A Merlen
email: merlen@univ-tln.fr

A Sangar

P Torchio

L N Deepak

D Grojo

O Uteza

P Delaporte

K L N Deepak

O Utéza

Multi

Keywords: Raman spectroscopy, Silicon, laser ablation

published or not. The documents may come

Introduction

Trapping light using nanostructures is a powerful approach to improve light-matter interaction. This could lead to many applications, particularly in the field of photovoltaics [START_REF] Cheng | Light trapping enhancements of inverted pyramidal structures with the tips for silicon solar cells[END_REF][START_REF] Fahim | Simultaneous broadband light trapping and fill factor enhancement in crystalline silicon solar cells induced by Ag nanoparticles and nanoshells[END_REF][START_REF] Vedraine | Optical characterization of organic blend films integrating metallic nanoparticles[END_REF]. Nowadays the huge development of optical nano-antennas [START_REF] Novotny | Optical antennas tuned to pitch[END_REF] is an intensive field of research. However the exact interaction between those nanostructures and light is not totally clearly understood due to the multiple mechanisms interfering. In many cases light scattering can be enhanced. This feature has led to the development of surface enhanced spectroscopy, such as surface enhanced Raman spectroscopy (SERS [START_REF] Etchegoin | Principles of surface-enhanced Raman spectroscopy and related plasmonic effects[END_REF]), surface enhanced fluorescence (SEF [START_REF] Fort | Surface enhanced fluorescence[END_REF]) or surface enhanced infra-red absorption (SEIRA [START_REF] Aroca | Surface-enhanced infrared spectroscopy[END_REF]). For all these techniques, the enhanced optical property of a nanostructured sample is used to detect a signal in conditions where this would normally not be possible due to a weak scattering cross section and a low amount of interacting molecules. For the above-mentioned techniques, the final signal intensity is an experimental sign of the enhanced scattering properties of the nanostructured sample. An enhancement factor (EF) can thus be calculated, even if it is absolutely necessary to be very cautious about its quantitative estimation and its real scientific meaning [START_REF] Etchegoin | Principles of surface-enhanced Raman spectroscopy and related plasmonic effects[END_REF]. These techniques are extremely powerful and a single molecule detection has been reported [START_REF] Nie | Probing single molecules and single nanoparticles by surfaceenhanced Raman scattering[END_REF][START_REF] Etchegoin | A perspective on single molecule SERS: current status and future challenges[END_REF][START_REF] Kneipp | Single molecule detection using surface-enhanced Raman scattering (SERS)[END_REF]. They are usually all based on the giant electromagnetic resonance in the near field of metallic nanostructures due to plasmonic resonance. This is the case for instance with silver, gold or copper nanostructures. Depending on the plasmonic resonance, the intensity of the enhanced signal can thus strongly depends on the wavelength of the excitation and the scattered light. Even if this mechanism leads to a very high enhancement, its plasmonic origin induces a severe limitation: enhancement occurs only if the incoming light wavelength is included in a plasmon absorption band of the nanostructured sample. For broadband optical applications, it is therefore necessary to build plasmonic nanostructures with a broad absorption band, which can require sophisticated protocols (fractal nanostructures, use of a mix of different metals etc…). Such mechanism is not the only one for the achievement of surface enhanced light scattering and several studies have already reported comparable enhancements without plasmonics. The use of silica microspheres enables also to obtain similar enhancements [START_REF] Anderson | Nonplasmonic surface enhanced Raman spectroscopy using silica microspheres[END_REF]. The precise mechanism is not fully explained but it appears that an optical resonance based on whispering-gallery mode plays a key role [START_REF] Ausman | Whispering-gallery mode resonators: Surface enhanced Raman scattering without plasmons[END_REF]. A lens effect can also occur [START_REF] Desmedt | Enhancement of the Raman scattering signal due to a nanolens effect[END_REF]. In the case of nanostructured silicon (nanoparticles or microcavity) enhancements are also reported [START_REF] Kuzik | Raman scattering enhancement in porous silicon microcavity[END_REF][START_REF] Liu | Enhanced-Raman scattering from silicon nanoparticle substrates[END_REF][START_REF] Sirleto | Giant Raman gain in silicon nanocrystals[END_REF]. A good experimental proof of this effect is the evolution of the main silicon Raman mode. Due to its mechanism, Raman scattering can be considered as an experimental signature of light-matter interaction and is extensively used to study the optical near field enhancement. As a consequence, the light trapping in nanostructured samples induces an increase of Raman modes compared to a flat classical substrate. For silicon, the final intensity of its main Raman mode can therefore be considered as an experimental and quantitative proof of light trapping efficiency. Such specificity is of considerable interest for the development of silicon nanophotonics, in particular for photovoltaics. Among all nanostructure architectures giving rise to an enhancement of light-matter interaction, the nanoholes appear as extremely promising [START_REF] Pellegrini | Enhanced optical-properties in porous silicon microcavities[END_REF]. In this paper we report Raman signal enhancement from silicon nanoholes. The silicon nanoholes are realized through laser ablation using silica nanospheres preliminary laid on the silicon substrate by means of the Langmuir-Blodgett technique. Those spheres act as lenses to focus the laser light. A network of nano-holes (typical diameter: 450 nm) is thus created on the silicon surface. Such a sample is studied by Raman spectroscopy using excitation wavelengths at 488 nm, 514.5 nm and 632.8 nm. For each wavelength, measurements are performed with various laser powers. A clear enhancement of the silicon Raman mode is observed, whatever the excitation wavelength and the laser power are. Our results confirm the possibility offered by such nanostructured sample to highly increase light-silicon interaction.

Experimental

Oriented (100) silicon wafers (from Siltronix) are used in the experiments. The wafers are polished on a single side and cut into ≈2×2 cm² area samples. They are sonicated in water and ethanol for 30 minutes, and further treated with a plasma torch (Acxys technologies) in order to remove any residual contamination and to increase the wettability of the surface. Afterwards, silica spheres with radii of 500 nm and low size dispersion (polydispersity index PDI < 0.2) are mixed with ethanol (40 mg/mL). A Langmuir-Blodgett (LB) film deposition machine (KSV-Nima, model Mini) equipped with a surface tension balance is employed to grow a close-packed sphere monolayer on the Si substrates. We choose to work with C18 functionalized commercial spheres (from Micromod) as they are strongly hydrophobic, enabling the self-assembly of spheres at the Air/Water Interface without the use of chemicals [START_REF] Goldenberg | Simple method for the preparation of colloidal particle monolayers at the water/alkane interface[END_REF][START_REF] Bardosova | The Langmuir-Blodgett approach to making colloidal photonic crystals from silica spheres[END_REF] that would modify the spheres or the LB subphase. The microsphere solution is carefully spread by small droplets (µL) at the surface of water which is then compressed until the surface pressure reaches 15 mN/m. The monolayer is then transferred to the silicon surface at a dipping speed of 5 mm/minute while the surface pressure is maintained at a constant level by progressive automated compression.

The microsphere 2D-arrays obtained on silicon are irradiated by laser pulses provided by an ArF (λlaser = 193 nm) laser source (from Lambda Physik, LPX220i). The laser pulse duration is 15 ns, and the experiments are performed at normal incidence. The spheres act as near-field focusing elements producing a periodic assembly of sub-diffraction limit light spots (also called photonic nanojets) hitting the silicon surface [START_REF] Grojo | Monitoring photonic nanojets from microsphere arrays by femtosecond laser ablation of thin films[END_REF]. When modest laser fluences are used, the substrate is ablated only locally at the tip of the photonic nanojets leaving behind an array of nanocraters in a simple, dry and fast single laser processing step [START_REF] Piglmayer | Laser-induced surface patterning by means of microspheres[END_REF]. Obviously the quality of the periodic structuring depends on the quality of the arrangement resulting from the sphere assembly, the level of monodispersity of the spheres and also on the laser dose impinging the substrate [START_REF] Grojo | Size scaling of mesoporous silica membranes produced by nanosphere mediated laser ablation[END_REF]. The laser-pulse energy was varied with the aid of a manually operated beam attenuator (from Optec, AT4030). In previous works [START_REF] Grojo | Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films[END_REF], parallel and long range (>> mm²) structuring at the mesoscopic scale (100 nm -1 µm) has been demonstrated on silicon samples showing the interest of this approach. We also studied the degree of control in surface structuring while varying the laser fluence and number of shots. The results relied on statistical analyses of the created features (in the range 100-1000 nm) observed by optical and scanning electron microscopy (SEM, JEOL JSM-6390) [START_REF] Grojo | Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films[END_REF]. In the present case, the templates are prepared with 10 laser shots at a laser fluence maintained at 0.55 J/cm 2 . This sequence is found to provide high quality structures for experiments performed at ambient air. The structures rely on the local ablations induced by the first pulse which ejects the microsphere and leaves a small hole behind it. Cleaning as well as local ablation of the resulting products around the holes is realized with the following laser pulses. Similar results could be likely obtained with a single laser pulse if the interaction were performed in a low pressure environment to avoid redeposition. The morphology of the silicon templates used for the optical experiments is analyzed using tapping-mode atomic force microscopy (AFM, PSIA XE-100). An AFM image of the laserimpacted silicon sample can be seen in figure 1. A hexagonal network of holes is formed after laser ablation. The pitch is 1 µm imposed by the sphere radius. The holes have a typical diameter of 440 nm and a deepness of 230 nm. Raman measurements are performed using a Horiba Labram HR800 spectrometer. Three different laser excitations wavelengths are used: 632.8 nm (from a HeNe laser), 514.5 nm and 488 nm (from an Ar laser). The laser beam is focused on the sample through a x100 objective (Numerical Aperture of 0.9). The laser spot size is similar to the typical diameter of the nanoholes. As a consequence, the signal relies on an individual nanohole for each Raman measurement. The use of different laser excitation wavelengths insures a rather broadband study of the optical response of the sample. The laser power varies between 1 μW and 3.6 mW. For the different laser excitation wavelengths, two kinds of Raman measurements were performed: one with the laser focused on a hole and the other one on a flat part of the sample between the holes. Despite the high power laser irradiation, the Raman signal coming from the flat part of the laser-impacted silicon sample is identical to the one from a standard flat silicon sample (position of the peak, full half-width maximum and intensity). The recorded Raman spectra of the laser-impacted silicon samples can be seen in figure 2. The single mode observed at 520.7 cm -1 arises from the first-order Raman scattering of the longitudinal optical (LO) and the transverse optical (TO) phonon modes which are degenerated at the -point (phonon wavevector q ≈ 0 near the center of the Brillouin zone). Surprisingly the position of the peak its profile are not changed after laser structuring of the sample surface. Such a feature suggests that the laser irradiation of the silicon surface induces little strain on the silicon lattice. Indeed it was reported that any significative strain should induce a shift and a broadening of the mode [START_REF] Kumar | Nanomechanical properties of silicon surfaces nanostructured by excimer laser[END_REF]. This is not observed here whatever the laser excitation wavelength is. It cannot be excluded that such stress could appear at the top surface, but that was not observed here due to the light penetration in silicon. If such stress occurs, it would happen at the top surface of silicon and would not be detected by classical Raman spectroscopy. Similar Raman measurements were performed on silicon samples ablated with much higher laser fluences for structuring compared to that used in the current study. The Raman mode of these high power impacted-laser silicon samples was shifted and strongly broadened, confirming the high sensitivity of our own measurement process to these issues. Such experimental feature has already been reported in previous studies [START_REF] Reif | Nanostructure formation upon femtosecond ablation from silicon: Effect of double pulses[END_REF][START_REF] Bonse | Modifying single-crystalline silicon by femtosecond laser pulses: an analysis by micro Raman spectroscopy, scanning laser microscopy and atomic force microscopy[END_REF]. Our structuring method is remarkable to prove that, despite the surface structuring is carried out in the ablation laser regime, no significant change in the silicon structure in nanoholes is detected by Raman spectroscopy.

Results and discussion

Even if the position of the Raman peak and its profile are not affected by laser structuring, it appears clearly that its intensity strongly depends on the structuring of the silicon surface: for any wavelengths the intensity is approximately twice as much for the nanostructured area compared to a flat zone. Depending on the experimental conditions (laser wavelength, laser power, light focalization, irradiated area) the intensity enhancement varies from 1.3 to 2.5 with a typical value around 2. In figure 3 the intensity ratio Inanohole/Iflat silicon for the three excitation laser wavelengths and different Raman laser powers can be seen. Except for low power laser illumination (for which the Raman signal is weak and the evaluation of Raman intensity is rather imprecise), this enhancement factor is almost independent of the laser power. In addition it does not vary significantly with the excitation wavelength value. The independence of the enhancement with laser excitation wavelength and laser power is remarkable. To our knowledge this is the first time that such behavior is reported. This result is an extremely important aspect to promote the use of nanoholes in the improvement of broadband silicon photonics. The origin of this enhancement needs to be explained. As silicon nanostructures do not involve any plasmonic resonance for such wavelengths, this enhancement obeys a nonplasmonic mechanism. This is confirmed by the independence of the enhancement versus wavelength: for plasmonic nanostructures with spectral narrow-band resonance, enhancement is strongly dependant with the excitation wavelength [START_REF] Merlen | Surface enhanced Raman spectroscopy of organic molecules deposited on gold sputtered substrates[END_REF].

In Raman spectroscopy (and generally for any light-matter interaction process), it is necessary to take into account the amount of probed matter when comparing two different samples. The surface ratio between a flat round and a curved round probed surface is 2, which is noticeable close to the enhancement factors reported in figures 2 and 3. This ratio can also vary depending on the exact shape of the hole. According to AFM images, the shape of our holes is not so far from a spherical hole. In addition to the surface increase, the whole volume where light-matter interaction occurs has to be considered to discuss the origin of the enhancement. Optical absorption length in silicon strongly depends on the wavelength used for Raman measurements: it is typically 495 nm at 488 nm, 680 nm at 514 nm and 2700 nm at 632 nm [START_REF] Palik | Handbook of optical constants of solids[END_REF]. But we demonstrated that the intensity ratio between the flat and the nanostructured sample is independent of the wavelength (see figures 2 and 3). This suggests that the increase of surface or volume interaction is not the only mechanism responsible for the enhancement of Raman scattering. Liu et al. [START_REF] Liu | Enhanced-Raman scattering from silicon nanoparticle substrates[END_REF] studied electrochemically roughened silicon substrates. They attributed the Raman scattering enhancement to two contributions: one due to an electromagnetic cavity resonance and the other to a resonance arising from silicon nanoparticles. In our Raman experiments, no significant shift and broadening of the silicon mode are observed suggesting that no silicon nanoparticles are formed the ablation laser process. Finally the enhancement observed herein very probably arises from the cavity resonance only. It must be noticed that such electromagnetic cavity resonance which induces a high Raman enhancement was also reported for silicon nanowires [29] and nanocones [START_REF] Cao | Enhanced raman scattering from individual semiconductor nanocones and nanowires[END_REF]. And this resonance mechanism occurring with silicon nanoholes should have broad-band properties as no significant difference in the final enhancement is observed among our three excitation wavelengths.

Another feature should also be taken into account: a light trapping effect. Multiple light reflections and light scattering could happen in the giving rise to multiple Raman scattering and an increase of the final intensity of the Raman mode. Such multiple reflections were already reported with inverted pyramidal nanostructures [START_REF] Cheng | Light trapping enhancements of inverted pyramidal structures with the tips for silicon solar cells[END_REF] which are considered as extremely promising anti-reflective treatment for silicon. Korsunska et al. [START_REF] Korsunska | Raman scattering characterization of macro-and nanoporous silicon[END_REF] explained also the Raman enhancement observed with macro-and nanoporous silicon by multiple absorption of the probe light beam scattered and reflected inside pores. This light trapping effect plays certainly a role in the final enhancement observed herein.

Conclusion

We demonstrated through Raman spectroscopy that the use of nanoholes can enhance lightsilicon interactions. Moreover this enhancement occurs without any clear sign of structural damages induced by laser irradiation during nanoholes fabrication. Microsphere mediated laser ablation can be an efficient method to prepare dense arrays of appropriate nanoholes. The enhancement of the Raman signal intensity coming from the structured area of a silicon substrate compared to the flat part is reported between 488 and 632.8 nm. This amplified intensity ratio is independent of the excitation wavelength and power of the Raman laser probe. Such an enhancement is induced by the combination of several mechanisms: a broadband optical resonance, an increase of the interaction surface and multiple scattering in cavities. The relative contribution of each phenomenon remains unclear and will require further study. The global enhancement of optical properties of nanostructured silicon with nanoholes is extremely promising for the future development of silicon Nanophotonics. This material engineering could also be applied to other optically active materials. It is confirmed that Raman spectroscopy is an efficient tool to study light-matter interaction in nanostructures.

Fig. 1 .

 1 Fig. 1. AFM image of a laser-impacted silicon sample. a) General 3D view and b) typical line profile of the nanoholes array.

Fig. 2 .

 2 Fig.2. Raman spectra of laser-impacted samples for the three laser excitations wavelengths: 488 nm, 514.5 nm and 632.8 nm. Two spectra are reported for each wavelength: one with the laser focused on a hole (black curve) and the other one on a flat part of the same sample (red curve).

Fig. 3 .

 3 Fig. 3. Evolution of Raman intensity ratio Inanohole/I flat silicon with Raman laser power. each excitation laser wavelength two series of measurements are reported.

Acknowledgements

The authors acknowledge the Agence Nationale de la Recherche (ANR) for financial support under the CARIOCA (number 2010-JCJC-918-01) and FELINS-ANR-10-BLAN-946 projects.


Nyoman Juniarta
email: nyoman.juniarta@loria.fr

Miguel Couceiro
email: miguel.couceiro@loria.fr

Amedeo Napoli
email: amedeo.napoli@loria.fr

A Unied Approach to Biclustering Based on Formal Concept Analysis and Interval Pattern Structure

Keywords: biclustering, FCA, gene expression, pattern structures

In a matrix representing a numerical dataset, a bicluster is a submatrix whose cells exhibit similar behavior. Biclustering is naturally related to Formal Concept Analysis (FCA) where concepts correspond to maximal and closed biclusters in a binary dataset. In this paper, a unied characterization of biclustering algorithms is proposed using FCA and pattern structures, an extension of FCA for dealing with numbers and other complex data. Several types of biclusters constant-column, constant-row, additive, and multiplicative and their relation to interval pattern structures is presented.

Introduction

Given a numerical dataset represented as a table or a matrix with objects in rows and attributes in columns, the objective of clustering is to group a set of objects according to all attributes using a similarity or distance measure. By contrast, biclustering simultaneously operates on the set of objects and the set of attributes, where a subset of objects can be grouped w.r.t. a subset of attributes, based on user-dened constraints such as having constant values, constant values within columns or rows. Then, if a cluster represents object relations at a global scale, a bicluster represents it at a local scale w.r.t. the set of attributes. More generally, biclustering searches in a data matrix for sub-matrices or biclusters composed of a subset of objects (rows) and a subset of attributes (columns) which exhibit a specic behavior w.r.t. some criteria.

Biclustering is an important tool in many domains, e.g. bioinformatics and gene expression data, recommendation and collaborative ltering, text mining, social networks, dimensionality reduction, etc. As surveyed in [START_REF] Madeira | Biclustering algorithms for biological data analysis: a survey[END_REF], biclustering received a lot of attention in biology, and especially, for analyzing gene expression data, where biologists are searching for a set of genes whose behavior is consistent across certain experiments/conditions [START_REF] Ben-Dor | Discovering local structure in gene expression data: the order-preserving submatrix problem[END_REF][START_REF] Cheng | Biclustering of expression data[END_REF][START_REF] Tanay | Discovering statistically signicant biclusters in gene expression data[END_REF]. Biclustering is still actively studied in biology [START_REF] Pio | A novel biclustering algorithm for the discovery of meaningful biological correlations between microRNAs and their target genes[END_REF][START_REF] Henriques | BicPAM: Pattern-based biclustering for biomedical data analysis[END_REF][START_REF] Pontes | Biclustering on expression data: A review[END_REF]. Biclustering is also actively studied in recommendation systems [START_REF] Ignatov | Concept-based biclustering for internet advertisement[END_REF][START_REF] Ignatov | Recommender system based on algorithm of bicluster analysis RecBi[END_REF], where the objective is to retrieve a set of users sharing similar interest across a subset of items instead of the set of all possible items. Following the lines of [START_REF] Henriques | BicPAM: Pattern-based biclustering for biomedical data analysis[END_REF][START_REF] Henriques | Bicspam: exible biclustering using sequential patterns[END_REF][START_REF] Henriques | Bicpams: software for biological data analysis with pattern-based biclustering[END_REF], in this paper we are interested in biclustering algorithms based on pattern-mining techniques [START_REF]Frequent Pattern Mining[END_REF]. These techniques allow an exhaustive and exible search with ecient algorithms. Moreover, authors in [START_REF] Henriques | BicPAM: Pattern-based biclustering for biomedical data analysis[END_REF] discuss the benets of using pattern-based biclustering w.r.t. scalability requirements, and mostly w.r.t. generality and diversity of the types of biclusters which are mined. In addition, they point out the fact that pattern-based biclustering algorithms can naturally take into account overlapping biclusters, and as well, additive, multiplicative and symmetric assumptions concerning biclusters.

In this paper, we revisit all these aspects and propose an alternative framework for pattern-based biclustering based on Formal Concept Analysis (FCA [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF]). In [START_REF] Veroneze | Enumerating all maximal biclusters in numerical datasets[END_REF], authors directly reuse the FCA framework and adapt the algorithms for biclustering. By contrast, in this paper, we go further and we consider the so-called pattern-structures, an extension of FCA for dealing with complex values such as numbers, sequences, or graphs [START_REF] Ganter | Pattern structures and their projections[END_REF]. We especially reuse interval pattern structures which are detailed in the following for dening a unique framework for pattern-based biclustering. In this way, we introduce an alternative approach than [START_REF] Henriques | BicPAM: Pattern-based biclustering for biomedical data analysis[END_REF], as we do not need to apply any scaling, discretization, or transformation procedures over the data to discover biclusters. This paper is organized as follows. First we describe some types of biclustering in Section 2 and basic denitions about FCA in Section 3. We then propose our approach of biclustering based on interval pattern structures in Section 4 and present the empirical experiments in Section 5. Finally, we conclude our work and give some future works in Section 6.

Biclustering

In this section, we recall the basic background and discuss illustrative examples of the dierent types of biclusters [START_REF] Madeira | Biclustering algorithms for biological data analysis: a survey[END_REF]. We consider that a dataset is a matrix (G, M ) where G is a set of objects and M is a set of attributes. The value of m ∈ M for object g ∈ G is written as m(g). In this paper, we work with numerical datasets. In such a dataset, it may be interesting to nd which subset of objects have the same values w.r.t. a subset of attributes. Regarding the matrix representation, this is equivalent to the problem of nding a submatrix where An additive bicluster is illustrated in Table 1c. Here we see that there is a constant dierence between any two columns. For example, each value in the second column is two more than the corresponding value in the fourth row. Therefore, Bicluster discovery is naturally related to FCA. In this paper, we show that an extension of FCA called partition pattern structures can be used for discovering biclusters. In the following section, we explain some basic theories about FCA and pattern structures.

given a dataset (G, M ), a pair (A, B) (where A ⊆ G, B ⊆ M ) is an additive bicluster i ∀g, h ∈ A, ∀m, n ∈ B, m(g) -n(g) = m(h) -n(h); or a multiplicative bicluster i ∀g, h ∈ A, ∀m, n ∈ B, m(g)/n(g) = m(h)/n(h).

FCA and Pattern Structure

In a binary matrix, FCA tries to nd maximal submatrices with a constant value across all of its cells. Therefore, a formal concept is a bicluster with constant value. More precisely, FCA is a mathematical framework based on lattice theory and used for classication, data analysis, and knowledge discovery [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF]. From a formal context, FCA detects all formal concepts, and arranges them in a concept lattice. FCA is restricted to specic datasets where each attribute is binary (e.g. has only yes/no value). This limitation prohibits FCA to work in more complex datasets, e.g. a user-rating matrix or a gene expression dataset, which are not binary. Therefore, FCA is then generalized into pattern structures [START_REF] Ganter | Pattern structures and their projections[END_REF].

A pattern structure is a triple (G, (D, ), δ), where G is a set of objects, (D, ) is a complete meet-semilattice (of descriptions), and δ : G → D maps an object to

c d = c ⇔ c d.
A description can be a number, a set, a sequence, a tree, a graph, or another complex structure. The Galois connection for a pattern structure (G, (D, ), δ)

is dened as:

A = g∈A δ(g), A ⊆ G, (1)
d = {g ∈ G|d δ(g)}, d ∈ D. (2)
A pattern concept is a pair (A, d), A ⊆ G and d ∈ D, where A = d and d = A.

FCA can be understood as a particular pattern structure. The description of an object is a set of attributes, and the operator between two description is the intersection of two sets of attributes.

Biclustering using Interval Pattern Structure

In gene expression data, we often have a numerical matrix. Biclustering in such matrix should nd submatrices whose cells present regularities, e.g. each column has similar value in the case of similar-column (SC) biclustering. SC biclustering task is similar to FCA in the sense that FCA also searches consistent submatrix.

But since SC biclustering works on a numerical matrix, we need to generalize FCA to a pattern structure. One such generalization is where the description of each object is a set of numerical values and the similarity between any two descriptions is the intervals that encompass those values. This kind of pattern structure is called an interval pattern structure.

Interval pattern structures (IPS) was introduced by Kaytoue et al. [START_REF] Kaytoue | Mining gene expression data with pattern structures in formal concept analysis[END_REF] to analyze gene expression data (GED). A GED is typically represented as a 2-D numerical matrix with genes as rows and conditions as columns, as shown in Table 2. In this matrix, the submatrix ({g 1 , g 2 , g 3 }, {m 1 , m 2 , m 3 , m 5 }) is an SC bicluster, dened by the parameter θ = 1. It means that the range of values of each column in the submatrix has the length of at most 1.

Interval Pattern Structure

In IPS, a description is several intervals describing the values of every column. For example, the description of g 1 denoted by δ(g

1 ) in Table 2 is [1, 1][2, 2][2, 2]
Table 4: Some interval pattern concepts with θ = 1 from Table 2. [START_REF] Ganter | Pattern structures and their projections[END_REF][START_REF] Ganter | Pattern structures and their projections[END_REF] . The similarity operator ( ) for IPS is dened as the convex hull of two intervals. Therefore, the similarity of δ(g 1 ) and δ(g 4 ) denoted by δ(g

Extent Intent {g1} [1, 1][2, 2][2, 2][1, 1][6, 6] {g1, g3} [1, 2][2, 2][1, 2] * [6, 6] {g1, g4} * * [2, 2] * [6, 7] {g1, g2, g3} [1, 2][1, 2][1, 2] * [6, 6] {g1, g2, g3, g4} * * [1, 2] * [6, 7] [1, 1]
1 ) δ(g 4 ) is [1, 8][2, 9][2, 2][1, 6][6, 7] .
Given a subset of objects A ⊆ G, Eq. 1 says that A is the similarity of the description of all objects in A. Therefore, in IPS the corresponding A is the convex hull of the descriptions of all objects in A. For example, with

A = {g 1 , g 2 , g 4 }, A = [1, 8][1, 9][1, 2][0, 6][6, 7] .
Furthermore, given a description d ∈ D, Eq. 2 indicates that d is the set of all objects whose description subsumes d. In IPS, a description d 1 is subsumed by another description d 2 denoted by d 1 d 2 if every interval in d 2 is a sub interval in the corresponding interval in d 1 . Notice that in IPS, a sub interval subsumes a larger interval. Therefore, if

d x = [1, 8][1, 9][1, 2][0, 6][6, 7] , then d x = {g 1 , g 2 , g 4 }. Since δ(g 3 ) = [2, 2][2, 2][1, 1][7, 7][6, 6] , g 3 is not included in d x because the fourth interval ([7, 7]) is not sub interval of the fourth interval of d x ([0, 6]).
Following the denition of a concept of any pattern structure (in Sec. 3), an interval pattern concept is a pair (A, d), for A ⊆ G and d ∈ D, where A = d and d = A. Furthermore, the set of interval pattern concepts are partially ordered, and can be depicted as a lattice. An interval pattern concept

(A 1 , d 1 ) is a subconcept of (A 2 , d 2 ) if A 1 ⊆ A 2 (equivalently d 2 d 1 ).

Similar-column biclustering

A similar-column (SC) bicluster can be found in an interval pattern concept by introducing a parameter θ. This parameter acts as the maximum dierence between any two values to be considered as similar. For example, with θ = 1, the value 1 is similar to 2, but not similar to 3.

In calculating the similarity between any two descriptions, if the length of an interval is larger than θ, then the star sign ( * ) is put as the interval. From From an interval pattern concept, an SC bicluster can be formed by the concept's extent and the set of columns where the interval is not * in the concept's intent. For example, from the concept ({g

1 , g 2 , g 3 }, [1, 2][1, 2][1, 2] * [6, 6] ), ({g 1 , g 2 , g 3 }, {m 2 , m 2 , m 3 , m 5 }) is an SC bicluster with θ = 1.
By using IPS with parameter θ, constant-column biclustering is a specic case of SC biclustering. It can be noticed that with θ = 0, we obtain intervals with length 0, and that corresponds to constant-column biclusters.

Additive and multiplicative biclustering

An additive bicluster is a submatrix where there is a constant (or similar) difference between any two columns across all of its rows (see Sec. 2). Constant (or similar) column biclustering is a specic case of additive biclustering. Using this fact, we can obtain additive biclusters by aligning (similar to [START_REF] Henriques | BicPAM: Pattern-based biclustering for biomedical data analysis[END_REF]) each column, and then nd interval pattern concepts on the alignments. Table 3 provides an example of column alignment for additive biclustering.

The original matrix is shown in Table 3a, having 4 rows and 4 columns. The submatrix ({g 1 , g 2 , g 3 }, {m 2 , m 3 , m 4 }) is an additive bicluster in the original matrix.

This bicluster can be found by applying constant-column or similar-column biclustering to the column alignments. Table 3b shows the rst column alignment, can be seen by the consistency of the rst column (m 1 ). In this example, each object value is converted such that its m 1 value is equal to the value of m 1 in g 1 . This means that the values 0, -2, 2, and 3 are added to g 1 , g 2 , g 3 , and g 4 respectively. This alignment is repeated for every column. Table 3c is the alignment of m 2 , by adding 0, -3, -2, and -5 to g 1 , g 2 , g 3 , and g 4 respectively.

Constant-column (or similar-column) biclustering is applied to every column alignment to nd additive biclusters. In the second column alignment (Table 3c), we obtain ({g 1 , g 2 , g 3 }, {m 2 , m 3 , m 4 }) as a constant-column bicluster. This corresponds to the additive bicluster ({g 1 , g 2 , g 3 }, {m 2 , m 3 , m 4 }) in the original matrix (Table 3a).

Multiplicative biclusters can also be obtained using similar column alignment. In multiplicative column alignment, instead of adding values to each row, we multiply each row such that a column has a constant value. Table 5b shows the second column alignment of the original matrix in Table 5a. Here, a constant value is achieved for m 2 by multiplying g 1 , g 2 , g 3 , and g 4 by 1, 1 and 1 2 respectively. Then, by applying IPS to each alignment, we can obtain the multiplicative biclusters. For example, constant-column biclustering using IPS in Table 5b returns ({g 1 , g 2 , g 3 }, {m 2 , m 3 , m 4 }), which is the corresponding multiplicative bicluster in Table 5a.

Concept mining

Being a generalization of FCA, the mining of interval pattern concepts can be performed using some existing algorithms that generate a complete list of formal concepts. In this paper, we use CloseByOne (CbO) [START_REF] Kuznetsov | Comparing performance of algorithms for generating concept lattices[END_REF] since it requires us to only dene the similarity ( ) and subsumption relation ( ) of any two descriptions.

In a given numerical matrix, we may obtain an exponential number of interval pattern concepts. To reduce the number of concepts, we should introduce some parameters that can lter out some uninteresting concepts.

The rst parameter, θ, is previously mentioned in Sec. 4.2. It limits the length of intervals, and later in Sec. 5 we demonstrate the eect of θ on the runtime and number of concepts.

The second parameter min_col is the minimum number of columns in the retrieved biclusters. The number of columns in a bicluster corresponds to the number of non-star intervals in the concept's intent. For example, the concept with intent * * [2, 2] * [START_REF] Ganter | Pattern structures and their projections[END_REF][START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF] gives us a bicluster with two columns (the third and the fth). To take into account the min_col parameter, it is necessary to modify the denition of similarity between any two descriptions. In addition to the denition of in Sec. objects whose description subsumes Y ) is performed only if the number of objects in Z (extent of the candidate concept) is at least min_row.

Experiments

In this section, we report some experimental results to show the scalability of IPS in the task of biclustering. By using CbO as concept miner, the space/time complexity of IPS follows CbO (see [START_REF] Kuznetsov | Comparing performance of algorithms for generating concept lattices[END_REF]). We use the synthetic datasets provided by Henriques and Madeira [START_REF] Henriques | BicPAM: Pattern-based biclustering for biomedical data analysis[END_REF]: 500×60 and 1000×100, with hidden SC biclusters.

First, we investigate the eect of θ on the runtime and the number of concepts. The results are illustrated in Fig. 1. The left gure conrms that the larger θ generates more interval pattern concepts, and generally longer runtime as it can be seen in the right gure. The θ = 0.4 requires longer runtime than θ = 0.5 to 0.9. This is normal since for similar number of concepts, the probability of smaller θ obtaining a concept is smaller than the larger θ. Using CbO with smaller θ, a candidate concept will have shorter intervals in its intent, hence smaller number of objects whose description subsumes this interval.

The eect of min_col is shown in Fig. 2 left. Lesser min_col produces more concepts, and therefore longer runtime. Similarly, Fig. 2 right shows that larger min_row generates more concepts.

In the previous experiments, the CbO was terminated until all interval pattern concepts were retrieved. In the following experiment, CbO is terminated until 500 concepts are found. We compare them to BicPAM [START_REF] Henriques | BicPAM: Pattern-based biclustering for biomedical data analysis[END_REF] that uses a discretization parameter (as a number of alphabet/items), while IPS uses the length of intervals as θ. 39 500

Furthermore, still from Table 6, the runtime of IPS is not exactly correlated with θ (especially with θ = 2), similar to our previous experiment shown in

Conclusion

In this paper, we propose an alternative method of biclustering in numerical datasets. Discretization is a general preprocessing step while working numerical values. Here we explore the possibility of working directly on numerical datasets without discretization. This can be achieved using interval pattern structures, where a bicluster can be found from any interval pattern concept.

To lter the number of concepts (which can be very large) it is necessary to provide some parameters, like the length of intervals, minimum number of rows and columns, or even minimum number of biclusters. Our experiments show that these parameters can reduce the computation to a reasonable runtime. Another way to reduce the number of biclusters is to develop post-processing techniques similar to BicPAM, which include merging, ltering, and extension.

We use the CbO algorithm, a formal concept generator that can be generalized to interval pattern structures. In-Close 2 [START_REF] Andrews | In-Close2, a high performance formal concept miner[END_REF] in particular is faster than CbO in formal concept mining, but its eciency in interval pattern concept mining should be studied. Another future research is to extend our FCA-based approach to other types of biclusters, e.g. coherent-evolution, coherent-sign-changes, etc.

Furthermore, the existence of missing values and/or outliers should be considered in improving the proposed biclustering method.

  Both additive and multiplicative biclusters were studied in the domain of gene expression dataset [4, 16, 5]. They represent a set of genes having similar expression patterns across a set of experiments.

Fig. 1 :

 1 Fig. 1: Eect of θ on a 500 × 60 dataset with min_col = 20 and min_row = 1.

Fig. 2 :

 2 Fig.2: Eect of min_col (with θ = 1 and min_row = 5) and min_row (with θ = 1 and min_col = 6) on a 500 × 60 dataset.

Fig. 1 .

 1 Fig. 1. Overall, with similar runtime, biclustering with IPS can return similar number of biclusters without discretization.

Table 1 :

 1 Examples of some bicluster types.

Table 2 :

 2 A numerical context and an SC bicluster in gray.

	G m1 m2 m3 m4 m5
	g1 1	2	2	1	6
	g2 2	1	1	0	6
	g3 2	2	1	7	6
	g4 8	9	2	6	7
	all elements have the same value. This task is called biclustering with constant
	values, which is a simultaneous clustering of the rows and columns of a matrix.
	Moreover, given a dataset (G, M ), a pair (A, B) (where A ⊆ G, B ⊆ M )
	is a constant-column (CC) bicluster i ∀m ∈ B, ∀g, h ∈ A, m(g) = m(h). An
	example of CC bicluster is illustrated in Table 1a. CC biclustering have more
	relaxed variations, namely similar-column (SC) biclustering. With these relax-
	ations, instead of nding biclusters with exactly constant columns, we can obtain
	biclusters whose columns have similar values as shown in Table 1b. These types
	of biclusters are widely used in recommendation systems to detect a set of users
	sharing similar preference over a set of items.

Table 3 :

 3 Example of additive column alignments. (a) Original table and the additive bicluster in gray, (b) alignment on m 1 , (c) alignment on m 2 .

	m1 m2 m3 m4	m1 m2 m3 m4	m1 m2 m3 m4
	g1 4	1	3	0	g1 4	1	3	0	g1 4	1	3	0
	g2 6	4	6	3	g2 4	2	4	1	g2 3	1	3	0
	g3 2	3	5	2	g3 4	5	7	4	g3 0	1	3	0
	g4 1	6	1	7	g4 4	9	4 10	g4 -4 1 -4 2
		(a)				(b)				(c)
	a description. The operator	is a similarity operation that returns the common
	elements between any two descriptions. It is veried that

Table 2 ,

 2 δ(g 2 ) δ(g 4 ) without θ is [2, 8][1, 9][1, 2][0, 6][START_REF] Ganter | Pattern structures and their projections[END_REF][START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF] , and with θ = 1 is For example, suppose that we have two descriptionsd x = [1, 1][2, 3] and d y = [2, 2] * . Then, d x d y = [1,2] * . This also means that * is subsumed by any other interval. Therefore, the description of each object in Table 2 subsumes * * [1, 2] * [6, 7] . With θ = 1, ({g 1 , g 2 , g 3 , g 4 }, * * [1, 2] * [6, 7] ) is an interval pattern concept. Some interval pattern concepts from Table 2 are listed in Table 4.

	The similarity	between * and any other interval is * .

* * [1, 2] * [6, 7] .

Table 5 :

 5 Example of multiplicative column alignments. (a) Original table and the multiplicative bicluster in gray, (b) alignment on m 2 .

	m1 m2 m3 m4	m1 m2 m3 m4
	g1 3	1	2	3	g1 3	1	2	3
	g2 1	3	6	9	g2 0.3 1	2	3
	g3 2	2	4	6	g3 1	1	2	3
	g4 1	2	6	8	g4 0.5 1	3	4
		(a)				(b)

  4.1, we verify if the number of non-star intervals in the description. The number of non-star intervals should be more than min col . If not, we skip the concept, by converting each interval to * . In Table 2 with θ = 1, g 1 g 4 is * * [2, 2] * [6, 7] . Using min_col = 3 for example, g 1 g 4 becomes * * * * * . Related to min_col is min_row, a parameter that put a constraint on the number of rows in a bicluster. It corresponds to the number of objects in a concept's extent. With the inclusion of min_row, the calculation of Y (all

		6	•10 4					2.5	•10 4
	Number of concepts	2 4					Number of concepts	1 1.5 2
								0.5
		0	6	7	8	9	10	0	6	8	10	12	14	16	18	20
				min_col							min_row

Table 6 :

 6 Comparison with BicPAM on 1000 × 100 dataset. For the IPS, the parameters min_row = 10 and min_col = 5 are used, with varying θ.

	After the mapping step (normalization, discretization, and miss-
	ing values and noise handling), BicPAM applies a pattern mining method (F2G
	[11] as default), and the closing step (extension, merging, and ltering) is per-
	formed. Results in Table 6 show a similar performance of both methods. It should
	be noted that the number of biclusters from BicPAM is lower due to the merging
	and/or ltering.


J M Prival

A Harris

L Gurioli

M O Chevrel

C Robustelli Test

S Kenderes

E Zanella

A Whittington

A history of deformation within Pietre Cotte rhyolite flow (Vulcano)

Geological setting

Pietre Cotte (literally "cooked stone") is a rhyolite flow located on the NW flank of the Fossa cone (Vulcano, Italy). It was emplaced around 1740.

The unit is ~380 m long, ~170 m wide, and up to ~15 m thick at the flow front. The lower half of the unit is well exposed through two lateral gullies, while the upper half has been buried under products of the 1888-1890 eruption.

Rationale and methods

Rhyolite flows are rarer than basalt flows. The only case observed while active was Cordón Caulle rhyolite flow (Chile) in 2011-2012. To better understand the dynamics of these flows, we thus need to study past, frozen flows. Pietre Cotte is an ideal case study: the flow is young and offers outstanding exposure in 3D.

We carried out structural mapping and facies analysis during two field campaigns. Orientated samples were also used to determine variations in flow direction through the anisotropy of magnetic susceptibility (AMS). The superimposition of structures, along with textural features, allowed us to build a five-step model of the rheological and structural evolution of the Pietre Cotte flow.

Step 3

Lava emplacement on a steep slope triggers collapse of lateral levees (to generate block and ash flows whose deposits crop out locally in the basal breccia). By lowering the lateral stress, these events generate local lateral spreading of the flow, as shown by AMS data. Cracks develop parallel to the main flow direction, extension cracks open at the surface and tension gashes on the sides. These openings promote outgassing, creating tuffisite veins. Extension leads to decompressiontriggered vesiculation, creating pumice pockets.

Step 4

The flow front slows down as the lava reaches flat ground and stagnates, while lava behind the flow front on steeper slopes applies stress. This compressive stress generates pluri-metric, "stairway-like" folds, with a right angle between a horizontal limb and a vertical limb, as well as microfolds. These overprint the step 3 structures all the way back to the vent.

Step 5

The now brittle folds break, with the horizontal limb ramping up on top of the vertical one. At the flow front, the lower half of the flow rolls over itself, while the upper part ramps up.

Step 1 Rhyolitic magma is extruded from a point at 270 m a.s.l. on the cone's outer flank just below the crater rim, with a sub-vertical foliation inherited from shearing along the conduit walls during ascent. rhyolitic magma La Fossa

Step 2

The lava starts flowing on the steep (~34°) upper slope. Brittle deformation occurs on the sides and at the base, where shearing is focused, generating a basal breccia and lateral rubble levees. Ductile deformation stretches vesicles parallel to the flow direction.

Conclusion

The Pietre Cotte flow presents a succession of structures revealing a complex emplacement history in a very high viscosity environment, with ductile and brittle deformation sometimes happening at the same time depending on local variation in stress conditions. Deformation events drive textural changes (e.g. shearing creating flow bands, extension promoting vesiculation), leading to the different facies observed today in the field. We will now complete this study with quantitative data from textural and geochemical analysis in order to properly define the emplacement model.

Acknowledgements

This research was carried out as part of the Laboratory of Excellence ClerVolc project.

Flow facies

As in Fink's (1983) model, the Pietre Cotte flow shows three main facies based on their texture: obsidian, lithoidal rhyolite, and pumice (either finely or coarsely vesicular). Obsidian and rhyolite are often intertwined, defining flow bands.


Jacques Fages
email: jacques.fages@enstimac.fr

Martial Sauceau

Jean-Jacques Letourneau

Élisabeth Rodier

Elisabeth Rodier

Supercritical carbon dioxide : an efficient tool for the production of ultra-fine particles for the food and pharmaceutical industries

Keywords: supercritical fluids, particle formation, RESS process, SAS process, PGSS process

come

Introduction: supercritical technology, a new route for the formation of ultrafine powders

As shown on the phase diagram (figure 1), supercritical fluids (SCF) are defined as being compounds placed simultaneously at a pressure and a temperature higher than those of their critical point.

Figure 1 -Phase diagram of a pure substance

They present properties (density, viscosity, mass transfer) intermediate between those of the liquids and gases. The other major characteristic of these fluids is that one can vary their properties in broad proportions by tuning pressure, a very convenient operating parameter. SCF are at the same time dense and compressible. These properties are at the origin of the discovery that it is possible to generate finely divided solids by abruptly depressurising a supercritical solution.

Among the usable compounds, carbon dioxide (CO 2 ), is the far most widely used supercritical fluid. Its Pc and Tc are relatively easy to reach. Its solvent power is relatively close to those of chlorinated solvents (CHCl 3 , CH 2 Cl 2 , CFC's) and normal alkanes like hexane which are already or might soon be banned. Nevertheless, CO 2 is a rather poor solvent for polar molecules. However, CO 2 can also solubilise highly polar compounds provided adequate CO 2 -soluble surfactants (like fluoro-ethers) are used.

In addition, it exhibits several other advantages: it is natural, cheap, easily available, gaseous at atmospheric pressure, non toxic and non corrosive. It is recycled from the fertiliser industry (it comes from the environment and will return to it after use) and therefore, in this way, does not add to the greenhouse effect.

Since the middle of the eighties, an intense activity of research has been developed which strongly intensified in the nineties. Several processes of elaboration and design of particulate • Production of ultra-fine powders, with a tight particle size distribution: The smaller the particles, the more the surface properties will override the volume properties. For example, the kinetics of dissolution of a powder depends directly on the particle size. A tightened distribution will ensure a reduced variability which is one of the key factors for a successful industrialization of a process.

• Production of composite particles: The microencapsulation and the coating of an active substance by a protective agent are techniques sought in many industrial sectors whose products require a protection or when a differed effect is desired.

Pharmacy, cosmetics, agrochemicals, inks and paintings industries, are large users of these composite particles.

• Control of crystal polymorphism: Undergoing crystallization, the molecules fit together according to an ordered structure. For complex organic molecules, it is frequent that there are several manners for the molecules of ordering in the crystal in formation, leading to the so-called polymorphs. Two polymorphs are two different substances, which can have different properties. Each polymorph will have a different thermodynamic stability, but the kinetics can lead first to a polymorph which is not the most thermodynamically stable. To obtain the desired form in a reproducible way is therefore a matter of concern. The fast and single-step supercritical processes often make it possible to control crystal polymorphism.

In this paper we will introduce the main supercritical particle generation processes followed by a brief presentation of two examples from our experience in the domains of the food and pharmaceutical engineering.

The three categories of particle generation process:

RESS and related processes

Collecting device

Extraction autoclave

Nozzle

The RESS process consists in a dissolution-recrystallization. As shown on figure 2, the solid of interest is dissolved in the FSC before being re-cristallised by an abrupt depressurisation through an expansion device. In the supercritical domain, a fluid is both dense and compressible. The density, connected to the solvent capacity will vary in great proportions, according to the conditions of temperature (T) and pressure (P). The great and sudden fall of pressure, accompanied by a simultaneous fall of the temperature, causes huge supersaturation ratio and the nucleation of crystals. Co-precipitation of two compounds is also possible provided they are both soluble in the SCF. For industrial applications, it is necessary to recycle the fluid. A pressure of a few MPa is therefore maintained in the expansion vessel. The depressurisation must however be sufficient to obtain a great fall of solubility, in spite of the reduction of supersaturation. When the solute is insufficiently soluble in the FSC, the latter can be used as anti-solvent. This is the rationale of the SAS process (Supercritical Anti Solvent) which is particularly adapted for molecules known for their very low solubility in supercritical CO2, such as sugars, proteins, polymers or metallic oxides. In addition to the compound to be micronised and the FSC, an organic solvent is necessary which must present at least the three following properties:

Anti-solvent: SAS and related processes

-to be a good solvent of the studied compound; -to have a good compatibility with the anti-solvent. Ideally, they must even be completely miscible; -to leave the minimum residues in the solid crystallized at the end of the process.

A semi-continuous operating mode is most often used (see figure 3): the contact of the solution (solute + organic solvent) with the supercritical anti-solvent occurs at the nozzle level. The supercritical fluid dissolves in the organic phase decreasing its density, and thus the solubility of the solute. At the same time, the solvent will evaporate in the supercritical phase, increasing the concentration of the solute. It is this bidirectional mass transfer which explains the rapid supersaturation leading to the solute nucleation.

Supercritical CO 2

Mixing device Collecting basket

Solution vessel

Except for the recovery of the particles, the remainder of the process can be operated continuously. The solvent can be recovered in separators downstream, while the anti-solvent can be recycled by compression and heating. A stage of washing or extraction with pure CO2 is recommended at the end of the process, in order to eliminate the last solvent traces which could remain within the crystallized particles. PGSS means "Particle from Gas Saturated Solutions". A gas under pressure can be solubilised in a liquid in large quantity: it is this property which is used here. A dense gas, generally CO 2 , is dissolved in a liquid in a first autoclave. This liquid can be a solution, a suspension or a molten polymer. The solution saturated with gas thus obtained is sent in an expansion vessel through a tube (see figure 4). The generation of particles takes place at the time of the drop in pressure and temperature and the formed particles are then recovered in the autoclave of expansion. In this process, CO 2 can also act as an activator of aerosolisation. This process was developed at first for polymers in which great quantities of CO 2 can be dissolved. This dissolution is accompanied by a depression in the glass transition and melting temperatures, and by the swelling and the plasticization of polymer. If a third component is previously dissolved or suspended in the polymer, the final depressurisation may lead to micro-spheres of polymer with an embedded substance. Moreover, CO 2 reduces viscosity considerably, which will facilitate the handling of these solutions/suspensions.

Solute: PGSS and related processes

Expansion chamber

Nozzle SC-CO 2

Organic solution or molten polymer

An example for the food industry: crystallisation of cocoa butter for improving the tempering step of the chocolate manufacturing process

Cocoa butter (CB), the fat of chocolate, is a complex mixture of triacylglycerol (TAGs), which melting points are in the "good" range to give the pleasant "melt in the mouth" sensation. The most abundant are monounsaturated TAGs, the unsaturated fatty acid (very often oleic acid) being in the 2-position. 1-palmitoyl-2-oleyl-3stearoylglycerol (POS) ; P-O-P and S-O-S account for 36-38 %, 14-16 % and 23-28 % respectively of the total TAGs. There are six different ways for the TAGs to arrange themselves when they crystallise. Six different polymorphs noted I (the less stable with the lowest melting point 290 K) to VI (the most stable with the highest melting point 307 K) can be obtained. The relative amount of these six polymorphs will condition several features of the end product. If the wrong type of fat is present the chocolate may exhibit a white powdery surface called fat bloom which is to be avoided because of its unpleasant appearance. The most desirable form is known as form V (or β2) having a melting temperature of 305-307 K. Form V is hard with a good snap and a glossy appearance and good resistance to bloom. The even more stable form VI is not desirable since it might lead to fat bloom as do the less stable forms. Form VI is only formed by solid-solid transformation and not directly from liquid cocoa butter. Chocolate with fat in form V may start to bloom after a period of months or even years [START_REF] Beckett | The science of chocolate, RSC paperbacks[END_REF].

To obtain predominantly form V a pre-crystallisation step also called tempering has been introduced in the chocolate manufacturing process. It is aimed at the formation of form V seed crystals (1 to 3 % of the total fat content), which in turn will force the bulk of the remaining fat to crystallise in this stable form. Two main operating routes are possible: § starting with hot liquid chocolate (318 -323 K) a succession of cooling to 300 K in order to form both stable and unstable crystals followed by re-heating to 303-305 K to melt only less stable crystals will lead to some form V crystals which, in turn, will seed the mass of chocolate. § seeding the mass of untempered chocolate by finely divided form V crystals of cocoa butter. This is usually carried out at 303 K. A mass fraction of 0.5% to 2% of ordinary cocoa butter seeds is mixed with the melted chocolate.

Experimental set up and configurations a) production of particles We have developed a RESS-derived process aimed at the production of fine formV cocoa butter crystals for tempering chocolate by seeding. The experiments were carried out in a stirred autoclave (Parr, USA) for the extraction coupled to a versatile pilot plant (Separex, France). The saturated mixture (CO 2 + CB) sprayed into the expansion vessel via the nozzle (fig. 2), is subjected to a sudden drop in pressure, resulting in supersaturated conditions and the crystallisation of the cocoa butter. This powder is collected in the bag inside the second vessel. Detailed experimental procedure can be found elsewhere [START_REF] Letourneau | Micronized cocoa butter particles produced by a supercritical process[END_REF]. Two major types of experiments have been achieved:

§ In the first one, the first vessel is let to slowly depressurise during its emptying through the nozzle. Therefore, particles of cocoa butter can also be formed in this vessel. The powders thus obtained were generally white, extremely light and sometimes pieces of original yellow cocoa butter could be found among the particles. § In the second, the pump feeds the vessel with CO 2 in order to compensate the emptying via the nozzle and to maintain a constant pressure. Powders collected in the bag of the expansion vessel were white, light, free-flowing with regular individual particles. The yield is directly correlated with the ΔP applied at the nozzle.

b) tempering experiments Tempering experiments were performed using a computerised tempermeter. The degree of pre-crystallisation can be evaluated by the following procedure:

The pre-crystallised chocolate sample is put in a special container and placed in the instrument cell at 281 K. A cooling curve in which the change in temperature is plotted against the time. In a typical tempering curve, the exothermic process of crystallisation generates an inflexion point. The slope at this point is used to determine if the chocolate is under-, well-or over-tempered. A well-tempered curve shows an horizontal plateau. Experiments were performed with a type A powder. The angular shape of these agglomerates may favour the seeding process. At 304 K -the usual temperature-and below, the chocolate was quickly over-tempered showing great potential for this powder. At 306 K and above, the seeding produced an under-tempered chocolate.

Main results a) characterization of the particles

A well-tempered chocolate was obtained at 305 K (32ºC), with a mass fraction of cocoa butter seeds of 0.1-0.2%. "Supercritically-produced " CB powder allows to seed liquid chocolate 2 K higher than common seeding and several degrees higher than most industrial temperers. Therefore, the chocolate is very much thinner and easier to handle because of its lower viscosity. In addition, the mass fraction used is at least 3-5 times lower than the classical quantity used. Time necessary to insure tempering is also much smaller than in the classical process The DSC measurements made on the final product showed that the whole mass of the fat has crystallised in form V. When an important overtempering or a very long duration was observed, some form VI crystals can be formed.

In summary, this new process shows great promises for producing seeds extremely efficient for chocolate tempering. By comparison with the classical seed tempering process, it allows to work:

• at higher temperature (305 K), the chocolate being less viscous which is advantageous in term of flowability and energy consumption.

• with less cocoa butter ( 0.1-0.2 %)

• quicker (about 5 minutes).

An example for the pharmaceutical industry: improving the bioavailability of an active pharmaceutical drug

The lack of bioavailability is one of the major actual problems of the pharmaceutical industry. A large number of new drugs are almost insoluble in the biological fluids leading to a reduced bioavailability. Improving their dissolution rate is therefore a goal for researchers. Various methods can be used to increase the dissolution rate like size reduction by comminution leading to finely divided powder or complexation with cyclodextrins. Cyclodextrins are cyclic oligosaccharides which can form inclusion complex with large organic molecules. The resulting complex may improve the stability, solubility and bioavailability of the host molecule. The most used techniques for achieving inclusion imply either an aqueous solution or a process in solid state like grinding or kneading [START_REF] Gil | Evolution of the interaction of a new chemical entity, eflucimibe with γ-cyclodextrin during kneading process[END_REF].

Supercritical carbon dioxide has recently appeared as an efficient complexation medium for the preparation of inclusion complexes using its favourable transport properties. Depending on the solubility of the host molecule different techniques can be applied: Extraction and percolation through a cyclodextrin packed bed, co-crystallisation by SAS process or complexation using a static method.

Eflucimibe is an active compound (M = 469,73 g.mol -1 ) provided by the Pierre Fabre company and poorly soluble in supercritical CO 2 (5.71 10 -7 mol/mol at 274 bars and 308 K) [START_REF] Sauceau | Solubility of eflucimibe in supercritical carbon dioxide with or without a co-solvent[END_REF]. It was therefore a good candidate to be processed by an anti-solvent method. We have developed a new process coupling an anti-solvent crystallisation of this drug with an inclusion in γ-cyclodextrin. It is composed of three consecutive steps, a co-crystallisation of both components by SAS, a batch step leaving the powder in a supercritical CO 2 medium and finally a solvent stripping step.

Experimental set up and configurations

The experiments were carried out in a versatile pilot plant (Separex, France). For the SAS experiments (fig. 3), the solvent used was DMSO and the antisolvent was CO 2 . Both compounds, γ-cyclodextrin, and eflucimibe were dissolved in DMSO. The solution was injected by an HPLC pump into the CO 2 stream.

The organic solution and the anti-solvent were contacted in the mixing chamber of a nozzle and the resulting mixture was sprayed into the expansion vessel. The powder formed was collected in a porous bag. Then, CO 2 -solvent mixture was depressurised and separated in cyclonic separators.

For the batch step configuration, 2.33 g of water were added to 7g of eflucimibe/cyclodextrin powder and placed in an autoclave. This vessel is then filled with supercritical CO 2 at the desired pressure and temperature and left during several hours. To finish this step, the vessel is slowly depressurised and the powder taken away.

During the solvent stripping step performed in the same autoclave, supercritical CO 2 flowing through a stainless steel basket containing the organic compound during two hours.

Dissolution tests were performed at 37°C. A mass of powder containing 100 mg of eflucimibe was put in 100 ml of aqueous solution containing sodium-dodecyl-sulfate, SDS (5% m/v). Periodically, 2 ml of solution were sampled and the concentration of dissolved drug was measured by HPLC.

Main results

a) The co-crystallisation process ensures a good and fine dispersion of the hydrophobic drug in the hydrophilic matrix of cyclodextrin. Moreover, a part of eflucimibe contained in the mixture after co-crystallisation was not visible by DSC suggesting the formation of an amorphous or microcrystalline form. The study of the influence of mass ratio of CO 2 to DMSO, varying between 50 and 300, on the composition and dissolution rate of the resulting powder revealed that this parameter had no effect. On the contrary, decreasing the molar ratio of eflucimibe to cyclodextrin in the initial mixture from 1/1 to 1/3 increased the drug yield in the resulting powder from 40% (w/w) to 70 % (w/w). The cyclodextrin may exert a stabilising effect on eflucimibe. After this step, the dissolution rate was increased in the first hours by comparison with the physical mixture with the same profile (Figure 6). In summary, co-crystallisation by SAS ensures a very fine and homogeneous dispersion of the hydrophobic drug among the hydrophilic CD powder. b) The second batch step, in presence of water, was carried out under supercritical conditions for a few hours, leading to a solid dispersion with an enhanced dissolution rate. This formation is favoured by the good mass transfer properties of supercritical CO 2 . The role of water is probably a plasticizing agent enhancing the dispersion degree between both eflucimibe and CD. In addition, water is a solvent for the mixture, which partially dissolves and recrystallises in other places, enhancing the dispersion of eflucimibe into CD. A strong modification of the drug dissolution profile can be observed with a peak having a maximum value as high as 500µg/ml (Figure 6). c) Finally, the stripping step extracts the residual solvent. The main drawback of this step is the simultaneous extraction of some active substance. On the dissolution rate the effect is even more pronounced with a much sharper and higher dissolution peak (figure 6) To explain the shapes of the dissolution curves we suggest the formation of a solid dispersion instead of a true inclusion complex. This hypothesis has been proposed based on more detailed analyses not presented here. First, the dissolving medium diffuses into the CD matrix containing eflucimibe. Then, eflucimibe is dispersed into the dissolving medium and is temporary stabilized into SDS micelles; this corresponds to the maximum of the peaks. Finally, a recrystallisation of eflucimibe occurs leading back, to the solubility of neat eflucimibe in this dissolving medium, that is around 100µg/ml. Finally, the acceleration of the dissolution kinetics after the stripping step may be due to the acceleration and amplification of the first dissolution step, that is the diffusion of the aqueous medium through the dehydrated CD matrix.

Finally, an in vivo bio-availability assay on dogs has been performed showing a 11-fold increase in bio-availability giving a clear confirmation of the interest of this new process.

  set up. They avoid the majority of the disadvantages of the traditional powder generation processes and have important specific advantages:

Figure 2 -

 2 Figure 2 -The RESS process

Figure 3 -

 3 Figure 3-The semi-continuous SAS process

Figure 4 -

 4 Figure 4-The PGSS process

Figure 5 .

 5 Figure 5. DSC patterns of original and processed cocoa butter

Figure 5 Fig. 3 .

 53 Figure5shows the DSC curves of original cocoa butter, compared with a powder obtained in the expansion vessel (sample A) and a powder collected in the extraction autoclave (sample B). Original cocoa butter was in form VI. Powder A characteristics (ΔH fus = 120.3 J.g-1; T melt = 304 K) are typical of form V. The particles obtained were made of an agglomeration of smaller particles For powder B, the two peaks are characteristic of both form IV and V. More geometric shaped particles were obtained in this case.

Figure 6 -

 6 Figure 6 -Dissolution curve of physical mixture (a); co-crystallised powder (b); powder after co-crystallisation and static batch step (c); powder after co-crystallisation, static batch step and stripping (d)


H Zhao1

G Gary

Behaviour Characterisations of Sheet Metals, Metallic Honeycombs and Foams at High and Medium Strain Rates

Keywords: Aluminium Foam, Honeycomb, Kolsky's Bar, Modelling, Strain Rate

This paper deals with dynamic testing and modelling of the metallic materials and soft structural material made of metals such aluminium honeycombs and foams. Tests are performed with the SHPB (Split Hopkinson Pressure Bar) system which is frequently used in a range of strain rates which correspond to those involved in most of civil industrial applications. Improvements of the SHPB were necessary in order to obtain reliable results on soft structural materials. Some experimental results and their modelling for metals as well as for aluminium honeycombs and foams wil1 be presented.

Material testing with SHPB

Recently, the industrial requirement of numeric simulations under dynamic loading condition make the characterisation of the materials behaviour under impact loading an important goal for engineers and researchers. The most efficient experimental tool in this field is the Split Hopkinson Pressure Bar (SHPB), or Kolsky's apparatus. Historically, the first use of a long thin bar to measure the pulse shape i nduced by an impact is considered due to Hopkinson [1]. This method has been well established after the critical work of Davies [2] The experimental setup with two long bars and a short specimen has been introduced by Kolsky [3].

Kolsky's original SHPB analysis is based on some basic assumptions. (i) The waves propagating in the bars can be described by the one-dimensional wave propagation theory. (ii) The stress and strain fields in the specimen are uniform in its axial direction. (iii) The specimen inertia effect is negligible. (iv) The friction effect in the compression test is also negligible.

Those assumptions have been extensively studied in past decades. Following Davies' works, a more accurate wave propagation theory has been used in the data processing. The oscillations due to wave dispersion effects observed in the average stress-strain curve have been diminished. The assumption of axial uniformity of stress and strain fields permits relating the average stress-strain curve to forces and velocities measured at both faces of specimens. Theoretical, experimental, and numerical investigations on this assumption have been reported. It has been proved then that stresses and strains are not axially uniform. However, the average stress-strain relation is quite reliable for metals. In order to minimise friction effects, an optimal length/diameter ratio of the specimen is recommended. Radial and longitudinal inertia effects should be taken into account. The correction, based on the assumption of the axial uniformity of fields, is also proposed. Related references can be found in [4].

Split Hopkinson Pressure Bar Test

A typical SHPB set-up is composed of long input and output bars with a short specimen placed between them (Fig. t ). The impact of the projectile at the free end of the input bar creates a compressive longitudinal incident wave, ei (t). Once it arrives at the bar-specimen interface, a reflected wave, Er(t), is developed in the input bar; a transmitted wave, Et(t), is induced in the output bar. From these basic experimental data (incident, reflected and transmitted waves), forces and velocities at both faces of the specimen can be deduced.

F OU!pU! (t) = sb Eb c[ (t) vou!pu! ( t) = ch £, ( t)
where Eb,Sb,Cb denote Young's modulus, cross-sectional area, and wave speed of the bar, respectively.

The Split Hopkinson pressure bar arrangement can give then quite accurate measurements of forces and velocities at both sample faces if the data processing is carefully performed. In order to relate material properties to measured forces and velocities at the two specimen faces, the classical analysis assumes the axial uniformity of stress and strain fields in the specimen. An average stress strain curve can be obtained from Eqn. 2a and 2b. which lead to the so-called two-waves analysis where E8(t),cr5(t),S5,ls are respectively average strain rate, stress, cross-sectional area and length of the specimen.

Application to metallic materials

Application of SHPB to metals deos not arise particular experimental problems today. However, The material behaviour cannot always be simply derived of the experimental data of SHPB test. For instance, in dynamic tests, strain and stress fields are not homogeneous in space and they are not constant in time [5]. The strain rate during a dynamic test is then not constant and a typical strain rate time history gives a variation of 20% around the average strain rate.

Dynamic tests are not isothermal. A significant temperature increase during the dynamic test is often observed [6]. If the material is thermally sensitive, experimental data (average stress-strain curve, for example) includes necessarily a thermal softening effect. This effect must be taken into account. The temperature increase could be estimated from the plastic work [7], a<>suming a quasi adiabatic condition and a quasi-complete conversion of plastic work into heat.

For the behaviour modelling of metals, the authors intend to combine the physical concepts with the phenomenological formulation. A simple and directly usable model (with several parameters to identify) is presented and it is associated with identification processes. Such a model is based on the work of Campbell and Ferguson (8] where they indicate that rate and temperature sensitivities of flow stresses can be divided into a few zones. Similar behaviours are observed in each zone for most of metals and alloys. For example, the flow stress increases slightly with the strain rates in zone II (low temperature, low strain rate) where the thermally activated flow mechanism is observed. A sharp increase of flow stress is found in zone IV (low temprature, high strain rate) where the viscous drag is dominant Tanimura [START_REF] Tanimura | Proc. ler Int. Symp. Impact Engng[END_REF] has proposed to model material behaviour of metals and alloys by the additive combination of those effects. Following this concept, we propose a model which is a combination of a quasi-static part 0'5(c P , T), a thermally activated flow part cr,(E P , € P , T) and a viscous drag part crv(E P , E:P ,T ). salt-rock, polymers [12] and polymeric foams [13]. Here our interest will be focused on soft structural materials such as aluminium honeycomb and aluminium foam. However, the use of viscoelastic bars in a SIIPB set-up introduces complications related to an important wave dispersion effect and requires a special calculation of stress and particle velocity in the bar from the measured strain. The correction of dispersion effects based on the analysis of the wave propagation in an infinite viscoelastic cylindrical bar (14] is used. Using careful data processin g, the Nylon bar provides a satisfa ctory measuring precision and has been used successfully in testing fibre-reinforced polymeric composite plates [ 15].

Another particular feature of soft material testing is the need to achieve a large maximum strain (up to 80%) for the study of the densification, associated with a significant increase of the stress. Ordinary SHPB arrangements can not measure till such strains, even if such strains arc easily reached because of the very low resistance of those materials. Indeed, measuring technique using bars is based on the superposition principle which implies that the stress, the strain and the particle velocity at any cross-section can be considered as the algebraic sum of those values associated with the two elementary wav es propagating in opposite directions at this cross-section (Eqn 1). Such a technique requires then a separate recording of ea ch single wave propagating in the b ar. This is cl assically realised by using a long bar and a short pulse to insure the existence of a particular cross section where those waves are not superimposed. The measuring duration 't of a given SHPB is then limited by the length of the bars ( t::; L/ C, C being the wave speed and L the length of the bar).

Consequently, the measurable crushing displacement is limited for a given impact velocity (�max .$ V0 t). To overcome this measuring limitation, one has to inv estigate the multiple reflections in b ars f 161. A two-gauge method has been reported to separate the two waves in elastic bars [17] and viscoelastic bar [18]. This two gauges method provides an unlimited measuring duration and gives consequently a su fficien t maximum strain.

Furthermore, particular situations in testing such soft materials lead to secondary effects more important than in testing metals. For example, the assumption of uniform stress and strain fields in the specimen can be not verified and an inverse method should be used to obtain a stress strain relation [START_REF] Gary | Proc. IUTAM symposium on Nonlinear wave in solids[END_REF] dynamic results are only found in the out-of-plane crushing [21]. According to Wierzbicki [22], the mean crushing pressure depends only on the flow stress and the so-called relative cell thickness. Increases in the mean crushing pressure could, then, be attributed to a different flow stress under dynamic loading. However, we consider that the factor primarily responsible for this enhancement of the crushing strength remains an open question. Possible factors have been mentioned in theoretical works in the open literature. For example, a dynamic buckling model of an elastoplastic column [23] has shown that the buckling mechanism is stabilised by effects due to lateral inertia, even when the initial imperfection is significant. There exist also a shock model based on structural inertia effects which has successfully explained a similar phenomenon observed in the results of woods [24], and a model based on air trapped in the cells [25]. For the aluminium foam, cylindrical specimens are machined from cylinder of diameter of 28 mm given by the industry. The experimental data shows an important dispersion. Post-test examination reveals holes in the middle of the specimen which appears to be due to manufacturing process. (Figure Sa). Anyway, the general tendency has been drawn for several foam of different cell size from 0.2mm to 0.6mm. Figure 5b show that the rate sensitivity is not influenced by this parameter.

F igure 1 .

 1 Sp ec ,men Out\t bar .__ __ • _ ___,II I � Typical Split Hopkinson Pressure Bar setup

F

  ;nvu•( t) = S b Eb (c;(t) + cr(t)) Vinpur (t) = C b (ei(t)-e,(t))and(1)

  crs(E P ,T) =(A+ B€:)(1-J..l �T) 01(E P ,E P ,T) = (C-Dc;1)1og(;_)(l-J..l �T) Eo crv(E P ,f: P , T) = E£�:(1-�T) cr; [A+ Be; + (C-De;) log(: , )+ E£' }1-,.W.T) (3 ) where A,B,C,D,E,k,m,n,ande0are coefficients of the model. The parameters of the model for a mild steels (XES) are then obtained (A==I45 Mpa, B=550 Mpa C=35 Mpa, D=47 Mpa, E=l85 Mpa; n=0.42, m=0.3, k=0.3) [10]. The quality of this set of parameters is shown in Figure 2: where the simulated strain-stress curves using the model with identified parameters and measured strain rate as input data are compared with corresponding experimental curves (the bold lines are experimental curves). . ..... ..... ..... ..... ..... . .... • :••• ..... ............................. ... :•• . .... ..... ..... ..... .......... ..... .... : . .... ..... .... � •-•• .... . ..... ..... ... . .

Figure 2 . 4 .

 24 Figure 2. Comparison between model and experimental data

4. 1

 1 Specific Problems in soft structural material Testing The measuring accuracy of SHPB depends on the amplitude of waves in the bars induced by the resistance of the specimen. Because the soft materials such as honeycomb or aluminium foam are quite weak, the induced strain in an ordinary steel Hopkinson bar is smaller than 10•6• In order to get an accurate measurement, the use of low impedance bars which are generally viscoelastic is proposed. The Nylon bars (Young's modulus 3.5xl09, density 1200 kg/m3) are used; they provide an improvement of the measuring sensitivity of about 200 times than that of a classical steel bar.

  .

4. 2 Figure 3 ,

 23 Figure 3, post test photograph of the honeycomb

Figure 4 .

 4 Figure 4. Rate sensitivity of honeycombs

Figure 5

 5 Figure 5 (a) specimen inhomogeneity, (b) rate sensitivity