Elias M Neeman
email: elias.neeman@univ-lille.fr

Juan Ramón

Avilés Moreno

Thérèse R Huet

Gas-phase hydration of nopinone: the interplay between theoretical methods and experiments unveils the conformational landscape

The structure of microsolvated nopinone formed in the supersonic jet expansion is investigated in the gas phase. The rotational spectra of nopinone•••(H2O)n (n=1,2,3) were analysed by means of Fourier transform microwave spectroscopy. In the present study, three monohydrates, two dihydrates as well as two trihydrates were observed and characterized. The observed structures are the lowest energy conformers predicted by quantum chemical calculations. In all the observed hydrates of nopinone, water was found to be linked to the ketone group (C=O) with a strong hydrogen bond (ONOP•••HW) and finishing with a dispersive one (OW•••HNOP). The structure of nopinone was found to alter the structure of water dimer and water trimer, which make nopinone to be surrounding with a chain of water molecules. A remarkable decreasing in the H-bonding length was observed when the number of attached water is increased. Different DFT and ab initio calculations at the equilibrium structure allowed the identification of the observed conformers. Evaluation of the B3LYP-D3 and ωB97X-D results revealed deficiencies to reproduce the structure of one observed monohydrated structure while MP2 and M06-2X reproduce all the three observed structures. A comparison with similar bicyclic ketone highlights how a small change in the bicyclic ring leads to different effects in the microsolvatation of Biogenic VOCs. This study presents the first step of molecular aggregation to understand the atmospheric formation of aerosols at the molecular scale.

Introduction

The volatile organic compounds (VOCs) are biogenic and anthropogenic particles suspended in the earth's atmosphere. Their tropospheric oxidations with ozone, nitrogen oxides and hydroxyl radicals contribute to the formation of atmospheric aerosols which represent a major environmental problem. They degrade visibility and affect human health. 1,2 They also directly or indirectly influence the climate by absorbing and reflecting solar radiation, or by modifying cloud formation.

The aerosol formation processes are still not well understood. [START_REF] Kanakidou | Organic aerosol and global climate modelling: a review[END_REF][START_REF] Vander Wall | [END_REF] The pre-nucleation clusters process is the first step. A molecular complex forms the critical nucleus and then grows to a detectable size. 3,6 Hydrogen-bond interactions are one of the several elements responsible for the formation of atmospheric relevant species. 7 Several experimental studies 8-10 were performed in the laboratory to understand the formation of secondary organic aerosols (SOA) from VOCs and its relation with water. Those studies have shown that SOA formation is enhanced by the water presence. [8][9][10] Actually, the aerosol formation process is directly related to relative humidity (RH) and the aerosol quantity increases with the increasing of the RH. [8][9][10] Moreover several chemical reactions in the atmosphere start by the formation of non-covalent complexes. So far Hbonding is one of the most important non-covalent interactions, which plays also a critical role in many biological and chemical processes. Please do not adjust margins

Please do not adjust margins

This information evidences the relevance of studying Hbonding interactions between water and oxygenated VOCs. High-resolution Fourier transform microwave (FTMW) spectroscopy combined with theoretical calculations is a powerful tool to study complex systems in the gas phase. This combination has allowed the identification of flexible conformational landscape [13][14][15] and the favourable hydration sites. [16][17][18][19][20][21][22][23] Recently, microsolvated clusters, which present the first step of hydration have been object of experimental 27 In a

previous study, we have demonstrated that during the oxidation process the structure of the precursor β-pinene and of the oxidized product Nopinone (labelled as NOP from now on) is maintained in the gas phase. 28 Different theoretical methods were performed in order to guide the identification of the observed hydrates species. This study is relevant to understand the hydration processes at the molecular scale, as well as its contribution to the formation of secondary organic aerosols.

Methods

Quantum chemical calculations

Quantum chemical calculations were performed using the Gaussian 09 [START_REF] Frisch | Gaussian 09[END_REF] and 16 [START_REF] Frisch | [END_REF] software package. All the conformers were optimized using ab initio calculations at MP2 (Møller-Plesset second order perturbation theory) level of the theory 32 and ωB97X-D, 33,34 DFT, with the Pople split-valence triple-zeta basis set augmented with diffuse and polarization functions on all atoms (the 6-311++G(d,p) basis set). Also the two DFT methods B3LYP 35,36 including Grimme's dispersion correction (D3), 37 and the M06-2X 38,39 were used to optimize the structure of the possible structure of NOP-water complexes with the 6-311++G(2df,p) basis set.

The equilibrium rotational constants (A, B, C) and the quartic centrifugal distortion parameters (D J , D JK , D K , d 1 , d 2 ) as well as the electric dipole moment components have been calculated for the lowest energy conformers. Their values are presented in Tables 1-3 and compared to the obtained experimental parameters. All calculated conformers were optimized and confirmed to be true minima without negative harmonic vibrational frequencies. Three monohydrates were optimized by MP2 and M06-2X methods, see Figures 1 and2. Surprisingly, only two monohydrates were optimized using B3LYP-D3 and ωB97X-D methods, the conformer 1w(ii) was Please do not adjust margins not found because it converges to conformer 1w(iii). Six dihydrates and six trihydrates were optimized by the theoretical methods (MP2, M06-2X, B3LYP-D3 and ωB97X-D).

Only the observed conformers are presented in Figure 2, Figure 3 and Figure 4. The molecular parameters of the nonobserved dihydrates and trihydrates conformers of NOP•••water are reported in the Supplementary Information. An analysis of the relevant NBOs involved in the intermolecular charge transfer responsible of the relative stability of the lowest energy conformer of nopinone•••water, 1w(i) 2w(i) and 3w(i), was also performed 40 using M06-2X/6-311++G(2df,p). The same method was used to plot the map of the noncovalent interactions (NCI) 41 involved in the stabilisation of the observed conformers. This analysis was carried out using the Multiwfn software. 42 Further structure optimization using the MN15-L/6-311++G(d,p) 43 was performed, to better describe the structure of the 1w(ii) conformer. This method provides better description for noncovalent interaction and allow a better description of the rotational constants of conformer 1w(ii).

Experimental

The jet-cooled pure rotational (2-18 GHz) spectra of NOPwater complexes were recorded (see Figure 5) using Lille FP-FTMW spectrometer 44,45 coupled to supersonic jet. Nopinone (98%) was purchased as liquid samples from Sigma-Aldrich and was used without any further purification. The supersonic expansion was generated by expansion of neon (Ne) at stagnation pressure of 4 bar. Nopinone was placed in a nozzle and heated at about 343 K. Water was inserted in the gas line just before the nozzle. Then the mixture was introduced into the Fabry-Pérot cavity along the optical axis, through a 1 mm diameter pinhole of a pulsed nozzle at a repetition rate of 1.5 Hz. The rotational temperature (T Rot ) of the supersonic jet is estimated to be about a few Kelvin. Microwave power pulses of 2 μs duration were used to polarize the molecules. The Free Induction Decay (FID) signal was recorded using heterodyne detection at 30 MHz and digitized at a repetition rate of 120 MHz. FID signals were accumulated between 50 and 1000 times, depending on the line intensity to obtain a good signalto-noise ratio. After a fast Fourier transformation of the time domain signals, lines were observed as Doppler doublets (≈10 kHz full width at half maximum, FWHM) due to the coaxial arrangement of the jet and the Fabry-Pérot cavity. Each resonance frequency was measured as the average frequency of the two Doppler components. The spectral resolution is depending on the number of recorded points. The frequency grid was set to 1.84 kHz, which was found sufficient since the Doppler width of the lines should be of several kHz.

Results and discussion

The main characteristic of rotational spectroscopy is the capability to provide accurate spectroscopic rotational parameters and when compared with those obtained by theoretical calculation leads to an unambiguous identification of the observed conformers. Theoretical calculations are used to help the analysis of the rotational spectra. According to those calculations, all the predicted conformers of NOP-water are asymmetric rotors and exhibit at least of one non-zero dipole moment component. The rotational spectra of all A wide scan has been performed to detect the possible conformers surviving in the adiabatic supersonic expansion. As shown in Figure 5, strong lines were assigned to complexes containing one water molecule; less intense lines were observed for cluster containing two water molecules and finally the weakest two spectra were assigned to the trihydrated conformers. For all observed conformers, transitions were measured experimentally at high-resolution and fitted to the experimental precision using Pickett's program 46 with a Watson's semi-rigid rotor Hamiltonian [START_REF] Watson | Vibrational spectra and structure[END_REF] in the representation and in the -reduction.

As shown in Table 1, three different sets of experimental parameters have been assigned to the monohydrates, a total of 82, 59 and 48 transitions have been measured at high resolution. The calculated parameters predicted by MP2/6-311++G(d,p) and M06-2X/6-311++G(2df,p) calculations, are in good agreement with the experimental results.

For the dihydrates 74 and 39 lines have been fitted together to the two dihydrated conformers of NOP-(H 2 O) 2 . In the same way, two sets of rotational constants have been identified for the trihydrates by measuring 60 and 66 transitions for those species. The results are shown in Tables 2 and3.

The confirmation of the identity of the observed conformers can be done by comparing the calculated rotational constants to those observed experimentally. Moreover, for the observed clusters, the calculated dipole moments should be in good agreement with observed transition types, which led to an unambiguous identification of the observed clusters.

For the monohydrates it is evident that all the predicted conformers by MP2 and M06-2X have been observed. The calculated dipole moments are in good agreements with those predicted by MP2/6-311++G(d,p) (see SI). Unfortunately, despite the good parameters predicted by B3LYP-D3 and ωB97X-D for the 1w(i) and 1w(iii) (see SI), these two methods fail to predict the conformer 1w(ii) which exhibits an intense rotational spectrum. The complete comparisons of the experimental parameter to all the theoretical calculations are provided in the SI in the tables S1-S7. In all the monohydrates, water is attached to NOP by a strong H-bonding interaction (O w -H w •••O NOP ) and a weaker dispersive interaction between O w and H NOP (see Fig. 2). Interestingly, a good agreement between experimental results and MP2 calculations for 1w(i) and 1w(iii) but for the 1w(ii) conformer the error on the A rotational constant is quite high. The problem of such a discrepancy between experiment and theoretical calculation may become from the incomplete description of the non-covalent interactions between water oxygen and NOP hydrogen involved in the stabilization of the conformer 1w(ii). It has been reported that this MN15-L method 43 predicts better the weak non-covalent interaction.

To test if any structural enhancement could be obtained using this method, we performed additional calculation using the MN15-L/6-311++G(d,p). Surprisingly, the calculated parameters by MN15-L/6-311++G(d,p) are in good agreement with those obtained experimentally. The discrepancy between the experimental (1442.2 MHz) and theoretical value at MN15-L (1534 MHz) of the A constant decreases significantly. The obtained rotational constants are compared to the experimental values in the supporting information (Table S8). Another approach could be used by modifying the ab initio structure of 1w(ii) to better match the observed experimental constants. Since the rotational constants depend directly on the mass distribution of the atoms around the rotational axis (a,b,c). In order to decrease the error on the A constant, the position of the water molecule was modified by scanning both the dihedral angle O-O-C-C from 35.9° to 67.7° and the angle O-O-C from 104.4° to 119° for the conformer 1w(ii). With this scan, the error on the A constant was reduced remarkably Please do not adjust margins which allows to better describe the observed structure of 1w(ii) monohydrate. In the modified structure, water molecule was found to be closer to the methyl group. This position allows the water molecule to take advantage of the stronger dispersive interaction offered by the methyl group. The modified structure is presented in the supporting information (Table S8).

For the dihydrated clusters, six different structures have been calculated by all the used theoretical methods. The identification of the lowest energy conformer 2w(i) is directly confirmed thanks to the good agreement between the observed and calculated rotational constants. The second set of the observed rotational constants matches both conformers 2w(ii) and 2w(iii) (see SI), but their dipole moment components are different. Therefore, the assignment of the experimental data to the conformer 2w(ii) is straightforward due to the non-observation of any c-type transitions that should be observed for the conformer 2w(iii).

As for the monohydrated cluster, each dihydrate has the first water molecule forming a strong H-bonding with the NOP oxygen (O w -H w •••O NOP ). Likewise, the oxygen of the second water for all dihydrates is stabilized in the mid-position between two hydrogens of NOP molecules by a bifurcated dispersive-bond. The structure of the two observed conformers is shown in Figure 3.

Concerning the trihydrated clusters, among the six calculated species (see the SI), only the two lowest energy conformers have been detected (see Figure 4 The oxidation of β-pinene to NOP does not affect the bicycle structure. 28 Although NOP affects the structure of the pure water dimer and trimer. Indeed, intermolecular forces act in those clusters (H-bonding and dispersive interactions) and led to their deformation. For the dihydrates, the hydrogen between the two water molecules in 2w(i) and 2w(ii) complexes is shortened by 0.1 Å compared to the water dimer and the dihedral angle (∠H w2 -O w2 -H w2 -O w1 ) value changes to 150.5° and -159.1° for both 2w(i) and 2w(ii), respectively, instead of 180° for the pure water dimer. For the trihydrate, the structure of those complexed to NOP is a chain and not a cycle but preserving a quasi-cyclic structure opening by around (∠O w1 -O w2 -O w3 ) 40° more than in the pure water trimer. The arrangement of the water molecules in those clusters is characterized by a remarkable decreasing in the H-bonding length. We can also observe that H-bond length decreases when the number of water molecule increases. However, the presence of three water molecules promotes the cooperativity between H-bonds of the water molecules over stabilizing the species. As the three water molecules are not forming a close cycle, the H-bond interaction (O w -H w •••O NOP ) together with the dispersive one break that cycle and oblige the water molecules to adapt their structure on NOP.

The stability of the energy clusters of NOP•••water, 1w(i), 2w(i) and 3w(i), can be traced back to a large extent to the strong inter-molecular interactions achieved in those complexes. Natural bond orbital (NBO) analysis provides an intuitive framework to rationalize the transfer of electronic charge associated with this interaction. 40 The NBO analysis were performed at M06-2X/6-311++G(2df,p) method for the lowest energy conformers NOP-water complexes (1w(i), 2w(i) and 3w(i)). The most relevant electronic transfers are shown in Another interesting analysis was performed using Non-Covalent Interactions (NCI) method 41 in order to visualize the interactions involved in the stabilization of the observed species of NOP•••water complexes. The NCI analysis is in good agreement with the NBO analysis and confirms the existence of a strong hydrogen bond and two additional weak hydrogen bonds interaction between water and NOP. As shown in the Figure 9 the dark blue iso-surface indicate strong hydrogen bonds. The weak attractive interactions are shown in the green iso-surface. These interactions play an important key role in the stabilization and the preference of the observed conformer of NOP•••water complexes. As can be observed in the Figure 9., for the trihydrates 3w(i) and 3w(ii) the map of the dispersive interaction is very important and involve both O w2 and O w3 participating directly to their stabilization. To plot the map of the interactions involved in the stabilisation of the observed conformer, noncovalent interaction (NCI) analysis was carried out using the Multiwfn software 42 The favourite hydration sites obtained in this study could also be compared to the favourite hydrogen atom abstraction obtained from a computational study [START_REF] Lewis | [END_REF] using high-level calculations in order to simulate the atmospheric oxidation of NOP with OH radical. Surprisingly, these hydrogen atom abstractions match with the favourite hydration sites found in our study and where the dispersive interactions stabilize the oxygen of the water molecule of the observed conformers.

Conclusions

In summary, the synergic combination of FP-FTMW spectrometer with theoretical calculations have led to the identification of the hydrated NOP structure up to three water molecules. The optimisation of monohydrate conformers with the B3LYP-D3 and ωB97X-D methods missed one monohydrate, while the MP2 and M06-2X methods reproduce qualitatively all the three observed monohydrates. For the 1w(ii) conformer the structure of water molecule obtained at MP2/6-311++G(d,p) was modified in order to better reproduce the experimental data. For the dihydrates and trihydrates the observed experimental structures have been perfectly predicted with all the theoretical methods performed in this study. The observed structures of the di-and trihydrate complexes are stabilized by strong inter-molecular interactions. The formation of the hydrogen bond with the NOP molecule alters the structure of the water dimer and trimer. The formation of hydrogen bond was also discussed by performing NBO and NCI analysis. In addition, this study emphasizes the importance of the molecular structure which gives rise to additional dispersive interactions and increase the number of the observed hydrates. This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 7

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These results could be useful as starting point to study atmospheric oxidation of BVOC's toward understanding the mechanism of the aerosol formation starting from the formation of H-bonding.

11 , 12 2

 1112 | J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xx

  the possibility to surround ketone molecules by a chain of three water molecules. For example, in the case of both verbenone (C 10 H 14 O, 4,6,6trimethylbicyclo[3.1.1]hept-3-en-2-one) and camphor, (C 10 H 16 O, 1,7,7-trimethylbicyclo[2.2.1]hepta-2-one) two monohydrates, two dihydrates and one trihydrate have been observed and characterized. In the case of alcohol monoterpenoid, fenchol, 13 only monohydrate cluster was observed. 14 Coming back to bicyclic ketones, it is interesting to evaluate how the change of the bicyclic ketone unit can affect the formation of the hydrated conformational behaviour: would the structure of ketone play an important role in the stabilization of hydrate structures, and how it contributes to the stabilization of further conformers? With this aim, we report on the analysis of nopinone-water clusters using impulse Fabry-Pérot Fourier-transform microwave spectroscopy technique coupled to a pulsed supersonic jet expansion (FP-FTMW spectrometer). Nopinone (C 9 H 14 O, 6,6dimethylbicyclo[3.1.1]heptan-2-one) represents the major oxidation product of β-pinene (C 10 H 16 , 6,6-dimethyl-2methylenebicyclo[3.1.1]heptane) in the atmosphere.

Figure 1 .

 1 Figure 1. Illustration of the monohydrate clusters, the position of NOP is fixed to help visualization. As it shown the position of water conformers 1w(ii) and 1w(iii) is close. This fact can explain the non-convergence of some DFT methods for those structures.

Figure 3 .

 3 Figure 3. Optimized geometries at MP2/6-311++G(d,p) of the observed lowest energy conformers of NOP•••(H2O)2. Bond length OH in (Å) and angle HOH in (°).

Figure 2 .

 2 Figure 2. Optimized geometries at MP2/6-311++G(d,p) of observed three lowest energy conformers of NOP•••H2O. Bond length OH in (Å) and angle HOH in (°).

Figure 4 .

 4 Figure 4. Optimized geometries at MP2/6-311++G(d,p) of the observed lowest energy conformers of NOP•••(H2O)3 .The water chain in the two conformers have a quasimirror structure. Bond length OH in (Å) and angle HOH in (°).

Figure 5 .

 5 Figure 5. A portion of the experimental spectrum showing rotational lines of all the NOP-water clusters. The low-resolution scan mode of the FP-FTMW spectrometer was used. Intensity is in arbitrary units.

  ), they have the smallest experimental signals. The structures of the trihydrated complex are stabilized by similar intermolecular interactions: a strong H-bond between the first water molecule and NOP (O w -H w •••O NOP ) and finishing with a dispersive interaction H w -O w •••H NOP , analogously to the other hydrates.

Figures 6 , 7 and 8 .Figure 6 .

 686 Figure 6. Most relevant NBOs involved in the inter-molecular charge transfer responsible of the relative stability of the lowest energy monohydrate 1w(i) calculated at the M06-2X/6-311++G(2df,p) level of theory.

Figure 8 .

 8 Figure 8. Most relevant NBOs involved in the inter-molecular charge transfer responsible of the relative stability of the lowest energy trihydrate 3w(i) calculated at the M06-2X/6-311++G(2df,p) level of theory.

Figure 7 .

 7 Figure 7. Most relevant NBOs involved in the inter-molecular charge transfer responsible of the relative stability of the lowest energy dihydrate 2w(i) calculated at the M06-2X/6-311++G(2df,p) level of theory.

Figure 9 .

 9 Figure 9. NCI plots showing the most relevant interactions responsible of the relative stability of the observed conformers of NOP•••(H2O)n (n=1,2,3).

Table 1 .

 1 Experimental spectroscopic parameters for the three observed monohydrates of NOP compared with those predicted from ab initio MP2/6-311++G(d,p)

	and DFT M06-2X/6-311++G(2df,p) calculations.
		1w(i)			1w(ii)	1w(iii)
	Parameters a	Exp.	Calc. e	Exp.	Calc. e	Exp.	Calc. e
	A	1322.42948(26)	1328│1335	1442.1291(18)	1637│1578	1748.772(10)	1758│1792
	B	1005.490818(68)	1038│1076	768.49995(19)	769│836	705.91270(13)	708│716
	C	756.541470(46)	774│797	719.00172(15)	720│772	669.09363(11)	671│681
	DJ	0.30252(48)	0.23643│0.2093	1.93749(56)	3.7413│0.2091	0.76314(25)	0.6470│2.55
	DJK	-0.0376(24)	-0.1709│-0.1954	0.0808(30)	-18.09│-0.5855	-3.0402(37)	-2.135│-8.621
	DK	-0.166(16)	-0.025│0.059	-	26.1│0.893	-	8.287│34.40
	d1	-0.08892(24)	-0.0424│0.0587	-0.67501(73)	-0.5419│-0.033	-0.05412(35)	0.0587│-0.220
	d2	-0.00518(20)	0.01941│0.0188	-0.07650(43)	0.00956│-0.00005	0.00670(20)	-1.115│0.030
	N b	82 f	-	59 f	-	48 g	-
	σ c	1.2	-	2.0	-	1.3	-
	ΔE d	-	0	-	3.5	-	3.8

a A, B, C are the rotational constants given in MHz; DJ, DJk, Dk, d1 and d2 are the quartic centrifugal distortion given in kHz; b Number of fitted transitions; c rms deviation of the fit in kHz; d ΔE represents the relative electronic energies in kJ mol -1 with respect to the global minima calculated at MP2/6-311++G(d,p); e calculated parameters at MP2/6-311++G(d,p) │ M06-2X/6-311++G(2df,p); f a-and b-type transitions; g a-type transitions.

Table 2 .

 2 Experimental spectroscopic parameters for the two observed dihydrates of NOP compared with those predicted from ab initio MP2/6-311++G(d,p) calculations.

		2w(i)		2w(ii)
	Para. a	Exp.	Calc. e	Exp.	Calc. e
	A	1086.12505(96)	1098│1099	1263.5956(30)	1277│1267
	B	(-1.1%) 685.249819(87)	690│719	(-1.1%) 561.264028(53)	569│599
	C	(-0.7%) 520.101739(56)	524│546	(-1.4%) 529.294393(43)	541│ 564
	DJ	(-1.0%) 0.06498(41)	-│0.0681	(-2.3%) 0.13759(10)	-│0.14010
	DJK	0.7815(32)	-│0.4653	-0.1264(34)	-│-0.2844
	DK	-0.733(43)	-│-0.445	-	-│0.6929
	d1	-0.01398(19)	-│-0.0102	-0.00840(11)	-│-0.00557
	d2	-0.01222(19)	-│-0.0101	-	-│-0.00016
	N b	74 f		39 f
	σ c	1.8		0.6
	ΔE d		0		1.3
	a A, B and C are the rotational constants given in MHz; DJ, DJk, Dk, d1 and d2 are
	the quartic centrifugal distortion given in kHz; b Number of fitted transitions; c
	rms deviation of the fit in kHz; d ΔE represents the relative electronic energies in
	kJ mol -1 with respect to the global minima calculated at MP2/6-311++G(d,p); e
	calculated parameters at MP2/6-311++G(d,p) │ M06-2X/6-311++G(2df,p) level of
	theory;

f a-type transitions.

Table 3 .

 3 Experimental spectroscopic parameters for the two observed trihydrates of

	NOP compared with those predicted from ab initio MP2/6-311++G(d,p) and DFT M06-
	X/6-311++G(2df,p) calculations.
		3w(i)		3w(ii)
	Para. a	Exp.	Calc. e	Exp.	Calc. e
	A	872.0385(11)	885│892	1106.17164(22)	1147│1149
	B	501.860939(57)	504│537	410.527409(34)	414│448
	C	446.355960(53)	451│482	359.971675(23)	370│404
	DJ	0.14100(12)	-│0.09372	0.098855(53)	-│0.0420
	DJK	-0.0541(14)	-│-0.1123	0.1371(11)	-│-0.05491
	DK	-	-│0.1479	0.756(19)	-│0.2805
	d1	-0.01360(13)	-│-0.01880	-0.000192(55)	-│-0.00616
	d2	-0.002691(95)	-│-0.0020	-0.000524(28)	-│-0.00033
	N b	60 f		66 g
	σ c	0.8		0.6
	ΔE d	-	0	-	1.3

a A, B and C are the rotational constants given in MHz; DJ, DJk, Dk, d1 and d2 are the quartic centrifugal distortion given in kHz; b Number of fitted transitions; c rms deviation of the fit in kHz; d ΔE represents the relative electronic energies in kJ mol -1 with respect to the global minima calculated at MP2/6-311++G(d,p). e calculated parameters at MP2/6-311++G(d,p)│M062-2X/6-311++G(2df,p); f atype transitions. g a-type and c-type transitions.

  ,26 In the present study, the unique structure of NOP, that its structure is slightly different to verbenone and camphor, we observed further one monohydrate and one trihydrate. This could be explained by the dispersive interactions offered by the NOP molecule which enhanced the stabilization of further one monohydrate and one trihydrate clusters. Therefore, this study highlights how the structural changes in the NOP compared to camphor and verbenone molecules leads to a richer conformational landscape in the case of NOP•••water. By comparing the water chain structure in the observed trihydrates of our study to that observed in the case of camphor and verbenone we can observe that ∠O w1 -O w2 -O w3 of water molecules are closer to the trihydrate of camphor than the verbenone. For 3w(i) and 3w(ii), the calculated values at MP2/6-311++G(d,p) level of theory of ∠O w1 -O w2 -O w3 are 105.1° and 100.8°, respectively. Their values (∠O w1 -O w2 -O w3 ) at the same method MP2/6-311++G(d,p) in the case of verbenone and camphor are 122.4° and 102.2°, respectively. This observation is very important and explains how the water chain adapts its structure to the ketone molecule. In the same way, the dihydrates of NOP adapt similar structure to those observed in the case of camphor and verbenone in which the water dimer adjusts its structure on the host molecule.

				from the M06-
	2X/6-311++G(2df,p) level output.
	Regarding similar hydrated bicyclic ketones in the literature,
	such as camphor	19 and verbenone	26 , NOP	28 has a similar
	bicyclic structure to verbenone, the only difference that in the
	case of verbenone the carbon C 4 is attached to a methyl group
	and an endocyclic double bond links C 3 =C 4 . The bicyclic unite
	of camphor (Bicyclo[2.2.1]heptane) molecule is different than

that of NOP (Bicyclo[3.1.1]heptane). In both cases NOP has one carbon less than verbenone and camphor. For both camphor and verbenone, two dihydrated conformers have been observed, but only two monohydrated and one trihydrated conformers have been reported.

19

Acknowledgements

The present work was funded by the French ANR Labex CaPPA through the PIA under contract ANR-11-LABX-0005-01, by the Regional Council Hauts-de-France and by the European Funds for Regional Economic Development (FEDER). It is a contribution to the scientific project CPER CLIMIBIO. E.M.N. would like to thank Dr. V. Vallet and Dr. F. Réal for the remote use of the PhLAM Linux clusters to perform the calculations.

Conflicts of interest

There are no conflicts to declare


Fabrice R A Onofri
email: fabrice.onofri@univ-amu.fr

Fabrice Lamadie

Digital in-line holography and the photonic jet method: routines and graphical user interfaces to generate numerical standards with the Lorenz-Mie theory

Keywords: Holography, Lorenz-Mie theory, Computation, Particle, Size, Composition

A set of MATAB® routines and a graphical user interface (GUI) to model holograms of homogeneous, coated and multilayered particles are introduced. Based, on Lorenz-Mie theory, they are thought to be useful to the community working on the characterization of multiphase flows with Digital In-line Holography (DIH). Likewise, they can be used for the development of alternative approaches requiring numerical standards to evaluate their accuracy.

Introduction

DIH is a volumetric imaging technique allowing measuring the 3D position, dynamics, shape, and size of flowing micrometer-to millimeter-sized particles (e.g., bubbles, droplets, biological cells, aggregates, colloids, etc.) [START_REF] Goodman | Introduction to Fourier Optic[END_REF][START_REF] Hinsch | Holographic Particle Image Velocimetry[END_REF][START_REF] Pan | Digital holography of particle fields: reconstruction by use of complex amplitude[END_REF][START_REF] Sentis | Bubbles, drops, and solid particles recognition from real or virtual photonic jets reconstructed by digital in-line holography[END_REF][START_REF] Thompson | Holographic particle sizing techniques[END_REF]. Despite its success, hologram processing times are still penalizing, and the resolution obtained on the particle axial position and particle size are not always sufficient. Furthermore, when operated as an imaging technique, conventioal DIH processing schemes does not provide any information on the particle composition, which is a critical parameter for many fields of research and industrial.

Basically, there exit two types of approaches -direct and inverse -to retrieve the particle field properties from recorded holograms. The first approach, called 'back-propagation', is a direct approach where the whole recorded hologram is back-propagated with diffraction integrals to obtain in focus 2D images of all the particles within the probe volume [START_REF] Dubois | Focus plane detection criteria in digital holography microscopy by amplitude analysis[END_REF][START_REF] Memmolo | Refocusing criterion via sparsity measurements in digital holography[END_REF][START_REF] Pan | Digital holography of particle fields: reconstruction by use of complex amplitude[END_REF][START_REF] Sentis | Digital in-line holography for the characterization of flowing particles in astigmatic optical systems[END_REF]. The particle properties are determined by image processing techniques, and thus limited by the corresponding issues (threshold criteria and pixel size). But this approach does not require any assumption on the particle morphology and composition. The second approach, called 'regularization', is an inverse approach where the 20th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 11 -14, 2022 particle properties are inferred thanks to a complex minimization process between experimental and numerically simulated holograms, e.g. [START_REF] Jolivet | Regularized reconstruction of absorbing and phase objects from a single in-line hologram, application to fluid mechanics and micro-biology[END_REF][START_REF] Mallery | Regularized inverse holographic volume reconstruction for 3D particle tracking[END_REF][START_REF] Méès | Evaporating droplet hologram simulation for digital in-line holography setup with divergent beam[END_REF]. This approach requires much more computational resources than the direct one. It demands also a particle model as one of the regularization inputs. Yet, its resolution in terms of particle axial position and particle size is potentially higher because it is not intrinsically limited by the pixel size in the particle plane. In some specific cases, this inverse approach can also infer the phase of the hologram [START_REF] Latychevskaia | Iterative phase retrieval for digital holography: tutorial[END_REF] as well as the particle refractive index [START_REF] Lee | Characterizing and tracking single colloidal particles with video holographic microscopy[END_REF]. For large particles, the particle model is a simple opaque disk propagated with diffraction integrals, while for micrometer particles data base of holograms computed with electromagnetic light scattering theories are preferred, e.g. [START_REF] Lee | Characterizing and tracking single colloidal particles with video holographic microscopy[END_REF][START_REF] Wu | Holography and micro-holography of particle fields: A numerical standard[END_REF]. In practice, it turns out that the inverse approach is limited to the characterization of simple objects in dilute flows, while direct approach can handle more complex particle shape and flows (like sprays with deformed droplets and ligaments for instance, e.g. [START_REF] Gao | Quantitative, three-dimensional diagnostics of multiphase drop fragmentation via digital in-line holography[END_REF]).

The present authors have recently proposed to analyze the photonic jet (also called Airy disk or caustic spot), formed by large spherical, transparent and homogeneous particle (see Fig. 1). It allows to enhance DIH capabilities in terms of particle axial location, in free space (Sentis, Onofri, & Lamadie, 2017;[START_REF] Sentis | Bubbles, drops, and solid particles recognition from real or virtual photonic jets reconstructed by digital in-line holography[END_REF] or in pipe flows [START_REF] Sentis | Digital in-line holography for the characterization of flowing particles in astigmatic optical systems[END_REF], as well as for particle material recognition (Sentis, Onofri, et al., 2017;[START_REF] Sentis | Bubbles, drops, and solid particles recognition from real or virtual photonic jets reconstructed by digital in-line holography[END_REF]. This method is an hybrid approach coupling holograms back-propagation and processing analyze assuming a particle model. In pioneer works [START_REF] Choi | High-accuracy three-dimensional position measurement of tens of micrometers size transparent microspheres using digital in-line holographic microscopy[END_REF][START_REF] Sentis | Digital in-line holography for the characterization of flowing particles in astigmatic optical systems[END_REF]Sentis, Onofri, et al., 2017;[START_REF] Tian | Digital holographic imaging of multiphase flows[END_REF] the particle model was a simple geometrical optics approximation only valid within the paraxial limit. But the concept of the method was also tested and proven with physical Optics [START_REF] Sentis | Bubbles, drops, and solid particles recognition from real or virtual photonic jets reconstructed by digital in-line holography[END_REF] and electromagnetic calculations [START_REF] Sentis | Digital in-line holography for the characterization of flowing particles in astigmatic optical systems[END_REF]Sentis, Onofri, et al., 2017), together with experiments carried on drops, bubbles and glass spheres [START_REF] Sentis | Bubbles, drops, and solid particles recognition from real or virtual photonic jets reconstructed by digital in-line holography[END_REF]. To go further in this process, and make DIH technique fully applicable, by reducing its computation time and improving its resolution, the development of more sophisticated particle models as well as the availability of numerical standards are thought to be fundamental. And this, whether a direct or inverse approach is employed to process the holograms.

In this perspective, this paper presents a work in progress on the development of numerical tools allowing to produce hologram standards and analyse the particle near-field characteristics.

Following this introduction, section 2 summarizes the mathematical and physical background of the numerical routines developed, while section 3 presents the GUI (called 'DIHMie 1.1. ') implemented under Matlab® environment. Section 4 provides some examples that illustrate the capabilities and possible applications of these numerical tools. Section 5 is an overall conclusion.

2. Electromagnetic standards of holograms of homogeneous, coated and multilayred particles

Lorenz-Mie calculations

The routines implemented to model holograms with the Lorenz-Mie theory (LMT) are based on previous Fortran codes developed by [START_REF] Barber | Light Scattering by particles : computational methods[END_REF] for homogeneous spheres and that were translated in MATLAB® and extended to coated and multilayer spheres by Onofri et al. [START_REF] Onofri | Electromagnetic scattering from a multilayered sphere located in an arbitrary beam[END_REF] . Hankel functions [START_REF] Barber | Light Scattering by particles : computational methods[END_REF][START_REF] Bohren | Absorption and scattering of light by small particles[END_REF][START_REF] Onofri | Electromagnetic scattering from a multilayered sphere located in an arbitrary beam[END_REF].

These series are truncated for expansion parameters such as  stop nn [START_REF] Wiscombe | Improved Mie scattering algorithms[END_REF], with  , . For a cloud of particles, the routines allow accounting for the interferences between the fields scattered by all particles as if their where alone.

However, classically, the perturbation of the incident wave by each particle is not taken into account, in other words a dilute (or single scattering) optical regime is assumed.

Interpolation method and asymptotic approximation to speed up electromagnetic calculations

Even with the parallelization option, the calculations can be rather lengthy for high resolution holograms of large particles, and more specially when the particles are far from the CCD sensor.

The latter limitation comes from the difficulty to calculate accurately the Hankel functions at large distances from the particle. To reduce the computational efforts, four solutions have been implemented. They allow to reduce significantly the total computational time without noticeable loss of precision (see Fig. 2 and Fig. 4): • First, for large distances, the calculation of the Hankel function is performed with asymptotic expressions [START_REF] Abramowitz | Handbook of mathematical Functions[END_REF] rather than downward recurrences.

20th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 11 -14, 2022

• Second, since the hologram of rotationally symmetric particles is also rotationally symmetric, a special numerical scheme was developed to interpolate the electromagnetic field componenetson the whole sensor surface from its values calculed along the CCD sensor diagonal. The latter is over sampled to improve accuracy in term of pixel location (i.e. scattering angle), see Fig. 2 a, b (and Fig. 4).

• Third, to account for the lateral shift of the particle coordinates ( ) p xy , with respect to the CCD coordinate system ( ) c xy , , the electromagnetic calculations for each particle are performed only for the diagonal of the rectangle which overlasp with the CCD sensor area (see Fig. 2 c).

• Fourth, the three complex components of each electrical field are interpolated with a bilinear interpolation method prior to calculate with the Poynting's vector the local electromagnetic intensity. 3 Graphical user interface : "DIHMie", version 1.1.

All routines can be used independently, however a MATLAB ® apps (called GUI here) was also developed for a more user-friendly usage, illustrated in Fig. 3. The first window (a) allows the user to manage the calculations, select the interpolation resolution, the output type (scattered or total field), etc. The second window allows generating (or import) different types of particle size distributions, as well as the particle spatial distribution. The third window allows controling the particle material properties, while the fourth one allows to set the other parameters of the setup:

CCD distance, pixel size and number,… The zone highlighted in red is used to control the particle lateral dispersion regarding the CCD area and calculate the particle concentration.

Example of results

The set of routines allows calculating hologram standards, as well as various quantities of interest for the development of DIH: intensity profile of the electromagnetic intensity in the Photonic jet region, the near-field intensity and electromagnetic field phase inside and around the particle, or in the whole xz-plane. 10 cm with a high resolution CCD camera, the maximum deviation is only of 0.25% for an interpolation of 1/2 pixel, and 0.02% for 1/8 pixel.

A basic ouput of DIHMie 1.1 is illustrated with as well as for illustrative purpose. This full hologram was obtained in 4 minutes with a labtop (Intel i7-10875H 2.3GHz 64 Go).

These routines allows also calculating other quantities of interest like the near-field intensity and phase inside and outside a single particle, see Fig. 6. The latter emphasizes on the extreme localisation and intensity of the photonic jet, as well as the quadratic shape of the diverging wave front [START_REF] Duan | Generalized rainbow patterns of oblate drops simulated by a ray model in three dimensions[END_REF] generated by refracting droplet in the near-forward direction.

Conclusion

The numerical tools introduced in this paper allow the exact calculation of holograms standards as well as the electromagnetic near-field intensity and phase inside and outside a homogeneous, coated or multilayered sphere. Available from the corresponding author on reasonable request, they are are thought to be useful to the community working on the characterization of multiphase flows. Likewise, they can be used for the development of alternative approaches requiring numerical standards to evaluate their accuracy.

20th

  International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 11 -14, 2022

Fig. 1

 1 Fig. 1 Particle morphologies taken into account: homogeneous, coated and multilayered spheres.In this model, the three components ( )  r,,

  for the particle size parameter in the considered medium (refractive index ext m ) and for the incident wavelength in the embedding medium  = particle properties are their 3D coordinates ( ) p xyz ,, with respect to a laboratory coordinate system ( ) Oxyz , a single complex refractive index ( ) mk , and diameter D for homogeneous spheres, 20th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 11 -, see Fig. 1.The calculations can be sequential or parallel (on the CPU workers). Basically, the main output is the intensity of the electromagnetic field on a plane of the CCD sensor that is perpendicular to z-axis and at located at distance c z from the laboratory coordinate center. The CCD sensor, quadratic, is defined by its pixels size and number. The considered electromagnetic field can be either the incident (reference wave) or, more interesting, the scattered or the total (scattered plus incident) field of a single particle or a cloud of p

Fig. 2

 2 Fig. 2 Some windows of DIHMie 1.1. (a-b) Residual error on the true pixel location for an interpolation increment of 0.5 pixel: (c) Chart exemplifying all individual particle (colored disks) location and corresponding holograms to be calculated (boundaries in color) to retrieve the total field generated on the CCD sensor (area filled in gray).

Fig. 3

 3 Fig. 3 Main windows of the GUI DIHMie 1.1 under Matlab® environement: (a) Selection of the electromagnetic output type (e.g. incident, scattered, full field), resolution of the interpolation scheme, excecution and stop...; (b) Parameters of the particle characteristic distribution in size and in space...; (c) Particle model (homogeneous, coated or multilayered) and refractive index...; (d) Incoming light characteristics and CCD specification...

  Fig. 4 (a) Comparison of the radial intensity profile of the hologram of a water droplet in air, with diameter 125µm, versus the resolution of the interpolation method used to calculate the complex electromagnetic field across the whole CCD: 1/2, 1/4, 1/8 or 1/16 pixels. (b) corresponding relative error. Other parameters:

Fig. 5

 5 Fig. 5 Typical output of DIHMie 1.1: hologram produced by a cloud of

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  Fig. 6 Near field (a-b) intensity and (c-d) unwrapped phase inside and around an oil droplet (m 0 =1.4437) in water (m ext =1.3317) with diameter D=2R=125µm, illuminated by a plane wave (=0.532m, unpolarized).The accuracy of the interpolation method for the hologram radial intensity profile can be evaluated from Fig.2and Fig.4. For the typical case considered here, a water droplet of 125µm observed at

  Fig. 5 which shows a hologram produced by a cloud of = p n 20 particles with a Log-Normal particle size distribution (PSD) randomly distributed in a probe volume with dimensions 3x3x8 mm 3 . The particle projected area is used to assess the particle concentration ( )


Weixuan Xiao

Gil De Sousa
email: gil.de-sousa@inrae.fr

Nancy El Rachkidy

Alexandre Guitton
email: alexandre.guitton@uca.fr

An Open-Source GNU Radio Framework for LoRa Physical Layer and Collision Resolution

LoRa (Long Range) is a physical layer designed for low-power wide area networks. It is widely used to provide long range connectivity to Internet of Things devices. In order to improve the limited throughput of LoRa, researchers have proposed several collision resolution algorithms. However, a common software framework to compare these algorithms is lacking. In this paper, we propose an open-source framework using GNU Radio, mainly designed to test and compare collision resolution algorithms, as well as physical layer algorithms. Our framework can help optimizing the parameters of algorithms according to channel conditions such as very low signal to noise ratio for instance. We also discuss technical implementation issues of existing collision resolution algorithms. Finally, we show how our framework can be used for either real experiments on USRPs, or for simulations with a large number of nodes.

I. INTRODUCTION

A Low-Power Wide Area Network (LPWAN) is a network where battery-powered devices are distributed in a large area, typically of tens or hundreds of square kilometers. LPWANs have recently obtained a lot of attention from industrial and academic communities thanks to recent wireless technologies such as LoRa (Long Range) [START_REF]AN1200.22 LoRa Modulation Basics[END_REF], enabling one-hop communications over several kilometers. LoRa is widely used for environmental monitoring of forests [START_REF] Park | Forestry monitoring system using LoRa and drone[END_REF] or volcanoes [START_REF] Awadallah | An Internet of Things (IoT) application on volcano monitoring[END_REF], smart buildings, smart cities [START_REF] Lachtar | Elderly monitoring system in a smart city environment using LoRa and MQTT[END_REF], the Internet of Things [START_REF] Wang | Performance of LoRa-based IoT applications on campus[END_REF], etc.

The main drawback of LoRa is its very small throughput, which is typically a few hundred bits per second. This throughput is further reduced by the overhead of the LoRaWAN protocol, the most common MAC protocol for LoRa, and by country-dependent regulations on the ISM sub-GHz band used by LoRa. For instance, in Europe, LoRaWAN devices cannot transmit for more than 1% of the time, which reduces the actual throughput to a few bits per second.

As the network density increases, more transmissions occur, and these transmissions might collide. When a collision occurs, the devices typically retransmit their frames, which increases the energy consumption and the end-to-end delay, while further reducing the throughput. Thus, several researchers have proposed algorithms to resolve LoRa collisions.

The performance of LoRa collision resolution algorithms greatly depends on several physical parameters, including the signal-to-noise ratio (SNR) of the LoRa signals (which can even be negative), the presence of a carrier frequency offset (CFO), the presence of a sample time offset (STO) in transmissions, etc. These parameters are not studied consistently in the literature, which makes the existing algorithms hard to compare. Moreover, existing algorithms generally use sophisticated signal processing techniques, which are often programmed in different environments (ex: Matlab, Python, etc.) or not described in the publications. They are difficult to re-implement, and their details might be missing in the papers.

Setting up a real test bed is also difficult, because it is often hindered by the duty-cycle on the ISM band. In order to study collisions, researchers either need a large number of devices (e.g., more than 20) or configure the devices to exceed the maximum duty-cycle of 1% in order to produce enough collisions. Additional collisions might also occur due to pre-deployed nodes from other networks or from concurrent technologies on the ISM band. In addition, the CFO, the STO and the noise are often hard to predict or to control, and their randomness make the experiments difficult to realize.

To the best of our knowledge, there is no framework that compares existing collision resolution protocols with all the parameters of a physical LoRa signal such as the noise strength, the CFO and the STO. Thus, in this paper, we present our framework on GNU Radio in which we implemented several existing collision resolution protocols. This framework can be used to optimize the parameters of an individual collision resolution protocol, or to compare all the protocols in similar conditions. To do so, our contributions can be seen as follows.

•

We propose an open-source framework using GNU Radio, aimed at comparing LoRa collision resolution algorithms, or more generally, LoRa algorithms. Our framework is extensible to new algorithms and features, and it enables homogeneous comparisons between algorithms, based either on real USRP hardware or on simulations. • We carry on with a technical discussion on some implementation issues of existing algorithms, when these issues are missing from the original description. These issues include CFO correction, STO correction, and computational complexity.

• We use our framework to compare the existing algorithms in a common scenario. We also include an analysis on the memory requirement and on the time complexity of our implementation of these algorithms. The remainder of this paper is organized as follows. Section II gives details on the LoRa physical layer and presents several representative LoRa collision resolution algorithms. Section III explains our proposed open-source framework. Section IV gives our simulation results in an ideal scenario first, and then in a more realistic scenario. Finally, Section V concludes our work.

II. STATE OF THE ART

In this section, we first describe the LoRa physical layer, including LoRa modulation, demodulation and coding. Then, we present several representative collision resolution algorithms, which are implemented in our framework.

A. LoRa

LoRa modulation: LoRa is a physical layer for LPWANs based on a chirp spread spectrum modulation. A chirp is a linear frequency sweep over a fixed bandwidth (BW). The chirp is called upchirp when the frequency increases over time, and downchirp otherwise. The initial frequency of a chirp encodes its value. A LoRa uplink frame starts with a preamble composed of 8 upchirps encoding value 0, a network identifier of 2 upchirps, and a delimiter of 2.25 downchirps. The frame continues with the payload encoded with upchirps and including a cyclic redundancy check. A LoRa downlink frame follows the same structure except that upchirps are replaced with downchirps, and conversely.

The duration SD of a chirp depends on a parameter called spreading factor (SF), and is equal to 2 SF /BW . SF also defines the number of bits encoding the value of the chirp. A large SF improves the sensitivity of the receiver, thereby increasing the communication range, but also increases the chirp duration, thereby reducing the throughput.

LoRa demodulation: To demodulate a frame, the receiver uses the preamble to synchronize itself with the transmitter. Then, the receiver captures the time-varying signal f m (t) for each chirp of value m. The signal is multiplied by an inverse chirp f ′ (t) encoding value 0, which removes the time-variant. The multiplication on phase is translated into an addition in frequency, as follows (for the case of an upchirp):

f m (t)+f ′ (t) = BW SD τ m , t ∈ [-SD 2 , SD 2 -τ m ) BW SD τ m -BW, t ∈ [ SD 2 -τ m , SD 2 ) (1)
where τ m = (m × SD)/2 SF encodes the m-th value. One of the two cases of Equation 1 is outside of the bandwidth. However, when BW is used as the sample rate, the aliasing makes the two cases identical in terms of frequency. Once the time-variant part of the signal is removed and the signal contains a single constant frequency, the receiver performs a Fast Fourier Transform (FFT) on the result, and finds the strongest FFT peak. This peak translates into the chirp value m, encoded with SF bits. After having computing the value of each chirp of the payload, the frame is completely demodulated and can be thus decoded. An open-source LoRa demodulation module1 for GNU Radio is described in [START_REF] Robyns | A Multi-Channel Software Decoder for the LoRa Modulation Scheme[END_REF].

LoRa coding: LoRa introduces four channel coding steps. First, LoRa uses a Hamming code in order to add redundancy in the bits of the application payload. According to the coding rate (CR) parameter, from CR=1 to CR=4 bits of redundancy are added every 4 bits of payload. Second, LoRa uses a whitening sequence in order to reduce the probability of long sequences of identical bits. Third, LoRa uses a diagonal interleaver to distribute consecutive payload bits into different codewords, and consequently into different transmitted symbols. This allows to distribute the potential errors of a symbol on distant bits, which can be detected more easily, and even corrected if a large coding rate (CR=3 or CR=4) is used. Fourth, a Gray decoding is used for the transmission. This ensures that a decoding error of one in the symbol value translates into a single bit error.

B. Existing collision resolution algorithms

CHOIR [START_REF] Eletreby | Empowering Low-Power Wide Area Networks in Urban Settings[END_REF] leverages tiny frequency offsets in symbols that occur due to hardware imperfections of real devices. These tiny offsets are shown to be stable over the entire duration of a frame. By having the receiver perform an FFT on a large window (typically, ten times the original window size), these tiny frequency offsets can be extracted and can help to identify the transmitter, even when frame collisions occur.

FTrack [START_REF] Xia | FTrack: Parallel Decoding for LoRa Transmissions[END_REF] uses the time offset among colliding frames to decode them. After the LoRa downchirp multiplication, each symbol becomes a series of constant frequencies over SD, called a track. FTrack uses these tracks to match the symbols to the transmitters. FTrack requires that the symbol frontiers of colliding frames are separated by at least SD/10.

OCT [START_REF] Wang | Online concurrent transmissions at LoRa gateway[END_REF] also uses the time offset to match the symbols to the transmitters. The receiver computes segments according to the symbol edges of the colliding LoRa frames, and performs an FFT on each segment. It stores all the peaks of each FFT, and determines the sent symbols based on the symbols that are repeated on all the corresponding segments. When the colliding frames are quasi-synchronized, the previous approach cannot be used and OCT instead uses the power level of symbols to match them to the correct frame.

CIC [START_REF] Shahid | Concurrent interference cancellation: decoding multi-packet collisions in LoRa[END_REF] is similar to OCT, and also leverages the time offset to divide each SD into segments, with an FFT being performed on each segment. CIC computes the intersection of the set of symbols of contiguous segments to extract the correct symbol. It is important to notice that CIC is one of the few collision resolution algorithms that has been tested under low SNR [START_REF] Shahid | Concurrent interference cancellation: decoding multi-packet collisions in LoRa[END_REF].

In Successive Interference Cancellation (SIC) approaches such as in [START_REF] Laporte-Fauret | An Enhanced LoRa-Like Receiver for the Simultaneous Reception of Two Interfering Signals[END_REF], a strong LoRa signal can be demodulated even if it collides with weaker signals, thanks to the capture effect. By reconstructing the stronger signal and removing it from the superposed signals, it is possible to attempt to iteratively demodulate the next stronger signal. This requires the signal-interference ratio to be above 6 dB, where the signal interference ratio is formally defined as:

SIR = 10 log 10 S ref S int = 20 log 10 A ref A int , (2)
where S ref and A ref are the energy and amplitude of the reference signal, and S int and A int are the energy and amplitude of the interfered signal.

In [START_REF] Xiao | Recovering Colliding LoRa Frames from Uncertainties Using LoRa Coding[END_REF], we showed that frame collisions might cause symbol ambiguities in most collision resolution algorithms. These ambiguities come from noise, close symbol edges in timebased approaches, or similar receive power in power-based approaches. We proposed to benefit from the LoRa coding, and especially from the diagonal interleaver and the Hamming code, in order to remove most symbol ambiguities.

III. FRAMEWORK ARCHITECTURE IN GNU RADIO

In this section, we first describe the GNU Radio programming environment. Second, we present the architecture and the general modules of our framework. Third, we present our implementation of a few existing collision resolution algorithms, as well as how to integrate a new algorithm into our framework. Finally, we explain how to add extra features.

A. GNU Radio primer

GNU Radio [13] is a free and open-source software providing signal processing blocks for software-defined radios.

The basic element in GNU Radio is the block. A block usually contains input and output ports, from which it can respectively receive and send data to other blocks. The data is either a continuous stream (e.g., a signal is usually a stream of complex float numbers representing the in-phase (I) and quadrature (Q) components of the signal) or a discrete message containing data with polymorphic type (i.e., a generic structure holding a variety of data such as a number, a string, a tuple, a list or a dictionary). Streamed data is also able to carry discrete messages, named tags in GNU Radio. Each block can be considered as a state machine with some initial parameters. A block consumes the data from its input ports, processes it, and produces data for the output ports. The blocks do not require to load all the data at once. GNU Radio has an intuitive graphical interface where blocks are interconnected through links between output and input ports, as shown on Figure 1.

In the example of Figure 1, the output of the "Throttle" block is connected to the input of the "Resampler" block. Blue ports use a stream of complex float numbers, while white ports represent messages.

B. Architecture of the proposed framework

Our proposed open-source framework is available at Zenodo 2 . The core of our framework is composed of two parts: a decoder and a node abstraction. The decoder contains an optional LoRa preamble detector, a demodulator (which can implement a collision resolution algorithm) that retrieves 2 Available at https://doi.org/10.5281/zenodo.6421804 demodulated symbols from a signal, and a LoRa decoder that transforms symbols into application bits. The node abstraction is capable of generating superposed LoRa signals with different parameters from a simulated network.

1) Decoder: The decoder part implements a single LoRa demodulator with a known SF. Figure 1 shows the structure of the entire receiver. The receiver takes the samples from a file (block "File Source") or from a connected SDR hardware (block "UHD: USRP Source"). The "LoRa Preamble Detect" block processes the samples to find a preamble. The samples, combined with preamble information of the detected frame, are then processed by the "Payload Decoder" block in order to synchronize the receiver with the transmitted symbols. In the example of Figure 1, the standard LoRa decoding algorithm is used. Then, the demodulated symbols of the entire frame are passed as a message to our "LoRa Decoding" block. This block decodes and outputs the original data from the symbols.

In a simulated scenario, the "LoRa Preamble Detect" block can be removed. In this case, the frame information is directly dispatched to the decoders in a tag, along with the samples of the base band signal.

Any GNU Radio block can be replaced by another block with the same input and output, as long as they implement similar features. Such decoupling can help to develop, evaluate and improve the preamble detection algorithm, the payload demodulating algorithm (which is often the core of the collision resolution algorithms) or the decoding algorithm.

2) Node abstraction: Figure 2 presents two node abstractions in our framework, depicted here as Node 1 and Node 2. Node abstractions are used to generate application data. This data can be randomly generated in GNU Radio for test purpose (as in Node 1) or from an external application (as in Node 2 for an example with UDP) including a network simulator. Note that the communication between the external application and GNU Radio can be made through TCP, UDP or ZeroMQ connections [13]. The generated data is packed as a message and passed to our LoRa Encoding block, which performs encoding and modulation, and produces LoRa signals.

The signals generated by our LoRa Encoding block have a normalized amplitude and are perfectly synchronized at the sample level. It is possible to implement a propagation model by delaying the signals (see the "Delay" blocks on Figure 2, which can be configured with a given number of samples) and by reducing the strength of the received signal with a given factor (see the "Multiply Const" blocks on Figure 2). Then, the signals from the two nodes are added, and the output is a colliding signal. Note that a very similar setup can be configured to evaluate the impact of low SNR, by adding a strong random noise to a weak signal. Also notice that this delay is also used to implement desynchronized transmissions, required by several existing collision resolution algorithms.

For instance, let us consider that we deploy a LoRa gateway which can receive the signals of two nodes, with a relative power difference of 20 dB. From Equation 2, we have Multiple gateways can be implemented by having as many preamble detector and payload decoder blocks as there are gateways (or more precisely, as there are demodulators). For each gateway, the parameters corresponding to the propagation model are likely to be different, as the signals received at each gateway will experience different channel losses. Therefore, we can set different parameters in the propagation models by calculating them in advance using the relative distances between the nodes and gateways.

A ref = A int •10

C. Implementations of collision resolution algorithms

We hereby discuss the implementations of various algorithms in our framework.

Our implementation of the default LoRa demodulation consumes 2 SF samples during SD, multiplies it with a downchirp, performs an FFT on the samples, and finds the strongest FFT bin. The index of this bin is the obtained symbol value.

In CHOIR, the decoder takes 2 SF samples from the synchronized frame during SD. CHOIR adds a zero padding to expand the FFT size to ten times its default value. Then, it stores the strongest FFT bin peaks into ten lists. The symbols are taken from these lists when decoding.

FTrack requires parameters whose values are not given in the reference article [START_REF] Xia | FTrack: Parallel Decoding for LoRa Transmissions[END_REF]: the number n step of steps between each FFT to extract a track, and the size N win of an FFT window. Our block uses this sliding window on N win samples to calculate the FFTs. The window advances by steps of n step samples. To extract the track of an entire symbol, we thus need to review 2 SF + N win samples.

OCT aims at a real-time decoding of LoRa collisions. To do so, OCT divides each symbol duration SD into several segments according to the time offset between the frames. We implemented OCT with a minimal resolution of 1/8-th of SD, that is 0.128 ms for SF = 7 and BW = 125 kHz. The decoder needs to perform at most as many FFT within every SD as there are collided frames.

CIC was not implemented in our framework, as the authors of CIC already made their code available in both Python and Matlab.

All these algorithms have in common the computation of several FFTs. This allows us to compare them depending on how many FFTs they perform, and on the size of these FFTs. The space complexity also mainly depends on the number of samples that a receiver has to review. Table I summarizes the complexities of each algorithm. We use the following notations: N is the size of each FFT (which depends on the algorithm), n sym is the number of symbols in the collided frames, and n c is the number of frames in collision. Note that in the case of CIC, the space complexity is n sym ×N according to the code published in [START_REF] Shahid | Concurrent interference cancellation: decoding multi-packet collisions in LoRa[END_REF], but it could be reduced to N by considering only the samples during an SF for each FFT.

SIC-based algorithms for LoRa need to store the samples during the entire superposition (at least n sym × 2 SF samples), so as to do the subtraction and recover a weaker signal. We believe the SIC-approach needs to be implemented by storing all the samples and analyzing them several times, in order to be compliant with our framework (and more generally, with the GNU Radio design).

D. Extra characterizations of LoRa signals

Noise is considered as an Additive Gaussian White Noise (AGWN) with zero mean [START_REF] Georgiou | Low Power Wide Area Network Analysis: Can LoRa Scale?[END_REF]. In our framework, such noise can be simulated by the "Noise Source" and added to the superposed signals. The relative amplitude is computed according to Equation 2:

A noise = A ref × 10 -SN R 20 (3)
To simulate SN R = 0 dB, A noise is set to the value of A ref .

Carrier frequency offset (CFO) is not negligible in a real transmission. To simulate CFO in the base band signal in our framework, it is possible to add a frequency offset in the "LoRa Encoding" block as part of the node abstraction. The CFO is translated into a time offset in the synchronization by the "LoRa Encoding" block.

Sample time offset (STO) can occur when the re-sampler in the receiver side is not perfectly synchronized with the original

N = 2 SF O(N log N ) N CHOIR 2 SF N = 10 × 2 SF O(N log N ) N FTrack various (≤ 2 SF ) N = 2 SF nstep × O(N log N ) N + N win OCT various (≤ 2 SF ) N = 2 SF nc × O(N log N ) N CIC various (≤ 2 SF ) N = 2 SF nc × O(N log N ) nsym × N signals.
To simulate the STO in our framework, we can simply configure the interpolation parameter in the "LoRa Encoding" block to generate the signals with a higher sample rate, and re-sample the signal with sample-level offsets.

Large networks are usually difficult to simulate as they require creating and interconnecting several blocks, which is tedious and error-prone. To facilitate this, we provide a script with an optional configuration file to automatically generate a complex network. Suppose that 100 nodes are deployed around a LoRa gateway, following a uniform distribution between 500 and 1000 meters. The path loss model is a log distance model. The noise amplitude A noise equals to the amplitude of the weakest signal A signal . Such a scenario can be described as follows:

IV. SIMULATION RESULTS

In this section, we use our framework to evaluate the collision resolution algorithms under different scenarios. Our goal is not to identify the best algorithm, but rather to show that they can be compared in an homogeneous setup. We also try to highlight results that are unknown (e.g., because the algorithms were not systematically tested with negative SNR) or inconsistent with previous works.

A. Scenario 1: impact of the duty-cycle

In the first scenario, we evaluate the ideal performance of different decoding algorithms with different duty-cycle constraints (or, equivalently, with different traffic load) without noise. The network consists of one gateway and a varying numbers of nodes. All the nodes have the same relative distance to the gateway, which is the worst-case for the algorithms based on power level. For the FTrack algorithm, we set n step to 8 and a sliding window of SD. Each node periodically sends at least five frames of 20 random bytes. The time on air of the frames is 57.3 ms for SF7 and 1.34 s for SF12. We set the duty-cycle to 1% and 10%. We add a random waiting time between each transmission in order to desynchronize the nodes. The maximum waiting time is set to 100 times the time on air for 1% duty-cycle, which is 5.7 s and 135 s for SF7 and SF12 respectively, and 10 times the time on air for 10% duty-cycle, which is 0.57 s for SF7 and 13.5 s for SF12. Therefore, the simulation time is 57 s and 1350 s for 1% duty-cycle, 5.7 s and 135 s for 10% duty-cycle. The results are averaged over 20 simulations.

Figure 3 shows the throughput for a varying number of nodes, for SF7 (on the left) and SF12 (on the right), for four algorithms and two possible duty-cycles (1% or 10%). Our results show that the setup with 1% duty-cycle does not generate enough collisions: collision resolution algorithms have similar performance as LoRa. LoRa saturation is not reached with 100 nodes at 1% duty-cycle, while it is reached at about 30 nodes with 10% duty-cycle. It can be seen that FTrack performs the best in all setups, which is not consistent with [START_REF] Shahid | Concurrent interference cancellation: decoding multi-packet collisions in LoRa[END_REF]. Indeed, the original paper for FTrack [START_REF] Xia | FTrack: Parallel Decoding for LoRa Transmissions[END_REF] mentions that they use FTrack to decode the frames with similar power levels and that the capture effect can be leveraged to decode frames with different power levels. In [START_REF] Shahid | Concurrent interference cancellation: decoding multi-packet collisions in LoRa[END_REF] however, FTrack is evaluated with collided frames of different power levels though, which degrades the results.

B. Scenario 2: impact of the SNR

In this scenario, we set the duty-cycle to 10% and deploy 50 nodes in the network for guaranteed collisions. The nodes are randomly deployed in a ring between 500 m and 1000 m around the gateway. The relative receiving power levels are calculated through a log distance loss model. The path loss L in dB is L = L 0 + 10nlog 10 ( d d0 ), where n = 3.76 is the path loss exponent, d 0 = 1 is the reference distance in meter, L 0 = 7.7 is the path loss at reference distance, and d is the distance between a node and the gateway. We vary the strength of the noise so that the SNR varies from 20 dB to -10 dB (recall that a negative SNR means that the LoRa signals are weaker than the noise). We also test two sliding windows for FTrack: N win = 2 SF and N win = 2 SF /2.

Figure 4 shows the throughput for a varying SNR, for SF7 (on the left) or SF12 (on the right), for four algorithms, 50 nodes, and 1% duty-cycle. Our results show that FTrack with full FFT window performs the best with SF7, while OCT reaches the best performance with SF12. The performance of most algorithms appears to be slightly impacted by the SNR, except for CHOIR, and for FTrack on full window with SF12. At SF7, throughput degradation can be seen from SNR=0 dB and lower for CHOIR, FTrack with full window, and OCT. At SF12, such degradation appears from SNR=10 dB for CHOIR and FTrack with full window, and from SNR=5 dB for OCT. Note that the impact of the SNR on the decoding algorithms was only known for LoRa and for CIC. V. CONCLUSION LoRa technology is used in a large variety of LPWAN applications. Several sophisticated collision resolution algorithms have been proposed in order to increase the throughput of LoRa. However, a common framework to compare these algorithms, or to test new ones, is lacking. In this paper, we described our open-source platform for GNU Radio. We discussed how we implemented several existing collision resolution algorithms, as well as specific features in a realistic LoRa network. We used our framework to compare the performance of several algorithms in various scenarios, and with various metrics including their time and space complexity. As an example, we use the ability of our framework to finely control LoRa signal features in order to study the impact of SNR. We believe that our framework can be used and improved by the community to implement and evaluate new collision resolution algorithms or new signal processing techniques.

SIR 20 .Fig. 1 .Fig. 2 .

 2012 Fig. 1. Decoding LoRa frames from a signal sample, obtained from a file or from a USRP SDR hardware.

Fig. 3 .

 3 Fig.3. Study of the throughput as a function of the number of nodes, with 1% or 10% duty-cycle, and with SF7 (on the left) and SF12 (on the right).

Fig. 4 .

 4 Fig. 4. Study of the throughput as a function of the SNR, for 50 nodes and a duty-cycle of 10%, and with SF7 (on the left) and SF12 (on the right).

TABLE I COMPLEXITIES

 I OF THE IMPLEMENTED COLLISION RESOLUTION ALGORITHMS.

	Algorithm Number of samples	FFT size	Time complexity of FFT per symbol Space complexity of FFT per symbol
	LoRa	2 SF

The module from[START_REF] Robyns | A Multi-Channel Software Decoder for the LoRa Modulation Scheme[END_REF] is available at https://github.com/rpp0/gr-lora.