the Barcelona basic model (BBM, see in Alonso et al., 1990) in order to interpret these results. The authors explained the swelling pressure variations, specially its decrease, by the yield stress decrease. This can be also used to explain the results obtained in tests E03 and E06 where the vertical stress (initially high) subsequently decreased, increased, and then decreased to reach stabilization (Fig. 6).

It has been observed that flooding the finer soil under constant-volume conditions at the initial σ v of 0.5 MPa induced a higher final value of σ v (for soil B, σ v equal to 5.08 MPa, test E04; for soil A, it equals 4.23 MPa for test E05; see Fig. 6). On the other hand, flooding the finer soil under constant-volume conditions at the initial σ v of 7 MPa equally induced a higher final value of σ v (for soil B, σ v equals 7.2 MPa, tests E03 and E06; for soil A, it equals 5.3 MPa, test E01). Furthermore, the swelling pressure determined by the swell-consolidation method was equally higher for the finer soil (Fig. 7b, 5.0 MPa for soil B, E08; 2.5 MPa for soil A, E07). It can be then concluded that the finer soil has a higher swelling potential than the coarser one. This effect of the grain size distribution may be explained by the following mechanisms: when crushing the argillite to a finer material, the initial structure of illite-smectite interstratified clay minerals may be more destructured and hence give more spaces for water penetration, increasing the swelling potential. However, it should be noted that, for a given soil, different methods represent different boundary conditions and can lead to different swelling pressures. For example, the swellconsolidation method can give higher swelling pressure than other methods in which the material is kept in a constant volume cell [START_REF] Al-Shamrani | Experimental study of lateral restraint effects on the potential heave of expansive soils[END_REF]. From a practical point of view, the boundary conditions of the backfilling material in the repository are highly complicated [START_REF] Siemens | Evaluation of the influence of boundary confinement on the behaviour of unsaturated swelling clay soils[END_REF], choosing an appropriate swelling testing technique to simulate the field conditions is essential for the design. In the present study, only two typical boundary conditions were considered, and further systematic investigation on the swelling pressure and the coupled hydro-mechanical behaviour of COx argillite will be carried in the next stage of the study.

MIP tests are usually performed to analyse the microstructure changes upon mechanical loading (Delage & Lefebvre, 1984; Lloret et al., 2003; Romero et al., 2003; Hoffmann et al., 2007; Tarantino & De Col, 2008). But this method usually results in underestimation of porosity due to the bottle neck effect [START_REF] Romero | Microstructure investigation in unsaturated soils: a review with special attention to contribution of mercury intrusion porosimetry and environmental scanning electron microscopy[END_REF]. It has been observed that the decrease of soil porosity during compaction is mainly attributed to the decrease in the size of inter-aggregates pores. This is in agreement with the results obtained in the present work (Fig. 8 and Fig. 9). As far as the microstructure changes upon wetting, few works have been carried out. Cui et al. (2002) analysed the microstructure changes of compacted expansive soil during wetting under constant-volume conditions. The results clearly show the decrease in inter-aggregates pores size with decrease in soil suction. Ferber et al. (2008) performed MIP tests on a compacted silty clay and a compacted high-plasticity clay subjected to flooding under a constant vertical stress σ v of 3 and 200 kPa. The results show that, for the silty clay, wetting induced collapse of inter-aggregates pores but did not affect the intra-aggregate pores. Nevertheless, for the high-plasticity clay, wetting induced not only the collapse of inter-aggregates pores but also the swelling of intra-aggregate pores.

Flooding resulted in inter-aggregates pores size decrease for both soil A and soil B. On the other hand, intra-aggregate pores were only observed on soil A; for the finer soil B, this population of intra-aggregate pores would be hidden by inter-aggregates pores having similar size because of the large amount of fine grains created during crushing. For soil A, the intra-aggregate pores was not influenced by the compaction; its size changed slightly after flooding. As explained previously, the size of intra-aggregate pores is closely related to the distance between the clay particles and thus to the clay water content, w. For the intact soil sample having w equal to 7 %, the intra-aggregate pores population was close to 0.05 µm (Fig. 8f and Fig. 9f). [START_REF] Fouché | Fabric control on strain and rupture of heterogeneous shale samples by using a non-conventional mechanical test[END_REF] also obtained an intra-aggregate pore population at 0.05 µm on an intact COx argillite sample freeze-dried from an initial water content w equal to 7.2%. For the compacted soil sample before flooding, it was air-dried and the water content was lower (2.4 % for soil B and 2.8% for soil A). The intraaggregate pores size observed was close to 0.03 µm, which is little smaller than that of the intact sample (0.05µm). These results are similar to that observed by other authors on oven-dried COx argillite: Sammartino et al. (2003) identified a population of intraaggregate pores at 0.02 µm on samples oven-dried at 105°C; [START_REF] Boulin | Pore gas connectivity analysis in Callovo-Oxfordian argillite[END_REF] also observed an intra-aggregate pore population at 0.02 µm on samples oven-dried at 70 °C for one week. It indicates that the decrease of water content can result in volume shrinkage and decreasing of intra-aggregate pores size. Note however that the water content of the compacted sample at ρ d equal to 2.0 Mg/m 3 after water flooding was 13%, much higher than that of the intact COx argillite, showing likely that water content is not a direct indicator of the intra-aggregate pore population. This slight effect of water flooding on the size of intraaggregate pores can be explained by the following two mechanisms:

(i) the swelling of the soil particles was confined under constant-volume conditions, the increase of the intra-aggregate pore sizes may be minimized. As suggested by Saiyouri et al. (1998), the constant volume condition does not allow the clay layer to absorb more than two layers of water. However, it should be noted that different results were obtained by [START_REF] Devineau | In situ neutron diffraction analysis of the influence of geomet-ric confinement on crystalline swelling of montmorillonite[END_REF] and [START_REF] Villar | Water retention of two natural compacted bentonites[END_REF]. In the test of [START_REF] Devineau | In situ neutron diffraction analysis of the influence of geomet-ric confinement on crystalline swelling of montmorillonite[END_REF], they found that the confined bentonite reached a three-layer hydrate under high relative humidity (RH=98%). The results of [START_REF] Villar | Water retention of two natural compacted bentonites[END_REF] indicated that the interlayer swelling is not hindered by external confinement. Differences of findings on the effect of confinement on soil hydration may derive from different test conditions, methods and materials.

(ii) the flooding duration (50-120 hours) was probably not long enough for the water entering inside the clay particles, since the permeability of COx argillite is quite low: 10 -21 m 2 after Homand et al., (2004). Delage et al. (2006) observed that the microstructure of compacted expansive clay was still changing after 90 days under constant-volume conditions.

The microstructure changes of compacted soils during wetting have been observed on SEM by Cui et al. (2002) and [START_REF] Komine | Experimental study on swelling characteristics of sandbentonite mixture for nuclear waste disposal[END_REF]: wetting induced separation of clay particles from the aggregates and as a result, filling of inter-aggregates pores. This interaggregates pores clogging can lead to a decrease in soil permeability [START_REF] Cui | Determining the unsaturated hydraulic conductivity of a compacted sand-bentonite under constant-volume and free-swell conditions[END_REF][START_REF] Ye | An experimental study of the water transfer through compacted GMZ bentonite[END_REF]. This is consistent with the observation made in the present work (see Fig. 10 and Fig. 11). Using the Environmental Scanning Electron Microscope (ESEM), [START_REF] Montes | Structural modifications of Callovo-Oxfordian argillite under hydration.dehydration conditions[END_REF] evidenced the swelling and the disaggregation of clay aggregates when wetting the COx argillite. The SEM pictures presented by Boulin et al. (2008), Fouché et al. (2004) and [START_REF] Wright | Altération et rupture d'échantillon d'argilite lors de sollicitations thermomécaniques en conditions humides contrôlées[END_REF] on intact COx argillite show a intra-aggregate pores size comprised between 0.02 and 0.1 µm; that is in agreement with the MIP results obtained in this study (see Fig. 8f and Fig. 9f).

The comparison between the SEM pictures after water flooding for the two soils (A and B) show that there is no significant difference between them (see Fig. 10 and Fig. 11 c,d,e,f). MIP tests also show similar soil structures (see Fig. 8 and Fig. 9 d,e). The compression indexes (C c and C s ) at saturated state are also similar for the two soils (Fig. 7), showing that the effect of grain size distribution of crushed soils was minimized after water flooding because of the disaggregation of clay aggregates. This is in agreement with the observation of complete disaggregation of clay aggregates after soaking shown in Fig. 1. Note also that the results in Fig. 1 were obtained after soaking the soil in water using dispersing agent while the results in Fig. 7 were obtained by soaking the soil in the oedometer cell. This can explain why the results in Fig. 1 of the two samples are similar while those in Fig. 7 Uniaxial compression tests to a dry density of 2.0 Mg/m 3 showed that the compression curves were strongly dependent on the grain size distribution: (i) higher stresses were needed for the finer crushed material to reach the same dry density; (ii) the compression index at the initial water content C cwi of the finer soil was larger. In addition, the swelling index at the initial water content C swi increased with the dry density and seemed to be independent of the grain size distribution.

The effect of the grain size distribution on the hydro-mechanical behaviour was also observed; the finer soil exhibited higher swelling potential when water flooded: (i) water flooding under constant-volume conditions under a low initial vertical stress of 0.5 MPa induced a final vertical stress of 4 MPa for the coarser soil (soil A) and 5 MPa for the finer soil (soil B); (ii) water flooding under constant-volume conditions under a high initial vertical stress of 7 MPa induced either swelling or collapse. The final vertical stress was equal to 5 MPa for the coarser soil and 7 MPa for the finer soil. (iii) the swelling pressure determined following the swell-consolidation procedure on the finer soil is equal to 5 MPa, i.e. twice that of the coarser soil (2.5 MPa). From a practical point of view, the swelling pressures of the finer crushed material (soil B) are high enough to ensure the backfill and seal function and low enough (< 7 MPa) to avoid any damage to the host-rock.

The MIP tests performed show that the soil volume changes during compaction was mainly due to the decrease of the sizes of inter-aggregate pores. Water flooding under constant-volume conditions also induced decrease in the sizes of inter-aggregates pores. SEM observations performed on the samples before and after water flooding show that water flooding separated the clay particles from the aggregates. The decrease in inter-aggregates pores size can be related to both the collapse of the aggregated-structure under mechanical stresses and the filling of clay particles separated from hydrated aggregates. Inter-aggregates pores

List of Tables

List of Figures

A

  . These results are in agreement with the observations by Attom et al. (2001), Omar et al. (2003) and Ferber et al.

  argillite has been considered as a possible sealing/backfilling material in deep radioactive waste disposal in France. The hydro-mechanical behaviour of the samples compacted from two crushed materials of the COx argillite has been investigated by performing uniaxial compression tests. The two crushed materials have different grain size distributions. The microstructure changes of the samples under different test conditions were characterized through MIP and SEM observations.

Fig. 1 .

 1 Fig. 1. Grain size distribution curves of the two soils, A and B. Fig. 2. Schematic view of the 1-D compression test. Fig. 3. Compression behaviour : dry density versus vertical stress. Fig. 4. Compression behaviour: void ratio versus vertical stress. Fig. 5. Compression behaviour: swelling index at initial water content versus dry density. Fig. 6. Vertical stress and void ratio changes during flooding under constant-volume conditions. Fig. 7. Oedometers tests on resaturated samples. (a) Void ratio versus time and (b) logarithm of vertical stress. Fig. 8. Results of MIP tests on soil A and intact COx argillite. Fig. 9. Results of MIP tests on soil B and intact COx argillite. Fig. 10. SEM pictures of soil samples having ρ d = 2.0 Mg/m 3 (Dimension of pictures: 125×100 µm).

Fig. 11 .

 11 Fig. 11. SEM pictures of soil samples having ρ d = 2.0 Mg/m 3 (Dimension of pictures: 25×20 µm).

Fig. 7 .Fig. 8 .Fig. 9 .

 789 Fig. 1. Grain size distribution curves of the two soils, A and B.

Fig

  Fig. 11. SEM pictures of soil samples having ρ ρ ρ ρ d = 2.0 Mg/m 3 (Dimension of pictures: 25× × × ×20 µ µ µ µm).

Table 1 .

 1 Test program

Table 1 . Test program

 1 Compacted at constant rate of 0.05 mm/min to ρ d equal to 2.0 Mg/m 3 and then

	Test Soil Procedure
	E01 A E02 A E03 B E04 B	Compacted at constant rate of 0.05 mm/min to ρ d equal to 2.0 Mg/m 3 with unloading/reloading cycles at lower ρ d of 1.7, 1.8, 1.9 Mg/m 3 , then flooded at σ v equal to 7 MPa Compacted at constant rate of 0.05 mm/min to ρ d equal to 2.0 Mg/m 3 with unloading/reloading cycles at lower ρ d of 1.7, 1.8, 1.9 Mg/m 3 , then flooded at σ v equal to 0.5 MPa Compacted at constant rate of 0.05 mm/min to ρ d equal to 2.0 Mg/m 3 with unloading/reloading cycles at lower ρ d of 1.7, 1.8, 1.9 Mg/m 3 , then flooded at σ v equal to 7 MPa Compacted at constant rate of 0.05 mm/min to ρ d equal to 2.0 Mg/m 3 with unloading/reloading cycles at lower ρ d of 1.7, 1.8, 1.9 Mg/m 3 , then flooded at σ v equal to 0.5 MPa
	E05 A

ACKNOWLEDGEMENTS

The authors wish to acknowledge ANDRA (the French Radioactive Waste Management Agency) for its financial support. The views expressed in this paper are that of the authors and do not engage ANDRA in any matter.

Clay aggregate

Inter-aggregates pores Inter-aggregates pores


Andreas Vogelsang
email: vogelsang@inp-greifswald.de

Andreas Ohl

Rüdiger Foest

Karsten Schröder

Klaus-Dieter Weltmann

Hydrophobic coatings deposited with an atmospheric pressure microplasma jet

Keywords: Number 52 micro plasma, plasma jet, thin film, deposition, fluorocarbon, XPS

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Introduction

Bulk polytetrafluoroethylene (PTFE) has excellent thermal and electrical properties as well as high chemical stability. The surface properties of PTFE, like hydrophobicity, oleophobicity and chemical inertness have particular importance both in food industry and in medical technology. Nowadays fluorocarbon films have many applications ranging from integrated circuits to waterproof coatings of textiles or in biocompatible medical devices [1] . Especially the exceptional low surface energy of CF x functional groups, which expresses in a very low wettability of the surface, makes it a ideal platform for further surface modifications, such as passivation of biomaterials by albumin [2] . To deposit films from C:F monomers, different chemical and physical methods exist. The plasma enhanced chemical vapor deposition (PE-CVD) utilizing low temperature plasma processes is one of the most appealing among these methods, as it allows the treatment of almost any material. For example, Favia et al. [3] , deposited hydrophobic films with large water contact angles from a modulated RF glow discharge fed with C 2 F 4 . PE-CVD is usually performed by high frequency or microwave discharges in the pressure ranges between 10 -4 and 1 mbar. These deposition systems offer plasmas with large-scale volumina and reactive species of long reach, but as there are vacuum systems required, they are expensive and complex. Therefore, atmospheric pressure plasmas have been of increasing interest in the past years.

Dielectric barrier discharges (DBD) are among those most used for this purpose [4][5][6] . However, this work focuses on the deposition by means of an atmospheric pressure plasma jet. The plasma source design in comparison to other atmospheric plasma sources is better suited for the precise localized treatment of complex 3D forms, e.g. trenches, inner walls or cavities, as the treated surfaces have not to be placed between the electrodes. The possibility to scale the dimensions of the discharge from centimetres down to the sub millimetre region enhances the applicable options.

Interestingly it appeared, that the deposition of C:F-films from non-thermal jet plasmas is not that simple. First trials with parameters based upon successful deposition experiments of SiO x thin films with a jet of similar design [7] and with octafluoropropane (C 3 F 8 ) as precursor gas, showed that no deposition could be achieved at all. Neither the admixture of H 2 to the ambient N 2 atmosphere, nor the change of the monomer to the more widely used octafluorocyclobutane (c-C 4 F 8 ) led to thin film formation in the predicted parameter range for afterglow deposition. After increasing the power coupled into the jet and the working gas flow, the active plasma zone was extended and came into contact with the substrate. Surprisingly, only in this case a deposition could be achieved.

The experiments were carried out under two different conditions regarding the ambient gas conditions: first N 2 instead of air, as oxygen is known to shift the C:F plasma regime from depositing to etching [8-10] , and second normal air, to gain a possible deeper understanding of whether a physical or a chemical mechanism is dominating in this effect.

In this paper results are presented for the first successful deposition of hydrophobic C:F thin films with a RF capillary atmospheric pressure microplasma jet. Deposition rate, chemical structure of the film, water contact angle, and the influence of the gaseous surrounding on the deposition process were investigated. Reasons for the deposition behaviour are discussed.

Experimental

Miniaturized atmospheric pressure plasma jet (APPJ)

Figure 1. Schematic of the basic jet setup.

The jet device consists of two concentrically arranged capillaries and two cylindrical electrodes (figure 1). The electrodes (stainless steel, width 5 mm, distance 4,5 mm) are attached to the outside of the outer quartz capillary (d out = 6 mm, d in = 4 mm) thus insulating the discharge from the electrode surface. The RF power is supplied by a Dressler Cesar RF generator, working at 27.12 MHz, which is capacitively coupled via a matching network to the upper electrode. The lower electrode, which is connected to ground potential, is arranged 3 mm above the end of the outer capillary. The monomer is fed into the jet through the inner capillary (d out = 1.8 mm, d in = 1.2 mm, Al 2 O 3 ). This design ensures that the monomer gas is introduced downstream of the discharge, which suppresses film deposition inside the jet and minimizes the influence of this feed on the stability of the discharge.

The discharge device is mounted on an aluminium ground plate, in which feedthroughs for gas lines and a vacuum flange are embedded. Connected to the flange are a pressure gauge (CMR 261, Pfeiffer Vacuum GmbH, Germany) and a membrane pump (MD 40, Vacuubrand GmbH, Germany). Upon this construction a glass vessel (d in 30 cm, height 42 cm) is placed on an O-ring seal. Thus, by evacuating and purging the set-up a defined ambient atmosphere during the deposition processes is ensured along with a protection against possible toxic reaction products.

Deposition conditions:

Throughout the experiments, the RF power was varied from 10 W up to 90 W (generator output reading). The source was operated with argon (5.0 purity, Linde AG, Germany) as working gas at a flow rate of 6 slm and 10 -100 sccm admixtures of c-C 4 F 8 (> 99,99 Mass.% purity, Air Liquide, Germany). The deposition experiments were carried out with the plasma jet pointing perpendicularly to the samples, which were prepared on a non-grounded support plate at a distance of 4 mm. Squares cut from silicon wafers (18 mm * 18 mm, 100 orientation, p doted, 100 Ohm/cm, University Wafer, USA) and strips of aluminium foil (18 mm * 40 mm, 99% purity, Carl Roth GmbH, Germany) served as substrate. The gases that created the different environmental conditions were N 2 (5.0 purity, Linde AG, Germany) and normal air (25 % humidity). Prior to the experiments in N 2 environment, the glass vessel was evacuated to 50 mbar and purged three times with N 2 , resulting in a residual oxygen content of about 0.0025 %. Afterwards, a flow of 6 slm of the respective gas through the vessel was initiated, to ensure static process conditions. The deposition experiment was preceded by a conditioning step where the plasma was ignited and the monomer admixture was switched on. Only after a delay time of 5 min the deposition started by moving the sample into position under the jet for the duration of the treatment time.

X-ray Photoelectron Spectroscopy (XPS)

The XPS device used was an Axis Ultra (Kratos, UK) with a conventional hemispherical analyzer. XPS spectra were recorded using a monochromatic Al K α source at 1486.6 eV with 150 W X-ray power in a standard configuration. For the photoelectron spectra of the element content and the highly resolved measurements, the data acquisition parameter were 80 eV and 10 eV pass energy, respectively. Charge compensation was applied. Data acquisition and processing were performed with vision 2.1.3 software (operating software Kratos, UK). XPS spectra were generally recorded for 1s electrons from C, O, N, F, 2p electrons from Si and 2s electrons from Al. Si and Al signals were taken in order to ensure sufficient thickness of the film. The C1s Peak, C-C and C-H components, was adjusted to 285.0 eV. The peak fitting procedure was carried out with the help of CasaXPS software version 2.3.15dev59 (Casa Software Ltd, UK). Gauss-Lorentz (30 % Lorentz) distribution, Tougaardbaseline, and a fixed FWHM between 1.1 and 1.5 eV were applied. All values referring to the elemental composition are given in XPS atomic-% and ratios thereof, as H is not detected by this method.

Water contact angle measurements

A DIGIDROP (GBX Instrumentation Scientifique, France) was used for the contact angle measurements utilizing the sessile drop method, i.e., a drop of distilled water with a defined volume (0.5 µl) was placed onto the horizontally aligned substrate surfaces with the help of a stainless-steel needle of a motorized syringe. The contact angle of the resting drop was determined utilizing the computational software of the meter. The values are reproducible within +/-3°for repeated preparation and measurement.

Profilometry

The shown spatial profile of the film thickness was measured by profilometry (Dektak 3 ST (Veeco Instruments Inc., USA, stylus force: 30 mg, scan velocity 100 µm/s). An uncoated reference point on the substrate is needed for absolute thickness values. Therefore, the measured samples were partially masked with a film of cellulose acetate prior to the deposition process. Removing the film after deposition leaves a sharp coating edge with a height that is readily measured with a profiler and a scan direction perpendicular to this edge. This method generates only one thickness value at a particular spot per scan. Thus, it is time-consuming to produce a spatial thickness profile, but reduces possible errors caused by a bending of the substrate surface.

Scanning Electron Microscopy (SEM)

The C:F coatings were inspected with a SEM device (JSM 7500F, Jeol Ltd., Japan) operated in 'gentle beam mode'. This mode utilizes low accelerating voltages thus circumventing the charging up of nonconductive samples. Hence, insulators needn't be sputter-coated with a metal prior to analysis. The resolution of the secondary electron image was about 1.4 nm for an accelerating voltage of 1 kV. The magnification was between x30000 and x75000 for a working distance of 4.7 mm. Magnification correction was set to automatic for acceleration voltage and working distance.

Optical Emission Spectroscopy (OES)

The optical emission of the plasma zone at the substrate was imaged side on through a lens to the entrance of an optical fibre and transferred to a monochromator (0.5 m length, grating 2400 lines/mm, Acton Spectra Pro 2500i, Princeton Instruments, USA). A CCD-Camera (PI-MAX, Princeton Instruments) at the exit port of the monochromator served as detector. The examined spectral range lies between 200 and 400 nm. Exposure times between 10 and 100 ms with a gain of 100 and 20 accumulations were chosen, depending on the intensity of the discharge.

Results and Discussion

Unlike other processes previously reported for jets of this design [7] , the deposition takes place only if the plasma plume is in tight contact with the substrate. The jet has to be driven with higher working gas flows compared to the former work [7] , and moderate to high powers to achieve these operational conditions. The discharge then develops three spatially relatively immobile filaments, which are in contact with the substrate and give the impression of constricted, high current 'hot' filaments with a high plasma density. They can be adjusted to have equal distance to each other. The deposition process is comparably fast. After only 10 s the film has reached a thickness sufficient to be visible with the naked eye. The deposition zone is formed by the foot prints of these filaments on the substrate (figure 2). A spatial profile of the foot print was measured by profilometry. Figure 3 shows the profile along a line crossing two of the three deposition patches. As can be seen, a maximum film thickness of 7.8 µm after 180 s of deposition was achieved, which leads to a local deposition rate of 43 nm/s for treatment in N 2 atmosphere. Because of this very inhomogeneous deposition profile, the quantification of the deposition rate is presented as mass difference (µg/s) instead of film growth rate (nm/s).

It has been pointed out [7] , that films deposited locally with jet plasmas tend to display local variations in their chemical structure, for example attributed to turbulences or inhomogeneous plasma conditions [11] . Hence, XPS scans along a line across the sample were carried out in order to substantiate a present axial and azimuthal variation in chemical composition and structure for the films reported here, too. Scans were recorded for three different samples treated in N 2 atmosphere. The distance between measurement points was 0.5 mm. Figure 4 shows a typical example of these measurements. Except for the outer, small boundary region, where the elemental composition is determined mainly by silicon (45 %) from the substrate which is sensed through the film and less fluorine (5 %) content, the chemical composition of the main diameter of the deposition foot print itself remains practically constant, independent on the film thickness. The films are mainly constituent of fluorine (60 %) and carbon (40 %), with small contributions of oxygen (1.5 %) just above the detection limit. With this local distribution in mind, further XPS analyses could be restricted to four points per film only. The results were averaged and the standard deviation is given as error bar. The desired main property of the coating is hydrophobicity, thus the wettability of the surface was determined by static water contact angle measurements. Values between 100°and 135°were obtained depending on the film thickness. In the middle of the sample the value was lower than on the deposition maxima. SEM photographs (figure 5) can facilitate the interpretation of this observation.

The microstructure of the thinner film is in the range of 100 nm, whereas it is in the range of micrometres in the area of the deposition maxima. It is supposed that this increased roughness further enhances the basic hydrophobic properties of the coating [12,13] and can create the so called super hydrophobic surfaces with contact angles approaching 180°. The applied power can have a strong effect on the fragmentation of the monomer and hence on the crosslinking of the thin film [START_REF] Yasuda | Plasma polymerization[END_REF][START_REF] Agostino | Plasma deposition, treatment, and etching of polymers[END_REF] . Experiments with different rf powers and fixed working gas flow and sample distance were carried out on aluminium substrates to quantify this effect. Nevertheless, with the exception of low powers (10 W and 20 W) the chemical composition remains unaffected by varying rf power (figure 6). The deviation at low powers can be explained by evaluating the fits for the highly resolved measured C 1s peak (figure 7). It demonstrates, that the existing carbon is bound to oxygen and that no CF 2 -or CF 3 -bindings are found. This indicates that all of the detected fluorine is directly bound to the aluminium surface forming aluminium fluoride (AlF 3 ). The presence of oxygen in the plasma, even under N 2 environment, caused by working under atmospheric pressure conditions and the layer of oxidized aluminium on the surface, enables reaction channels that lead to the creation of carbonic fluorides (COF x ), through many reactions originating from different fragments of C 4 F 8, which are reviewed in [START_REF] Losada | [END_REF] , for example:

C 2 F 4 + O COF 2 + CF 2 (1) CF 3 + O 2 COF 3 + O (2) CF 3 + O COF 2 + F (3) CF 2 + O 2 COF 2 + O (4) CF 2 + O COF + F (5) CF + O 2 COF + O (6)
These molecules then react with the aluminium surface, forming AlF with various byproducts.

The main intermediates in this reaction scheme are CF 2 , CF 3 , CF, COF and AlF, the resulting stable products CF 4 , COF 2 , CO, CO 2 and AlF 3 . The summarized reaction scheme is given by:

(C 2 F 4 ) n + O 2 + Al AlF 3 + CO + CO 2 (7)
As the presence of water cannot be neglected, another reaction leading to AlF 3 is possible:

CF 4 + H 2 O COF 2 + 2 HF (8) COF 2 + H 2 O CO 2 + 2 HF (9) Al 2 O 3 + 6 HF 2 AlF 3 + 3 H 2 O (10)
The most interesting effect here is, that only above 30 W a deposition occurred, which is indicated by the vanishing of the aluminium signal and the rise of the fluorine and carbon content. For even higher powers the content of elements changes only in the range of 1-2 %. The composition of the coating and also the degree of crosslinking is therefore independent on the power coupled into the jet, as can be seen in figure 9. This whole behaviour is due to the fact that for higher powers the active plasma zone of the jet is extended and at 30 W comes into contact with the substrate. At this point the discharge contracts itself, the deposition starts and the filaments form the three deposition patches.

This behaviour resembles a transition from αto γ-mode for the discharge zone between orifice of the jet and substrate surface. In fact, many similarities can be found. The α-mode is prevailing in the effluent plasma of the (capacitively) coupled rf capillary discharge. When this mode suddenly changes to γ-mode at about 30 W, the substrate would become the grounded electrode and the jet plasma the 'driven electrode', respectively. This discontinuous transition of the modi was first observed in moderate pressure rf-discharges and calculated for these parameters by Raizer et al [START_REF] Raizer | Radio-frequency capacitive discharges[END_REF] . For discharges at atmospheric pressure an expansion of these calculations was reported by Odrobina and Kando [START_REF] Odrobina | [END_REF] .

They too found a similar behaviour of the transition and calculated that for an argon discharge and a secondary electron emission coefficient of 0.01 such a discontinuous form of transition would be possible, as long as the applied frequency satisfied the condition:

( )

γ cr f p f < / (11)
with f cr = 300kHz/Torr. The discharge in this work with f/p = 35.68 kHz/Torr remains well within this range for P=760 Torr and f=27.12 MHz. This discontinuous transition was also observed for an atmospheric pressure rf-discharge in Helium [18,19] .

Other characteristics of a γ-mode can be concluded from the successful and fast deposition process after the transition, like the local increase of power density near the electrode surface (in this case the substrate) and an increase of electron density in the near surface region by up to an order of magnitude, which are the most probable causes for an increased concentration of very reactive and thus short lived CF x -radicals, which as they are created in close vicinity to the substrate can polymerise fast.

Based on the reaction mechanisms proposed for the fluorination of aluminium one could assume that an increased concentration of CF x -radicals caused by the higher power density would just result in a preferential formation of volatile components like COF x , CO 2 , etc. instead of polymerisation, as these are the main competing channels to the polymerisation reaction. The simplified reaction schemes are as follows:

CF x + CF x (C y F z ) n (Polymerisation) (12)
CF x + O, H 2 O COF x , CO 2 , etc. (Volatile components) (13)
The summarized reactions indicate that the polymerisation reaction runs roughly quadraticly with C CFx * C CFx * R pol , whereas the competing channel only runs linearly with C CFx * C O, H2O * R vol , which results in a net thin film forming.

Evidence for the existence of CF x -radicals during the deposition was provided by complementary conducted optical emission spectroscopy of the active plasma zone on the substrate. The detected emission in the range from 200 to 360 nm (figure 8) can be allocated to CF (220 -265 nm) [20,21] , CF 2 and CF 2 + (240 -360 nm) [START_REF] Pearse | The identification of molecular spectra[END_REF][START_REF] King | [END_REF] , OH (300 -320 nm) [START_REF] Pearse | The identification of molecular spectra[END_REF] and the 2 nd pos. system of N 2 (330 -460 nm) [22] .

Figure 8. Optical emission spectrum of the deposition process, 60 W, N 2 atmosphere. As main parts of the C1s peak of the thin film, the termination element CF 3 , the chain element CF 2 and the crosslinking elements CF, C-CF, CF-CF and C-C were fitted [START_REF] Agostino | Plasma deposition, treatment, and etching of polymers[END_REF] . The contribution of CF 2 and CF 3 amounts to 32 % and 18 %, respectively. This indicates a short chain structure, as for each two chain links one termination element exists. The total amount of all crosslinking elements is 40 % and independent of the deposition power. Deposition with different treatment times was carried out as another approach for a possible variation of surface properties. Additionally, these experiments were carried out for air as ambient atmosphere.

Despite the fact that the oxygen content during the deposition process is about four orders of magnitude higher then for the experiments at N 2 environment, no noticeable change in the film composition was observed. The content of elements is comparable to the deposition with N 2 atmosphere and varies only in the range of one percent. However, the differences in the deposition rate show clearly the influence of the oxygen. Figure 11 shows the respective time dependencies. For the treatment with N 2 environment it rises from 0.5 µg/s for 10 s treatment time up to 2.0 µg/s for times of 60 s and longer. The overall deposition rate drops about half an order, when the ambient atmosphere is air. After a rise of the rates within the first 60 s, which we attribute to substrate heating, stationary conditions are obtained. These allow the comparison of the processes. Interestingly, the air atmosphere does not reduce the CF x /F ratio enough to inhibit deposition completely. This might be a result of the comparatively high working gas flow, which can effectively keep the oxygen away from the deposition process.

As stated before, the deposition profile is very inhomogeneous, but assuming a homogeneous film thickness with an area of 1 cm² and an average density of 1 g/cm³ a growth rate of 12 nm/s can be derived from the maximum mass deposition rate of 2 µg/s. The results shown up to now describe an astonishingly robust process. Neither increase of applied rf power by up to 300 %, nor increase of treatment time or the deposition at air, alters the composition or degree of crosslinking of the coating in a significant way. It seems that the contraction of the glow discharge, the forming of filaments with high energy density on the substrate, i.e. direct plasma contact, and a stable footprint are the decisive factors for the deposition conditions. A physical explanation for these effects shall be given here using the equivalent circuit diagram of the discharge. The circuit diagram of the system of jet and substrate shows a simplified scheme of the main components that contribute to the electric relations and is given in figure 12. C 1 and C 2 are formed by the sum of the capacitances of the dielectric barrier and the plasma sheath, and are determined as of the same dimension. R 1 and R 2 are the resistances of the respective plasma zones between the electrodes and between jet orifice and substrate surface, whereas R 3 is the resistance of the substrate.

C 3 is the series connection of the capacitances of the plasma sheath on the substrate, the substrate itself and the stray capacitance between substrate and ground.

All decisive elements can be calculated or directly measured. The capacitances C 1 and C 2 are calculated as cylindrical capacitors with either thickness of the capillary or debye length as width. For an average electron density of 3*10 13 cm -3 and an electron temperature of 2 eV, the debye length is calculated as 1.9 µm, which leads to a capacitance of the plasma sheath in the range of 300 pF. The upstream connected capacitance of the dielectric is 2.5 pF, so that C 1 and C 2 are both about 2.48 pF. C 3 is dominated by the stray capacitance which was measured to be 1.9 pF. The value for the aluminium substrate is some 10 pF and for the plasma sheath on the sample again about 300 pF.

The stray capacitance forms an impedance of about 3*10 3 Ω for the frequency used in these experiments and thus is large enough to allow a charge flow. This offers the potential for the discharge to contract itself on the substrate if the resistance of the substrate R 3 (contact resistance) is not too high. Indeed, during experiments with silicon substrates an effective RF potential of about 150 V on the substrate was measured. If one assumes an average diameter of a filament of 0.1 mm and a resistance of the contact area on the silicon substrate of about 0.1 Ω, a current density of about 6*10 2

A/cm 2 can be derived. This current density characterises the discharge as diffuse arcing [START_REF] Raizer | Gas discharge physics[END_REF] . An even higher current density can be assumed for aluminium with a resistance in the range of 10 -6 Ω. In contrast, the resistance of a polystyrene substrate is about 10 10 Ω and thus too high to allow such an effect. Hence, the choice of substrate material does influence the potential of the jet to form a discharge regime capable of deposition and the energy density of the filament. Please note that, the exact geometric factors of the contact resistance are not taken into account and the substrate resistances were roughly approximated using the specific resistance of the material and the thickness of the respective substrate. Therefore the values rather represent upper limits and their exact values are most likely smaller. This hypothesis is supported by the observations made during experiments with different substrates and is in concordance with the observed power dependency of the deposition rates.

These rates are generally higher for aluminium than for silicon (figure 13). With polystyrene substrates no deposition could be achieved at all. Furthermore, it can be assumed from the progression of the curves that the monomer deficient regime is reached in the case on an aluminium substrate. For silicon this is not case, which is in accordance to the proposed model.

Conclusion

Conditions were discovered to deposit C:F-films with an atmospheric pressure plasma jet. The dependencies from parameters like power or treatment time were investigated. Additionally the impact of the ambient atmosphere on this process was evaluated by using N 2 and air environment. The deposition regime found provides very high film growth rates (up to 43 nm/s with N 2 atmosphere) and creates coatings with a distinctive deposition pattern. The chemical composition and structure was determined by XPS and found to be laterally homogeneous, with a [F]/[C] of about 1.4 and a very low content of nitrogen and oxygen. Fits of the highly resolved measured C 1s peak show a degree of crosslinking of about 40% and a [CF 2 ]/[CF 3 ] of about 1.8. The static water contact angle for these films was determined to be in the range of 100°to 135°. The use of normal air as ambient atmosphere has no impact on the film chemistry, but reduces the deposition rate by a factor of four.

The formation of a special discharge regime, namely the constricted and localized arc-like discharge, is mandatory to deposit C:F films with this device. The discharge mode is considered to be an atmospheric pressure γ-mode discharge with the substrate acting as second grounded electrode.

Figure 2 .

 2 Figure 2. Example for growth of C:F films: photograph showing deposition pattern caused by localized filaments on aluminium (left), micrograph with a top view of the deposition pattern on silicon (right).

Figure 3 .

 3 Figure 3. Spatial profile of thickness of a C:F polymer film, 60 W, 4 mm distance, 180 s treatment

Figure 4 .

 4 Figure 4. XPS linescan of a C:F film on silicon substrate, 60 W, 4mm 180 s treatment time, N 2 atmosphere.

Figure 5 .

 5 Figure 5. SEM pictures of a C:F coating, thin film in the middle of the substrate (left), thick film at a

Figure 6 .

 6 Figure 6. XPS atomic percentage for different applied rf powers, aluminium substrate, 4 mm distance,

Figure 7 .

 7 Figure 7. XPS: Fit of a highly resolved measured C 1s peak for 20 W.

Figure 9 .

 9 Figure 9. XPS: Comparison of the fits of highly resolved measured C 1s peaks for different powers, 30

Figure 10 .

 10 Figure 10. XPS atomic percentage for different treatment times and environment (air or N 2 ) (silicon substrate, 60 W, 4 mm distance).

Figure 11 .

 11 Figure 11. Mass deposition rate for different treatment times, aluminium substrate, 60 W, 4 mm

Figure 12 .

 12 Figure 12. Equivalent circuit diagram of the system jet -substrate.

Figure 13 .

 13 Figure 13. Mass deposition rate for different jet powers and substrates, 4 mm distance, 180 s treatment

Table 1 .

 1 XPS: Fitted components of the C 1s Peaks, their respective binding energies and percentage of area for the two peaks at different applied powers.

	Peak	binding energy [eV]	% area [30 W]	% area [90 W]
	C-C, C-H	285.0	2.91	1.23
	C-CF	287.1	12.01	14.13
	C-CF x	288.1	3.21	3.30
	-CF	289.6	21.01	22.46
	CF-CF	290.6	7.27	6.69
	-CF 2	291.8	32.22	32.13
	CF 2 -O	292.8	2.74	1.32
	-CF 3	293.9	18.63	18.72

However, prerequisite is a sufficient conductivity of the substrate. A simplified scheme of the electric properties of the system jet -substrate is given which clarifies the observed dependence of the discharge regime on the choice of substrate material.

Regarding the useful Teflon-like film properties and very high local growth rates, it would be interesting to further investigate this discharge regime to identify setups resulting in a practically useful deposition pattern.


J J Verbeek

N Vlassis

B J A Kröse

Self-Organization by Optimizing Free-Energy

Keywords: self-organizing map, mixture modeling, variational EM

We present a variational Expectation-Maximization algorithm to learn probabilistic mixture models. The algorithm is similar to Kohonen's Self-Organizing Map algorithm and not limited to Gaussian mixtures. We maximize the variational free-energy that sums data loglikelihood and Kullback-Leibler divergence between a normalized neighborhood function and the posterior distribution on the components, given data. We illustrate the algorithm with an application on word clustering.

Introduction

Kohonen's Self-Organizing Map (SOM) [START_REF] Kohonen | Self-Organizing Maps[END_REF] is a data analysis method that combines vector quantization with topology preservation. With each quantizer we associate a fixed location in a 'latent' space. The latent space is of much lower dimension (typically just two) than the data space. The SOM algorithm finds locations for the quantizers in the data space such that the summed squared distance from data to closest node is small and simultaneously topology is preserved. Topology preservation means that nearby components in latent space are also nearby in the data space. As a consequence, data points associated with nearby components in latent space are coming from similar locations in the data space. Hence, to check if data points are 'close' in the data space, we can check whether their associated components are close in the latent space. Topology preservation makes SOMs useful for visualization of high dimensional data since it reduces the data dimensionality while preserving small distances.

We present a constrained or variational Expectation-Maximization (EM) learning algorithm to learn probabilistic mixture models similar to SOM. The algorithm applies to a wide class of mixture models.

In the next section we discuss a Gaussian mixture model that will serve as the example mixture model throughout the paper and also briefly discuss the EM algorithm. In section 3 we present our Self-Organizing Mixtures algorithm. We compare our algorithm to several closely related algorithms and present our conclusions in Section 4.

As generative model consider the Mixture of Gaussians (MoG), given by :

p(x) = 1 k k s=1 p(x | s) for x ∈ IR D ,
where p(x|s) is an isotropic Gaussian with inverse variance β and mean µ s . We use this simple MoG to illustrate the similarity with SOM and because it keeps notation simple. Given data x = {x 1 , . . . , x N }, and an initial parameter vector θ = {β, µ 1 , . . . , µ k } the EM algorithm finds a local maximizer θ of the loglikelihood L(θ). Assuming data are independent and identically distributed:

L(θ) = N n=1 log p(x n ; θ) = n log 1 k s p(x n | s).
Learning is facilitated by introducing N hidden variables. Each hidden variable indicates which of the k mixture components generated the corresponding data point. The EM algorithm maximizes the negative free-energy [START_REF] Neal | A view of the EM algorithm that justifies incremental, sparse, and other variants[END_REF]:

F (Q, θ) = E Q log p(x, s; θ) + H(Q) = L(θ) -D KL (Q p(s | x; θ)), (1)
where we used H and D KL to denote respectively entropy and Kullback-Leibler (KL) divergence Q = n q n is a distribution over the hidden variables, q n is a distribution on the mixture components for data point n. Note that, due to the non-negativity of the KL-divergence, for any Q we have:

F (Q, θ) ≤ L(θ).
The two forms in which we expanded F are associated with the M-step and the E-step of EM. In the M-step we change θ as to maximize, or at least increase, F (Q, θ). The first decomposition includes H(Q) which is a constant w.r.t. θ, so we essentially maximize the joint log-likelihood in the M-step. In the E-step we maximize F w.r.t. Q. Since the second decomposition includes L(θ) which is constant w.r.t. Q, what remains is a KL divergence which is the effective objective function in the E-step of EM.

In standard applications of EM (e.g. mixture modeling) Q is unconstrained, which results in setting q n = p(s | x n ; θ) in the E-step since the non-negative KL divergence equals zero if and only if both arguments are equal. Therefore, after each E-step F (Q, θ) = L(θ). Variational methods are for example used when optimization over the unconstrained Q is intractable, in which case Q is restricted to a certain class Q of distributions allowing for tractable computations. Variational EM maximizes F instead of L, the objective sums log-likelihood and a penalty which is high if the true posterior is far from any member of Q.

Using free-energy for self-organization

By appropriately constraining Q we can enforce a topological ordering of the mixture components, as we explain below. By using the right class Q, the penalty term measures topological ordering. This is much like the approach taken in [START_REF] Roweis | Global coordination of local linear models[END_REF], where constraints on Q are used to globally align the local coordinate systems of the components of a probabilistic mixture of factor analyzers.

We associate with each mixture component s a latent coordinate g s . It is convenient to take the components as located on a regular grid in the latent space. We then restrict the distributions q n to be discretized isotropic Gaussians in the latent space, centered on one of the components. The distributions q n can be regarded as normalized neighborhood function of Kohonen's SOM. We thus put the constraint that q n ∈ Q = {p 1 , . . . , p k }, where:

p r (s) = exp (-λ g s -g r 2 )
t exp (-λ g tg r 2 )

.

A small λ corresponds to a broad distribution, and for large λ the distribution p r becomes more peaked. For fixed λ we can always increase or keep constant the objective by performing EM steps, since in the E-step we can always keep Q fixed or change Q if this increases the objective function.

Since the objective function might have local optima and EM is only guaranteed to give locally optimal solutions, good initialization of the parameters of the mixture model is essential to finding a good solution. Analogous to the method of shrinking the extent of the neighborhood function with the SOM, we can start with a small λ (broad neighborhood function) and increase it iteratively until a desired value is reached. In implementations we started with λ such that the p r are close to uniform over the components, then we run the EM algorithm until convergence. Note that if the p r are almost uniform the initialization of θ becomes irrelevant. After convergence we set λ new ← ηλ old with η > 1 (typically η is close to unity). In order to initialize the EM procedure with λ new , we initialize θ with the value found in running EM with λ old .

Using q ns = q n (s), for our MoG case F can be rewritten as:

F (Q, θ) = 1 2 N D log β - ns q ns β x n -µ s 2 /2 + log q ns .
For fixed λ an EM algorithm can be derived by differentiation:

• E: Determine (by means of exhaustive or sparse search in Q, see below) for each x n the distribution p r * ∈ Q that maximizes F , set q n = p r * .

• M: Set: µ s = n q ns x n / n q ns and β = N D/ ns q ns x nµ s 2 .

The component r * on which p r * is centered for data point n, is referred to as the 'winner' for x n . The computational cost of the E-step is O(N k 2 ), a factor k slower than SOM and prohibitive in large-scale applications. However, by restricting the search for a winner in the E-step to a limited number of candidate winners we can obtain an O(N k) algorithm. A straightforward choice is to use the l components with the largest joint likelihood p(x, s) as candidates, corresponding for our MoG to smallest Euclidean distance to the data point. If none of the candidates yields a higher value of F (Q, θ) we keep the winner of the previous step, in this way we are guaranteed to increase the objective in every step. We found l = 1 to work well and fast in practice, in this case we only check whether the winner from the previous round should be replaced with the closest node.

Let us consider why our algorithm yields topology preservation. Consider the second decomposition of the objective function [START_REF] Anouar | Probabilistic self organized map: application to classification[END_REF], and two mixtures, parameterized by θ and θ , that yield equal data log-likelihood L(θ) = L(θ ). The objective function prefers the mixture for which the KL-divergences are the smallest, i.e. the mixture for which the posterior on the mixture components looks like a discretized Gaussian in the latent space. This implies that nearby components in the latent space model similar data. Moreover, we see the same effect as in SOM: for each data item a winning mixture component is selected and next the winning component and its neighbors in the latent space (to a lesser degree) are tuned toward that data item. Note that these observations hold for any mixture model and that the algorithm is readily generalized to any mixture model for which we have a maximum-likelihood EM algorithm.

Discussion, illustration, and conclusions

Discussion. Our algorithm is very close to SOM when applied on the simple MoG. If we use only l = 1 candidate winner in the E-step the difference with Kohonen's winner selection is that we only accept the closest node as a winner when it increases the energy and keep the previous winner otherwise. Our M-step coincides exactly with the update rule of the batch SOM.

In [START_REF] Graepel | Self-organ. maps: generalizations and new optimiz[END_REF][START_REF] Heskes | Self-organizing maps, vector quantization, and mixture modeling[END_REF] another SOM-like algorithm is proposed with objective function:

- ns q ns [β r h sr x n -µ r /2 + log q ns ].
There, the neighborhood function, implemented by the h sr , is fixed, but the winner assignment is soft. Instead of selecting one 'winner', an unconstrained distribution q ns over the components is used. The β is used for annealing: for very small β the entropy term, with only one global optimum, becomes dominant, whereas for large β the quantization error, with many local optima, becomes dominant. By gradually increasing β more and more structure is added to the objective function.

Our work differs in several ways. We use one localized distribution in the latent space to constrain q n , as opposed to a mixture of localized distributions in the latent space. As a consequence the easy speed-up of our algorithm does not apply to the algorithms in [START_REF] Graepel | Self-organ. maps: generalizations and new optimiz[END_REF][START_REF] Heskes | Self-organizing maps, vector quantization, and mixture modeling[END_REF]. We use the neighborhood function width (controlled by λ) for annealing as opposed to using β. Note that although both λ and β can be used for annealing, only β can be optimized efficiently as a free parameter. Both our objective function and the one of [START_REF] Graepel | Self-organ. maps: generalizations and new optimiz[END_REF][START_REF] Heskes | Self-organizing maps, vector quantization, and mixture modeling[END_REF] can be interpreted as a log-likelihood plus a penalty term for non-topology preserving configurations. We believe the interpretation provided in this work is more direct and applies trivially to any mixture model, whereas for the interpretation of [START_REF] Graepel | Self-organ. maps: generalizations and new optimiz[END_REF][START_REF] Heskes | Self-organizing maps, vector quantization, and mixture modeling[END_REF] it is not clear whether it applies to any mixture model.

Another similar model is presented in [START_REF] Anouar | Probabilistic self organized map: application to classification[END_REF]. There, in the E-step we set: The M-step finds a new parameter vector that maximizes:

q n = arg max
n log s q n (s)p(x n |s).
This algorithm does not optimize a single likelihood function, even if we keep the neighborhood function width fixed, since the mixing weights vary for each data item and change throughout the iterations of the algorithm. The algorithm has run-time O(nk 2 ), but can benefit from the same speed-up used here. The Generative Topographic Map (GTM) [START_REF] Bishop | GTM: The generative topographic mapping[END_REF] achieves topographic organization in a quite different manner. In GTM mixture components are parameterized by a linear combination of nonlinear basis functions of the locations of the components in the latent space. The parameters of the linear map are learned from data by maximizing the data log-likelihood. The number of basis functions and their smoothness have to be fixed in advance or are found by model selection procedures. Other maximum-likelihood approaches achieve topographic organization by a smoothness prior [START_REF] Utsugi | Density estimation by mixture models with smoothing priors[END_REF] on the parameters. However, for this method is it not clear whether it generalizes directly any mixture model and whether speed-ups can be applied to avoid O(nk 2 ) run-time.

Illustration. Figure 1 shows results of modeling word occurrence data on a part of the 20 newsgroup data set, word x n is shown at g n = s p(s|x n )g s . Each of the 100 words is a data item with 16242 binary features indicating its occurrence in each of 16242 documents. We learned a mixture with our algorithm, where each mixture component is a product of Bernoulli distributions:

p(x n |s) = 16242 i=1 p s,i x i n (1 -p s,i ) (1-x i n ) .
Due to the huge dimensionality of this data, the posteriors are rather peaked, resulting in g n almost equal to one of the g s . Therefore, we did not use the actual posterior but the distribution p (s|x n ) ∝ p(s|x n ) α for α < 1 such that the entropies of the p (s|x n ) were close to two bits, giving a smoother visualization.

Conclusions. We presented a penalized log-likelihood probabilistic mixture modeling method, similar to Kohonen's SOM. The probabilistic formulation offers several benefits: (i) The method is directly applicable to any mixture model for i.i.d. data for which we can write down an EM algorithm. (ii) The generative model allows seamless embedding in larger probabilistic systems. It is for example straightforward to learn mixtures of Self-Organizing maps. (iii) There is a direct and clear link to the mixture model data log-likelihood.

p

  r (s)p(x n |s).

Figure 1 :

 1 Figure 1: Self-organizing Bernoulli models, k = 25. Right plot zooms left.