Activité antimicrobienne ;

Activité antifongique des fractions actives pures ont été déterminées contre des levures, des champignons filamenteux et des bactéries, dont la plupart sont pathogènes ou toxinogènes pour l'homme et multirésistants aux antibiotiques. L'activité la plus importante a été observée contre Candida albicans M3 et Bacillus subtilis ATCC 6633. Les structures chimiques des molécules ont été déterminées par analyses spectroscopiques (UV-visible et spectres RMN du 1 H et du 13 C) et analyse spectrométrique (spectre de masse). Les composés P41A et P41B ont été identifiés comme étant les saquayamycines A et C, respectivement. Ces composés appartiennent aux antibiotiques du groupe des aquayamycines, connus dans la littérature pour leur activité anticancéreuse et antibactérienne.

showed that this strain belonged to the genus Streptomyces. Two bioactive compounds, named P41A and P41B, were extracted by dichloromethane from the cell-free supernatant broth of strain PAL114 and were purified by HPLC. Minimum inhibitory concentrations of the pure antibiotics were determined against yeasts, filamentous fungi and bacteria, most of which are pathogenic or toxigenic for human and multiresistant to antibiotics. The strongest activities were observed against Candida albicans M3 and Bacillus subtilis ATCC 6633. The chemical structures of the compounds were determined by spectroscopic analysis of UV-visible and 1 H and 13 C NMR spectra and spectrometric analysis of mass spectrum. The compounds P41A and P41B were identified as saquayamycins A and C, respectively. These compounds belong to the aquayamycin-group antibiotics, which are known in the literature for their anticancer and antibacterial activities.

Résumé Une nouvelle souche d'actinomycète désignée PAL114, produisant des composés antimicrobiens, a été isolée à partir du sol saharien de Ghardaïa, Algérie. Des études morphologiques et chimiques ont permis de rattacher cette souche au genre Streptomyces. Deux composés actifs, nommés P41A et P41B, ont été extraits par le dichlorométhane à partir du filtrat de culture de la souche PAL114 et purifiés par HPLC. Les concentrations minimales inhibitrices

Introduction

Considering the increasing resistance of pathogens to antibiotics, research has intensified in order to discover new bioactive molecules. Among these pathogenic microorganisms, fungi and yeasts cause serious diseases that affect humans. Mycoses have increased dramatically during the last decade. They rank fourth among nosocomial infections [START_REF] Abdallaoui | Mycoses nosocomiales systémiques à Trichosporon asahii : à propos de trois cas au CHU Ibn Rochd de Casablanca (Maroc)[END_REF]. Bacterial infections pose also clinical problems, since many works in the world and in Algeria have reported the emergence of new bacterial strains resistant to multiple antibiotics, including some cephalosporins of third and fourth generation [START_REF] Kumarasamy | Emergence of a new antibiotic resistance mechanism in India, Pakistan, and the UK: a molecular, biological, and epidemiological study[END_REF][START_REF] Messai | Prevalence and characterization of extended-spectrum b-lactamases in Klebsiella pneumoniae in Algiers hospitals (Algeria)[END_REF][START_REF] Sekhsokh | Fréquence et sensibilité aux antibiotiques des bactéries isolées dans les urines[END_REF].

Because of still perceptible deficiencies, which are observed during the treatment of fungal and bacterial diseases, and considering the resistance of microorganisms to antibiotics, the current works are oriented toward the search for new bioactive molecules. Among the molecules with antimicrobial activity, the range of the antifungal compounds is much more limited than those of the antibacterial compounds [START_REF] Billaud | Interactions métaboliques des antifongiques azolés[END_REF]. This is why new non-toxic antifungal molecules should be sought.

Actinomycetes are Gram-positive bacteria with a percentage of guanine-cytosine higher than 55%, and most of them produce mycelia. They are particularly interesting due to their capacity to produce secondary metabolites with diverse chemical structures [START_REF] Valan Arasu | Antimicrobial activity of Streptomyces sp. ERI-26 recovered from Western Ghats of Tamil Nadu[END_REF]. They are especially known for the production of antibacterial and antifungal antibiotics. It has been estimated that about two-third of the natural antibiotics have been isolated from actinomycetes [START_REF] Solanki | Bioactive compounds from marine actinomycetes[END_REF]. The genus Streptomyces is known as the producer of the largest number of antibiotics. It produces about 80% of the antibiotics secreted by actinomycetes [START_REF] Demain | From natural products discovery to commercialization: a success story[END_REF].

Our previous work showed the richness of antibiotic-producing actinomycetes in the Algerian Saharan soil [START_REF] Sabaou | Les sols des oasis du Sahara algérien, source d'actinomycètes rares producteurs d'antibiotiques[END_REF]. The present work aims to study the antifungal and antibacterial activities of a new strain of actinomycete against several pathogenic and toxigenic microorganisms to human, and to determine the chemical structure of the bioactive molecules produced by this strain.

Materials and methods

Isolation and features of the actinomycete strain

A new strain of actinomycete, designated PAL114, was isolated from a Saharan soil collected from Béni-isguen (latitude, 32827 0 N; longitude, 03840 0 E; altitude, 468 m), Ghardaïa province (South of Algeria). The dry soil sample was suspended in sterile deionized water and diluted. Aliquots (0.2 mL) of each dilution were spread onto chitin-vitamin agar [START_REF] Hayakawa | Humic acid-vitamins agar, a new medium for the selective isolation of soil actinomycetes[END_REF]. The medium was supplemented with cycloheximide (80 mg/L) to inhibit the growth of any fungi. The plates were incubated at 30 8C for two weeks.

The strain PAL114 was identified at the genus level on the basis of morphological and chemical features. Both of these criteria are widely used to identify the genera of actinomycetes [START_REF] Kämpfer | The genus Streptomyces[END_REF].

The morphological and cultural characteristics were observed by naked-eye examination of 14 day-old cultures grown on various International Streptomyces Project (ISP) media: yeast extract-malt extract agar (ISP-2), oatmeal agar (ISP-3), inorganic salts-starch agar (ISP-4) and glycerol-asparagine agar (ISP-5) [START_REF] Shirling | Methods for characterization of Streptomyces species[END_REF], and also on Bennett medium [START_REF] Waksman | Classification identification and descriptions of genera and species[END_REF]. The micromorphology was observed by light microscopy.

For chemotaxonomic analyses, biomass was obtained by growing strain PAL114 in ISP-2 broth in shake flasks [START_REF] Shirling | Methods for characterization of Streptomyces species[END_REF] and incubated at 30 8C for 4 days. Diaminopimelic acid isomers and whole-cell sugar pattern were analyzed according to the methods of Becker et al. [START_REF] Becker | Rapid differentiation between Nocardia and Streptomyces by paper chromatography of whole-cell hydrolysates[END_REF] and Lechevalier and Lechevalier [START_REF] Lechevalier | Chemical composition as a criterion in the classification of aerobic actinomycetes[END_REF], respectively.

Target microorganisms

Except strains with ATCC numbers, all other target microorganisms were isolated from diseased patients undergoing antibiotic therapy, from medical equipment and from the hospital environment sampled in five Algerian hospitals located in Algiers and Béjaïa. The used target microorganisms were yeasts (Candida albicans M1, M2, M3, IPA200 and Saccharomyces cerevisiae ATCC 4226), filamentous fungi (Aspergillus carbonarius M333, Aspergillus flavus AF3, Fusarium culmorum FC200 and Penicillium glabrum PG1) and bacteria (Bacillus subtilis ATCC 6633, Staphylococcus aureus S1 and Klebsiella pneumoniae E40). These microorganisms were mostly pathogenic or toxigenic for humans and multiresistant to antibiotics. All strains of C. albicans are resistant to cycloheximide, itraconazole, nystatin, terbinafine and thioconazole, and sensitive to amphotericin B. Saccharomyces cerevisiae ATCC 4226 resists to the same antifungal compounds, except cycloheximide and amphotericin B. Aspergillus carbonarius M333, A. flavus AF3 and Penicillium glabrum PG1 resist to cycloheximide but Fusarium culmorum FC200 resists to cycloheximide, itraconazole, nystatin, terbinafine and amphotericin B. Bacillus subtilis ATCC 6633 is a sensitive bacterium and resists only to neomycin. Staphylococcus aureus S1 resists to gentamicin, kanamycin, neomycin, spiramycin and vancomycin. Klebsiella pneumoniae E40 resists to cefotaxime and ceftazidime.

Production and purification of bioactive compounds

The production of bioactive compounds was conducted in ISP-2 broth. This medium was chosen among several others because of its very good results in our preliminary work. A seed culture was prepared with the same medium and used to inoculate 20 Erlenmeyer flasks of 500 mL, each containing 100 mL of medium. The cultures were incubated on a rotary shaker (250 rpm) at 30 8C. The extraction of the active compounds was carried out on the fifth day (previously determined as being the day of the optimal production) by centrifugation (5000 g, 20 min) of the culture broth to eliminate cells. The cell-free supernatant was extracted with an equal volume of dichloromethane. The organic extract was concentrated to dryness by a rotary evaporator under a vacuum at a temperature lower than 40 8C. The resulting dry extract was recuperated in 0.5 mL of methanol and bioassayed against Candida albicans M3, Aspergillus carbonarius M333 and Bacillus subtilis ATCC 6633 by paper disk diffusion method (disk diameter, 6 mm) at the rate of 30 mL per disk. These three target microorganisms were selected from the twelve previously cited because they represent the three groups of microorganisms studied (bacteria, yeast and filamentous fungi), and also because of their higher sensitivity to the antibiotic activity (noticed in preliminary work), thus, facilitating the detection of bioactive compounds.

Preparative chromatography with silica gel plates (Merck Art. 5735, Kiesselgel 60HF 254-366; 20 Â 20 cm) was employed for the partial purification of antimicrobial products. TLC plates were developed in the solvent system ethyl acetate-methanol (100:15 v/v). The developed TLC plates were air dried overnight to remove all traces of the solvents. The separated compounds were visualized under UV at 254 nm (absorbance) and at 365 nm (fluorescence), and the bioactive spot was detected by bioautography [START_REF] Betina | Bioautography in paper and thin layer chromatography and its scope in the antibiotic field[END_REF]. The retention factor (R f ) of the bioactive spot was measured.

The final purification of bioactive compounds was carried out by Waters reverse phase HPLC using an XBridge C18 (5 mm) column (200 Â 10 mm, WATERS) with a continuous linear gradient solvent system from 20 to 100% methanol in water, a flow rate of 2 mL/min and UV detection at 220 nm and 254 nm. All the peaks fraction were collected and tested against Candida albicans M3, Aspergillus carbonarius M333 and Bacillus subtilis ATCC 6633, in order to detect the active fractions and distinguish them from the non-active fractions. The choice of these three target microorganisms was already justified (as described above).

Structure determination of the active compounds

These analyses were carried out with the antimicrobial compounds purified by HPLC. The UV spectrum was given with a SHIMADZU UV1605 spectrophotometer. The mass spectrum was recorded on an LCQ ion-trap mass spectrometer (Finnigan MAT, San Jose, CA) with nanospray ion electro-spray ionization (ESI) source (positive and negative ion mode). 1 H and 13 C NMR spectroscopy were used for the characterization of the active molecules. NMR sample was prepared by dissolving 3 mg of each purified compound in 600 mL of CD 3 OD. All spectra were recorded on a Bruker Avance 500 spectrometer equipped with a 5 mm triple resonance inverse Z-gradient probe (TBI 1 H, 31 P, BB). All the chemical shifts for 1 H and 13 C are relative to TMS using 1 H (residual) or 13 C chemical shifts of the solvent as a secondary standard. The temperature was set at 298 K. All the 1 H and 13 C signals were assigned on the basis of chemical shifts, spin-spin coupling constants, splitting patterns and signal intensities, and by using 1 H-1 H COSY45, 1 H- 13 C HSQC and 1 H- 13 C HMBC experiments. Gradient-enhanced 1 H COSY45 was realized included 36 scans per increment. 1 H- 13 C correlation spectra using a gradient-enhanced HSQC sequence (delay was optimised for 1 J CH of 145 Hz) was obtained with 200 scans per increment. Gradient-enhanced HMBC experiment was performed allowing 62.5 ms for long-range coupling evolution (340 scans were accumulated). Typically, 2048 t2 data points were collected for 256 t1 increments.

Determination of minimum inhibitory concentrations

Minimum inhibitory concentrations (MICs) of pure bioactive compounds were performed using conventional agar dilution method [START_REF] Oki | Pradimicins A, B and C: new antifungal antibiotics II. In vitro and in vivo biological activities[END_REF] on the twelve target microorganisms cited above (see paragraph about target microorganisms). These latter were inoculated onto Mueller Hinton medium for bacteria and Sabouraud medium for yeasts and filamentous fungi, containing different concentrations of active compounds (1, 2, 5, 10, 20, 30, 50, 75 and 100 mg/mL). After a growth period of 24-48 h at 37 8C for bacteria and 48-72 h at 28 8C for fungi, the plates were examined for growth and the lowest antibiotic concentration that inhibited the growth of each organism was determined. Mueller Hinton and Sabouraud media without active compounds and inoculated with target microorganisms was used as a control treatment.

Results

Identification of actinomycete to the genus level

The actinomycete strain PAL114 formed a well-developed and ramified aerial mycelium with long spiraled chains of spores carried by sporophores. The spores were oval and 1-1.5 Â 0.6-1 mm in size. The sporulation was good on all the media used. The substrate mycelium was non-fragmented. The strain showed good growth on ISP-2, ISP-3, ISP-4, ISP-5 and Bennett media. The aerial and substrate mycelia were light to medium gray and light brown, respectively. Diffusible pigment with light brown color was produced on ISP-2, ISP-3 and Bennett media. The chemotaxonomic study showed the presence of LL-diaminopimelic acid isomer and glycin in the cell wall and the presence of galactose, glucose and ribose in the whole-cell hydrolysates.

Detection and purification of bioactive compounds

The dichloromethane extract of the culture filtrate, previously dried and then redissolved in 0.5 mL of methanol, was inoculated on paper disks (6 mm diameter) at a rate of 30 mL per disk. The diameters of inhibition obtained were 20 mm for Bacillus subtilis ATCC 6633, 20 mm for Candida albicans M3 and 12 mm for Aspergillus carbonarius M333 (disk diameter included).

On silica gel thin-layer chromatogram, the dichloromethane extract migrated and gave one bioautographic fraction, which was active against Candida albicans M3, Aspergillus carbonarius M333 and Bacillus subtilis ATCC 6633. This fraction (R f = 0.9 in ethyl acetate-methanol, 100:15 v/v) was injected in HPLC apparatus. Two active compounds against these three microorganisms, named P41A and P41B, were detected and eluted with 80% of methanol in water at a retention times of 41.05 min and 41.56 min, respectively. A quantity of 3 mg of each compound was obtained from 2 L of the culture filtrate.

Structure determination of bioactive compounds

The UV-visible spectra in methanol (data not shown) exhibited a maximal absorption at 218, 317 and 425 nm for P41A and at 218, 316 and 425 nm for P41B. The ESI-MS spectra contained an ion peak at m/z 819.1 [MÀH] À for P41A and at m/z 823.2 [MÀH] À for P41B (Fig. 1). Thus, the molecular weights of these two compounds were M = 820 for P41A and M = 824 for P41B.

The 1 H and 13 C chemical shifts of P41A and P41B compounds are given in Table 1 and their structures can be seen in Figs. 2 and3.

The ) for P41b compound. The hydrogens of the hydroxyl group are not observed due to the rapid exchange with MeOD. The 2D 1 H-1 H and 1 H-13 C (see Figs. 2 and3) permitted to establish the connectivity between all the groups of the P41A and P41B molecules.

Minimum inhibitory concentrations

Minimum inhibitory concentrations of antimicrobial compounds purified by HPLC are summarized in Table 2. The MIC values obtained against the target microorganisms the same for both active compounds. They were comprised between 30-50 mg/mL for yeasts, 75-100 mg/mL for filamentous fungi and 30-50 mg/mL for Gram-positive bacteria. Gram-negative bacterium tested was resistant. The most sensitive microorganisms were Bacillus subtilis ATCC 6633 (30 mg/mL) and Candida albicans M3 (30 mg/mL).

[ ( F i g u r e _ 1 ) T D $ F I G ]

[ ( F i g u r e _ 2 ) T D $ F I G ]

Discussion

The strain PAL114 has an aerial mycelium producing chains of spores, each carried by sporophores, and non-fragmented substrate mycelium. This strain possesses in its wall, the LL-diaminopimelic acid isomer and glycin, but does not have in its cells diagnostic sugars, such as arabinose-galactose, xylose-arabinose and rhamnose-galactose couples, or madurose. These results indicate that strain PAL114 has a chemotype IC as defined by Lechevalier and Lechevalier [START_REF] Lechevalier | Chemical composition as a criterion in the classification of aerobic actinomycetes[END_REF].

Based on the morphological and chemical characteristics, strain PAL114 belonged to the genus Streptomyces [START_REF] Holt | Bergey's Manual of Determinative Bacteriology[END_REF].

Species of the genus Streptomyces are mainly known for the production of bioactive molecules [START_REF] Solanki | Bioactive compounds from marine actinomycetes[END_REF]. The antibiotics secreted by the species of this genus may be antibacterial and/or antifungal (aminoglycosides, aromatics, glycopeptides, macrolides, polyenes, etc.) or, have cytostatic and antitumor properties (anthracyclines, such as adriamycin and daunomycin for example), and many of them have found a fairly significant therapeutic application [START_REF] Jose | Non-streptomycete actinomycetes nourish the current microbial antibiotic drug discovery[END_REF]. The same species of actinomycete can produce several antibiotics (sometimes belonging to different classes); however, the same antibiotic can be produced by very different species and sometimes even different genera [START_REF] Kämpfer | The genus Streptomyces[END_REF][START_REF] Solanki | Bioactive compounds from marine actinomycetes[END_REF]. Therefore, this shows that the correlation between the production of bioactive molecules and taxonomy is weak.

The structure of natural antifungal compounds produced by Streptomyces (and other microbial genera) can be polyenic or non-polyenic. However, the number of these molecules used in therapy is quite limited because of their toxicity and sometimes their instability and insolubility [START_REF] Berdy | Bioactive microbial metabolites[END_REF], hence, there is a need to continue the search for interesting new antifungal molecules.

The strain PAL114 produces two molecules with antifungal and antibacterial activities. The structure of these two molecules, named P41A and P41B, was determined by NMR and mass spectrometry, and they appear to be saquayamycin A and C, respectively. They are antibiotics belonging to the family of anthracycline, class of angucycline and group of aquayamycin [START_REF] Matsumoto | Synthetic study of aquayamycin. Part 1: synthesis of 3-(phenylsulfonyl)phthalides possessing a b-C-olivoside[END_REF]. Among aquayamycin-group antibiotics, saquayamycins are the most closely related to vineomycin A1 [START_REF] Uchida | Saquayamycins, new aquayamycin-group antibiotics[END_REF]. They are known to be produced by actinomycetes, especially Streptomyces nodosus, which produces saquayamycins A, B, C and D [START_REF] Uchida | Saquayamycins, new aquayamycin-group antibiotics[END_REF]. It has been reported that saquayamycin Z was produced by some strains of the genus Micromonospora [START_REF] Antal | Retymicin, galtamycin B, saquayamycin Z and ribofuranosyllumichrome, novel secondary metabolites from Micromonospora sp. Tü 6368. I. Taxonomy, fermentation, isolation and biological activities[END_REF].

Saquayamycins are known as inhibitors of farnesyl-protein transferase, an enzyme involved in cell division [START_REF] Lingham | Farsenyl-protein Transferase: A new paradigm for cancer chemotherapy advances in discovery and development of natural product inhibitors[END_REF]. Therefore, they are cytotoxic, and used as anticancer agents in chemotherapy [START_REF] Abdelfattah | Moromycins A and B isolation and structure elucidation of C-glycosylangucycline-type antibiotics from Streptomyces sp. KY002[END_REF][START_REF] Lingham | Farsenyl-protein Transferase: A new paradigm for cancer chemotherapy advances in discovery and development of natural product inhibitors[END_REF]. Squayamycins A, B, C and D are active against adriamycin-sensitive and adriamycin-resistant sublines of P388 leukemia cells, and it has been reported that saquayamycin A is particularly very active against cancerous prostate cells [START_REF] Shaaban | Saquayamycins G-K, cytotoxic angucyclines from Streptomyces sp. including two analogues bearing the aminosugar rednose[END_REF][START_REF] Uchida | Saquayamycins, new aquayamycin-group antibiotics[END_REF]. Adriamycin is an anticancer compound developed in the 1960s, belonging to the anthracycline family, produced by actinobacteria of the genus Streptomyces [START_REF] Weiss | The anthracyclines: will we ever find a better doxorubicin?[END_REF]. All saquayamycins are also active against Grampositive bacteria [START_REF] Antal | Retymicin, galtamycin B, saquayamycin Z and ribofuranosyllumichrome, novel secondary metabolites from Micromonospora sp. Tü 6368. I. Taxonomy, fermentation, isolation and biological activities[END_REF][START_REF] Uchida | Saquayamycins, new aquayamycin-group antibiotics[END_REF], but no antifungal activity was reported in the literature. This is the first time that the antifungal activity (especially against yeasts) of saquayamycins is highlighted.

The results obtained in this work are interesting because of the resistance of strains of C. albicans (M1, M2, M3 and IPA200) used in our experimentation to cycloheximide, itraconazole, nystatin, terbinafine and thioconazole.

However, it should be noted that although the antifungal activity of compounds P41A and P41B is interesting in vitro, it is necessary to continue in vivo studies to really evaluate their effectiveness, and it is only from these results that the prospects for clinical application could be considered.

  HSQC and HMBC spectra show 36 carbon signals for P41A and P41B molecules. It was possible to discern 4 ketone groups (d c 182.4 to 205.2), 3 hydroxyl groups (d c 70.9 to 157.8), 7 ether functions (d c 66.7 to 98.9) 4 sp 2 -hybridized carbons (d c from 116.4 to 145.2) and 11 sp 3 -hybridized carbons (d c 13s ˙9 to 42.6) for P41a molecule, 5 ketone groups (d c 182.4 to 205.2), 7 ether functions (d c 66.7 to 95.0) and 2 hydroxyl groups (d c 70.9 to 157.8) 6 sp 2 -hybridized carbons (d c from 116.4 to 145.2) and 9 sp 3 -hybridized carbons (d c 13s ˙9 to 42.6

Figure 1

 1 Figure 1 Mass spectrometry profile of P41A (a) and P41B (b) compounds produced by the actinomycete strain PAL114. Spectre de masse des compose ´s P41A et P41B produits par la souche d'actinomyce `te PAL114.

Figure 2

 2 Figure 2 Structure of antibiotic P41A (A) and COSY correlations (B). Structure de l'antibiotique P41A (A) et corre ´lations COSY (B).

Figure 3

 3 Figure 3 Structure of antibiotic P41B (A) and COSY correlations (B). Structure de l'antibiotique P41B (A) et corre ´lations COSY (B).

Table 1 1

 1 H and13 C NMR data assignments of P41A and P41B compounds in CD 3 OD at 298 K. See Figs.2 and 3for numbering of hydrogen and carbon atoms. Donne ´es RMN 1 H and13 C des compose ´s P41A et P41B dissous dans du CD 3 OD a `298 K.

	1 H and 13 C number	1 H chemical shift (ppm)		13 C chemical shift (ppm)
		P41A	P41B	P41A	P41B
	1	1.33	1.33	13.91	13.91
	2	4.61	4.61	70.10	70.10
	3	-	-	197.27	197.27
	4	7.03	7.03	141.1	141.1
	5	6.08	6.08	126.1	126.1
	6	5.35	5.35	95.0	95.0
	7	1.25	1.25	15.9	15.9
	8	4.24	4.24	66.7	66.7
	9	3.71	3.71	76.4	76.4
	10	1.50	1.50	24.0	24.0
	11	1.98	1.98	24.0	24.0
	12	5.25	5.25	91.8	91.8
	13	2.39	2.39	24.8	24.8
	14	-	-	81.9	81.9
	15	2.00-2.32	2.00-2.32	42.61	42.61
	16	-	-	79.8	79.8
	17	-	-	79.5	79.5
	18	-	-	205.2	205.2
	19	2.73-2.93	2.73-2.93	50.1	50.1
	20	6.91	6.91	116.4	116.4
	21	6.41	6.41	145.2	145.2
	22	-	-	139.3	139.3
	23	-	-	138.5	138.5
	24	-	-	182.4	182.4
	25	-	-	137.7	137.7
	26	-	-	131.0	131.0
	27	-	-	182.4	182.4
	28	-	-	157.8	157.8
	29	-	-	114.0	114.0
	30	7.89	7.89	132.8	132.8
	31	7.62	7.62	118.6	118.6
	32	1.38	1.38	17.4	17.4
	33	3.57	3.57	74.9	74.9
	34	3.14	3.14	86.8	86.8
	35	3.83	3.83	71.0	71.0
	36	1.41-2.49	1.41-2.49	38.9	38.9
	37	4.95	4.95	71.0	71.0
	38	1.26	1.33	16.7	13.9
	39	3.92	4.61	70.5	70.1
	40	3.24	-	70.9	197.3
	41	1.84	7.03	26.7	144.1
	42	1.31-1.94	6.08	29.4	126.1
	43	5.02	5.35	98.9	95.0

Table 2

 2 Minimum inhibitory concentrations (MICs) of the antimicrobial compounds produced by the strain PAL114 against several target microorganisms. CMI des compose ´s antimicrobiens produits par la souche PAL114 contre plusieurs microorganismes-cibles.

	Target microorganisms	MIC (mg/mL) a
		P41A	P41B
	Saccharomyces cerevisiae ATCC 4226	30	30
	Candida albicans M1	50	50
	C. albicans M2	50	50
	C. albicans M3	30	30
	C. albicans IPA200	50	50
	Aspergillus carbonarius M333	75	75
	A. flavus AF3	100	100
	Fusarium culmorum FC200	75	75
	Penicillium glabrum PG1	75	75
	Bacillus subtilis ATCC 6633	30	30
	Staphylococcus aureus S1	50	50
	Klebsiella pneumoniae E40	> 100	> 100

a MIC values represent the average of two repetitions.

Disclosure of interest

The authors declare that they have no conflicts of interest concerning this article.


Benjamin Rotenberg
email: benjamin.rotenberg@upmc.fr

Olivier Bernard

Jean-Pierre Hansen

Underscreening in ionic liquids: a first principles analysis

Keywords:

An attempt is made to understand the underscreening effect, observed in concentrated electrolyte solutions or melts, on the basis of simple, admittedly crude models involving charged (for the ions) and neutral (for the solvent molecules) hard spheres. The thermodynamic and structural properties of these "primitive" and "semiprimitive" models are calculated within the mean spherical approximation (MSA), which provides the basic input required to determine the partial density response functions. The screening length λ S , which is unambiguously defined in terms of the wave-number-dependent response functions, exhibits a cross-over from a low density, Debye-like regime, to a regime where λ S increases with density beyond a critical density at which the Debye length λ D becomes comparable to the ion diameter. In this high density regime the ratio λ S /λ D increases according to a power law, in qualitative agreement with experimental measurements, albeit at a much slower rate.

Introduction

The theoretical understanding of the microscopic structure and dynamics of ionic solutions and melts, both in the bulk and at interfaces, has made enormous progress since the pioneering work of Gouy [START_REF] Gouy | Sur la constitution de la charge électrique à la surface d'un électrolyte[END_REF], Chapman [START_REF] Chapman | A contribution to the theory of electrocapillarity[END_REF], Debye and Hückel [START_REF] Debye | The theory of electrolytes I. The lowering of the freezing point and related occurrences[END_REF][START_REF] Debye | The theory of electrolytes II. The border law for electrical conductivity[END_REF], Onsager (see [START_REF] Hemmer | The collected works of Lars Onsager[END_REF]) and many others. The "Coulomb singularity", due to the infinite range of Coulombic forces, entails a very specific behaviour, in particular the "screening" mechanism, first examined, under bulk conditions, by Debye and Hückel [START_REF] Debye | The theory of electrolytes I. The lowering of the freezing point and related occurrences[END_REF][START_REF] Debye | The theory of electrolytes II. The border law for electrical conductivity[END_REF] within a mean-field framework. Rigorous statistical mechanics investigation have led to a wealth of "exact results", including sum rules which provide useful guides for systematic approximation schemes (for an exhaustive review see [START_REF] Ph | Sum rules in charged fluids[END_REF]). The adsorption of ions near a charged surface, first suggested by Stern [START_REF] Stern | Zur Theorie Der Elektrolytischen Doppelschicht[END_REF] leads to layering of ions of alternative charge; at high concentration, this layering can entail an "overscreening" phenomenon, whereby successive layers can overshoot local charge neutrality, and can lead to an effective attraction between equally charged parallel surfaces for short separations (for a review, see [START_REF] Hansen | Effective Interactions Between Electric Double Layers[END_REF]).

Recently, a number of surface force measurements have revealed a new and unexpected behaviour, namely "underscreening" in ionic melts and concentrated electrolyte solutions. [START_REF] Gebbie | Long-range electrostatic screening in ionic liquids[END_REF][START_REF] Smith | The Electrostatic Screening Length in Concentrated Electrolytes Increases with Concentration[END_REF][START_REF] Lee | Underscreening in concentrated electrolytes[END_REF] Careful measurements of the force acting between charged mica sheets plunged into an electrolyte show that it decays exponentially with the separation D between sheets, beyond a short-range layering regime. The variation of the screening length λ S (characterizing the exponential decay) with ion concentration crosses over from a Debye regime, where λ S ∼ λ D , the Debye length which decreases with concentration, to a regime where λ S increases with concentration to reach values much larger than λ D . The cross-over occurs when λ D becomes comparable to the mean ion diameter σ. Moreover, this behaviour is found to be "universal", i.e. the ratio λ S /λ D obeys a simple scaling, viz:

λ S /λ D = 1 , λ D > σ = (σ/λ D ) 3 , λ D < σ (1)
which holds for a wide range of pure (solvent-free) ionic liquids and of ionic solutions. [START_REF] Lee | Underscreening in concentrated electrolytes[END_REF] A number of ingenious phenomenological interpretations of the scaling relations (1) have been put forward [START_REF] Lee | Underscreening in concentrated electrolytes[END_REF][START_REF] Lee | Scaling Analysis of the Screening Length in Concentrated Electrolytes[END_REF][START_REF] Zachary | Underscreening, overscreening and double-layer capacitance[END_REF], but in the present paper we adopt a "first principles" approach. We consider primitive and semi-primitive models of ionic solutions, whereby ions are taken to be charged hard spheres, while the solvent is either implicit, providing a dielectric permittivity (primitive model), or taken to be neutral hard spheres in addition to providing a given (semi-primitive model). The primitive model with = 1 naturally applies to the solvent-free case. These models are admittedly very crude since they neglect, inter alia, the dipole and higher order moments carried by the solvent molecules, as well as the polarizability of the solvent and ions. They have however the advantage that their thermodynamic and structural properties can be readily calculated using standard tools of liquid state theory, in particular the mean-spherical approximation (MSA) and its extensions [START_REF] Hansen | Theory of Simple Liquids[END_REF], giving unambiguous access to the concentration dependence of the screening length λ S .

The outline of the paper is the following: In section 2, Landau fluctuation theory is applied to calculate the partial structure factors of the primitive and semiprimitive models as functions of concentration; the required thermodynamic properties are calculated within MSA. In section 3, the Landau approach is reformulated and generalized in terms of wavenumber-dependent response functions. The asymptotic behaviour of the partial structure factors is determined by the poles in the plane of complex wavenumbers k = k + ik , along the lines of Ref. [START_REF] Leote De Carvalho | The decay of correlations in ionic fluids[END_REF]. Section 4 presents the MSA-based numerical results for the screening length λ S of both primitive and semiprimitive models. Conclusions are drawn in Section 5, together with plans for future work.

Landau fluctuation theory

The semi-primitive model is a three component hard-sphere system of cations (charge q 1 = z 1 e, diameter σ 1 ), anions (charge q 2 = z 2 e, diameter σ 2 ) and solvent molecules (charge q 3 = 0, diameter σ 3 ). The number densities are ρ α = N α /V (α = 1, 2, 3), while global charge neutrality implies:

z 1 ρ 1 + z 2 ρ 2 = 0. Let: f (ρ 1 , ρ 2 , ρ 3 , T ) = F (N 1 , N 2 , N 3 , V, T ) V (2)
be the Helmholtz free energy per unit volume. Similarly one defines the internal energy per unit volume, u, and the entropy per unit volume, s. The chemical potentials are defined by:

µ α = ∂f ∂ρ α ρᾱ,T (3)
while:

δs = 1 T δu - 1 T α µ α δρ α . (4)
Consider local fluctuations of the variables T and ρ α around their equilibrium values, and expand the entropy fluctuation δs(r) to second order in δT (r) and δρ α (r), along the lines of Landau fluctuation theory [START_REF] Landau | Statistical Physics[END_REF]:

δs(r) = 1 T δu(r) - 1 T α µ α δρ α (r) - 1 2 c V T 2 [δT (r)] 2 + 1 T α β ∂µ α ∂ρ β δρ α (r)δρ β (r) (5)
where c V = ∂u ∂T {ρα} is the specific heat per unit volume. We temporarily restrict the discussion to the solvent-free case (ρ 3 = 0) and generalize the calculation of Ref. [START_REF] Vieillefosse | Statistical mechanics of dense ionized matter. V. Hydrodynamic limit and transport coefficients of the classical one-component plasma[END_REF], valid for the one-component plasma (OCP: point ions in a uniform neutralizing background of opposite charge), to the primitive model.

The local fluctuation in internal energy is the sum of the kinetic energy and electrostatic contributions:

δu(r) = u(r) -u 0 = - 1 2 ρ m |v(r)| 2 - 8π |E(r)| 2 (6)
where v(r) is the local fluid velocity, ρ m = m 1 ρ 1 + m 2 ρ 2 is the global mass density, is the dielectric permittivity and E(r) is the local electric field, which satisfies Poisson's equation:

∇ • E(r) = 4πe [z 1 δρ 1 (r) + z 2 δρ 2 (r)] . (7)
Switching to Fourier space representation, e.g.:

δρ α (r) = 1 V k ρ kα e -ik•r (8)
and integrating both sides of Eq. ( 5) over the volume V , one arrives at:

S k B = 1 k B s(r)dr = S 0 k B + 1 k B δs(r)dr = S 0 k B - ρ m 2k B T 0 V k v k • v -k - c 0 V 2k B T 2 0 V k T k T -k - 1 2k B T 0 V k 4πe 2 k 2 z 2 1 |ρ k1 | 2 + z 2 2 |ρ k2 | 2 + 2z 1 z 2 (ρ k1 ρ -k2 ) - 1 2k B T 0 V k χ -1 11 |ρ k1 | 2 + χ -1 22 |ρ k2 | 2 + 2χ -1 12 (ρ k1 ρ -k2 ) (9)
where the subscript 0 refers to equilibrium quantities, refers to the real part of a complex argument, k B is Boltzmann's constant and:

χ -1 αβ = ∂µ α ∂ρ β 0 = ∂ 2 f ∂ρ α ∂ρ β 0 (10)
The probability of a fluctuation is:

P ({v k , T k , ρ k1 , ρ k2 }) ∼ exp ∆S k B ∼ k P (v k , T k , ρ k1 , ρ k2 ) (11)
with ∆S = S -S 0 the associated change in entropy. Eq. ( 9) shows that the velocity and temperature fluctuations are decoupled from the density fluctuation.

The key objective is the calculation of the charge-charge structure factor:

S ZZ (k) = 1 V ρ Z k ρ Z -k = P(ρ k1 , ρ k2 )ρ Z k ρ Z -k dρ k1 dρ k2 P(ρ k1 , ρ k2 ) dρ k1 dρ k2 (12)
A lengthy but straightforward calculation leads to the required expression for the dimensionless charge-charge structure factor:

S ZZ (k) α ρ α z 2 α = λ 2 D k 2 1 + λ 2 S k 2 (13)
which is valid for all k at this level of theory (Gaussian fluctuations), but is also a good approximation of more accurate ones in the small k limit, and where the Debye length is given by:

λ D = 4πl B (ρ 1 z 2 1 + ρ 2 z 2 2 ) -1/2 (14)
with l B = e 2 / k B T the Bjerrum length, while the screening length λ S is defined by: [START_REF] Leote De Carvalho | The decay of correlations in ionic fluids[END_REF] where

λ S = 1 4πl B χ -1 11 χ -1 22 -χ -2 12 (Z 1 + Z 2 ) 2 ρ 2 0 χ T k B T 0 1/2
Z α = |z α | and χ T is the isothermal compressibility χ T = -1 V ∂V ∂P T , i.e.: χ -1 T = ρ 2 0 ∂ 2 f ∂ρ 2 0 T = ρ 2 0 (Z 1 + Z 2 ) 2 Z 2 2 χ -1 11 + Z 2 1 χ -1 22 + 2Z 1 Z 2 χ -1 12 . ( 16
)
Note that λ 2 S can be expressed as the ratio -I 4 /I 2 where I 2 and I 4 are the coefficients of k 2 and k 4 terms in the small k expansion of S ZZ (k); these coefficients have been derived from second and fourth moment sum rules within linear response theory [START_REF] Alastuey | Fourth Moment Sum Rule for the Charge Correlations of a Two-Component Classical Plasma[END_REF] and lead back to the result [START_REF] Leote De Carvalho | The decay of correlations in ionic fluids[END_REF].

λ S can now be estimated by calculating χ T and the χ -1 αβ , defined by Eq. ( 10), using the MSA expression for the free energy of the primitive model both for the symmetric case (σ 1 = σ 2 , z 1 = -z 2 ) [START_REF] Waisman | Mean spherical model integral equation for charged hard spheres I. Method of solution[END_REF] and for asymmetric cases [START_REF] Blum | Mean spherical model for asymmetric electrolytes[END_REF][START_REF] Blum | Mean spherical model for asymmetric electrolytes. 2. Thermodynamic properties and the pair correlation function[END_REF]. Results for the symmetric case will be presented in Section 4. The main conclusion is that fluctuation theory, using the thermodynamic functions χ -1 αβ corresponding to the MSA free energy, predicts a screening length λ S which is systematically smaller than the Debye length λ D , and hence cannot account for underscreening.

For the sake of completeness, we also quote the fluctuation theory results for the number-charge and number-number structure factors S N Z (k) and S N N (k):

S N Z (k) = 1 V ρ kN ρ -kZ = ρ 0 λ 2 N Z k 2 1 + λ 2 S k 2 (17)
where ρ 0 = ρ 1 + ρ 2 = N V and:

λ 2 N Z = ρ 0 χ T 4πl B z 2 χ -1 11 + z 1 χ -1 22 -(z 1 + z 2 )χ -1 12 (Z 1 + Z 2 ) 2 . ( 18
)
As expected S N Z (k) = 0 in the fully symmetric case (z 1 = -z 2 and χ -1 11 = χ -1 22 , i.e. charge and number densities are uncorrelated). Note also that lim k→0 S N Z (k) = 0, as is the case for S ZZ (k). Finally:

S N N (k) = 1 V ρ kN ρ -kN = ρ 2 0 k B T 0 χ T [1 + λ 2 N k 2 ] 1 + λ 2 S k 2 (19)
where:

λ 2 N = 1 4πl B k B T 0 χ -1 11 + χ -1 22 -2χ -1 12 (Z 1 + Z 2 ) 2 . ( 20
) As expected, lim k→0 S N N (k) = ρ 2 0 k B T 0 χ T = ρ 0 χ T χ id T
. Note that the expressions ( 13), ( 17) and ( 19) of the structure factors imply that the corresponding correlation functions h ZZ (r), h N Z (r) and h N N (r) ultimately decay on the same length scale, as expected [START_REF] Leote De Carvalho | The decay of correlations in ionic fluids[END_REF].

Screening in terms of partial density response functions

The failure of fluctuation theory, which is only adapted to the k → 0 limit, suggests that finite wavevectors k must be included in a realistic description of screening incorporating excluded volume effects, as suggested by the expreimental evidence of a cross-over in screening behaviour when kσ ≈ 1. [START_REF] Lee | Underscreening in concentrated electrolytes[END_REF] We hence reformulate the theory in terms of k-dependent partial response functions χαβ , defined by:

χ-1 αβ (k) = δ αβ ρ α -ĉαβ (k) (21)
which occur naturallly in the description of the linear response to an external field [START_REF] Hansen | Theory of Simple Liquids[END_REF]; the ĉαβ (k) are the Fourier transforms of the direct correlation functions c αβ (r). The relations between the χ-1 αβ (k) and the partial structure factors:

S αβ (k) = 1 V ρ kα ρ -kβ = ρ α δ αβ + ρ α ρ β ĥαβ (k) ( 22
)
where ĥαβ (k) are the Fourier transforms of the pair correlation functions, are easily established via the Ornstein-Zernike (OZ) relations between the ĉαβ (k) and the ĥαβ (k). [START_REF] Hansen | Theory of Simple Liquids[END_REF] Restricting once more the subsequent calculations to the primitive (i.e. solventfree) model, the coupled Orstein-Zernike equations lead directly to the following matrix relation between the S αβ (k) and the χ-1 αβ (k):

S 11 S 12 S 21 S 22 = 1 χ-1 11 χ-1 22 -χ-1 12 χ-1 21 χ-1 22 -χ-1 21 -χ-1 12 χ-1 11 (23)
and hence:

S ZZ (k) = α β z α z β S αβ (k) = z 2 1 χ-1 22 (k) + z 2 2 χ-1 11 (k) -2z 1 z 2 χ-1 12 (k) χ-1 11 (k) χ-1 22 (k) -χ-1 12 (k) χ-1 21 (k) (24)
Note that the k → 0 limit of χ-1 αβ (k) is related to the thermodynamic derivatives defined in Eq. [START_REF] Smith | The Electrostatic Screening Length in Concentrated Electrolytes Increases with Concentration[END_REF], thus pointing to the link between the fluctuation theory of Section 2 and the present extension to finite k. In charged systems, the ĉαβ (k) split into a Coulombic and short-range contribution:

ĉαβ (k) = - 4πl B k 2 z α z β + ĉS αβ (k) (25)
Using the definition [START_REF] Blum | Mean spherical model for asymmetric electrolytes. 2. Thermodynamic properties and the pair correlation function[END_REF] and substituting into [START_REF] Henderson | Liquid-state integral equations at high density: On the mathematical origin of infinite-range oscillatory solutions[END_REF], one arrives at the final expression for S ZZ (k):

S ZZ (k) = k 2 A(k) k 2 + 4πl B A(k) (26a) A(k) = ρ 1 z 2 1 [1 -ρ 2 ĉS 22 (k)] + ρ 2 z 2 2 [1 -ρ 1 ĉS 11 (k)] + 2ρ 1 ρ 2 z 1 z 2 ĉS 12 (k) [1 -ρ 1 ĉS 11 (k)][1 -ρ 2 ĉS 22 (k)] -ρ 1 ρ 2 [ĉ S 12 (k)] 2 (26b)
Since the c S αβ (r) are short-ranged functions, one may expand their Fourier transforms, as well as the resulting A(k), into even powers of k [START_REF] Evans | A density functional theory for inhomogeneous charged fluids[END_REF]:

ĉS αβ (k) = c (0) αβ + c (2) αβ k 2 + O(k 4 ) (27a) A(k) = A (0) + A (2) k 2 + O(k 4 ) (27b)
where:

A (0) = ρ 1 z 2 1 [1 -ρ 2 c (0) 22 ] + ρ 2 z 2 2 [1 -ρ 1 c (0) 11 ] + 2ρ 1 ρ 2 z 1 z 2 c (0) 12 [1 -ρ 1 c (0) 11 ][1 -ρ 2 c (0) 22 ] -ρ 1 ρ 2 [c (0) 12 ] 2 (28)
The corresponding small k expansion of the charge-charge structure factor is:

S ZZ (k) α ρ α z 2 α = k 2 λ 2 D - k 4 λ 2 D 4πl B A (0) (29)
Minus the ratio of the coefficients of k 4 and k 2 leads back to an expression of λ 2 S identical to that provided by Eq. ( 15), as expected. The thermodynamic derivatives

χ -1 αβ = k B T χ-1 αβ (k = 0) = k B T δ αβ ρα -c (0) αβ
can be calculated analytically in the symmetric case within the MSA, with the result for the screening length:

λ M SA S 2 = 1 -2Γ 2 σ 2 1 + 2 3 Γσ + 1 6 Γ 2 σ 2 λ 2 D ( 30
)
where Γ -1 = λ D 1 + 1 + 2σ/λ D . The result (30) is similar, although not identical to an expression derived by Attard [START_REF] Attard | Asymptotic analysis of primitive model electrolytes and the electrical double layer[END_REF] based on second moment considerations. As pointed out in Section 2, the resulting λ M SA S are always less than λ D . To make further progress we must return to the exact expression [START_REF] Wertheim | Exact Solution of the Mean Spherical Model for Fluids of Hard Spheres with Permanent Electric Dipole Moments[END_REF] for S ZZ (k), and exploit the full k-dependence of the response function χ-1 αβ (k), or equivalently of the ĉS αβ (k). The long-range behaviour of the charge density correlation function:

h ZZ (r) = α β z α z β [ρ α δ αβ + ρ α ρ β h αβ (r)] (31)
is determined by the poles of its Fourier transform S ZZ (k) in the complex wavenumbers k = k +ik , i.e. by the zeros of the denominator: 26). [START_REF] Leote De Carvalho | The decay of correlations in ionic fluids[END_REF] The dominant (longest) decay length is controlled by the pole of S ZZ (k) closest to the real axis in the complex wavenumber plane. If this pole lies on the imaginary axis k , as occurs at low enough concentrations, the decay of h ZZ (r) is purely exponential, while for poles with a non-zero real part k , the charge density correlations exhibit an oscillatory behaviour, at least at short distances r between ions, due to layering (for details, see Ref. [START_REF] Leote De Carvalho | The decay of correlations in ionic fluids[END_REF]). The complex roots of D(k) are determined numerically and the key results of this numerical analysis are summarized in Section 4. Before discussing our results for the bulk correlation function h ZZ (r), it is important to recall why these results are relevant for the interpretation of the experimental data which apply to a liquid film confined between two surfaces. The key reason is that the asymptotic decay of the density profile of such an inhomogeneous fluid is governed by a correlation (in this case screening) length which is identical to that characterizing the static correlations in a bulk fluid at a given density and temperature [START_REF] Henderson | Liquid-state integral equations at high density: On the mathematical origin of infinite-range oscillatory solutions[END_REF][START_REF] Evans | Asymptotic decay of liquid structure: oscillatory liquid-vapour density profiles and the Fisher-Widom line[END_REF].

D(k) = k 2 +4πl B A(k) in Eq. (

MSA results for underscreening at high concentrations

Some of the numerical results based on the expressions derived in Sections 2 and 3 for the solvent-free primitive model and for the three-component semi-primitive model are summarized in Figures 1 to 3. In all cases examined so far, the ion and solvent diameters are equal (σ 1 = σ 2 = σ 3 = σ), while cations and anions carry opposite charges (z 1 = -z 2 ). The diameter and valencies are chosen with the case of sodium chloride (NaCl) in mind, namely σ = 0.3 nm and z 1,2 = ±1. αβ . The symbols are calculated from the complex poles of the structure factor S ZZ (k), Eq. ( 26), using once more the MSA for the calculation of the k-depedent ĉS αβ (k). Calculations correspond to ion diameter σ = 3 Å and charges ±e, in water at room temperature ( = 78, T = 300 K) for l B /σ = 2.3 (black symbols), while the value 4.6 corresponds to = 39 (red symbols).

Figure 1 shows the variation of the reduced screening length λ S /σ vs the reduced density ρσ 3 , for two values of the reduced Bjerrum length l B /σ = e 2 / k B T σ. For the above-mentioned case of NaCl, the value l B /σ = 2.3 corresponds to room temperature water conditions ( = 78, T = 300 K), while the value 4.6 corresponds to = 39. The continuous curves correspond to the Debye prediction λ S = λ D , while the dashed curves are the Landau fluctuation theory results (30) using the MSA predictions for the thermodynamic quantities χ -1

αβ . The symbols are calculated from the complex poles of the structure factor S ZZ (k), Eq. ( 26), using once more the MSA for the calculation of the k-depedent ĉS αβ (k). While λ D decreases steadily like 1/ ρσ 3 , the Landau fluctuation theory, which applies only to small k, predicts that λ S decreases much faster and goes to zero when ρσ 3 ≈ 0.03 (corresponding approximately to 2 mol/L with the above parameters) under water at room temperature conditions, and at ρσ 3 ≈ 0.06 for the lower . On the other hand, the MSA predictions within the density response formalism of Section 3 point to a clear-cut cross-over behaviour from a rapid drop of λ S with density, close to the predictions of Landau fluctuation theory, up to ρσ 3 ≈ 0.02 (l B /σ = 2.3) or ρσ 3 ≈ 0.03 (l B /σ = 4.6), followed by an increase of λ S beyond those cross-over densities. This cross-over corresponds precisely to the underscreening observed experimentally at high densities or concentrations [START_REF] Lee | Underscreening in concentrated electrolytes[END_REF], even though the order of magnitude of λ S remains here comparable to the ion diameter in this concentration regime. Note that the minimum of λ S vs ρσ 3 is in fact a cusp, due to the cross-over from a pure exponential to a damped oscillatory decay of the correlations [START_REF] Leote De Carvalho | The decay of correlations in ionic fluids[END_REF]. In the vicinity of the cusp (which occurs at σ/λ D ≈ 1.228 within the MSA [START_REF] Leote De Carvalho | The decay of correlations in ionic fluids[END_REF]), the wavelength of the oscillations becomes very long and hence practically unobservable, which explains why no oscillations are observed in the force measured in that range [START_REF] Lee | Underscreening in concentrated electrolytes[END_REF]. From Eq. ( 26)

Figure 2. Ratio of the screening length over the Debye length as a function of the ratio of the ion diameter over the Debye length. The horizontal line corresponds to the Debye prediction λ S = λ D , while the dashed line is the Landau theory prediction (30) using the MSA results for the thermodynamic quantities χ -1 αβ . The symbols are calculated from the complex poles of the structure factor S ZZ (k), Eq. ( 26), using once more the MSA for the calculation of the k-depedent ĉS αβ (k). The results correspond to the same parameters as in Figure 1 (•, ) as well as upon changing the permittivity of the solvent according to the experimental values ( ) reported in Ref. [START_REF] Smith | The Electrostatic Screening Length in Concentrated Electrolytes Increases with Concentration[END_REF][START_REF] Lee | Underscreening in concentrated electrolytes[END_REF][START_REF] Lee | Scaling Analysis of the Screening Length in Concentrated Electrolytes[END_REF].

To make contact with the experimental scaling relation [START_REF] Gouy | Sur la constitution de la charge électrique à la surface d'un électrolyte[END_REF] [START_REF] Smith | The Electrostatic Screening Length in Concentrated Electrolytes Increases with Concentration[END_REF][START_REF] Lee | Underscreening in concentrated electrolytes[END_REF][START_REF] Lee | Scaling Analysis of the Screening Length in Concentrated Electrolytes[END_REF], we plot the ratio λ S /λ D vs σ/λ D on a log-log scale in Figure 2, for the solvent-free (primitive model) case, once again for the two values of l B /σ, namely 2.3 and 4.6. We also indicate on this plot the predictions of the same theory when changing the permittivity of the medium upon increasing concentration according to the experimental values reported in Ref. [START_REF] Smith | The Electrostatic Screening Length in Concentrated Electrolytes Increases with Concentration[END_REF][START_REF] Lee | Underscreening in concentrated electrolytes[END_REF][START_REF] Lee | Scaling Analysis of the Screening Length in Concentrated Electrolytes[END_REF]. The Landau fluctuation theory results drop rapidly below the Debye prediction λ S /λ D = 1 when σ/λ D approaches 1. The predictions based on the density response formalism, Eq. ( 26), using MSA results for ĉS αβ (k), again clearly indicate a sharp cross-over from the Landau fluctuation prediction, at σ/λ D close to (and slightly above) 1 towards a rapid increase for σ/λ D > 1 , in qualitative agreement with experimental data. As emphasized in Ref. [START_REF] Leote De Carvalho | The decay of correlations in ionic fluids[END_REF], the MSA necessarily leads to a "universal" scaling relation (at least for fixed permittivity ), as illustrated in Fig. 2 where the black and red dots fall on the same straight line. More unexpectedly, even when is allowed to vary with concentration (triangles in Fig. 2), the results obey the same scaling. The slope of the log-log plot in Figure 2 suggests a scaling λ S /λ D ∝ (σ/λ D ) ν with ν = 3/2 which is well below the value ν = 3 suggested by the experimental data [START_REF] Lee | Scaling Analysis of the Screening Length in Concentrated Electrolytes[END_REF], i.e. the theoretical screening length increases much less rapidly with density than observed experimentally. In other words, underscreening is indeed predicted by our theory, but is less dramatic than suggested by the experimental data. Note also the dip in λ S /λ D = 1 predicted just before the cross-over density, which is not observed in the experimental measurements.

Finally, Figure 3 shows a similar λ S /λ D vs σ/λ D plot, comparing results for the two-component primitive model and the three-component semi-primitive model with a hard sphere solvent of diameter equal to that of the ions. For the latter, the total volume fraction η = π 6 (ρ 1 + ρ 2 + ρ 3 )σ 3 is held fixed at a value η = 0.47, which corresponds approximately to that of liquid water for a diameter σ = 0.3 nm, meaning that the ion concentration increases, while the solvent concentration decreases, as σ/λ D increases. In the latter case the screening length is computed as the imaginary part of the pole of S ZZ (k) closest to the real axis, determined numerically from the roots of the denominator:

D 3 (k) = k 2 [1 -ρ 1 ĉ11 (k)] [1 -ρ 2 ĉ22 (k)] -ρ 1 ρ 2 [ĉ 12 (k)] 2 [1 -ρ 3 ĉ33 (k)] -ρ 1 ρ 3 [ĉ 13 (k)] 2 [1 -ρ 2 ĉ22 (k)] -ρ 2 ρ 3 [ĉ 23 (k)] 2 [1 -ρ 1 ĉ11 (k)] -2ρ 1 ρ 2 ρ 3 ĉ12 (k)ĉ 23 (k)ĉ 31 (k)] (32)
In both cases, the cross-over occurs as already shown in Figure 2 for the primitive model, but the scaling exponent ν is smaller, close to ν = 1, in the case of the semi-primitive model. Even though this exponent is further from the experimental one compared to the primitive model case, the present calculations based on simple models of ionic solutions confirm that the cross-over behaviour and subsequent underscreening regime are driven by excluded volume effects which become predominant for λ D σ. However improvements of the microscopic model are required to explain the scaling law and large screening length observed experimentally.

Conclusion

We have examined the recently discovered underscreening phenomenon in concentrated ionic solutions and melts [START_REF] Lee | Underscreening in concentrated electrolytes[END_REF] by considering simple models involving charged and/or neutral hard spheres, namely the so-called primitive and semi-primitive models. The charge-charge structure factor S ZZ (k) was calculated within Landau's gaussian fluctuation theory, appropriate for small wavenumbers k, and within the density response function formalism which extends the validity of the Landau fluctuation theory to arbitrary k. The required thermodynamic quantities and structural properties of the models were evaluated over a range of concentrations, using the MSA approximation, which is reasonably accurate over a wide range of hard sphere packing fractions. The hard sphere diameter used in the MSA calculations is an effective diameter taking into account the "softness" of the true ion-ion interactions; this effective diameter is (weakly) density-dependent [START_REF] Hansen | Theory of Simple Liquids[END_REF], so that the packing fraction itself should be regarded as statedependent in more realistic calculations.

Our calculations are capable of reproducing, at least qualitatively, the cross-over from a low concentration regime, where the screening length λ S remains close to the Debye length λ D , to a high concentration regime, where λ S increases with concentration, contrary to λ D , thus confirming the underscreening phenomenon. In that regime the growth of λ S /λ D obeys a simple scaling relation similar to that observed experimentally (see Eq. ( 1)), but with an exponent ν which is considerably smaller (ν ≈ 3/2 compared to ν = 3); in other words underscreening is underestimated.

This situation is most likely due to the crude representation of the solvent, which is either implicit (primitive model) or reduced to neutral hard spheres (semi-primitive model). Clearly a "civilized" model of the solvent is required, whereby the solvent hard spheres carry an electric dipole and higher multipoles. Analytic solutions of the MSA for dipolar hard spheres are available [START_REF] Wertheim | Exact Solution of the Mean Spherical Model for Fluids of Hard Spheres with Permanent Electric Dipole Moments[END_REF][START_REF] Stell | Dielectric Constants of Fluid Models: Statistical Mechanical Theory and its Quantitative Implementation[END_REF], although the dielectric properties calculated from the MSA approximation become highly unreliable for high dipole moments. The polarizability of solvent molecules and ions also play an important role in realistic descriptions of ionic solutions [START_REF] Tazi | A transferable ab initio based force field for aqueous ions[END_REF], which is totally absent in the oversimplified models considered in the present work. In future work we plan to generalize the present analysis of underscreening to asymmetric cases, involving unequal particle diameters σ 1 = σ 2 = σ 3 and ion valences (Z 1 = Z 2 ) to investigate the effect of these asymmetries on the scaling relations discussed in Section 4 and illustrated in Figures 2 and3. It is finally worth stressing that the present calculation is for bulk electrolytes. In the experimental set-up, the electrolyte is confined between parallel charged planes. It is conceivable that the charge fluctuations are affected by the confinement and could lead to a classical Casimir effect, well known in the theory of critical fluctuations (for a review, see [START_REF] Gambassi | The Casimir effect: From quantum to critical fluctuations[END_REF]).

Figure 1 .

 1 Figure 1. Reduced screening length λ S /σ as a function of the reduced density ρσ 3 , for two values of the reduced Bjerrum length l B /σ = e 2 / k B T σ. The solid lines correspond to the Debye prediction λ S = λ D , while the dashed lines are the Landau fluctuation theory results (30) using the MSA predictions for the thermodynamic quantities χ -1αβ . The symbols are calculated from the complex poles of the structure factor S ZZ (k), Eq. (26), using once more the MSA for the calculation of the k-depedent ĉS αβ (k). Calculations correspond to ion diameter σ = 3 Å and charges ±e, in water at room temperature ( = 78, T = 300 K) for l B /σ = 2.3 (black symbols), while the value 4.6 corresponds to = 39 (red symbols).

Figure 3 .

 3 Figure 3. Ratio between the screening length and the Debye screening length as a function of the ratio between the ion diameter and the Debye screening length, for l B /σ = 2.3. The solid lines correspond to the Debye prediction λ S = λ D , while the dashed lines are the Landau fluctuation theory results (30) using the MSA predictions for the thermodynamic quantities χ -1 αβ . The symbols are calculated from the complex poles of the structure factor S ZZ (k) for the two-component primitive model (•) and the three component semi-primitive model ( ).

Acknowledgments

The authors are grateful to Susan Perkin, Alpha Lee and Bob Evans for helpful discussions.


Edito IJMAV special issue IMAV 2017

The International Micro Air Vehicle Conference and Competition (IMAV) is a yearly event that aims at fostering key technologies for the development of micro-air vehicles. It combines a scientific conference and a flight competition intended to all research groups around the world. The conference part aims at presenting innovations in different areas related to drones : design, fluid mechanics, navigation and control, sensors, data processing, simulation,…. On the other hand, the flight competition showcases novel and/or efficient designs through a set of challenges, while different prizes recognize different possible achievements. After Delft, The Netherlands (IMAV 2014), Aachen, Germany (IMAV 2015) and Beijing, China (IMAV 2016), the 9 th edition IMAV 2017 has been held in Toulouse, France, from Sept. 18 to Sept. 21, 2017.

IMAV 2017 has gathered more than 280 participants from 30 different countries including Asia, North, Central and South America, Europe, and Australia. During the two-day flight competition, 30 international teams have taken part to the indoor or the outdoor event. As for the conference, more than 40 papers have been selected for oral presentations from a total number of 60 submitted papers. This edition was, again, an occasion to put forward the methodological and technological advances in this rapidly growing field. The impressive variety of application cases, related needs, and breakthroughs has been represented through the selected papers gathered in this special issue.

The papers selected in this special issue have been nominated for the "Best Paper Award" which has been given to the paper entitled "Quad-thopter: Tailless Flapping Wing Robot with 4 Pairs of Wings", by C. De Wagter et al. They went through a double review process before the conference and after the oral presentation. They received the highest scores through this process, and were thus selected to be part of this special issue.

The 2017 edition of the IMAV conference and flight competition was intended to emphasize different aspects of drones. In particular, the conference intended at setting forward novel designs : the papers selected illustrate this aspect with designs dedicated to challenging environments such as the windy urban environment (R. Gigcaz et al., "Exploring Tandem Wing UAS designs for Operation in Turbulent Urban Environments") or the Martian atmosphere (T. Désert et al., "Numerical and Experimental investigation of Airfoil design for a Martian micro rotorcraft").

Following the example provided by natural flyers, with flapping-wing configurations or bioinspired aerodynamic sensors have been considered to extend the flight envelope of drones and to enhance their endurance by dynamic soaring (N. Gavrilovic et al., "Bioinspired wind field estimation-part 1: Angle of attack measurements through surface pressure distribution").

Finally, the issue of acoustic covertness has attracted the interest of the scientific community and is also addressed in the present special issue (N. Gourdain et al., "Application of a Lattice Boltzmann Method to some challenges related to Micro Air Vehicles").

Jean-Marc Moschetta, Gautier Hattenberger, Henry de Plinval, Thierry Jardin IJMAV guest editors


Antoine Bouchain

José Picheral

Elisabeth Lahalle

Agathe Vercoutter

Arnaud Talon

Sparse Method for Tip-Timing Signals Analysis with Non Stationary Engine Rotation Frequency

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

Compressors and turbines blades vibrations are a key element in the turbomachinery test certification campaign. Indeed, in operating conditions, blades are excited by alternating aerodynamic turbulences that may cause mechanical resonances. Compressors and turbines test aim at identifying and quantifying blade mechanical responses under such excitation sources. Blades vibrations are traditionally measured by strain gauges. However, due to the high implementation cost and complexity, only a few blades are generally instrumented. Besides, sticking a gauge and its wire on a blade slightly modifies its mechanical response. All these gauges drawbacks made popular an other measurement method, less intrusive and easier to set up, called tip-timing. The acquisition system is composed by a set of C optical probes mounted on the stator and monitoring the blade tip passing times, see Fig. 1. When a blade is passing in front of the laser beam, the optical probe detects the blade and records the corresponding time. This time can vary if the blade is vibrating and the delay can be interpreted as a blade tip displacement, knowing the rotation frequency. As shown Fig. 1, the sampling times, indicated by the sampling function e(t), are irregular because probes are non-equidistant on the stator. Moreover, the irregular sampling pattern is repeated at each revolution, then this time sampling pattern is irregular and periodic. Thus, the frequency sampling pattern consists of peaks located at every multiple of the rotation frequency. Most often, the spectral frequency components are largely higher than the rotation frequency. Consequently, signals are sub-sampled and the spectrum contains aliased components generated by the periodic sampling patterns.

For the last 30 years, several solutions have been developed to face these difficulties [START_REF] Heath | An improved single parameter method for turbomachinery blade vibration measurements using optical laser probes[END_REF]- [START_REF] Heath | A Survey of Blade Tip-Timing Measurement Techniques for Turbomachinery Vibration[END_REF]. But with those methods, the multi-components spectral analysis remains delicate. Furthermore, most of them lie on strong hypotheses about blade mechanical behavior, which are not necessarily confirmed in real functioning. Recently, new approaches that do not require such hypotheses have been proposed. Minimum Variance Spectral Estimator (MVSE) [START_REF] Vercoutter | Tip Timing Spectral Estimation Method For Aeroelastic Vibrations of Turbomachinery Blades[END_REF], is an iterative method to estimate the covariance matrix for Capon method [START_REF] Capon | High-resolution frequency-wavenumber spectrum analysis[END_REF] in the case of tip-timing signals. Nevertheless, this method involves a high computation time. A compressed sensing approach has been developed in [START_REF] Lin | Sparse reconstruction of blade tip-timing signals for multi-mode blade vibration monitoring[END_REF] based on a Sparse Reconstruction Method (denoted as SRM in this paper). An other ℓ 1 -regularized method, developped in [START_REF] Bouchain | Méthode parcimonieuse pour l'analyse spectrale de signaux tip-timing[END_REF] by the authors, has been proposed for spectral analysis of tip-timing signals. Note that, inherent hypothesis of [START_REF] Lin | Sparse reconstruction of blade tip-timing signals for multi-mode blade vibration monitoring[END_REF] and [START_REF] Bouchain | Méthode parcimonieuse pour l'analyse spectrale de signaux tip-timing[END_REF] methods do not allow engine speed variations. Consequently, on real data, the rotation speed variations limit the length of the observation window and can lead to erroneous amplitudes estimations. To this day, to our knowledge, no method using engine speed variations has been developed for tip-timing analysis. Therefore, this paper presents a new fast sparse method that makes use of the engine rotation variations to eliminate artefacts in tip timing spectral analysis. The use of sparsity in the method is justified with the a priori knowledge that blades present a limited number of vibrations modes on the engine range of use. Thus, the vibration signal spectrum consists of a very limited number of frequency lines. As far as line spectra of non uniform sampled signals are concerned, they have already been studied with sparse methods (see [START_REF] Bourguignon | A Sparsity-Based Method for the Estimation of Spectral Lines From Irregularly Sampled Data[END_REF], for instance). Nonetheless, in the context of tip-timing signals, the issue is slightly different since the sampling pattern, see Fig. 1, leads to numerous aliasings artifacts in the signal spectrum on the considered frequency range.

This paper is organized as follow: section II introduces the tip-timing signal model and the proposed estimator. Section III exposes the experimental results and performances in terms of mean squared error (MSE), spectrum sparsity and computation time. Finally, section IV concludes the paper and discusses future studies.

II. SPARSE MODELS OF TIP-TIMING SIGNAL

Assuming that the vibration signal of a unique blade is x(t), the measured signal y(t) is extracted from a set of C probes by revolution over N t revolutions, with non stationary rotation frequency F r (n) = 1/T r (n), where 1 ≤ ≤ N t is the revolution number. The model of the time continuous sampled signal over N points is:

y(t) = N k=1 x(t k )δ(t -t k ). (1)
Besides, probes are placed around the stator at arbitrary angular positions θ c , 1 ≤ c ≤ C. Sampling pattern is then irregular. For a given revolution n, the sampling times are:

t k = θ c T r (n)/2π + n-1 i=1 T r (i) (2)
with k = c + C(n -1).

Vibrations are supposed to be sinusoidal and frequencybounded on [0, f max ]. The signal x(t) is then approximated by ξ(t) such as x(t) = ξ(t) + b(t), where b(t) includes the model error and the noise measurement. The model ξ(t) can be written as a sum of M cosines whose frequency components are f m = m M f max of respective amplitudes α m and phases ϕ m :

ξ(t) = M m=1 α m cos(2πf m t + ϕ m ), (3a)
and its Fourier transform is given by:

ξ(f ) = 1 2 M m=1 α m e jϕm δ(f -f m ) +α m e -jϕm δ(f + f m ). (3b)
Blades are only vibrating on a limited number of modes, hence, most of the amplitudes α m are zero. This particular sparsity property about the α m will be exploited further. The signal model presented in this section leads to two sampling models: the model of irregular periodic sampling (abbreviated as MIPS) which uses the F r periodicity and the presented model of absolute sampling (abbreviated as MASS) when F r is not constant.

A. Model of Irregular Periodic Sampling: MIPS

The model of irregular periodic sampling has been developed in [START_REF] Bouchain | Méthode parcimonieuse pour l'analyse spectrale de signaux tip-timing[END_REF] and some concepts are recalled here. Assuming the rotating frequency is not varying during runs, the sampling is periodic of period T r = T r (n), ∀1 ≤ n ≤ N t . Thus, the sampled vibration signal can be written as:

y(t) = x(t)e(t), (4)
with e(t) the sampling function. This function can be written as the convolution between a Dirac comb of period T r , noted X Tr (t), and a tip-timing sampling pattern e T T (t) linked to the irregular probe distribution:

e(t) = e T T (t) * X Tr (t) = C c=1 δ(t -θ c T r /2π) * +∞ n=-∞ δ(t -nT r ). (5)
A spectral estimation of x(t) is given by the Non Uniform Fourier Transform (NUFT) ỹ(f ) = N k=1 x(t k )e -j2πf t k of y(t) and can be modeled by:

ζ(f ) = F r +∞ n=0 ( ξ * Π[0,NtTr] )(f -nF r )ẽ T T (nF r ). ( 6
)
This spectrum is the convolution of ξ(f ) with the observation window spectrum Π(f ) and replicated on the multiples of the rotation frequency. Unlike the regular case, each refolding is weighted by ẽT T (nF r ), related to the probes positions. With (3b) and an even number of revolutions, (6) can be rewritten as:

ζ(f ) = M m=1 a m g + m (f ) + a * m g - m (f ) (7)
where

g ± m (f ) = N t 2 e -j2πf N t Fr ∞ n=0 ẽT T (nF r ) × sinc N t F r (f ± f m -nF r ) e -jπ(±fm) N t Fr , a m = α m e jϕm
and a * m the complex conjugate of a m . By splitting the quantities into real and imaginary parts, respectively marked with the exponent R and I , (7) becomes:

ζR (f ) = M m=1 a R m (g +R m + g -R m )(f ) + a I m (-g +I m + g -I m )(f ) ζI (f ) = M m=1 a R m (g +I m + g -I m )(f ) + a I m (g +R m -g -R m )(f ).
With the interval

I discretised into (f k ), 1 ≤ k ≤ N f values, (7)
can then be written as a linear system:

ζ = Gθ (8) = G 1 G 2 G 3 G 4 θ where ζ = [ ζR (f 1 ) ... ζR (f N f ) ζI (f 1 ) ... ζI (f N f )] T , of size [2N f ×1], G 1 = [(g +R M + g -R M )(f k ) ... (g +R 1 + g -R 1 )(f k )], G 2 = [(-g +I 1 + g -I 1 )(f k ) ... (-g +I M + g -I M )(f k )], G 3 = [(g +I M + g -I M )(f k ) ... (g +I 1 + g -I 1 )(f k )], G 4 = [(g +R 1 -g -R 1 )(f k ) ... (g +R M -g -R M )(f k )] of sizes [ N f ×M ], θ = [a R M ... a R 1 a I 1 ... a I M ] T of size [2M × 1]
, and [.] T is the transposition operator. Since {f m } m=1,...,M are usually choosen to sample uniformly the frequency range of interest, θ provides an estimation of the amplitudes a m of each f m and thus a spectral representation of the signal. In the case of line spectrum and from the (3b) comment, most of the a m are null so that θ is sparse. This motivate us to introduce sparsity prior in the regularization term. In order to solve [START_REF] Bourguignon | A Sparsity-Based Method for the Estimation of Spectral Lines From Irregularly Sampled Data[END_REF] with sparse a priori, a ℓ 1 -norm penalisation has been introduced on θ, with a regularisation parameter λ. Consequently, a LASSO [START_REF] Tibshirani | Regression Shrinkage and Selection via the Lasso[END_REF] problem is obtained: 9) is solved with the Alternating Direction Method of Multipliers (ADMM) [START_REF] Boyd | Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers[END_REF] algorithm.

J(θ) = min θ ||ỹ -Gθ|| 2 2 + λ||θ|| 1 . (9)
with ỹ = [ỹ R (f 1 ) ... ỹR (f N f ) ỹI (f 1 ) ... ỹI (f N f )] T of size [2N f ×1]. Criterion (

B. Proposed model : Model of AbSolute Sampling, MASS

The MIPS previously mentioned is no longer accurate in the case of non stationary rotation frequency. Thus, in this paper we propose a generalized model called Model of AbSolute Sampling (MASS), that takes into account the real rotation frequency fluctuations. On the considered observation windows, the frequencies f m are supposed asynchronous and are neither rotation frequency nor time dependent. With (1) and (3a), the sampled vibration signal is rewritten as follows:

y(t) = M m=1 a R m N k=1 cos(2πf m t)δ(t -t k ) - M m=1 a I m N k=1 sin(2πf m t)δ(t -t k ) + b(t). ( 10
)
Similarly to (8), a linear system is obtained:

y = Eθ + b (11)
where

y = [y(t 1 ) y(t 2 ) ... y(t N )] T of size [N × 1], b = [b(t 1 ) b(t 2 ) ... b(t N )] T of size [N × 1], E = [E c E s ],
where E c n,m = cos(2πf M-m+1 t n ) and E s n,m = -sin(2πf m t n ), E c and E s are both of sizes N × M and θ remains the same as in [START_REF] Bourguignon | A Sparsity-Based Method for the Estimation of Spectral Lines From Irregularly Sampled Data[END_REF]. To solve [START_REF] Stoica | Spectral Analysis of Signals[END_REF], a LASSO problem is obtained:

P (θ) = min θ ||y -Eθ|| 2 2 + λ||θ|| 1 . (12)
Criterion ( 12) is also solved with the ADMM [START_REF] Boyd | Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers[END_REF] algorithm. The MASS model is more generic than the MIPS one. In consequence, when the speed rotation is constant, the main difference is due to the size of the matrices E and G but the results are quite similar.

III. EXPERIMENTAL RESULTS

As a reminder, due to the irregular sub-sampling, spectrum consists of numerous aliased components that require adapted methods. Such methods must enable to distinguish real components among aliased ones and to estimate their amplitude accurately, with reasonable computation times. The following section presents the estimation performances of the proposed method in terms of MSE, frequency identification capacity and computation time.

A. Synthetic signal

Let us consider a synthetic signal denoted as "Signal A" whose characteristics are coherent in comparison to real test cases. It is composed of three frequency components,

f A 1 = 213Hz, f A 2 = 1626Hz and f A 3 = 1931Hz.
A white Gaussian noise is added whose variance σ 2 is:

σ 2 = σ 2 f A 2 10 -SN R/10 , (13)
where the signal to noise ratio (SNR) is -10dB and with σ 2 f A 2 the power of f A 2 component. The signal is generated with C = 5 probes at angular position 3.6 • , 39.6 • , 144 • , 291.6 • and 313.2 • during N t = 200 revolutions (the signal duration is 0.4s). Rotation frequency is established with a p = 12 autoregressive (AR) process [START_REF] Stoica | Spectral Analysis of Signals[END_REF] to which an acceleration trend is added (δ Fr = 0.005Hz by revolution) starting from F r = 470 Hz to end at 471Hz. It is important to notice that f A 1 aliasings will be located on

f A 2 since f A 2 -f A 1 = 3 × 471Hz.
The spectral estimation provided by the proposed method (based on MASS) is compared with the MVSE estimations in Fig. 2. The MIPS and SRM spectra are also given in Fig. 2. The regularization parameter for the LASSO problem is set to λ = 1. MVSE was set with 5 iterations with 3 windows of 199 revolutions. For SRM, 359 spots and 10 -10 as threshold were used. To represent the frequency range, [0, 2499]Hz, a grid of M = 2500 points is used with a step of 1Hz. Note that 1Hz step is a sufficient resolution for this application, since blades physical responses are spread among a few hertz. The estimated amplitudes are summarized Table I. MASS and MVSE methods are quite similar in terms of amplitude estimation. As expected in the rotation variation context, MIPS and SRM fail: on MIPS spectrum, numerous aliasings are not removed and amplitudes are largely underestimated; on SRM spectrum, one component is missing and estimations are more erroneous than MASS and MVSE. The tests have been conducted in terms of MSE and spectrum sparsity for 100 realizations of Signal A with various signal durations. MSE is calculated separately for each frequency component and results are given Fig. 3. In the legend, the subscripts indexes, after the name of the method, indicate the number of revolutions used. Due to excessive computation time, MVSE has been tested only for 100 and 200 revolutions. MVSE has better performances than MASS when comparing them for the same number of revolutions. Nevertheless, increasing the number of samples improves the MASS performances. With MASS, a large number of revolutions can be exploited because the model takes into account the rotation speed variations allowing an increase of the observation window. For 500 revolutions, the MSE of MASS is slightly better than the one of MVSE over 200 revolutions, especially for f A 2 component despite f A 1 aliasings. The performances of the frequency components identification are estimated by S ỹ , which is the ℓ 2 -norm of the spectrum background (ie. by removing the three frequency components) and expressed as follows: where âm is the estimation of a and m k the index of the frequency components {f A k } k=1,2,3 of the simulated signals. Results are given Fig. 4. As expected, spectrum background decreases when SNR increases. Furthermore, the MVSE spectrum is less sparse than the MASS one. Beyond a certain number of revolutions, 200 in this example, the MASS spectrum background does not decrease anymore but slightly increases. Consequently, a compromise must be found between sparsity and estimation accuracy.

f A 1 = 213Hz f A 2 = 1626Hz f A 3 = 1931Hz
S ỹ = M m=1 |â m | 2 - 3 k=1 |â m k | 2 , (14)
The different execution time of MASS, MVSE and MIPS has been calculated and data are summarized Table II. They have been gathered under MATLAB R with a four core Intel R Core i7 processor unit cadenced at 2.5GHz. It can be observed that the execution time of MASS increases practically linearly with the number of revolutions whereas it increases exponentially for MVSE. Notice that MASS 500 is about 10 times faster than MVSE 200 and provides more accurate estimation. As for MIPS, its time decreasing is due to its number of iterations falling. Nevertheless, MASS is unquestionably the fastest method whatever the number of revolutions.

B. Real signal

The real signal, referenced as "Signal B" is extracted from data of an experimental compressor test rig. Signal B was recorded over 200 revolutions and is composed of a unique frequency component at 1.179. Fig. 5 shows the fluctuation on the rotation frequency and the results of the methods are given Fig. 6 with normalized amplitudes. For ADMM, the regularization parameter is set to λ = 10 -4 . MVSE was set with 3 windows of 199 revolutions over 5 iterations. The frequency range, [0, 3999]Hz, is computed with M = 4000. Similarly to synthetic signals, MIPS presents numerous aliased components and is not well adapted to the signal due to the rotation frequency variations. MASS and MVSE both provide an estimate of the frequency component but with about 20% difference on estimated amplitudes.

Regarding the execution time, MASS took 4s and MIPS 33s with K MASS = K MIP S = 228 iterations whereas MVSE took 89s with K MV SE = 5 iterations. Comparing to signal A, the time difference is due to the increase of the range frequency. However for MASS, this increase is negligible and MASS is about twenty times faster than MVSE. It can be explained by the complexity of the algorithms which are O(2M N 2 + 4K MASS (M N + N 2 ) + N 3 /3) for MASS and O((4K MV SE + 1)M N 2 + 6K MV SE N 3 ) for MVSE.

IV. CONCLUSION

The non stationarity of the rotation speed needs to be taken into account for spectral analysis of tip-timing signals. Taking the rotation speed into consideration allows the presented method to extend the observation windows, which is an essential element to reduce aliased components and to improve amplitudes estimation on the spectra. The proposed method yields promising results. For long observation windows, it performs as well as the acknowledged method MVSE and even surpasses it in terms of spectrum background sparsity and MSE with a much lower computation time. Its quasi-linearity with the revolution number makes it very fast compared to previous methods. Some perspectives are currently investigated about the choice of the minimization algorithm as well as the optimization of the probes angular positions.

ig. 1 .

 1 Blade vibration measurement with tip-timing technology.

Fig. 2 .

 2 Fig. 2. Signal A spectra with MASS (a), MVSE (b), MIPS (c) and SRM (d).

Fig. 3 .

 3 Fig. 3. MSE of MASS and MVSE for 3 components.

Fig. 4 .

 4 Fig. 4. Spectrum background of MASS and MVSE

Fig. 5 .Fig. 6 .

 56 Fig. 5. Engine rotation frequency of Signal B.

TABLE I AMPLITUDE

 I ESTIMATIONS.

TABLE II COMPUTATION

 II TIME (IN SECONDS) FOR MASS, MVSE AND MIPS.

	Method	Revolutions		100		200	500
		MASS					0.8		1.7	5.5
		MVSE					11		48	361
		MIPS					11.4		10.7	9.5
		0.1667
	Normalized rotation frequency	0.1663 0.1664 0.1665 0.1666
		0.1662
		20	40	60	80	100	120	140	160	180
					Revolutions