Jadwiga Handzlik
email: jhandzli@cm-uj.krakow.pl

Ewa Szymańska

Sandrine Alibert

Jacqueline Chevalier

Ewa Otrębska

Elżbieta Pękala

Jean-Marie Pagès

Katarzyna Kieć-Kononowicz

Ewa Szymanska

Ewa Otre ˛bska

Elzbieta Pe ˛kala

Katarzyna Kiec-Kononowicz

Search for new tools to combat Gram-negative resistant bacteria among amine derivatives of 5-arylidenehydantoin

Keywords: Arylideneimidazolidine-2, 4-dione Arylidenehydantoin, Enterobacter aerogenes Multidrug, resistance, Efflux pump, AcrAB-TolC

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Introduction

Multidrug resistance (MDR) is a serious problem in therapy of bacterial-, [1][2][3][4][5][6][7][8][9] fungal-10 and cancer diseases. [11][12][13] In each case, one of the main mechanisms of MDR involves efflux pumps, for example, membrane transport proteins which are able to expel drugs or antibiotics before they can reach their intracellular targets. The MDR efflux pumps belong to transporter families, including two main superfamilies: major facilitator superfamily (MFS) and the ATP-binding cassette (ABC) superfamily 8 playing an important role in cancer MDR. [11][12][13] Furthermore, two sub-classes of efflux transport proteins are described: the small multidrug resistance family aeruginosa (the active pump component being AcrB or MexB, respectively). 3,8 With the continuous increase of reports describing MDR bacteria and the involvement of efflux pumps in clinical resistant isolates, strategies to annihilate the pump mechanism to preserve an appropriate intracellular concentration of antibiotics are urgently needed. 3,5 To this aim, different ways have been proposed: (i) inhibiting the activity of efflux pump by selective competition, (ii) blocking the membrane channel by original plugs or (iii) collapsing the energy driving force used by efflux mechanism. 3,5 During last decade, new compounds, so called efflux pump inhibitors/ modulators (EPIs), have been described for structural-activity studies on MDR mechanism at the molecular level and for their capacity to improve the activity of antibiotics and chemotherapeutics. 3,5 During the search for new tools to combat MDR among bacterial pathogens, a special attention is paid to Gram-negative bacteria due to their double-membrane cells and the complex membrane (SMR) and the resistance/nodulation/division family (RND). 7 barrier for antibiotic penetration corresponding to outer mem-The RND transporters are particularly widespread among Gram-brane structure. Among resistant Gram-negative bacteria, Enteronegative bacteria as it is illustrated by the tripartite efflux system bacter aerogenes is frequently described. This bacterium, AcrAB-TolC in Escherichia coli or MexAB-OprM in Pseudomonas responsible for hospital-acquired respiratory tract infections, exhibits a significant decrease of antibiotic susceptibility. 2 Studies on MDR mechanism in E. aerogenes indicated that tripartite efflux pump AcrAB-TolC is a major MDR system in clinical isolates. Recent lines of evidence [3][4][5][6] have described several groups of chemical compounds with EPI-properties in Gram-negative bacteria, including peptidomimetics, arylpiperazines, quinolines or pyridopyrimidines (Fig. 1a). For most of the compounds, however, a therapeutic usage as 'adjuvant' is rather impossible because of their toxicity and possible adverse side effects. Thus, a search for new EPIs in different chemical groups is a current question of medicinal chemistry.

On the other hand, hydantoin derivatives, that are in the area of interest in our research group, display various biological actions, particularly GPCR-ligand properties [14][15][16] and/or hypotensive, antiarrhythmic or anticonvulsant activities in both in vitro and in vivo assays. [17][18][19] Our previous studies, 9 focused on hydantoin derivatives with amine-alkyl substituents at position N1, showed gate relationships between molecular structures and their influence on antibiotic susceptibility. Indeed, the hydantoin skeleton could be viewed as a cyclic peptide sequence similar to the open sequence located in the central part of PAbN structure (Fig. 1b). This semi-rigid structure allows to modify the nature and the position of substituents and facilitates molecular modeling to guide rational synthesis. The 5,5-diphenylhydantoin compounds described previously 9 displayed weak or moderate ability to increase the activity of nalidixic acid in E. aerogenes. Among 28 tested compounds with the same scaffold, only two 2-methoxyphenylpiperazine derivatives with acetic acid motif (compounds 1 and 2, Fig. 1b) displayed significant anti-MDR effect at a dose range close to that of PAbN. On the basis of those not satisfying results, in the present work we decided to introduce significant changes into the structure of hydantoins comparing to the initial series. 9 In this context, chemical modifications of the 5,5-diphenylhydantoins were designed to improve activity by: (1) a replacement of 5,5-diphenylsubstituents with 5-arylidene one, (2) an exchange of position of amine-alkyl substituents from N1 into N3 or (3) an introduction of carbonyl or hydroxyl motif into the alkyl chain (Table 1). Thus, the present studies concern nineteen amine derivatives of 5-arylidenehydantoin 3-21 (Table 1). Synthesis of new compounds and microbiological assays to evaluate their anti-MDR properties in 2. Results and discussion

Synthesis

The presented groups of 5-arylidenehydantoin derivatives 3-21 were divided into three sub-groups, including (A) amide hydroxyethylpiperazine derivatives 3 and 4, (B) derivatives of secondaryor tertiary non-aromatic amine 5-16, and (C) derivatives of that the hydantoin could be a new appropriate scaffold to investiphenylpiperazine 17-21 (Table 1). Synthesis routes of the arylidenehydantoins (3-21) are shown in the Schemes 1 and 2. Synthesis of compounds 3, 4 and 17-21 is described elsewhere. 19,24 Compounds 5-9 and 11-16 were synthesized within three-step synthesis starting from hydantoin (Scheme 1) that was condensed with suitable aromatic aldehydes to give 5-arylidenehydantoins 23-29. The condensations went as typical Knoevenagel reactions in basic conditions 19,[24][25][26] by the use of sodium acetate in acetic acid (23-25, 27-29) 19,24 or equimolar natrium carbonate and alanine in water (26). 26 The resulted 5-arylidenehydantoins (23-29) occurred at their (Z)-configuration, which is preferable in the case of Knoevenagel condensation between N1-unsubstituted hydantoin and the considered benzaldehydes. [27][28][29] In the next step, the arylidenehydantoins 23-29 were N-alkylated at position 3 within Mitsunobu reactions to give oxiran derivatives 30-36. The reactions were performed using racemate of oxiran-2-ylmethanol (1.5 equiv), triphenylphosphine (1.0 equiv) and diethyl azodicarboxylate (1.0 equiv) in dry DMF 24 (Scheme 1). In the last step, the oxiran derivatives 31-35 were used as alkylating agents to react with primary-or secondary amines. In the case of compound 33, the alkylation was performed in solvent-free conditions under microwave irradiation 14,18 fragment at position 3 of hydantoin, was obtained by acidic hydrolysis of acetylpiperazine derivative 8 in 15% HCl (Scheme 2).

Pharmacology

Compounds 3-21 were tested in microbiological assays for their ability to increase antibiotics effectiveness in two strains of E. aerogenes: a reference strain (ATCC-13048) and the derivative CM-64 which over-produces the AcrAB-TolC efflux pump.

Intrinsic antibacterial activity

In the first step of the biological assays, direct antibacterial activity of the hydantoin derivatives was evaluated. Minimal inhibitory concentration (MIC) of the compounds 3-21 for both strains was determined (Table 2). The tested compounds displayed very weak direct antibacterial activities. The lowest MICs observed for hydroxyethylpiperazine derivatives 3, 14 and 16 were in the range of 125 lM whereas all derivatives of 4-methoxybenzylidenehydantoin (5-10) did not inhibit the growth of both strains of E. aerogenes at 2 mM. This low intrinsic effect allowed us to test these molecules in combination with usual antibiotics.

Influence on antibiotic MIC

In the next step of the microbiological assays, the compounds 3-21 were tested for their ability to influence the activity of the three following antibiotics: nalidixic acid (NAL), sparfloxacin (SFX) and chloramphenicol (CHL) in the tested bacteria strains. All the compounds have been assayed at a concentration that have no intrinsic effect, usually 1 4 of their respective MIC (Table 2), in combination with usual antibiotics and a control was systematically

O O OH N HN N N R 2 R 1 O O R 1 O OH R 1 HN N N HN N Amine O N OH O Group A Group B Group C Compound Group R 1 R 2 Amine 3 A H - 4 A 4-Br - - - 5 B 4-MeO - N N CH 3 6 B 4-MeO - HN N N CH 3 7 B 4-MeO - N 8 B 4-MeO - N N C O 9 B 4-MeO - N O 10 B 4-MeO - N NH 11 B 4-Cl - N H OH 12 B 4-Cl - N N 13 B 4-Br - N H OH 14 B 4-NEt 2 - N N 15 B 2,4-diMeO - N H OH 16 B 2,4-diMeO - N N 17 C H 2-MeO - 18 C 4-Cl 2-MeO - 19 C 4-Cl 2-EtO - 20 C 3,4-diMeO 2-MeO - 21 C 3,4-diMeO H -
carried out without antibiotic. For most of the compounds, the used concentration (63 lM, 6MIC/4) was in the range of the active concentration used for PAbN (50 lM). The concentration was established for the tested compounds as a suitable one to compare their activities to those of previous hydantoins tested at the concentration of 63 lM, as well. 9 Weakly active phenylpiperazine compounds 17-21 were tested at their concentration (100-200 lM) which ensured a lack of a direct antibacterial effect. The 'activity gain' (A) 9 was calculated according to the equation presented in Figure 2. The tested compounds showed moderate or low chemosensitizing properties when conjointly added to each antibiotic, displaying the highest activity gains at the range of 4.

In the case of nalidixic acid assay (Fig. 2a), only twofold decrease of MIC values was observed for relatively most efficient compounds (6, 14, 15, 17, 18 and 21) in reference strain of E. aerogenes, whereas eight arylidene hydantoins (3, 4, 12, 14-16, 18 and19 The results obtained during the assays with nalidixic acid and chloramphenicol indicated that ability of the hydantoin derivatives (3-21) to improve antibiotic efficacy is distinctly higher in CM-64, the strain of E. aerogenes which over-produces AcrAB-TolC efflux pump, than that in the reference strain (Table 3). This suggests that their way to modulate the bacterial resistance is connected to the tripartite efflux pump action. Four compounds (6, 11, 12 and 15) involve some additional anti-MDR mechanism distinct from Ac-rAB-TolC pump as they cause decrease of MIC of sparfloxacin significantly higher in the reference strain of E. aerogenes. Particularly, it applies to compound 12 while the chemosensitizer potency of compounds 11 and 15 in CM-64 is noticeable as well. An analysis of direct MIC of the tested compounds in compari-son to their properties to enhance antibiotics indicate that the most active compounds 14 and 15 displayed relatively higher intrinsic antibacterial action (MIC = 250 lM). The compounds were tested at 1/4 of the MIC to ensure a lack of a direct antibacterial effect. As a role of drug-hydantoin combinations in the chemosensitizer action is concerned, the combination with nalidixic acid seems to be superior. Eight members of the tested population (3, 4, 12, 14-16, 18, 19) generated statistically different values of activity (A) comparing the over-producing MDR efflux pump strain (CM-64) to the wild one (ATCC 13048), significantly higher in the case of CM-64. The weakest hydantoin-drug collaboration was observed during the assays with sparfloxacin in the strain CM-64. Only one compound, the diethylaminobenzylidene derivative 14, generated statistically different higher A value for the strain over-producing MDR efflux pump AcrAB-TolC. In contrary, four compounds (12, 15 > 11 > 6) caused statistically different stronger increase of sparfloxacin action in the reference strain ATCC 13048.

) decreased O R 1 CHO R 1 O HN NH , i HN NH HN N R 1 O R 1 O O 37,
O Amine O HN N OH R 1 R 2 O OH N HN N N O O HN N N O N OH 3, 4 (Group A) 5-7, 9, 11-16 (Group B) O O C N NH H 3 C HN N R 1 R 1 O 37, 38 R 1 O iv HN N CH 2 COOC 2 H 5 O HN N CH 2 COOH O O 30-36 17-21 (Group C) HN N HO N O O N N R 2 viii OCH 3 H R 1 N R 1 O N , v
The results obtained within the assays using strain CM-64 allow to divide the series of hydantoin derivatives (3-21) into following classes of potential efflux pump inhibitors: (I) the promising compounds which caused fourfold decrease of MIC of at least one antibiotic (3, 4, 11, 12, 14-16) at the lower dose (63 lM);

(II) compounds with slight efflux-modulator properties (5-8, 10, 17-21) causing the twofold decrease at lower concentration or fourfold one at the higher concentrations (100-200 lM); and (III) inactive compounds showing no effect in the efflux pump overproducing strains (9 and 13). From the effect obtained on CM-64 (Table 3), we can conclude that 14 and 15 were the more attractive molecules.

Molecular modeling

Analysis of structural features and structure-activity relationship for the set of the obtained compounds was based on their low energy conformations, calculated for a free base form of 3-21. On the basis of various lines of evidence including crystallographic studies for our previous 5-arylidenehydantoins, [20][21][22][23][24][27][28][29] only (Z)-configuration was considered for whole investigated population. Our previous investigations performed for the arylpiperazines 17-21 showed that in case of molecules at the global minimum energy point the absolute configuration of the secondary alcohol carbon almost did not influence on calculated distances between crucial pharmacophoric elements. 24 Thus, this time we limited our investigations to one enantiomer with the (S)-configuration. The obtained lowest energy conformations for compounds 3-21 are presented in Figure 3. The superimposition of one of the most active chemosensitizer 14 with the rest of the tested population (3-13 and 15-21) is shown in respect to the three structural categories A-C.

From the structural point of view, all the studied molecules consist of a flat moiety of hydantoin, linked by methylidene bridge with a (un)substituted phenyl ring from one side, and by flexible hydroxypropyl or ethanone chain with a secondary or tertiary amine group from the other. In the energetic global minimum all the compounds adopted an extended conformation with the arylidene phenyl ring rotated either by c.a. 57° or ÿ57° with reference to the imidazol ring plane, regardless of the amine fragment (Fig. 3). It should be noticed here that both positions of the phenyl ring were almost equivalent energetically and differed from each other with not more than 0.05 kJ/mol.

Structure-activity relationship

The arylidenehydantoin derivatives (3-21) differ within three structural fragments, what gives an opportunity to evaluate an influence of the fragments on the compound properties to restore a susceptibility to investigated antibiotics. Considering an influence of the amine terminated fragments, non-aromatic amines (group A and B, Table 1) seem to be more profitable than phenylpiperazine (group C). Among phenylpiperazine derivatives (17-21), only two compounds (18 and 19) caused fourfold decrease of one antibiotic (NAL) but their action was identified at higher doses (100 lM) than that of corresponding non-aromatic amine derivatives (Table 3, Fig. 2). Two non-aromatic amine moieties seem to be beneficial, hydroxyethylpiperazine (3, 4, 12, 14 and 16) and isopropylamine (11 and 15). The hydroxyethylpiperazine fragment is particularly favorable to improve the antibacterial effect of nalidixic acid, whereas a role of this fragment is shared with isopropyl moiety in the case of assays with sparfloxacin and chloramphenicol. The compounds 14 and 15 may be identified as the most active chemosensitizers of the resistant E. aerogenes because of their fourfold MIC reduction for all the tested antibiotics (Table 3). The compounds contain an aromatic arylidene fragment which is significantly activated by electron donating substituents as follows: a strong activating group of diethylamine at position para (14) or two moderate activating groups of 2,4-dimethoxyl substituents (15). Considering the influence of substitution at the arylidene aromatic rings on the tested biological activities, it can be noted that 4-diethylamine-( 14), 2,4-dimethoxy-(15 and 16) and 4-chloro-(11 and 12) substituents are the most favorable. Furthermore, the substituent-free benzylidene fragment (3) seems to be more profitable than the 4-methoxybenzylidene (6-10). The role of 4bromide substituent is hard to evaluate since the compounds possessing this fragment (4, 13) caused an aggregation allowing to test them only with nalidixic acid. The 4-chlorobenzylidene fragment, present in compounds 11 and 12, improved efficacy of sparfloxacin similarly- (11) or higher (12) than PAbN in the reference strain of E. aerogenes. This suggests that the p-Cl substituent may promote some unknown mechanism of anti-MDR action which disappears when tripartite efflux pump AcrAB-TolC is the predominant resistance mechanism, as in strain CM-64. A role of linkers between amine and hydantoin is difficult to explain on the basis of the obtained results. Although the compounds possessing acyl linker (3 and 4) were relatively less active than compounds with the hydroxypropyl one (11, 14-16), the structural differences of the both groups involves other fragments (e.g., substitutuents at arylidene moieties). Chemical properties of the most active compounds (14, 15), assessed basing on their 3D-structure, allow to propose pharmacophore features which may be responsible for chemosensitizing action on the MDR pump AcrAB-TolC, observed among the tested hydantoins (Fig. 4). It could be seen that an arylidene moiety at position 5 (AR) as well as a basic positive ionizable area linked to position 3 of the hydantoin (PIs) are crucial for the activity. A non-aromatic hydrophilic substituent at the positive ionizable area (HYL) seems to improve reversal-MDR activity of compounds, particularly in the case of nalidixic acid, whereas electron donating substituents in para-and ortho-position of the aromatic ring (ED) enable to develop the activity for all tested antibiotics. An aromatic substituent at the piperazine area PI can be considered as an undesirable feature decreasing anti-MDR properties of the compounds. This can underline a role of both, HYL and PIs, since the hydrophobic aromatic substituent at piperazine nitrogen drastically changes its basic properties and causes a huge increase of lipophilicity of the whole substituent at position 3 of the hydantoin. Molecular modeling calculations allowed to estimate topology of four pharmacophoric areas (PI, HYL, AR, ED) of the promising compounds (3, 11, 12, 14-16) in respect to the centre (HC) of the hydantoin scaffold (Fig. 4). Analysis of the calculated distances (X1-X4) indicated that the rotation of the arylidene phenyl ring (AR) by 57° or ÿ57° with reference to the imidazol ring (HC) did not affect the distance X3 between both fragments. The distance was in range of 4.84-4.87 Å (Fig. 4) for all the tested compounds (3-21, Fig. 4). Only small differences can be noticed also in the case of the X1 value, expressed as a distance between the hydantoin ring and the closest basic positive ionizable centre (PI). Superimposition of 14 with the low-energy poses of compounds from coming B or C groups shows overlapping of the basic nitrogen atoms in piperidine/morpholine/piperazine as well as isopropylamine fragments. In the first case (10, 12, 14, 16, 17-21) the X1 distance was 6.00 Å (8.72-8.76 Å for the second basic nitrogen atom, if possible), while this distance was around 5.94 Å for isopropyl derivatives 11, 13 and 15. In the case of piperazinamides 3 and 4 from the group A, the distance X1 was extended to 7.68 Å (Fig. 3b).

Compound X1 (PI-HC) [Å] X2 (HYL-HC) [Å] X3 (AR-HC) [Å] X4 (ED-HC) [Å]

Conclusions

In summary, the work contributes to the search of new tools to combat multidrug resistant bacteria among chemical compounds possessing hydantoin scaffold. From our previous studies, 9 a new antibiotic efficacy in Gram-negative E. aerogenes bacteria overproducing AcrAB-TolC efflux pump. The SAR analysis was performed on basis of the microbiological assays and the molecular modeling calculation. In comparison to the previous series of N1aminealkyl derivatives of 5,5-diphenylhydantoin, 9 the line of chemical modifications of hydantoin described here (3-21) seems to be more favorable as it gave a higher number of new compounds displaying chemosensitizing action on the MDR bacteria. Seven of the tested compounds (3, 4, 11, 12, 14-16), used in the concentration closed to the active dose of PAbN, displayed significant fourfold increase of antibacterial activity of nalidixic acid (3, 4, 12, 14-16), chloramphenicol (11, 14-16) and/or sparfloxacin (11, 14, 15). The general SAR derived from the studies indicates that an pected chemosensitizing action on AcrAB-TolC, what can be expressed by 'antipodal' positions of both, the hydrophilic basic amine fragment and the hydrophobic aromatic one, in the case of the most active compounds. The SAR study also demonstrates that strong electron-donor properties of substituent(s) at the arylidenene aromatic ring, that is seen in the case of (Z)-5-(4-(diethyl-1-yl)propyl)imidazolidine-2,4-dione ( 14) and (Z)-5-(2,4-dimethoxybenzylidene)-3-(2-hydroxy-3-(isopropylamino)propyl)imidazolidine-2,4-dione (15), could be responsible for similar chemosensitizing properties observed in the assays with different antibiotics (NAL, CHL, SFX) in the strain over-producing MDR efflux efflux pump inhibitors, the most active compounds (14 and 15) displayed only moderate chemosensitizing properties, lower than that of peptidomimetics (PAbN). Thus, further modifications within the group of hydantoin derivatives are necessary to improve their activity. Results of the comprehensive studies performed here will be a base for rational design of new MDR efflux pump inhibitors possessing hydantoin scaffold. The most active compounds, detected here, will be investigated using other bacterial species and antibiotic classes to understand their exact effect on various bacterial mechanisms of resistance and their precise target, a special attention will be paid for tripartite AcrAB-TolC efflux pump described in the MDR Enterobacterial isolates. or DMSO-d 6 at ambient temperature using the solvent signal as an internal standard. IR spectra were recorded on a Jasco FT/IR-410 apparatus using KBr pellets and are reported in cm ÿ1 . Thin-layer chromatography was performed on pre-coated Merck silica gel 60 F 254 aluminium sheets, the used solvent systems were (I) CH 2 Cl 2 /acetone 17:3; CH 2 Cl 2 , (II) CH 2 Cl 2 /acetone 19:1; (III) CHCl 3 / i-PrOH/NH 3 9:11:3; (IV) MeOH/acetone/benzene 1:1:1; (V) toluene/acetone/methanol 15:5:1. Melting points were determined using Mel-Temp II apparatus and are uncorrected. Elemental analyses were within ±0.4% of the theoretical values unless stated otherwise. Syntheses under microwave irradiation were performed in household microwave oven Samsung M1618. Synthesis of compounds 3, 4, 17-21, 23-29, 31, and 34-38 has been described elsewhere. 19,24 4.1.1. General procedure for preparation of 5-arylidene-3series of N3-aminealkyl arylidenehydantoin derivatives was (oxiran-2-ylmethyl)imidazolidine-2,4-diones (30, 32 and 33) designed, synthesized and evaluated for their ability to improve 5-Arylideneimidazolidine-2,4-dione (10 mmol), oxiran-2ylmethanol (15 mmol, 1.11 g) and triphenylphosphine (10 mmol, 2.62 g) were added to 50 mL of dry DMF at 0 °C. The mixture was stirred on the ice-bath until the reactants were totally dissolved (20 min). A solution of diethyl azodicarboxylate DEAD (10 mmol, 1.74 g) in dry DMF (10 mL) was added to the mixture dropwise for 45 min and stirring was continued at room temperature for further 3 h. Then, the mixture was poured into 100 mL of cold water to precipitate and left at 0-4 °C overnight. After filtration, pure product was obtained from precipitate by column chromatography with CH 2 Cl 2 /acetone and additional crystallization with EtOH or MeOH. 33) (1.6-3.0 mmol) and appropriate amine (2.8-6.0 mmol) were irradiated in a standard household microwave oven at various power (600-750 W) and time intervals (16-24 min) of irradiation for each prepared compound (5-7), respectively. The reaction progress was controlled with TLC (toluene (15):acetone (5):methanol (1)). After irradiation, the glassy residue was crystallized and saturated with gaseous HCl to give a pure product in hydrochloride form. The precipitate of pure product (5-7) was separated by filtration. dazolidine-2,4-dione 33 (0.82 g) and 1-methylpiperazine (0,60 g) were irradiated at 600 W during 6 min and next at 750 W in 18 min (4 1 2 min, 2 1 3 min, 2 min, 2 1 1 min, 2 min). Methanol was used to crystallization. After conversion into the hydrochloride form crystallization with dry ethanol and next ethanol gave white crystals of 5 (0.18 g, 0.5 mmol, 17%), mp: 193-199 °C, R f (III): 0. (8, 9), respectively. The reaction progress was controlled with TLC (V). After irradiation, the glassy residue was crystallized to give a pure product in basic form. The precipitate of pure product (8, 9) was separated by filtration.

Molecular modeling methods

The 3D models of compounds 3-21 were built using Schrödinger Maestro 30 molecular modeling environment basing on the solved crystal structure of 19, crystallized in an unprotonated form and described previously. 24 Only (S)-isomers of alcohols 5-21 were taken into account. Compounds were submitted to further analysis as free bases. For each compound, a conformational search was 4.1.4.3.

(Z)-5-(4-bromobenzylidene)-3-(2-hydroxy-3-(isopro-then performed using the mixed torsional method as implemented pylamino)propyl) imidazolidine-2,4-dione (13).

(Z)-5-(4-in MacroModel 9.8 with MMFFs force field and TNCG (Truncated Newton Conjugate Gradient) method of energy minimization. The conformational analysis was carried out for aqueous solutions with continuum solvation treatment (Generalised Born/Solvent Accessible, GB/SA) and terminated when the root means square (RMS) of conjugate gradient was below 0.05 kJ mol ÿ1 Å ÿ1 . Found global minimum energy poses of the ligands were superimposed by a least-squares method; heavy atoms of hydantoin fragment (carbon, nitrogen and oxygen atoms) were chosen as fitting points (Fig. 3). The distances between structural fragments of molecules were measured in Maestro using centroids defined for a phenyl (AR) and a hydantoin ring (HC). For the graphic presentation of superimposed lowest energy conformations PyMOL software [START_REF][END_REF] was used. To determine minimal inhibitory concentrations (MICs), approximately 10 6 cells were inoculated into 1 mL of Mueller-Hinton broth containing twofold serial dilutions of nalidixic acid or chloramphenicol or sparfloxacin according to the Clinical and Laboratory Standards Institute (CLSI, http://www.clsi.org/) guidelines. Triplicate results were read after 18 h at 37 °C. To study synergistic activity, different fixed sub-inhibitory concentrations of the potential inhibitors, determined without antibiotic, were added during incubation with nalidixic acid or chloramphenicol or sparfloxacin. In addition, the reference efflux pump modulator, PAbN, was used at 0.050 mM. Control experiments were carried out without the different inhibitors. MIC values are means of three independent determinations.

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2 .Figure 2 .

 22 Scheme 2. Synthesis of products 3-21: (iv) EtOH/H2O, NaOH; (v) TEA, DMF, BOP; 19 (vi) microwave irradiation; (vii) MeOH, boiling temp, 3 h; (viii) H2O, H + .

Figure 3 .

 3 Figure 3. Molecular modeling results. The superimposition of the most active chemosensitizer 14 (purple) with the rest of the compounds in respect to the three structural groups: (a) compounds of the group A (yellow); (b) compounds of the group B (blue); (c) compounds of the group C (black). All conformations represent the global energy minimum point. The hydrogen atoms are not shown.

Figure 4 .

 4 Figure 4. Pharmacophore features proposed based on properties of a most promising compound (14); (a) HC-hydantoin core, AR-aromatic ring, ED-electron donating substituent at aromatic ring, PI(s)-area with positive ionizable center(s), HYL-hydrophilic non-aromatic substituent; (b) distances from the center of hydantoin core to PI (X1), HYL (X2), AR (X3) and ED (X4) estimated for the most promising compounds 3, 11, 12, 14-16.

  spectra were recorded on a Varian Mercury VX 300 MHz PFG instrument (Varian Inc., Palo Alto, CA, USA) in CDCl 3

4. 1

 1 .1.1. (Z)-5-benzylidene-3-(oxiran-2-ylmethyl)imidazolidineamphiphilic character of the compounds is beneficial for the ex-2,4-dione (30

  (m, 1H, CHO), 3.63-3.65 (d, J = 4.87 Hz, 2H, N3-CH 2 ), Ar-3,5-H), 7.60-7.63 (m, 2H, Ar-2,6-H), 10.72 (s, 1H, N1H). 4.1.2. General procedure for preparation of hydrochloride form of amine derivatives of 5-arylideneimidazolidine-2,4-diones (5-7) (Z)-5-(4-methoxybenzylidene)-3-(oxiran-2-ylmethyl)imidazolidine-2,4-dione (

( 2 -

 2 hydroxyethyl) piperazin-1-yl)propyl)imidazolidine-2,4-dione 4.3. Microbiological assays (14). (Z)-5-(4-(diethylamino)benzylidene)-3-(oxiran-2-ylmethyl)imidazolidine-2,4-dione 34 and 2-(piperazin-1-yl)ethanol were

Table 1

 1 Structure of the tested derivatives of 5-arylidenehydantoin 3-21

  38

					HN	N CH 2 COOC 2 H 5
	O		O	HO
	22	23, 30, 37: R 1 = H	23-29		O
		24, 31, 38: R 1 =4-Br
		25, 32: R 1 = 4-Cl
		26, 33: R 1 = 4-MeO
		27, 34: R 1 = 4-NEt2 28, 35: R 1 = 2,4-di MeO 29, 36: R 1 = 3,4-diMeO			O 30-36	O
	Scheme 1. Synthesis of intermediates 23-38: (i) Knoevenagel condensation CH3COONa/CH3COOH	19,24 or Na2CO3/alanine/H2O	26 ; (ii), DIAD or DEAD, TPP, dry DMF, 0 °C;	24 (iii)
	ClCH2COOEt, K2CO3, TEBA, acetone.	14,18

Table 2

 2 Intrinsic antibacterial activity for compounds 3-21, tested in two strains of E. aerogenes, ATCC 13048 (reference) and the derivative strain CM-64 (over-producing AcrAB-TolC)

	Compound	MIC (mM) EA ATCC 13048	MIC (mM) EA CM-64
	4, 13	>1.25	>1.25
	5-10	>2	>2
	20			1	1
	17-19	0.5	1
	21			0.5	0.5
	11, 12, 15	0.25	0.25
	3, 14, 16	0.125	0. 25
	NAL	a	0.008	0.128
	SFX	b	6 1 10 ÿ5	0.001
	CHL	c	0.004	0.256
	PAbN	5	5

a Nalidixic acid. b Sparfloxacin. c Chloramphenicol. 17, 18, 20, 21) caused twofold reduction of chloramphenicol MIC in the efflux over-producer CM-64.

Table 3

 3 Effect of hydantoin derivatives(3-21) on the antibiotic susceptibility of E. aerogenes ATCC-13048 and CM-64 strains

	Compound	a	EA ATCC-13048			EA CM-64
		NAL MIC (lg/mL)	SFX MIC (lg/mL)	CHL MIC (lg/mL)	NAL MIC (lg/mL)	SFX MIC (lg/mL)	CHL MIC (lg/mL)
	No	8	0.06	4	128	1	512
	3	8	0.06	8	32	0.5	256
	4	8	b	8	32	b	b
	5	8	0.03	b	64	0.5	b
	6	4	0.015	2	64	1	256
	7	8	b	b	64	b	b
	8	8	0.03	4	64	1 1	512
	9	8	0.06	8	128	0.5	512
	10	8	0.06	8	128	0.25	512
	11	8	0.007	8	64	1	128
	12	8	0.003	8	32	b	512
	13	8	b	4	128		b
	14	4	0.06	8	32	0.25	128
	15	4	0.003	8	32	0.25	128
	16	8	0.06	8	32	0.5	128
	17	4	0.06	8	64	1	256
	18	4	0.06	8	32	1	256
	19	32	0.06	2	32	0.5	512
	20	16	0.06	8	64	1	256
	21	4	0.06	8	64	1	256

a Compounds concentrations according to the Figure

2

. b Compounds which induced aggregates of bacteria in the presence of sparfloxacin or chloramphenicol. These aggregates did not allow MICs measurement.

  -dione 25 (2.23 g) was used. Crystallization with isopropanol gave white crystals of 33 (750 mg, 2.6 mmol; 26.9%), mp: 207-212 °C, R f (I): 0.67. Anal. Calcd for C 13 H 11 ClN 2 O 3 : C, 56.03; H, 3.98; N, 10.05. Found: C, 56.20.85; H, 4.03; N, 10.01. 1 H

	). (1.88 g) was used. Crystallization with isopropanol gave white (Z)-5-benzylideneimidazolidine-2,4-dione 23 crystals of 30 (200 mg, 0.82 mmol, 8.2%), mp: 170-172 °C, R f (I): 0.74. Anal. Calcd for C 13 H 12 N 2 O 3 : C, 63.93; H, 4.95; N, 11.47. Found: C, 63.85; H, 5.10; N, 11.04. 1 H NMR (CDCl 3 ) d (ppm): 2.69 (dd, J 1 = 4.87 Hz, J 2 = 2.56 Hz, 1H, N3-CH 2 CHCH a (H)), 2.81-2.84 (m, 1H, N3-CH 2 CHCH b (H)), 3.22-3.28 (m, 1H, N3-CH 2 CHCH 2 ), 3.75 (dd, J 1 = 14.36 Hz, J 2 = 4.87 Hz, 1H, N3-CH c (H)), 3.90 (dd, J 1 = 14.36 Hz, J 2 = 5.13 Hz, 1H, N3-H d (H)), 6.78 (s, 1H, CH@C), 7.35-7.48 (m, 5H, Ar-2,3,4,5,6-H), 8.37 (s, 1H, N1H). IR KBr (cm ÿ1 ): 3276 (N1H), 1764 (C2@O), 1711 (C4@O), 1656 (CH@C), 1573 (Ar). 4.1.1.2. (Z)-5-(4-chlorobenzylidene)-3-(oxiran-2-ylmethyl)imi-dazolidine-2,4-dione (32). (Z)-5-(4-chlorobenzylideneimi-dazolidine-2,44.8 mmol, 49%), mp: 176-179 °C, R f (II): 0.59. Anal. Calcd for amino)benzylidene)-3-(2-hydroxy-3-(4-(2-hydroxyethyl)piperazin-pump AcrAB-TolC. Although the studies gave a promising group of potential MDR C 14 H 14 N 2 O 4 : C, 61.31; H, 5.14; N, 10.21. Found: C, 61.44; H, 5.25; N, 10.28. 1 H NMR (DMSO-d 6 ) d (ppm): 2.52-2.55 (m, 2H, CH 2 O), 3.15-3.17

  13. Anal. Calcd for C 19 H 28 N 4 O 4 Cl 2 1 2.5 H 2 O: C, 48.28; H, 6.16; OCH 3 ), 3.91 (m, 1H, CHOH), 4.87-4.88 (m, 1H, OH), 6.43 (s, 1H, CH@C), 6.93-6.96 (m, 2H, Ar-3,5-H), 7.59-7.62 (m, 2H, Ar-2,6-H), 10.59 (s, 1H, N1H). 4.1.3. General procedure for preparation of amine derivatives of 5-arylideneimidazolidine-2,4-diones (8, 9) (Z)-5-(4-methoxybenzylidene)-3-(oxiran-2-ylmethyl)imidazolidine-2,4-dione 33 (2.2-5 mmol) and appropriate amine (2.2-5 mmol) were irradiated in a standard household microwave oven at various power (450-750 W) and time intervals (16-28 min) of irradiation for each prepared compound

	N, 11.81. Found: C, 48.11; H, 6.59; N, 11.81. 1 H NMR (DMSO-d 6 ) d (ppm): 2.20-2.40 (m, 3H, N-CH 3 ), 2.72-3.12 (m, 8H, 2 1 Pp-2,3,5,6-H), 3.39-3.46 (m, 2H, N3-CH 2 ), 3.78 (s, 3H, OCH 3 ), 3.90-3.98 (br s, 1H, CHOH), 4.95-5.05 (br s, 1H, OH), 6.49 (s, 1H, CH@C), 6.95-9.98 (m, 2H, Ar-3,5-H), 7.59-7.62 (m, 2H, Ar-2,6-H), 10.65 (m, 1H, N1H). methylpiperazin-1-yl)propan-1-amine (0,50 g) were irradiated at 600 W during 2 min, next at 750 W in 6 min (3 1 2 min) and next at 600 W in 8 min (4 1 2 min). Methanol was used to crystalliza-tion. After converting into the hydrochloride form, crystallizing with ethanol with water and precipitating with dry ethanol gave CH 2 CHCH a (H)), 2.83-2.86 (m, 1H, N3-CH 2 CHCH b (H)), 3.23-3.29 (m, 1H, N3-CH 2 CHCH 2 ), 3.76-3.89 (m, 2H, N3-CH 2 ), 6.72 (s, 1H, CH@C), 7.38-7.44 (m, 4H, Ar-2,3,5,6-H), 8.86 (s, 1H, N1H). 4.1.1.3. (Z)-5-(4-methoxybenzylidene)-3-(oxiran-2-ylmethyl)-imidazolidine-2,4-dione (33). (Z)-5-(4-methoxybenzylidene -imidazolidine-2,4-dione 26 (2.18 g) was used. Purification by column chromatography gave white crystals of 33 (1.34 g, white crystals of 6 (0.15 g, 0.35 mmol, 22%), mp: 213-219 °C, R NMR (CDCl 3 ) d (ppm): 2.69 (dd, J 1 = 4.87 Hz, J 2 = 2.57 Hz, 1H, N3-3.79 (s, 3H, OCH 3 ), 6.52 (s, 1H, CH@C), 6.95-6.98 (m, 2H,	4.1.2.3. (Z)-5-(4-methoxybenzylidene)-3-(2-hydroxy-3-(piperi-din-1-yl)propyl)imidazolidine-2,4-dione hydrochloride (7). (Z)-5-(4-methoxybenzylidene)-3-(oxiran-2-ylmethyl)imi-dazolidine-2,4-dione 33 (0.77 g) and 1-methylpiperazine (0,24 g) were irradiated at 600 W during 6 min and next at 750 W in 18 min (4 1 2 min, 2 1 3 min, 2 min, 2 1 1 min, 2 min). Methanol was used to crystallization. After conversion into the hydrochloride form crystallization with dry ethanol and next ethanol gave white crystals of 7 (0.04 g, 0.11 mmol, 4%), mp: 171-176 °C, R f (III): 0.26. Anal. Calcd for C 19 H 26 N 3 O 4 Cl: C, 57.64; H, 6.62; N, 10.61; Found: C, 57.81; H, 6.34; N, 10.21. 1 H NMR (DMSO-d 6 ) d (ppm): 1.28-1.42 (m, 6H, Pip-3,4,5-H), 2.22-2.32 (m, 4H, Pip-2,6-H), 3.40-3.52 (m, 4H, N3-CH 2 , Pip-CH 2 ), 3.78 (s, 3H,

f (III): 0.33. Anal. Calcd for C 22 H 35 N 5 O 4 Cl 2 1 3H 2 O: C, 47.10; H, 7.32; N, 12.52. Found: C, 47.31; H, 7.40; N, 12.54. 1 H NMR (DMSO-d 6 ) d (ppm): 1.95-2.05 (m, 2H, Pp-CH 2 -CH 2 ), 2.47-2.50 (s, 7.63 (d, J = 8.98 Hz, 2H, Ar-2,6-H), 8.65-8.90 (m, 2H, NH 2 + ), 10.71 (s, 1H, N1H).

Acknowledgements

Partly supported by grants: 501/N-COST/2009/0 (K/PMN/ 000031), COST action BM0701, Polonium 8127/2010 and ANR-11-BS07-019-02 'IBEF'.

.


Hadrien Kurkjian

Yvan Castin

Alice Sinatra

Landau-Khalatnikov phonon damping in strongly interacting Fermi gases

Keywords: 03.75.Kk -Dynamic properties of condensates, collective and hydrodynamic excitations, superfluid flow 67.85.Lm -Degenerate Fermi gases 47.37.+q -Hydrodynamic aspects of superfluidity, quantum fluids

published or not. The documents may come

Landau-Khalatnikov phonon damping in strongly interacting Fermi gases

Introduction. -Phonons, sound waves, low energy normal modes or gapless collective excitations are ubiquitous in physics. In uniform weakly-excited quantum many-body systems with short-range interactions, they are described as quasiparticles characterized by a dispersion relation approximately linear at low wavenumber, ω q ∼cq with c the speed of sound, and by a damping rate much smaller than the angular eigenfrequency Γ q ≪ ω q . Phonon damping plays a central role in transport phenomena such as thermal conduction in dielectric solids, and in hydrodynamic properties such as temperature dependent viscosity and attenuation of sound in liquid helium [1,2]. It is also crucial for macroscopic coherence properties, since it determines the intrinsic coherence time of bosonic and fermionic gases in the condensed or paircondensed regime [3][4][5]. In the absence of impurities the damping of low-energy phonons is determined by phononphonon interactions that conserve energy and momentum and it crucially depends on the curvature of the phonon dispersion relation [6,7]. For a concave dispersion relation, 1 ↔ 2 Beliaev-Landau processes involving three phonons are not resonant and the 2 ↔ 2 Landau-Khalatnikov process involving four quasiparticles dominates at low q.

In this paper we consider an unpolarized gas of spin-1/2 fermions prepared in thermal equilibrium at a temperature T below the critical temperature, where a macroscopic coherence between pairs of opposite spin fermions appears. Compared to other many-body fermionic systems, atomic gases offer the unique possibility to tune the interaction strength with an external magnetic field close to a so-called Feshbach resonance. This allows experimentalists to explore the crossover between the Bose-Einstein Condensate (BEC) and Bardeen-Cooper-Schrieffer (BCS) regimes [8][9][10][11][12][13][14][15][16]. The dispersion relation of low energy excitations, describing the collective motion of the pair center of mass, has a phononic start at small wavenumbers [17][18][19][20][21][22] and changes from convex to concave in the BEC-BCS crossover, close to the strongly interacting unitary limit [20,22]. Therefore, the damping caused by the 2 ↔ 2 processes should be directly observable in cold Fermi gases, contrarily to weakly-interacting Bose gases where the convex Bogoliubov dispersion relation supports Landau-Beliaev damping. On the theoretical side, the original study by Landau and Khalatnikov of the 2 ↔ 2 damping rate [1] is limited to the case where one of the colliding phonons has a small wavenumber compared to the other and it performs as we shall see an unjustified approximation on the coupling amplitude. Here, we give the general expression of the phonon damping rate in the concave dispersion relation regime at low temperature, where it is dominated by the 2 ↔ 2 processes, correcting and extending the original calculation of reference [1]. In the whole paper we restrict to the so-called collisionless regime where the phonon angular frequency times the typical collision time in the gas is much larger than one, ω q τ c ≫ 1 [23,24]. This is in general the case in superfluid gases at low temperature1 .

Effective 2 ↔ 2 phonon coupling. -The theoretical framework we use is the irrotational quantum hydrodynamics of Landau and Khalatnikov [1]. Quantum hydrodynamics is an effective low-energy theory that relies only on the equation of state and can thus be applied in all interaction regimes. For fermions, it neglects from the start the internal fermionic degrees of freedom and the corresponding gapped BCS excitation spectrum, treating the pairs of fermions at large spatial scales as a bosonic field. It is expected to give exact results to leading order in the low temperature T , at least for observables involving low energy scales. The quantum hydrodynamics Hamiltonian 2 reads

Ĥ = d 3 r 2 2m ∇ φ • ρ ∇ φ + e 0 (ρ) (1)
where e 0 (ρ) is the ground state energy density, m is the mass of a particle, and the superfluid velocity field operator v(r, t) = m ∇ φ(r, t) is the gradient of the phase field operator, canonically conjugated to the density field operator ρ(r, t), so that [ρ(r, t), φ(r ′ , t)] = iδ(rr ′ ). Assuming small deviations of ρ(r, t) and φ(r, t) from their uniform spatial averages, one finds the normal modes Fourier components δ ρq ∝ q 1/2 ( bq + b † -q ) and δ φq ∝ -iq -1/2 ( bqb † -q ), where the annihilation and creation operators of quasiparticles bq and b † q obey bosonic commutation relations. One inserts the expansion of ρ and φ over these modes in the Hamiltonian (1), which results in the series Ĥ = E 0 + Ĥ2 + Ĥ3 + Ĥ4 + . . . , where the index refers to the total degree in bq and b † q , each term being written in normal order.

In the concave dispersion relation regime considered in this paper, the direct phonon coupling due to Ĥ3 is not resonant, and the leading resonant coupling is a four-phonon process. An effective interaction Hamiltonian Ĥeff coupling an initial Fock state of quasiparticles |i of energy E i to a final one |f of same energy E f = E i , where two wavevectors q 1 and q 2 are annihilated and two other wavevectors q 3 and q 4 are created, can then be derived in second order perturbation theory by considering the direct coupling by Ĥ4 to first order and the indirect coupling (involving a non-resonant intermediate state |λ ) by Ĥ3 to second order,

f | Ĥeff |i ≃ f | Ĥ4 |i + λ f | Ĥ3 |λ λ| Ĥ3 |i E i -E λ ≡ A i→f (2)
The reader will notice that, for the purely linear dispersion relation ω q = cq predicted by Ĥ2 , the denominators in (2) vanish for aligned wavevectors because the intermediate processes become resonant, for example E i -E λ = ω 1 + ω 2 -ω q1+q2 = 0, with the short-hand notation which ensures the collisionless regime. 2 In principle, this Hamiltonian has to be regularized by introducing an ultraviolet momentum cut-off or by discretizing the real space on a lattice as in reference [5]. This however does not play a role here.

ω i ≡ ω qi . Following Landau and Khalatnikov, we regularize the resulting divergence in A i→f by including the actual curvature of the spectrum [20,22] in the energy denominators3

ω q = q→0 cq 1 + γ 8 q mc 2 + O q mc 4 . (3)
Here the speed of sound c is related to the gas density ρ and the ground state chemical potential µ by

mc 2 = ρ dµ dρ (4)
whereas the dimensionless curvature parameter γ < 0 must be measured or determined from a microscopic theory. By introducing the dimensionless and stateindependent effective coupling amplitude A eff ,

A i→f = n q1 n q2 (1 + n q3 )(1 + n q4 ) 4mc 2 ρL 3 A eff (5)
where n qi are the phonon occupation numbers in the initial Fock state |i , and by considering in eq. ( 2) the six possible intermediate states |λ where a virtual phonon is created and reabsorbed (or absorbed and recreated in the six corresponding finite temperature diagrams, in such a way that the temperature dependence disappears) we find A eff (q 1 , q 2 ; q 3 , q 4 ) = 1 16

4 ω 1 ω 2 ω 3 ω 4 m 4 c 8 × Σ F + (ω 1 + ω 2 ) 2 A 1234 + ω 2 q1+q2 B 1234 (ω 1 + ω 2 ) 2 -ω 2 q1+q2 + (ω 1 -ω 3 ) 2 A 1324 + ω 2 q1-q3 B 1324 (ω 1 -ω 3 ) 2 -ω 2 q1-q3 + (ω 1 -ω 4 ) 2 A 1423 + ω 2 q1-q4 B 1423 (ω 1 -ω 4 ) 2 -ω 2 q1-q4 (6)
We introduced the angle-dependent coefficients

A ijkl = (3Λ F + u ij )(1 + u kl ) + (3Λ F + u kl )(1 + u ij ) + (1 + u ij )(1 + u kl )(7) B ijkl = (3Λ F + u ij )(3Λ F + u kl ) (8)
with u ij = q i • q j /q i q j and the thermodynamic quantities

Σ F ≡ ρ 3 mc 2 d 3 µ dρ 3 (9) Λ F ≡ ρ 3 d 2 µ dρ 2 dµ dρ -1 (10)
From the coupling amplitude A eff we finally obtain the sought low-energy effective Hamiltonian for the 2 ↔ 2 phonon process in a cubic quantization volume of size L:

Ĥeff = mc 2 ρL 3 q1,q2,q3,q4 q1+q2=q3+q4 A eff (q 1 , q 2 ; q 3 , q 4 ) b † q3 b † q4 bq1 bq2 (11)
The phonon damping rate. -To calculate the phonon damping rate from the effective Hamiltonian (11) we use as in reference [26] a master equation approach for a harmonic oscillator bq , b † q coupled to a thermal bath, see complement B IV in reference [START_REF] Cohen-Tannoudji | Processus d'interaction entre photons et atomes[END_REF]: the phonon mode of momentum q is linearly coupled to the thermal reservoir containing all the other phonon modes by an interaction Hamiltonian Ĥint = R †b q + Rb † q , where R † collects terms in b † q3 b † q4 bq2 from Ĥeff . With respect to Fermi-goldenrule-based rate equations or linearised kinetic equations as in reference [START_REF] Landau | [END_REF], the master equation gives a complete picture of the quantum dynamics of the phonon mode q, including the case to come of an initial Glauber coherent state, while of course giving the same damping rate for the phonon occupation number. In the Born-Markov approximation [START_REF] Cohen-Tannoudji | Processus d'interaction entre photons et atomes[END_REF], the master equation for the density operator σ of the phonon mode q takes the usual form [START_REF] Cohen-Tannoudji | Processus d'interaction entre photons et atomes[END_REF]:

d dt σ = 1 i [ ω q b † q bq , σ] + Γ - q bq σb † q + Γ + q b † q σb q - 1 2 Γ - q b † q bq + Γ + q bq b † q , σ (12)
where [, ] and {, } are the commutator and the anticommutator of two operators, and the raising and lowering rates Γ ± q are given as in [START_REF] Cohen-Tannoudji | Processus d'interaction entre photons et atomes[END_REF] (see also equations (A6) and (A7) of [26]) by integrals of free temporal correlation functions of R and R † in the equilibrium state of the reservoir. As the reservoir is in thermal equilibrium, the two rates are related by Γ - q = exp( ω q /k B T )Γ + q [START_REF] Cohen-Tannoudji | Processus d'interaction entre photons et atomes[END_REF] and can be expressed in terms of their difference Γ q ≡ Γ - q -Γ + q , with

Γ q = +∞ -∞ dt 2 e -iωqt [ R(0), R † free (t)] th (13)
Here . . . th is a thermal average, and the free evolution is governed by the quadratic part of the Hamiltonian, R † free (t) = exp(i Ĥ2 t/ ) R † (0) exp(-i Ĥ2 t/ ). As explained in [START_REF] Cohen-Tannoudji | Processus d'interaction entre photons et atomes[END_REF], Γ q gives both the exponential relaxation rate of the mean occupation number of the mode q towards its thermal equilibrium value nq and of the mean phonon amplitude in mode q towards zero:

d dt b † q (t) bq (t) = -Γ q ( b † q (t) bq (t) -nq ) (14) d dt bq (t) = -iω q + Γ q 2 bq (t) (15)
Calculating the commutator in the expression (13) gives

Γ q = (mc 2 ) 2 4π 5 2 ρ 2 d 3 q 2 d 3 q 3 |A eff (q, q 2 ; q 3 , q 4 )| 2 × δ(ω 3 + ω 4 -ω 2 -ω q ) [n 2 (1 + n3 + n4 ) -n3 n4 ] (16)
where q 4 = q + q 2q 3 due to momentum conservation and we have used the notation ni ≡ nqi to indicate the phonon occupation numbers in thermal equilibrium 4 . The result ( 6),( 16) goes beyond that of Ref. [1] where the authors concentrated on a single diagram for the intermediate processes. This will have a significant impact on the behavior of the damping rate as a function of q as we will see. To go further analytically, we restrict to sufficiently low temperature

ǫ ≡ k B T mc 2 ≪ 1 ( 17
)
so that only the (almost) linear region of the phonon dispersion relation is explored and the integral ( 16) is dominated by configurations in which the four involved wavevectors are almost aligned. By introducing the rescaled quantities

qi = cq i k B T and θi = θ i ǫ|γ| 1/2 (18)
with θ i the angles between q i and q and γ the curvature parameter in the spectrum (3), and by expanding the coupling amplitude (6) for ǫ → 0 at fixed qi and θi , we obtain the central result of this paper:

Γ q ǫ F q fixed ∼ ǫ→0 K |γ| T T F 7 Γ(q) (19)
In this formula ǫ F = k B T F = 2 k 2 F /2m is the Fermi energy, K is an interaction-dependent thermodynamic quantity

K = 2 3 4 6 ǫ F mc 2 3 (1 + Λ F ) 4 (20)
and the rescaled phonon damping rate Γ(q), shown in fig. 1, is a universal, monotonically increasing function of the dimensionless wavenumber. Explicitly

Γ(q) = ∞ 0 dq 2 q+q2 0 dq 3 q3 2 q3 3 (q + q2 -q3 ) q|v| × [1 + f (q 2 )]f (q 3 )f (q + q2 -q3 ) f (q) × π 0 dφ π/2 0 dα sin α cos α Θ - v u |A red | 2 (21)
4

The expression involving the ni between the square brackets in equation ( 16) results from the simplification of the extended form n2 (1 + n3 )(1 + n4 ) -(1 + n2 )n 3 n4 . The extended form is physically more transparent as it reveals the spontaneous and stimulated factors appearing in the direct q + q 2 → q 3 + q 4 and reverse q 3 + q 4 → q + q 2 processes. Using (1 + ni )/n i = exp( ω i /k B T ) and energy conservation, it can also be rewritten as in equation (7.4) of reference [START_REF] Landau | [END_REF], that is (1 + n2 )n 3 n4 /nq. We shall use this last form in equation (21). Γ ~(q ~)/q ~2 Fig. 1: Rescaled phonon damping rate due to 2 ↔ 2 phonon processes in a superfluid spin-1/2 Fermi gas, as a function of the rescaled wavenumber q = cq/kBT , or more precisely of its inverse hyperbolic sine argsh q = ln(q + 1 + q2 ). Due to these rescalings, see eq. ( 19), the result is universal and applies at sufficiently low temperature in the whole region of the BEC-BCS crossover where the dispersion relation is concave at low q. The dashed lines show the limiting behaviors ( 25) and ( 26) and the dot-dashed curve at large q is a fit of Γ(q)/q 2 by an affine function of 1/q.

where Θ(x ≥ 0) = 1, Θ(x < 0) = 0 is the Heaviside function, f (x) = 1/(e x -1) originates from the Bose law, u and v are the following functions of q, q2 , q3 , α = arctan θ 3 /θ 2 , and of the relative azimuthal angle φ of q 2 and q 3 :

u = q(q 3 sin 2 α -q2 cos 2 α) + q2 q3 (1 -sin 2α cos φ) q + q2 -q3 (22) v = 1 4 q3 + q3 2 -q3 3 -(q + q2 -q3 ) 3 (23)
We introduced in eq. ( 21) the reduced coupling amplitude

A red = 1 q2 cos 2 α (q+q2) 2 -3u 4v - 1 q3 sin 2 α (q-q3) 2 -3u 4v - 1 q2 cos 2 α-q3 sin 2 α+u (q2-q3) 2 -3u 4v (q + q2 -q3 ) (24)
The limiting behaviors of the normalized universal Landau-Khalatnikov damping rate ( 21)

Γ(q) = q→0 16π 5 135 q3 + O(q 4 ) (25) Γ(q) = q→∞ 16πζ(5) 3 q2 + O(q) (26)
are shown in fig. 1 as dashed lines. Here ζ is the Riemann zeta function. Equations ( 25)-( 26) disagree with Eqs.(7.6) and (7.12) in reference [1], even for the order in q, our results being subleading by two orders. It was already noted in reference [6] that the diagrams neglected by Landau and Khalatnikov in the calculation of the effective amplitude A eff are comparable in magnitude with the one that they keep. As our explicit calculations show, the failure of this approximation is striking in the low q and in the high q behaviour of Γ(q), where the neglected diagrams turn out to interfere destructively with the supposedly leading diagram kept in reference [1], lowering the final leading order in q.

There is an interaction strength in the BEC-BCS crossover where the curvature parameter γ vanishes, since γ is positive in the BEC limit [19] and γ is negative in the BCS limit [18]. Close to this point, on the side γ < 0, one may expect from the factor 1/|γ| in eq. ( 19) that there is a strong enhancement of the damping rate. This is however not the case. The result (19) can only be used at low enough temperature k B T < mc 2 |γ/η| 1/2 , such that the cubic term cq γ 8 ( q/mc) 2 in the dispersion relation ( 3) is large as compared to the quintic one cq η 16 ( q/mc) 4 for typical thermal wave numbers q ≈ k B T / c. For the same reason, one must also have q < mc|γ/η| 1/2 where q is the wave number of the phonon mode whose damping is under consideration. As a result, within the validity range of eq. ( 19), and due to Γ(q) = O(q 2 ), one has Γ q /ǫ F = O(|γ|5/2 /|η| 7/2 ). In practice, at the point γ = 0 or very close to it, a new calculation should be performed using a linear plus quintic dispersion relation. Note that, according to the RPA, γ = 0 for 1/k F a ≃ -0.144, in which case η ≃ -0.0428, the dispersion relation remains concave at low q and the Landau-Khalatnikov process is still the leading damping mechanism [22]. 5 Physical discussion and observability. -We now discuss the main result (19) and the possibility to test it in state-of-the-art experiments with cold fermionic gases. In fig. 2 we use the measured equation of state of the spin-1/2 Fermi gas [14,[START_REF] Navon | Thermodynamique des gaz de bosons et fermions ultrafroids[END_REF] to plot the thermodynamic quantity K in eq. ( 20) as a function of 1/k F a. The curvature parameter γ appearing in the phonon dispersion relation (3) has not yet been measured in strongly interacting Fermi gases. We plot in fig. 3 the prediction of reference [22] which relies on the Random Phase Approximation and coincides with that of other approximate theories [18,20]. We plot it separately so that, once an experimental value will be available, it can be used in eq. ( 19). To give a numerical example, we choose the interaction strength such that the coefficient η of the quintic term in the expansion (3) vanishes, 1/k F a ≃ -0.39 according to reference [22]. From the measured equation of state, the value of the sound velocity is c ≃ 0.43 k F /m and the parameter Λ F obeys 1 + Λ F ≃ 0.866, leading to the thermodynamic constant K ≃ 4. 20) as a function of 1/kF a where kF = (3π 2 ρ) 1/3 is the Fermi wavenumber and a the s-wave scattering length, from the measured zerotemperature equation of state of the spin-1/2 Fermi gas [14,[START_REF] Navon | Thermodynamique des gaz de bosons et fermions ultrafroids[END_REF].

curvature parameter is γ ≃ -0.30 [22]. The temperature should be sufficiently low and the wavenumber q much smaller than 2∆/ c to avoid the excitation of the fermionic branch [START_REF] Schirotzek | [END_REF]31]. By choosing T = 0.073 T F , larger than the temperature already achieved in reference [32], and q ≡ cq/k B T = 5/2, one obtains ǫ ≡ k B T /mc 2 ≃ 0.20, q/mc ≃ 0.50, q ≃ 0.21k F , cq/2∆ ≈ 0.2 and Γ ≃ 265. For the typical value T F = 1 µK with 6 Li atoms, one obtains 2π/q ≃ 6 µm, ω q /2π ≃ 3.8 kHz and c ≃ 2.2 cm/s. Eq. ( 19) then predicts Γ q ≃ 5 s -1 , that is a phonon lifetime Γ -1 q ≃ 190 ms and a mode quality factor ω q /Γ q ≃ 4600. 6 Quality factors of this order of magnitude have been observed for the transverse monopole mode of an atomic Bose-Einstein condensate [33].

The possibility of trapping cold atoms in flat bottom potentials [34] opens the way to a direct test of our prediction in a spatially homogeneous system. In the box trapping potential, a Glauber coherent state of phonons in a standing-wave mode with a well-defined wavevector q along a trap axis can be created by laser Bragg excitation of the condensate of pairs in the strongly interacting Fermi gas [35][36][37], on top of the preexisting background of thermal phonons. One simply matches the frequency difference and the wavevector difference of two laser standing waves to the angular eigenfrequency ω q and the wavevector q of the desired phonon mode. The subsequent decay of the phonon coherent state can be monitored by 6 As expected, this is in the collisionless regime since the angular frequency ωq is much larger than the thermalization rate, that we estimate by the damping rate at the typical thermal wavenumber q th = k B T / c: one finds Γ(1) ≃ 26 and Γq th /ωq ≃ 2 × 10 -5 . For comparison, we also give the Beliaev-Landau damping rate at the unitary limit (where γ RPA ≃ 0.084 > 0), Γ Bel-Lan q ≃ 140 s -1 for 6 Li atoms with the same values ǫ = 0.2, q = 5/2 and T F = 1 µK. measuring in situ the spatial modulation of the density at wavevector q using the bosonizing imaging techniques of reference [13]. Note that in Bragg spectroscopy the density is usually determined after a time-of-flight which amounts to making a measurement in Fourier space. This is appropriate for q ≫ k F where the scattered atoms separate from the Fermi sea of unscattered atoms. Here on the contrary q ≪ k F and the measurement is best performed in real space. In weakly interacting Bose gases, damping times of a fraction of a second have been measured in experiments without being affected by extraneous damping mechanisms [33], and the Bragg technique has allowed to measure the Bogoliubov dispersion relation [38] and to observe the zero-temperature 1 → 2 Beliaev damping [39] with a successful comparison of their q-dependence to theory. To be complete, let us analyse in more detail the experimental proposal. In reference [34] the flat bottom trap has an elongated cylindric shape. For simplicity, we model it by an infinite square well potential in the three dimensions with widths L x = L y ≡ L ⊥ < L z of rounded up values L ⊥ = 50 µm and L z = 100 µm. 7 The 7 One may wonder if this is large enough for our infinite-system theory to apply. This will be the case if the typical spacing between the discrete values of ω 3 + ω 4 -ω 2 -ωq in the argument of the Dirac distribution in eq. ( 16) is ≪ Γq/2. In the decay process of an unstable state, the energy is indeed conserved within ± Γq/2. We estimate the typical spacing by 1/ρstates(ωq) with the density of states

ρstates(ω) = typ q 2 ,q 3 δ(ω 3 + ω 4 -ω 2 -ω)
where the typical values of q 2 and q 3 are at an angle at most ǫ with phonon mode functions are then products of sine functions

2 3/2 V 1/2
α=x,y,z sin(q α r α ), where q α L α /π = n α ∈ N * and V = L 2 ⊥ L z is the trap volume. During the short time interval 0 < t < τ , two retro-reflected far-offresonant Bragg laser beams illuminate the trapped gas. They induce a conservative lightshift potential W (r, t) = W 0 |E(r, t)| 2 where E(r, t) is the reduced, dimensionless positive-frequency part of the laser electric field. In the second quantized form this gives rise to the phonon-light coupling Hamiltonian ĤW = d 3 rW (r, t)δ ρ(r) [START_REF] Cohen-Tannoudji | Processus d'interaction entre photons et atomes[END_REF] Following the values of the physical parameters given above, we take the phonon mode to be excited in the transverse ground state n x = n y = 1 with a wavenumber q z ≃ 0.5mc/ along z, that is n z = 33. To optimally excite this mode, we choose E(r, t) = sin(k 1 • r)e -iω1t + i cos(k 2 • r)e -iω2t . Here k i and ω i , the wavevectors and the angular frequencies of the laser standing waves, obey k 2 = k 1 + q z e z and ω 2 = ω 1 -ω q , with ω q the angular frequency of the phonon mode and e z the unit vector along z. In practice the wavevectors k 1 and k 2 are in the optical domain, with submicronic wavelength, so k 1 ≃ k 2 ≫ q and the condition k 2 = k 1 + q z e z is achieved by introducing a small angle between them. In the resulting optical potential

W (r, t) = W 0 1 - 1 2 [cos(2k 1 • r) -cos(2k 2 • r)] + [sin(q z z) -sin((k 1 + k 2 ) • r)] sin(ω q t) (28)
the time-independent part is non-resonant and can be neglected. In the time dependent part, the second sine term excites a mode of wavevector k 1 + k 2 far from resonance and can also be omitted. One is left for 0 < t < τ with the effective Bragg Hamiltonian 8

Ĥres

W ≃ 2 Ω( bq + b † q ) sin(ω q t) (29)
respect to q. Also we impose q 2 < q th where c q th = π 4 30ζ(3) k B T is the mean thermal energy per phonon. The wavenumber q 3 is automatically limited by q + q 2 as in eq. ( 21). Replacing k by V d 3 k/(2π) 3 in the thermodynamical limit, and taking the ǫ → 0 limit as in Γq, we obtain ρstates(ωq)Γq/2 ∼ ( L/L 0 ) 6 with L3 = V the trap volume and

L 0 = 4π 3mc k F mc ǫ -7/3 π -1/3 (1 + Λ F ) -2/3 ( ΓI) -1/6
where the integral

I = π 4 30ζ(3) 0 q2 2 dq 2 q+ q2 0 q2 3 dq 3 × 1 0 R 3 dR π/2 0 sin α cos α dα π 0 dφ π δ(uR 2 + v)
is evaluated numerically, I ≃ 2.445. The thermodynamic limit is reached for L > L 0 . In our numerical example L ≃ 63 µm is indeed larger than L 0 ≃ 50 µm. 8 As the lightshift potential W 0 sin(qzz) sin(ωqt) is x-and yindependent, it actually couples to several transversally excited of Rabi frequency Ω = W 0 [ω q ρV /(16 mc 2 )] 1/2 . Integratthe Heisenberg equations of motion for the phonon annihilation operator bq and taking the expectation value in the initial t = 0 thermal state9 , one finds bq (τ ) = Ω[τ e -iωqτ -sin(ω q τ )/ω q ] where the second term, which is the off-resonant effect of the negative-frequency part of sin(ω q t), is zeroed by the choice τ = π/ω q ≃ 130µs. At the end of the Bragg pulse, this gives a z-modulation of the gas density that will subsequently decay to zero due to scattering of thermal phonons. Since absorption imaging gives in general access to an integrated density, we give the density modulation integrated along x and y:

δρ(z, t) = t>τ πW 0 ρL 2 ⊥ 2mc 2 sin(q z z) cos(ω q t)e -Γqt/2 (30)
Conclusion. -We have calculated the Landau-Khalatnikov low-temperature phonon damping rate in strongly interacting superfluid Fermi gases. Our expression, thanks to an appropriate rescaling of the wavenumber, takes a universal functional form, plotted in fig. 1, that applies in the whole region of the BEC-BCS crossover where the phononic dispersion relation has a concave start. This includes the strongly interacting regime and paves the way to an experimental observation with ultracold atomic gases. An alternative, not discussed in detail here, is to use superfluid liquid helium-4 at high pressure, whose excitation branch is concave at low q, while it is convex at low pressure [41]. The observation of the Landau-Khalatnikov phonon damping would be unprecedented and would open a new era in the exploration of lowtemperature dynamics of macroscopically coherent quantum many-body systems. nxny . In the main text we have presented a monomode calculation assuming |c 0,0 | 2 = 1 as if only the transversally fundamental mode was excited. In reality, the result in eq. ( 30) has to be weighted by |c 0,0 | 2 and the transversally excited modes give additional time dependent contributions to the integrated density modulation that dephase in a time of the order of t ⊥ = qL 2 ⊥ /(2π 2 c) ≃ 6 ms, that is ≃ 140/ωq and ≃ 0.03/Γq. Since sx,sy |cs x,sy | 2 converges rather rapidly to unity, the relevant nx and ny are indeed ≪ nz, and the angular frequency differences ω (nx,ny ,nz ) -ω (1,1,nz ) ≃ t -1 ⊥ [sx(sx + 1) + sy(sy + 1)], with the obvious notation ωq = ω (1,1,nz ) , etc. To summarize, eq. ( 30) becomes δρ(z, t) = |cs x ,sy | 2 e -i[sx(sx+1)+sy(sy +1)]t/t ⊥ Since ωqt ⊥ ≫ 1, f (t) plays the role of an envelope function and mainly its modulus matters. For our square well potential |c 0,0 | 2 ≃ 0.657 and |f (t)| < 0.343. This shows that the contributions of the excited modes and their Hamiltonian dephasing are rather harmless and cannot imitate a complete damping of the phonon mode.

2 .Fig. 2 :

 22 Fig.2: Dimensionless thermodynamic quantity(20) as a function of 1/kF a where kF = (3π 2 ρ) 1/3 is the Fermi wavenumber and a the s-wave scattering length, from the measured zerotemperature equation of state of the spin-1/2 Fermi gas[14,[START_REF] Navon | Thermodynamique des gaz de bosons et fermions ultrafroids[END_REF].

Fig. 3 :

 3 Fig.3: Curvature parameter γ obtained with the RPA in reference[22] for the phonon dispersion relation (3), as a function of 1/kF a where kF is the Fermi wavenumber and a the s-wave scattering length. The dashed lines mark the two points where either the cubic (1/kF a ≃ -0.144) or the quintic (1/kF a ≃ -0.389) correction to the linear dispersion relation in eq. (3) vanishes.

  phonon modes during the Bragg pulse. Only the even x-parity and even y-parity states are populated, nx = 2sx + 1 and ny = 2sy + 1, (sx, sy) ∈ N 2 , with normalised amplitudes cs x ,sy =

πW 0 ρL 2 ⊥ 2mc 2 sin

 22 (qzz){|c 0,0 | 2 cos(ωqt) + Re[f (t) exp(-iωqt)]}e -Γqt/2 with the periodic functionf (t) = (sx,sy )∈N 2 *

One can estimate τc ≃ 1/Γq th where cq th = k B T . Then for excitation frequencies scaling as k B T , as in eq. (18), the condition ωq ≫ Γq, satisfied for a weakly-excited gas, implies ωq ≫ Γq th

The quantum hydrodynamics Hamiltonian can be supplemented by terms leading to a curved dispersion relation[25]. Except in the energy denominator, this brings a negligible correction to the phonon damping rate at low temperature.

On the side γ small and positive, Beliaev-Landau decay is allowed only for wavevectors q < mc[8γ/(5|η|)] 1/2[22]; for higher wave numbers, the quintic term eventually dominates and the dispersion relation becomes concave. This situation resembles that of superfluid liquid helium-4[7].

A similar calculation, but with bosons and periodic boundary conditions, was done in reference[40].


Farhang Radjai

Multicontact dynamics of granular systems

We present numerical results concerning force distributions and force-texture correlations in granular media. We show that two modes of stress transmission coexist in a granular assembly, and discuss the robustness of this behavior with respect to material parameters, boundary conditions, and simulation methods.

Introduction

Granular materials are composed of macroscopic particles that interact only through contacts. Although non cohesive granular materials exhibit fluid-like and solid-like states, they behave in a very different way from molecular solids and fluids. Recent active research has uncovered interesting and often counterintuitive effects in the dynamics of granular media, and has raised deep questions as to the relevance of familiar concepts to this field [START_REF]Excellent reviews of the state of art can be found, for example[END_REF].

The aim of this short contribution is to discuss the origins and some important features of multicontact states of granular systems, i.e. those states that involve a network of contacts through which the momenta propagate more efficiently than by the motion of individual particles.

Multicontact states

Fundamentally, three microscopic features play a key role in the dynamics of granular media: mutual exclusion of particles, dissipative nature of interactions due to plastic deformation during collisions or sliding friction, and dynamically-induced randomness as the only source of thermodynamic disorder that sets by itself the energy scale. As a consequence of these microscopic features, arguments based on entropy may not work any more. An open dissipative system can in principle self-organize and give rise to structures. In fact, dense clusters appear even in strongly excited granular systems [START_REF] Goldhirsch | [END_REF]. This process may lead to the collapse of particles into multicontact sub-structures, where the interparticle contacts form a connected network through which the momenta can propagate over distances far larger than the particle size [3]. It is obvious that the evolution of such structures can not be described in terms of a sequence of binary collisions and thus thermodynamic description and any event-driven algorithm for their simulation breaks down. Each collision in the multicontact state is a multiple collision due to the propagation of momenta in the contact network.

The simplest example of a fully multicontact system is a granular packing in quasi-static flow or simply in static equilibrium. In traditional fields, such as soil mechanics, a longstanding effort has been spent in order to rationalize the mechanical response of granular materials to shear for specified boundary con-ditions. Elasto-plastic models describe in many respects the strain-stress behavior of granular materials in quasi-static flow. This macroscopic approach deals with granular medium as a continuum whose mechanical behavior is described in terms of a stress tensor, a strain-rate tensor and some internal variables. In the 60s, however, two important discoveries revived interest in the microscopic aspects of granular materials:

(1) The contact network is generically anisotropic.

This means that the contact normals are not randomly directed in space. In a properly sheared packing, the density of contacts E(θ) with direction θ is usually well fitted by a truncated Fourier expansion [4]:

E(θ) = c π 1 + A cos 2(θ -θ c ) , ( 1
)
where c is the total number of contacts, the parameter A represents the amplitude of anisotropy, and θ c is the direction for which the maximum of E is reached. Starting a system with an initially isotropic texture (A = 0), A increases with the shear strain and θ c coincides with the major principal direction of the strain-rate tensor. Fig. 1 shows a polar diagram of E for a system of 4070 particles subject to simple shear. The velocity field is displayed in Fig. 2. (2) In contrast to the highly uniform density of a close-packed assembly, the distribution of contact forces is highly heterogeneous [START_REF] Dantu | Proc. 4th Int. Conf. Soil Mechanics and Foundations Engineering[END_REF]. The forces can be as large as six times the average force in static equilibrium and nearly 60% of contacts carry a force below the average force [START_REF] Radjai | [END_REF]; see Fig. 3. These observations hint at a physics very different from that of dilute systems and, they also suggest that the mechanical state of the system may be governed by several internal variables associated with the texture and the modes of force transmission inside the system.

Numerical methods

For the investigation of multicontact states, we performed extensive numerical simulations of systems containing as large as 4000 circular particles. For these simulations, we used two different methods: contact dynamics (CD) and molecular dynamics (MD). The CD method prescribes two contact laws [7,[START_REF] Jean | Mechanics of Geometrical Interfaces[END_REF]:

(1) Unilaterallity condition: If the separation velocity v n of two particles is positive, then the normal force N is zero. If, on the other hand, v n is zero, i.e. the two particles stay in contact, then N can have a positive indefinitely large value so as to prevent interpenetration. This is shown as a graph, Signorini's graph, in Fig. 4(a). (2) Coulomb's friction law: If the sliding velocity v t is nonzero, then the friction force T resists sliding and its value is given by the coefficient of friction μ times the normal force N . If, on the other hand, v t is zero (non-sliding contact), then T can take any value in the interval [-μN, μN]. This is shown in Fig. 3(b) as a graph. Both Signorini's condition and Coulomb's law are nonsmooth in the sense that the two conjugate variables {v n , N} or {v t , T } belong to a continuous set of acceptable values which can not be represented as a mathematical function. It can be shown that dynamics removes locally this degeneracy and allows to calculate all forces and velocities with no resort to any regularization trick [7][START_REF] Jean | Mechanics of Geometrical Interfaces[END_REF][START_REF] Radjai | Physics of Granular Media[END_REF]. In the MD method, "regularized" forms of the above laws are implemented [START_REF] Schäfer | [END_REF]. In the simplest MD algorithms, the vertical branches of the above graphs (v n = 0 and v t = 0) are replaced by steep lines.

In this way, the main physical input in the two methods is essentially the same. Both methods have been applied with much success for the investigation of granular phenomena. We note, however, that the CD method does not imply a fine time resolution and is thus faster than the MD method. Moreover, it allows for a more precise handling of friction than in the MD method. On the other hand, the CD method does not prescribe the elasticity of contacts (particles are infinitely stiff) and hence cannot be used for the investigation of the phenomena, such as sound propagation, that involve the deformation of particles. The most significant results of our simulations can be summarized around two main topics: statistical distribution of forces and force-texture correlations.

Force distributions

Our simulations by MD and CD methods show that the statistical distribution P N of normal forces is generically a power law, with a negative exponent, for the forces below the mean normal force N , and a decreasing exponential law for larger forces [START_REF] Radjai | [END_REF][START_REF] Radjai | Powders and Grains 97[END_REF]:

P N (ξ ) = kξ -α ξ < 1, ke β(1-ξ ) ξ > 1, (2)
where ξ = N/ N is the force normalized by the mean force and k is the normalization factor. One example is shown in Fig. 5. The distribution P T (T ) of tangential forces has the same form as Eq. ( 2) with different exponents α and β . One might write the distribution law of forces as an exponentially decreasing function with a power-law prefactor, as well. This alternative writing, however, does not fit the transition part between the two behaviors better than Eq. ( 2). The writing of the force distribution in the form of Eq. ( 2), which completely separates the two parts of the distribution, is motivated by the fact that, as we shall see below, the contacts that carry a force below the mean force, to which we will refer as "weak contacts" in the following, do not play the same role with respect to the overall resistance of the system to shear as the contacts carrying a force larger than the mean force ("strong contacts"). As the average force N = ∞ 0 NP N (N) dN is the point of separation between the two parts of the distribution, the exponents α and β are related by β 2 = (1α)(2α). These exponents change only with the level η of kinetic agitation and the average coordination number. α increases from zero with η. In a quasistatic flow, the distribution of weak forces is nearly uniform, i.e. α 0 and β 1.4 (for tangential forces α 0.3 and β 1). Interestingly, our numerical simulations show the same distribution (2) also in 3D packings with exponents α 0 and β 1.4 at static equilibrium. These exponents change very slightly with the coefficient of friction. The size-dispersion of particles does not seem to influence the force distributions either. This distribution with almost the same values of exponents has been recently confirmed by careful experiments by means of the carbon paper technique for forces of a granular packing at the contacts with the walls of its container [START_REF] Mueth | Force distribution in a granular medium[END_REF].

The distribution P N of normal forces in Eq. ( 2) plays the same role with respect to the static pressure of a granular system as the Maxwell-Boltzmann distribution of particle velocities in a gas with respect to the kinetic pressure. The two distributions have the same exponential tail, but the important difference between the two distributions is that in P N the weak forces are at least as frequent as the average force.

Force-texture correlations

In the above statistical analysis of contact forces, we considered all forces independently of directions of the contacts through which they are transmitted. There is, however, an interesting correlation between forces and contact directions, which determine the anisotropy of the texture. A simple way to study this correlation is Fig. 6. Amplitude of anisotropy A of the ξ -network as a function of the force cutoff ξ in the system of Fig. 2; see text for the definition of the ξ -network.

to consider the subset of contacts which carry a force below a given cutoff ξ . We shall refer to this subset as the "ξ -network". The variation of a quantity such as the anisotropy evaluated for the "ξ -network" as ξ is varied from 0 to the maximal force in the system, allows then to estimate its correlation with the contact force.

Fig. 6 shows the amplitude of anisotropy A as a function of ξ in a simply sheared system at a given moment of deformation. Surprisingly, the direction of anisotropy is orthogonal to the major principal direction of the strain-rate tensor (negative values) as long as ξ is below the average force [START_REF] Radjai | [END_REF]. In other words, the direction of the anisotropy of the weak network is orthogonal to the axis of compression, whereas that of the strong network is parallel.

It is also possible to evaluate the contribution of the ξ -network to the overall stress inside the system. We found that the shear stress Q for ξ < 1 is negligibly small compared to the total deviation load sustained by the system. Those forces only contribute 28% of the average pressure in the medium. This means that the weak network behaves essentially like an interstitial liquid. The whole shear stress is sustained by the strong network. We also found that the whole dissipation due to sliding friction takes place inside the weak network, while all contacts in the strong network are non-sliding.

These properties, which are well reproduced both by CD and MD simulations, indicate a bimodal transmission of stress in multicontact systems. The texture is composed of two phases, a weak phase and a strong phase, that are intimately correlated. This means that for the description of stress transmission in a granular medium, two stress tensors are needed.

Conclusion

We discussed features of the multicontact states of granular systems at variance with the dilute states. After a brief account of the major physical inputs in numerical simulations of granular systems, we presented two sets of numerical observations concerning the statistical distribution of contact forces and the forcetexture correlations. These observations suggest that the texture of a granular material is composed of two complementary phases and raise the problem of a theoretical understanding of the origins of this bimodal behavior.

Fig. 1 .

 1 Fig. 1. Polar diagram of the number of contacts E(θ) as a function of their direction θ in the system of Fig. 2. The solid line is a fit according to Eq. (1).

Fig. 2 .

 2 Fig. 2. Velocity field during simple shearing of a system of 4070 particles.

Fig. 4 .

 4 Fig. 4. (a) Signorini's graph relating normal force N and separation velocity v n at a contact between two particles; (b) Coulomb's graph relating friction force T and sliding velocity v t at a contact.

Fig. 5 .

 5 Fig. 5. Probability density P N of normal forces ξ normalized with respect to the mean force.


Alfredo Taboada

Nicolas Estrada

Farhang Radjai

Farhang Radjaı

Additive Decomposition of Shear Strength in Cohesive Granular Media from Grain-Scale Interactions

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Cohesive granular materials are of particular interest to various fields of science and engineering such as soil mechanics, geology, and processing of fine powders and grains. Although cohesive interactions can have very different physicochemical origins, their common denominator is to freeze, up to a threshold in force or torque, the relative degrees of freedom between grains. Besides the Coulomb friction, which is also present in cohesionless materials as the main source of shear strength, well-known examples of cohesive bonding are mineral cementation, capillarity, and van der Waals forces in dense or finegrained materials [START_REF] Santamarina | Soil Behavior and Soil Ground Construction, A Symposium in Honor of Charles C. Ladd[END_REF][START_REF] Kohonen | [END_REF].

From a grain-scale viewpoint, an interesting and largely open issue is how local force thresholds scale up to macroscopic strength properties (friction angle and cohesion). However, the influence of local thresholds can hardly be investigated by direct experiments. The thresholds are material dependent and therefore cannot be isolated and varied methodically. A unique opportunity is offered by capillary cohesion which can be controlled by liquid content [3][4][5]. For example, several studies have focused on the effect of the adhesion force on the maximum angle of stability [6].

On the other hand, discrete element simulations provide a suitable tool to investigate large scale properties from local parameters. Several numerical studies have been reported focusing mostly on the compaction and flow properties of cohesive granular materials [7][8][9][10][11]. We are aware of few numerical studies dealing in a systematic manner with the macroscopic strength properties [12,13].

In this Letter, we present a detailed analysis of the shear strength in cohesive packs simulated by the contact dynamics method. Parametric study leads to an appealing picture depicting the precise role of each bond parameter in the cohesive behavior. The model of cohesive bonding in the simulations was designed to mimic cohesion by cementation as observed in rocks and soils. It incorporates the following features : (1) cohesive interactions are gov-erned by shear force and torque thresholds that are proportional to the normal force, as well as a tensile force threshold [Fig. 1(a)]; (2) cohesive bonding occurs inside a small zone between grains, representing in a way the cementing material [Fig. 1(b)]; (3) contact rupture leads to an irreversible loss of tensile strength (debonding) and the contact turns to purely frictional behavior.

In the following, we attribute positive values to compressive forces. By definition, the tensile force threshold ÿF a is the largest tensile force that can be supported by a pair of grains along the contact normal. It is also useful to introduce a tensile stress a F a =' where ' is the mean diameter of the grains in contact. For sliding friction, we use Coulomb's law in which the shear force threshold T max is given by

T max s N F a ; ( 1
)
where s is the coefficient of sliding friction and N is the normal force. The offset s F a can be interpreted as the effect of adhesion forces increasing the contact area and thus the resistance to sliding. Full cementation requires also torque transmission at the contact points. This is equivalent to a rolling resistance with possibly a torque threshold ÿ max . We assume that ÿ max obeys a Coulomb-like law [START_REF] Radjaı | Continuous and Discontinuous Modelling of Cohesive Frictional Materials[END_REF] ÿ max r 'N F a ;

(

) 2
where r is the coefficient of rolling friction. Obviously, the length scales involved in the description of rolling resistance are those of the cohesive bond. In grain-scale numerical simulations, however, it is not desirable to in- troduce small length scales compared to grain size. In Eq. ( 2), the presence of ' is simply meant to make r dimensionless. At the particle scale, s , r , and a represent the microscopic parameters of the model that express at this scale the underlying physicochemical properties of the contact interactions. The kinetics of contact loss and gain is governed by a creation length and an elliptic zone of weak extension located between the grains; see Fig. 1(b). The cohesive bonds are created initially between all grains such that the gap, i.e., the distance between their two closest material points P i and P j , is below . A cohesive bond persists as long as the point P j is located inside a small elliptical zone fixed to disk i. The parameter and the size of the elliptic zone simply control the scale of strain softening and they do not influence the results presented below as far as they remain small compared to the average particle size.

T max max , 1 µ l r N µ s,
The simulations are based on the contact dynamics method [START_REF] Moreau | [END_REF][START_REF] Jean | Mechanics of Geometrical Interfaces[END_REF]. Specific implementation for cohesionless materials can be found elsewhere [START_REF] Taboada | [END_REF]. Numerical samples are composed of 5000 disks with diameters uniformly distributed by volume fraction between 1 m and 4 m. The gravity is zero and, as for most geomaterials, the restitution coefficients are set to zero. The packing is built according to simple geometrical rules inside a rectangular cell and then uniaxially compacted with zero friction in order to produce a dense disordered configuration [START_REF] Taboada | [END_REF]. We get a volume fraction 0:84 and a coordination number Z 3:9.

Before shearing, the two vertical walls are removed and the confining pressure is applied directly on the disks located at the boundaries of the sample [START_REF] Taboada | [END_REF]. The packing is sheared as in a biaxial test by downward displacement of the top wall for different values of the confining pressure 3 . Simulations were performed for the following set of contact parameters: s 2 0:01; 0:1; . . . ; 0:6, r 2 0; 0:1; 0:25, a 2 0; 1; 2; 4; 10; 20 MPa.

Figure 2 displays typical stress-strain and volumetric strain plots at different levels of confinement. The stress ratio 1 = 3 , where 1 is the stress along the compression axis, first rises to a peak in a rigid fashion and then gradually declines with the shear strain q 1 -3 , where 1 , 3 are major and minor cumulative strains. The peak stress ratio is a decreasing function of the confining pressure, as also observed in soil mechanics [START_REF] Wood | Soil Behaviour and Critical State Soil Mechanics[END_REF]. This behavior reflects a decrease of the effective level of cohesion a = 3 . In all tests, the stress ratio declines to the same constant residual strength for q > 0:1, corresponding to a permanent loss of cohesion along the shear bands in this state. The plots of the volumetric strain p 1 3 show dilatant behavior due to the initially high solid fraction.

Our simulation data are fit well to the Mohr-Coulomb model. In other words, the failure envelope of Mohr circles constructed in the stress space (, ), where and are, respectively, the normal and shear stresses, is a straight line (see inset to Fig. 5). For each combination of bond pa-rameters, we determined the angle of friction P (the slope of the failure line) and the cohesion C P (intersection with the axis) at the peak state. We also determined the angle of friction R for the residual state (the cohesion C R being zero in this state), as well as the dilatancy angle sin ÿ1 p = q at the peak state.

Figure 3 shows P and R as a function of the sliding friction angle s tan ÿ1 s for three values of the coefficient of rolling friction r . Each symbol and the corresponding error bar represent the mean and extreme values of P and R for six different values of the tensile strength a . A remarkable feature of these plots is that the macroscopic friction angles P and R are independent of the tensile strength as the variability is small around the mean values. Both P and R increase with r .

As s tends to zero, both angles of friction tend to a finite value 0 lying between 4 and 7 . This shows that contact friction is not the only source of frictional behavior at the macroscopic scale. Energy dissipation in this ''frictionless'' limit can take place only as a result of inelastic collisions through unstable particle rearrangements. Dissipation by inelastic collisions being proportional to the mean stress, its effect appears as a Coulomb-like friction law with an apparent angle of friction 0 .

The residual angle of friction R increases with s from 0 and saturates to a value 1 that depends only on the rolling friction r . The data are well fit to an exponential function:

R 0 1 ÿ 0 1 ÿ e ÿ s ;

(3) with ' 0:1. Remark that, within statistical precision, is independent of r . The nonlinear dependence of R on s is consistent with other simulations reported in the past for r 0 [START_REF] Oger | [END_REF]20]. The saturation of R suggests that, as s increases, a transition occurs in the grain-scale phenomena underlying macroscopic friction. In particular, sliding and rolling appear to be the dominant deformation modes at low and high sliding frictions, respectively. It is a basic observation that the stress-strain behavior is strongly dependent on the initial value of the solid fraction. Only the so-called ''critical state,'' reached at large shear strains, is independent of the initial state [START_REF] Wood | Soil Behaviour and Critical State Soil Mechanics[END_REF]. This ''critical state'' is such that the stress ratio 1 = 3 is constant and the dilatancy angle is zero on average. Thus, the strength parameter R in the critical state should be considered as a material property.

On the other hand, the peak strength P reflects the initial compactness of the material: higher dilatancy and peak friction angles are expected for higher initial compactness of the material. To check this correlation directly from numerical data, we plot in Fig. 4(a) the dilatancy angle at the peak state as a function of the difference between peak and residual friction angles P ÿ R . We observe a nice linear dependence with a slope close to 1. This indicates that the difference between peak and residual friction angles is fully explained by dilatancy. The dilatancy angle for 0 corresponds to the macroscopic friction angle 0 for frictionless particles (see Fig. 3). Both, and , are nearly linear functions of s and are barely influenced by r [Fig. 4(b)]. Hence, with a good approximation, the dilatancy angle can be split into two parts:

0 0 k s ; (4)
with k ' 0:5. This remarkable correlation between friction and dilatancy angles, here verified for a cohesive granular material, is in agreement with Taylor's model proposed for cohesionless sand and known as stress-dilatancy relation [START_REF] Wood | Soil Behaviour and Critical State Soil Mechanics[END_REF]. We now consider the Coulomb cohesion C P with respect to the influence of bond parameters. For reasons that will become clear below, we focus on the ''theoretical'' tensile stress A C P = P obtained by extrapolating the Mohr-Coulomb envelope from the compressional regime to the tensile regime (see inset in Fig. 5) [21].

Figure 5 shows A as a function of the local tensile strength a . Each symbol and the corresponding error bar represent the mean and extreme values of A for the whole set of sliding and rolling frictions. We observe a linear dependence of the macroscopic tensile strength A on a : A ' a ;

(

with ' 1:4. Interestingly, A is independent of both sliding and rolling frictions. This feature may be explained by considering the stress on a plane perpendicular to the direction of extension and remarking that the friction forces and torques from contacts oriented along different directions cancel out along this direction [5,21]. This shows that the macroscopic strength parameters have different local origins. Local friction coefficients and inelastic collisions are responsible for the macroscopic friction, whereas the tensile strength at the contact scale seems to be the unique origin of macroscopic tensile strength.

The picture arising from this parametric study is both simple and rich. Equations ( 3) and ( 4) suggest an additive partition of the friction angles as illustrated in Fig. 6. In particular, the peak friction angle P is expressed as a sum of three terms of different origins:

P 0 : (6)
The offset 0 is a purely collisional contribution independent of rolling friction and tensile strength. The term R ÿ 0 represents a purely frictional contribution. It is an exponential function of s and saturates to 1 ÿ 0 which increases with rolling friction r . The term ÿ 0 represents the contribution of dilatancy to the shear strength and, according to Eq. ( 4), it is an approximately linear function of s .

In the same way, from Eqs. ( 5) and ( 6), the Coulomb cohesion can be expressed by a simple form:

C P a P a tan 0 k s : (7)
This writing shows that the Coulomb cohesion is an increasing function of all bond parameters, the basic contribution coming from the tensile strength as a multiplicative factor. It is noteworthy that the cohesion in the limit s r 0 appears here to be a consequence of the collisional friction angle 0 , i.e., C P a tan 0 . In the absence of this term, the packing would behave as a cohesionless material independently of the local tensile strength a . The constant is likely to depend strongly on the distribution of cohesive bonds in the initial state. In particular, the fraction of cohesive bonds decays in inverse proportion to the solid fraction, suggesting that will decrease dramatically in loose packings. The same is true for the parameter k relating to s . The dilatancy varies with the initial value of the solid fraction so that we expect lower values of k down to zero when the solid fraction is decreased to the critical state solid fraction.

The well-defined partition of the macroscopic angles of friction for a wide range of sliding and rolling friction coefficients at the contact scale in cohesive granular media is a key finding of the present investigation. In view of the large number of independent numerical tests performed, this result provides a solid basis for further insight into the behavior of cohesive granular media. Along these lines, a more detailed analysis of the respective roles of the sliding and rolling friction with respect to the frictional term merits further investigation. It is also important to consider less compact configurations than those studied in this work. Finally, the evolution of granular microstructure up to and beyond the peak state is a crucial issue for understanding the scale up of bond interactions. On more general grounds, some quantitative correlations discussed in this Letter might turn out to be closely related to the underlying model of cohesive bonding. But, we believe that the partition of the Mohr-Coulomb strength parameters P , R , and C P into collisional, frictional, and dilational parts, and the role of the local tensile strength as the unique source of the macroscopic tensile strength, are robust features of cohesive granular materials with respect to the model of cohesive bonding.

FIG. 1 .

 1 FIG. 1. (a) Shear and torque failure envelopes and (b) cohesive bond.

FIG. 2 (

 2 FIG.2(color online). Shear stress (a) and volumetric strain (b) as a function of shear strain at various confinement levels.

FIG. 4 .

 4 FIG. 4. (a) Dilatancy angle as a function of ; (b) and as a function of the angle of sliding friction s ; see the text for definitions.

We thank K.-J. Chang for contribution to numerical simulations and A. Delplanque for technical help.