Valentina Calabrese

Hélène Lavanant

Frédéric Rosu

Valérie Gabelica

Carlos Afonso

Collision Cross Sections of Phosphoric Acid Cluster Anions in Helium Measured by Drift Tube Ion Mobility Mass Spectrometry

Keywords: Ion mobility, mass spectrometry, phosphoric acid, clusters, hydrogen bonds, instrument performance evaluation, collision cross section

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Introduction

Ion mobility spectrometry (IMS) separates ionized species in the gas phase by exploiting the balance between the force exerted by a weak electric field (E) and the friction due to the collisions with a buffer gas. When the electric field is uniform and weak, the ions reach a steady-state velocity (vD) and take different times (tD) to cross a drift cell of length L. The resulting mobility (K= vD/E) is mainly dependent on the structure of the gaseous compounds and it gives access to a molecular and conformation descriptor known as collision cross section (CCS or ) [START_REF] Zeleny | On the ratio of the velocities of the two ions produced in gases by Röntgen radiation; and on some related phenomena[END_REF][2][START_REF] Gabelica | Fundamentals of ion mobility spectrometry[END_REF].

Along with the application as free-standing device, ion mobility spectrometry is now routinely coupled with mass spectrometry (IMS-MS) [START_REF] Kanu | Ion mobility-mass spectrometry[END_REF][START_REF] Campuzano | Ion mobility in the pharmaceutical industry: an established biophysical technique or still niche?[END_REF] and the release of commercial instruments has contributed progresses in multi-dimensional separation. Using ion mobilities and CCS for the characterization of complex mixtures is fast becoming common practice in many "-omic" areas, to obtain information on the three-dimensional structure, as well as to infer biomolecules folding [START_REF] Dwivedi | Metabolic Profiling of Human Blood by High Resolution Ion Mobility Mass Spectrometry (IM-MS)[END_REF][START_REF] Mclean | Ion mobility-mass spectrometry: a new paradigm for proteomics[END_REF][START_REF] Ahmed | Application of the Mason-Schamp equation and ion mobility mass spectrometry to identify structurally related compounds in crude oil[END_REF]. The drift tube ion mobility mass spectrometers (DTIMS) have been established for the longest time. [START_REF] Kanu | Ion mobility-mass spectrometry[END_REF][START_REF] May | Ion mobility-mass spectrometry: time-dispersive instrumentation[END_REF] In contrast with DTIMS that allows a direct determination of the CCS using the measured experimental ion mobilities, non-uniform electric field drift cells such as the traveling wave (TWIMS) and the trapped (TIMS) ion mobility cells, usually use a calibration procedure, with reference ions of known CCS or reduced mobility to determine CCS [START_REF] Shvartsburg | Fundamentals of traveling wave ion mobility spectrometry[END_REF][START_REF] Fernandez-Lima | Gas-phase separation using a trapped ion mobility spectrometer[END_REF]. Many macromolecules such as polymers, peptide mixtures or polysaccharides are commonly used as reference compounds for CCS determination [START_REF] Tao | A collision cross-section database of singlycharged peptide ions[END_REF][START_REF] Duez | Polymers for Traveling Wave Ion Mobility Spectrometry Calibration[END_REF][START_REF] Pagel | Ion mobility-mass spectrometry of complex carbohydrates: collision cross sections of sodiated N-linked glycans[END_REF][START_REF] Forsythe | Collision cross section calibrants for negative ion mode traveling wave ion mobility-mass spectrometry[END_REF]. However, the choice of the suitable calibrant is sometimes tricky and several factors must be considered. For example, it is recommended to match the chemical nature, the mass range and charge state of the analyte and the calibrant [START_REF] Gelb | A study of calibrant selection in measurement of carbohydrate and peptide ion-neutral collision cross sections by traveling wave ion mobility spectrometry[END_REF]. Richardson et al. [START_REF] Richardson | Fundamentals of travelling wave ion mobility revisited: I. Smoothly moving waves[END_REF] theorized that in TWIMS, because of velocity relaxation effects, the time-mobility calibration also depends on the ion mobility-m/z trends. Haler et al. [START_REF] Haler | Predicting Ion Mobility-Mass Spectrometry trends of polymers using the concept of apparent densities[END_REF] suggested that apparent densities matter. In order to obtain transferrable calibration curves, and to test these hypotheses, it is thus important to have mobility and CCS values of series of compounds having different apparent densities and mobility-m/z trends. Moreover, ion mobility measurements should be supported by an evaluation of the influence of instrumental settings on the CCS of the investigated compounds. Experimental conditions in the pre-IMS region could indeed influence ion conformation and hence the accuracy of the results. Reference compounds derived from peptides can be affected by collision induced unfolding (CIU) processes prior or into the IMS cell and undergo changes in their conformation and CCS [START_REF] Duez | Polymers for Traveling Wave Ion Mobility Spectrometry Calibration[END_REF][START_REF] Hopper | Collision induced unfolding of protein ions in the gas phase studied by ion mobility-mass spectrometry: the effect of ligand binding on conformational stability[END_REF]. Besides, a significant fragmentation of the calibrant due to activation after the ion mobility cell could lead to fragment ions with different m/z value generating inconveniences during the ion mobility peak identification [START_REF] Campbell | Fragmentation in the ion transfer optics after differential ion mobility spectrometry produces multiple artifact monomer peaks[END_REF].

To develop new reference materials, we previously performed IMS-MS measurements on phosphoric acid cluster anions, in nitrogen gas [START_REF] Lavanant | Traveling wave ion mobility mass spectrometry and ab initio calculations of phosphoric acid clusters[END_REF]. Organic and inorganic salts easily form cluster ions by direct infusion in an electrospray ionization source (ESI). Such clusters exist with a large charge state range and have long been used as mass calibrants [START_REF] Zhou | Formation of Sodium Cluster Ions in Electrospray Mass Spectrometry[END_REF][START_REF] Zhang | Doubly charged cluster ions [(NaCl)m(Na)2]2+: magic numbers, dissociation, and structure[END_REF][START_REF] Schroder | Ion clustering in electrospray mass spectrometry of brine and other electrolyte solutions[END_REF]. As phosphoric acid contains both proton donor and acceptor sites, its cluster ions are maintained by non-covalent interactions akin to those found in biological molecules susceptible to undergo collision-induced unfolding. Phosphoric acid cluster ions have been used on TWIMS ion mobility mass spectrometers to obtain experimental parameters that reduce fragmentation and retain hydrogen bonds [START_REF] Lavanant | Traveling wave ion mobility mass spectrometry and ab initio calculations of phosphoric acid clusters[END_REF]. Recently, using a Tofwerk DTIMS instrument operated in N2 at ~305 K we determined values of K0 and CCS (termed DT CCSN 2 ) for twentysix singly, doubly and triply charged phosphoric acid cluster anions [START_REF] Lavanant | Collision cross sections of negative cluster ions of phosphoric acid in N2 determined by drift tube ion mobility and their use in travelling wave ion mobility[END_REF]. These clusters were used as reference compounds for calibration and CCS determination on a TWIMS drift cell ( TW CCSN 2 ). Accuracy tests on ions derived from polyalanine, dextran and the hexakis(fluoroalkoxy)phosphazine at m/z 602 when the TWIMS was calibrated with phosphoric acid cluster anions, evidenced a bias of 9-11 Å 2 (between the CCS values in N2 ( TW CCSN 2 ) and and the published DT CCSN2 values [START_REF] Lavanant | Collision cross sections of negative cluster ions of phosphoric acid in N2 determined by drift tube ion mobility and their use in travelling wave ion mobility[END_REF].

Here, we determined new K0 and DT CCSHe values for phosphoric acid cluster anions with charge states between 1-and 4-using an Agilent DTIMS operated in He at 296 K. In addition, we monitored the DT CCSHe values of phosphoric acid cluster ions for different pre-and post-IMS voltage gradients on the 6560 IMS-Q-TOF. We observed how different pre-and post-IMS settings led to changes in the ion mobility and mass spectra and how their K0 and DT CCSHe values were affected.

MATERIALS AND METHODS

Sample preparation

Phosphoric acid (for analysis, 85% solution in water, density 1.680) was purchased from Acros Organics (Fisher Scientific France, Illkirch, France). Deionized water (18 M) was produced by a Milli-Q apparatus (Millipore, Bedford, MA, USA). Acetonitrile was purchased as LC-MS grade from VWR (Fontenay-sous-bois, France). Phosphoric acid samples were prepared at a concentration of 5 mM using a volume of 0.035% of the initial phosphoric acid solution in water/acetonitrile (50/50 v/v). This 5 mM solution of phosphoric acid was enriched with 50 µM ammonia to perform the collision induced experiments (CID) on the ammonia adducts of multiply charged phosphoric acid cluster anions. Ammonium hydroxide solution 29.3% NH3 was from Sigma Chemical (St. Louis, MO, USA). The mixture of hexakis-(fluooalkyloxy)-phosphazines (Tuning Mix, Agilent Technologies) was diluted to 1:4 with 95% acetonitrile in water and was used either alone or in mixture with phosphoric acid.

Drift tube ion mobility mass spectrometry measurements

Uniform electric field drift tube ion mobility mass spectrometry (DTIMS-MS) measurements were performed and replicated with an interval of eight months, on an Agilent 6560 (Santa Clara, CA). The instrument couples a drift tube to a quadrupole time-of-flight mass spectrometer (IMS-QTOF) and it has been described in detail elsewhere [START_REF] May | Conformational ordering of biomolecules in the gas phase: nitrogen collision cross sections measured on a prototype high resolution drift tube ion mobility-mass spectrometer[END_REF]. The samples were directly infused at a flow rate of 3 L/min in an electrospray ionization source (ESI) operated in the negative ion mode. The generated anions were transmitted and accumulated by two stages of ion funnels and separated in a drift tube with a length of 78.1 ± 0.2 cm. The standard uncertainty on the length takes into account the possible edge effects of the electric field. This was estimated from the study by Stow et al. [START_REF] Stow | An Interlaboratory Evaluation of Drift Tube Ion Mobility-Mass Spectrometry Collision Cross Section Measurements[END_REF] which determined the effective length of three other Agilent 6560 DTIMS instruments operated in N2 and under higher voltages to be 78.24 ± 0.16 cm. Ion mobility separations were performed using helium (He) as buffer gas. An in-house modification to the pumping system allows faster (a few minutes) equilibration of the pressures for the helium measurements: a second Tri-scroll 800 pump (Agilent) was connected to the source region (with an Edwards SP16K connected to the front pumping line), while the original Tri-scroll 800 pump is connected to the Q-TOF region.

To ensure appropriate pressure differentials on either side of the drift tube for collision cross section determination, the pressures were set to 3.76 Torr in the drift tube and 3.47 Torr in the trapping funnel before the flow controller was turned on. For all measurements, the flow controller was turned on, this resulted in a pressure of the gas in the drift tube of 3.894 ± 0.010 Torr (measured using an Agilent CDG-500 capacitance diaphragm gauge), and 3.63 ± 0.01 Torr in the trapping funnel. The correct gas flow and pressure difference in these regions were maintained through a feedback system (640B, MKS Instruments) based on the IMS pressure measured at the rear of the drift tube. We found that adjusting the helium pressure differential with the mass flow controller active was not robust enough to ensure 100% helium in the drift tube (for nitrogen measurements, this is not critical). The instrument was operated, with covers open, in a well air-conditioned room and the temperature of the drift cell was monitored by a K-type thermocouple (Omega Engineering). The estimated mean value was 296.6 K with a standard deviation of 0.5 K.

Table S1 provides a detailed list of the parameters used for all the measurements on phosphoric acid cluster anions. These conditions were optimized by Gabelica et al. to gather mainly the native state 7+ of ubiquitin on the same apparatus and the signs were reversed for the negative ion mode [START_REF] Gabelica | Optimizing Native Ion Mobility Q-TOF in Helium and Nitrogen for Very Fragile Noncovalent Structures[END_REF]. Both folded proteins and phosphoric acid clusters are maintained by hydrogen bonds in the gas phase. The enthalpy requirement for unfolding of gas phase ubiquitin ions has been evaluated around 0.4 eV [START_REF] Breuker | Detailed unfolding and folding of gaseous ubiquitin ions characterized by electron capture dissociation[END_REF]. This energy is lower than the enthalpy estimated for the dissociation of one phosphoric acid molecule from cluster ions [(H3PO4)n -H] -with one to six phosphoric acid molecules, and which was calculated to be around 1 eV [START_REF] Lavanant | Traveling wave ion mobility mass spectrometry and ab initio calculations of phosphoric acid clusters[END_REF]. Furthermore, the dissociation energy of one phosphoric acid molecule decreases with increasing aggregation number. The energy barrier for protein unfolding was thus expected to be in the same order of magnitude as the declustering of phosphoric acid clusters.

We chose to probe the influence of two different conditions based on different values of the voltage applied on the fragmentor, which represents the part of the ESI source that speeds up ions from the glass capillary to the high pressure funnel resulting in one of the principal cause of in-source collision induced dissociation (in-source CID). Fragmentor voltages of 300 V or 350 V (which are here equivalently named "soft") and 600 V ("harsh") were used. These values corresponded to a potential difference established between the fragmentor and the entrance of the high pressure funnel of 40 or 90 V and 340 V, respectively.

To obtain the reduced mobility values (K0) and determine CCS values, we used the stepped field (also called multi-field) method. Five different values of the electric field applied in the drift tube were imposed for 1 to 3 min each. The applied electric field  were 5.009, 6.280, 7.561, 8.841, 10.122 V/cm, in helium. Taking the pressure and temperature into account, the gas number density N was 1.268 10 23 m -3 , and this amounted to reduced electric field strengths (E/N) of 4 to 8 Td. The time values (t) obtained from the extracted ion mobility spectra (EIM) (also called arrival time distributions (ATD)) of the experiments at different electric fields were then fitted using a linear regression of time vs. the reciprocal of the field strength (1/E). Drift times were collected picking the apex of the peak of the intact phosphoric acid cluster anions. This procedure allowed to retrieve, from the intercept, the time spent by the ions outside the drift cell (t0) and the slope from which the value of the ion mobility (K) was inferred, according to the Eq.1.

𝑡 = (

𝐿 𝐾 ) • 1 𝐸 + 𝑡 0 (1)
We will refer to t as the drift time, although it is often called arrival time because of the nonnull value of t0. The K values were then converted to a reduced ion mobility K0 to scale the ion mobility into the number gas density value N0 at standard pressure and temperature (p0 and T0, 760 Torr and 273.15 K, respectively) (Eq. 2). We determined the DT CCS values in He using to the low-field limit equation (Eq. 3) where e is the elementary charge, k the Boltzmann constant, z the charge number and µ the reduced mass of the ion and helium gas [START_REF] Mason | Mobility of gaseous lons in weak electric fields[END_REF].

𝐾 0 = 𝑁 𝑁 0 𝐾 = 𝑝 𝑝 0 𝑇 0 𝑇 𝐾 (2
For the post-IMS transmission, we used the two experimental tunings described thoroughly by Gabelica et al. [START_REF] Gabelica | Optimizing Native Ion Mobility Q-TOF in Helium and Nitrogen for Very Fragile Noncovalent Structures[END_REF] that were developed using as benchmark the ammonium-bound bimolecular G-quadruplex, which easily undergoes ammonia loss. These tunings consist in different gradients of DC voltages between the exit of the ion mobility cell and the TOF analyser. We will refer to the softest gradient as "soft" and the other one as "harsh", although both tunings are gentle. A complete list of the applied voltages forming the gradients is presented in Table S1.

Data analysis

The ion mobility spectra acquired by the Agilent instrument were extracted through the Agilent IM-MS browser (version B.08.00, build 8.0.815.0). Classical linear regression was based on least squares minimization. We obtained the linear regressions of five points with uncertainties on the slope and intercept using the classical least squares regression analysis in OriginPro 2018 (version b.9.5.0.193) or in a Microsoft Excel spreadsheet (2016 MSO 16.0.4266.1001 64 bits). An example of linear regression with residual analysis and associated uncertainties with 95% level of confidence is given in figure S1 for the P#8 -phosphoric acid cluster anion ([(H3PO4)8 -H] -. In all cases, the determination coefficient R 2 was higher than 0.995. The slopes ranged from 6.6 to 13.6 s V cm -1 with a maximum relative standard deviation of 5.4%. The mean relative standard deviation for the slopes was 0.7 %.

The uncertainty on the DT CCSHe values was evaluated according to the current GUM guideline.

[31] First, a type A evaluation on collision cross sections obtained from replicates (n) provided the experimental standard deviation (σ) and the experimental standard deviation of the mean (u) (calculated as σ/√𝑛). Second, we also performed an estimation of the uncertainty on CCS with the propagation of errors from different components such as pressure, temperature, slope and length of the drift tube. As a type B standard uncertainty is derived from the probability that an event will occur, we performed a Monte Carlo simulation using the Crystal Ball add-in for Microsoft Excel (Oracle®, version 11.1.2.4.850) on the DT CCS (He) of the P#8 - cluster anion ([(H3PO4)8 -H] -). We obtained a normal distribution and the standard deviation.

A similar approach has been applied by Stow et al. [START_REF] Stow | An Interlaboratory Evaluation of Drift Tube Ion Mobility-Mass Spectrometry Collision Cross Section Measurements[END_REF] to propagate errors on the arrival times due to the uncertainty of the voltage ΔV. Errors were propagated by applying a uniform distribution both for the drift tube length and a normal distribution for the drift tube pressure and temperature. The uncertainties were estimated to ± 0.2 cm for the tube length (78.1 cm, relative uncertainty 0.3 %), ± 0.5 K for the temperature (296.6 K, relative uncertainty 0.2%), ± 0.010 Torr for the pressure (3.894 Torr, relative uncertainty 0.3%). A triangular distribution was used for the slope (L/K) using a maximum deviation of ± 0.5 s V m -1 on the 10.3 s V m -1 slope obtained from the P#8 -cluster anion. This ± 0.5 s V m -1 value was the maximum standard deviation observed for the slope for all ions.

Tandem mass spectrometry measurements

Tandem mass spectrometry measurements were performed on a Synapt G2 (Waters Corp., Manchester, UK). The 5 mM solution of phosphoric acid, supplemented with 50 µM of ammonium hydroxide was infused using the sample pump at a flow rate of 5 L/min to the ESI source coupled with the instrument and operated in negative ion mode. Conditions in the electrospray were: capillary voltage 2.7 kV, sampling cone voltage 60 V, extraction cone voltage 2 V, source temperature 90°C, desolvation temperature 250°C, desolvation gas flow 500 L/h, source gas flow 20 mL/min. Collision-induced dissociation (CID) experiments were performed by selecting the monoisotopic ion of NH3 adducts for sample phosphoric acid cluster ions in the quadrupole. We chose to fragment the selected clusters in the trap cell by applying a collision voltage of 5 to 15 V. A list of other relevant experimental parameters used to perform measurement on the Synapt G2 are gathered in Supplementary Table S2. The raw data were extracted using MassLynx™ (4.1 version).

RESULTS AND DISCUSSION

Drift tube ion mobility mass spectrometry of phosphoric acid cluster anions

Solutions of 5 mM phosphoric acid were investigated by DTIMS under four different combinations of experimental pre-IMS and post-IMS conditions. Figure 1 shows colour maps of m/z vs drift time from the electrospray-generated ions in the negative ion mode for two different sets of conditions in the ESI source (soft and harsh pre-IMS) and very soft post-IMS conditions, with a drift field of 5 V cm -1 and helium as drift gas. The maps clearly indicate the formation of singly and multiply charged clusters [(H3PO4)n -zH] z-covering an m/z range of 300 -3100 and well separated according to their charge states in diagonal lines, each corresponding to a different charge state up to 8-. The assignment of the charge for each line was based on the isotopic pattern of the extracted mass spectra. In addition to the pure phosphoric acid cluster ions, the mass spectra also included peaks of significant intensity corresponding to adducts formed by clusters with the residual ammonia, sodium and potassium present as contaminant in the ion source. While sodium or potassium adducts

[(H3PO4)n -zH + xNa] (z-x)-or [(H3PO4)n -zH + xK] (z-x)-
were present in low abundance for all the charge states, ammonia adducts [(H3PO4)n -zH + xNH4] (z-x)-were observed only for multiply charged states. The number of NH4 + and Na + (or K + ) included in the clusters appeared to be different depending on the charge state, with up to two cations for the doubly charged, three for the triply charged ions and four for quadruply charged clusters (Figure S2).

Figure 1a shows the presence of an additional population of signals at lower drift times, compared to figure 1b, that we annotated as metastable species. These signals correspond to ions with high charge states, and formed an appendix to the main more intense signals.

Similar phenomena exist in the ion mobility spectra of several ionic liquids [START_REF] Larriba-Andaluz | Gas molecule scattering & ion measurements for organic macro-ions in He versus N2 environments[END_REF][START_REF] Rus | IMS-MS studies based on coupling a differential mobility analyzer (DMA) to commercial API-MS systems[END_REF] and iodide salt cluster ions [START_REF] Ouyang | The collision cross sections of iodide salt cluster ions in air via differential mobility analysis-mass spectrometry[END_REF] obtained by a differential mobility analyser (DMA). The most intense signals include intact cluster ions as well as the product of neutral and ion evaporation events of higher charge state cluster ions that occur between the DMA and the detection system [START_REF] Hogan | Tandem ion mobility-mass spectrometry (IMS-MS) study of ion evaporation from ionic liquid-acetonitrile nanodrops[END_REF]. This additional population of signals was not present when harsher conditions were used in the ESI source, pointing out that these metastable species are formed during the electrospray process and maintained up to the IMS when very soft pre-IMS conditions are applied, but fragment during or after the IMS even with our very soft post-IMS conditions. We observed that the absolute intensities of these metastable ions decreased by a factor of 1.6 when the electric field in the ion mobility cell was 10 V cm -1 compared to the lowest values of 5 V cm -1 .

The other phosphoric acid clusters (that are observed in both ESI conditions) were 15 times more intense than the metastable population and their absolute intensities did not change with the variation in electric field (Figure 2). The decrease in the absolute intensities of metastable ions pointed towards a possible increase in fragmentation yields as the ions go faster through the drift tube. However, as reported elsewhere [START_REF] Hogan | Tandem ion mobility-mass spectrometry (IMS-MS) study of ion evaporation from ionic liquid-acetonitrile nanodrops[END_REF], dissociation events taking place during the mobility separation should be signalled by the presence of extended streaks at an intermediate mobility of the parent and the fragmentation product. Being for now uncertain as to the attribution of what we named metastable species, we chose to concentrate our attention on the main intense population of cluster ions that were reproducibly observed in all experimental conditions. When dissociation processes occur during the post-IMS transmission towards the TOF analyser, the extracted ion mobility (EIM) spectra, also called arrival time distribution (ATD), of a given m/z corresponding to a phosphoric acid cluster with aggregation number n, [(H3PO4)n -zH] z-(further noted P#n z-), may display several peaks. The presence of more than one peak for each ion should not be attributed to different conformations, but to ion mobility signals of larger phosphoric acid clusters (n+1, n+2, etc.) that dissociate after the ion mobility cell due to ion and neutral evaporation processes. In a previous work, we described extensively the method to assign the ion mobility peak to the correct phosphoric acid cluster ion [START_REF] Lavanant | Traveling wave ion mobility mass spectrometry and ab initio calculations of phosphoric acid clusters[END_REF]. This consists in overlaying and comparing the EIM of clusters with adjacent aggregation numbers, which are possible precursor ions. A peak that is present only in the EIM of the investigated cluster and not in the EIM of larger ones is then assigned to the intact ion.

As an example, Figure 3a shows the overlay of EIM for phosphoric acid clusters with m/z 782.80 (P#8 1-, black trace), 880.78 (P#9 1-, red trace), 978.76 (P#10 1-, blue trace) obtained from the measurements in helium at a drift field of 5 V cm -1 . A drift time value of 26.39 ms was attributed to the intact singly charged cluster P#8 -(black line). The signals at longer drift time (28.05 and 29.72 ms) correspond to the amount of P#8 -formed by neutral evaporation events of H3PO4 molecules from the clusters P#9 -and P#10 -after the ion mobility cell. Moreover, during the process of assignment of the drift times, one must consider that for a given m/z value, higher-order multimers could exist. Figure 3 shows the full EIM spectra of the m/z values 782.80, 880.78, 978.76 corresponding to the singly charged clusters P#8 -, P#9 -, P#10 -also contained the mobility peaks of the doubly charged phosphoric acid ions P#16 2-, P#18 2-and P#20 2-. Overlaying different EIM spectra is useful whenever one wants to avoid a wrong assignment of the drift time value. In fact, some clusters may completely be dissociated after the ion mobility separation. On figure 3a, the signals at lower drift times in the EIM of m/z 782.8 (black trace) include a broad peak at 21.35 ms which appeared with the same drift time value in the EIM of m/z 880.78 (red trace) with a larger intensity. We therefore could not attribute any drift time value to the intact cluster P#16 2-. Indeed, the most probable assignment for the peak at 21.35 ms corresponded to the neutral evaporation process of H3PO4 molecules from the doubly charged cluster P#18 2-after its mobility separation and no drift time could be assigned to an intact P#16 2-.

As the amount of the intact cluster ions depends on experimental conditions, we used the method of overlaying the EIM relative to a given phosphoric acid cluster and its potential precursor ions to monitor how pre-and post-IMS experimental settings affect the abundance of intact and precursor clusters. Figure 3 shows the overlaid EIM at a drift field of 5 V cm -1 in helium for m/z 782.80 (P#8 -and P#16 2-), m/z 880.78 (red) (P#9 -and P#18 3, which differ only in the pre-IMS conditions, reveals that soft pre-IMS (c and d) leads to a higher proportion of larger doubly charged cluster anions This is also visible is Figure S3 that overlays the EIM of m/z 880.8 in all experimental conditions. The comparison of the couples formed by panels (a) with (b) and (c) with (d), in Figure 3, reveals the effect of different post-IMS settings on the amount of cluster ions that undergo dissociation after the ion mobility cell. The peak positions for harsh post-IMS transmission were shifted to lower drift times because the higher speed of ions between the ion mobility and the TOF analyser led to lower t0 values (see also Figure S3). Peaks were better resolved with smaller peak widths due to a lower diffusion time in the post-IMS area.

The overlaid EIM obtained with the harsh post-IMS transmission settings (panel b and d) highlights that the assigned intact cluster ions P#8 -, P#9 -and P#10 -appeared only as low intensity shoulder peaks and that some dissociation occurred for the doubly charged clusters P#20 2-and P#21 2-(presence of two spikes for each peak). The two EIM spectra (a) and (c) obtained with the soft post-IMS transmission settings show larger relative intensities of intact cluster ions P#8 -, P#9 -and P#10 -. Harsh post-IMS transmission mode therefore induced many dissociations of hydrogen bonds after the ion mobility cell. The soft post-IMS transmission also presented some dissociation processes, but in a lower proportion. The attribution of ion mobility signals was always easier and more reliable with soft post-IMS transmission settings. For multiply charged cluster anions, the amount of dissociation was difficult to estimate as the signals of adjacent cluster ions overlapped. However, in most cases, we hypothesized the apex of the signals of the multiply charged cluster anions to be largely attributable to the intact cluster ions. Indeed, here as in our previous study [START_REF] Lavanant | Collision cross sections of negative cluster ions of phosphoric acid in N2 determined by drift tube ion mobility and their use in travelling wave ion mobility[END_REF], we observed that the relative proportion of dissociated precursor ions to intact cluster ions tended to decrease with the aggregation number. For several cluster anions, however, no reliable signal attribution could be found for the intact cluster. The singly charged cluster anions P#5 -, P#6 -and P#7 -had signals which overlapped the signals of unresolved multiply charged ions, and the signals of several ions, as the P#16 2-in Figure 3, we entirely attributed to fragmented doubly charged precursor ions. Also, in cluster ions with an aggregation number close to the charge state change (aggregation numbers 14 to 16 between charge states 1-and 2-and aggregation numbers 33 to 36 between charge states 2-to 3-), the intensities and signal over noise ratio were too low for any reliable attribution. Note that when the precursor ions are of the same charge state as the resulting fragment ion, they are expected at a drift time slightly higher than that of the intact cluster in the EIM. For ion evaporation processes, precursor and fragments have different charge states so that their drift times are more different. For example, a triply charged ion resulting from the ion evaporation of a "metastable" quadruply charged cluster, should appear with a drift time similar to that of quadruply charged ions, while the true and intact, triply charged should appear as a peak in the zone of the triply charged species of the EIM spectra. The omitted y axes correspond to the signal intensity in arbitrary units normalized to the highest peak on display.

As ammonium and sodium adducts were present, we wanted to assess whether they could be potential precursor ions to pure phosphoric acid cluster ions. While the loss of Na + from an [(H3PO4)n -zH + xNa] (z-x)-adducts is highly improbable as it would lead to the formation of a phosphoric acid cluster ions with a higher charge state, the loss of neutral NH3 from an [(H3PO4)n -zH + xNH4] (z-x)-was more likely. If ammonia adducts were precursors to pure deprotonated phosphoric acid clusters, fragmentation of ammonia adducts could lead to possible ambiguities in the drift time assignment, as their respective drift times were nearly identical (inset in figure 1a). In order to figure out how ammonia adducts dissociate, we performed collision induced experiments (CID) on several doubly and triply charged ammonia adducts chosen among the most intense species in the mass spectra. Fragmentations were carried out, after selection in the quadrupole and in the trap cell of a Waters Synapt G2 ion mobility mass spectrometer by applying a collision voltage of 5 V. The CID spectrum obtained for the triply charged adduct [(H3PO4)39 -4H + NH4] 3-(m/z 1278.37) (Figure S4) showed two preferred fragmentation pathways, and all adducts acted in a similar way. One fragmentation pathway corresponded to the loss of neutral molecules of H3PO4 to form triply charged ammonia adducts with lower aggregation numbers. The other fragmentation pathway involved the loss of singly charged phosphoric acid cluster anions with small aggregation numbers leaving as fragment ions doubly charged ammonia adducts. The neutral loss of NH3 from the precursor ion was detected only in very low intensity: the relative abundance of the triply charged phosphoric acid cluster fragment [(H3PO4)39 -3H] 3-represented less than 1% of the total intensity of all fragment ions. The ionic interactions formed by ammonia in the adducts could thus be considered as strong enough to ensure its retention in the clusters and ammonia adducts could be excluded as potential precursor ions of pure deprotonated phosphoric acid cluster ions.

Effect of experimental conditions on DT CCSHe values

To check the effect of different pre-IMS parameters on CCS values, we first performed multifield measurements on the calibrants of the Agilent tuning mix, for which DT CCSHe were published in 2017 [START_REF] Marchand | Drift Tube Ion Mobility: How to Reconstruct Collision Cross Section Distributions from Arrival Time Distributions?[END_REF]. Measurements were repeated eight months apart and replicated for different pre-IMS conditions using solutions of tuning mix either alone or in a mixture with phosphoric acid, with harsh post-IMS conditions. Figure 4 (and Table S3) presents the DT CCSHe values obtained for the tuning mix in our experiments and the standard deviation for the total sets of experiments (n = 6) along with the DT CCSHe values from Anal. Chem 2017 [START_REF] Marchand | Drift Tube Ion Mobility: How to Reconstruct Collision Cross Section Distributions from Arrival Time Distributions?[END_REF], represented on the right of each graph with its standard deviation. Regardless of the change in pre-IMS conditions, considering a 90% level of confidence (≈2σ) all tuning mix ions presented DT CCSHe values that were consistent with the Anal. Chem 2017 [START_REF] Marchand | Drift Tube Ion Mobility: How to Reconstruct Collision Cross Section Distributions from Arrival Time Distributions?[END_REF] values. The absence of any significant bias between previously published values and our determined values ensured the reliability of instrument performances for this study. In fact, the standard deviation of six replicates on DT CCSHe values obtained with different instrumental parameters ranged between 0.3 to 1.2 Å 2 (0.3 to 0.5% relative standard deviation (RSD)). These standard deviations were lower than the ones published in Anal. Chem 2017 which were obtained from three independent replicates (1.3 to 1.9 Å 2 i.e. 0.6-1.2% RSD). These standard deviation values are not far from the relative standard deviations of 0.22 % RSD (n = 36) reported in N2 [START_REF] Stow | An Interlaboratory Evaluation of Drift Tube Ion Mobility-Mass Spectrometry Collision Cross Section Measurements[END_REF] or the 0.7 % RSD (n = 5) reported in helium in the positive ion mode [START_REF] Morris | Evaluating Separation Selectivity and Collision Cross Section Measurement Reproducibility in Helium, Nitrogen, Argon, and Carbon Dioxide Drift Gases for Drift Tube Ion Mobility-Mass Spectrometry[END_REF]. We then evaluated if a change of the peak profiles and a variable proportion of dissociation affected the CCS values of phosphoric acid cluster anions. We used overlaid EIM spectra to perform careful peak assignment for the experiments carried out with harsh post-IMS conditions. We determined the CCS values of phosphoric acid cluster anions with charge state between 1-and 4-for which the attribution was unambiguous. Each CCS was determined from the slope of the linear correlation obtained with all five electric field values. The DT CCSHe values and the standard deviation (σ) of replicated experiments (n = 10) for a subset of singly, doubly, triply and quadruply charged phosphoric acid clusters are shown in Figure 5 (and Table S4). We obtained very consistent CCS values for phosphoric acid cluster anions in all explored experimental conditions also in presence of high extents of fragmentation. We hence treated all experimental conditions as repeated measurements and determined the DT CCSHe values for 75 phosphoric acid cluster anions with charge state up to 4-by averaging CCS from at least three different experiments at different experimental conditions (Table 1).

Table 1 gathers the mean reduced ion mobilities K0, the mean DT CCSHe and the type A experimental standard deviations (σ) obtained from three to ten replicates (n). The difference in the number of replicates n comes from the fact that some signals were not visible in every experiment, because the different voltage parameters applied in the pre-IMS and post-IMS areas led to differences in the relative proportion of the charge states and in the amount of dissociation. Only the cluster anions with reliable signal attribution were retained. We obtained a maximum relative standard deviation of 1.7%, and on average the RSD was 0.4%.

We also performed a type B evaluation of the uncertainty on a DT CCSHe value applying the propagation of errors on the result of the Mason-Schamp equation. Taking into account the stochastic nature of the Monte Carlo method and the impossibility to reach a direct control on the degree of freedom by simply choosing a priori the number of trials (nT), we set the program to select nT adaptively as the simulation proceeds in order to reach a 95% coverage interval, as suggested by Cox et al. [START_REF] Cox | Evaluation of Measurement Uncertainty Based on the Propagation of Distributions Using Monte Carlo Simulation[END_REF]. The simulation stopped at nT ̴ 6x10 6 and the normal distribution obtained for the DT CCSHe of the P#8 -cluster anion of 150.9 Ų yielded a standard deviation (equivalent to a combined uncertainty uc) of ̴ 3.0 Ų which amounts to a relative combined standard uncertainty of ̴ 2.0%. It makes sense that this value is slightly higher than the relative standard deviations observed from our replicate measurements (0.6 to 1.7 % RSD), because it takes into account more contributions to uncertainty.

The collected values of DT CCSHe of phosphoric acid cluster anions could be fitted to the power function 41.1 n 0.61 where n is the aggregation number. The determination coefficient R 2 of this fit obtained from seventy-five DT CCSHe values of cluster anions with aggregation numbers from 4 to 89 was 0.9992 (Figure 6). This power function growth was also observed for the DT CCS(N2) values determined before for which the function was 79.6n 0.51 [START_REF] Lavanant | Collision cross sections of negative cluster ions of phosphoric acid in N2 determined by drift tube ion mobility and their use in travelling wave ion mobility[END_REF]. Only sixteen clusters were common to the two studies, among the singly, doubly and triply charged cluster anions. The DT CCSHe values determined here and the DT CCSN 2 determined previously on the TOFWerk instrument showed an excellent linear correlation (Figure S5). Similar power function growth has been observed before for proteins [START_REF] Ruotolo | Ion mobility-mass spectrometry analysis of large protein complexes[END_REF] and, Bleiholder et al. [START_REF] Bleiholder | Ion mobility-mass spectrometry reveals a conformational conversion from random assembly to β-sheet in amyloid fibril formation[END_REF] reasoned that isotropic growth of the collision cross-section Ω should be related to the oligomeric state n by equation 4:

Ω = An 2/3 ( 4
)
where A is the monomer collision cross section. The power function observed for phosphoric acid cluster anions was 0.51 in nitrogen [START_REF] Lavanant | Collision cross sections of negative cluster ions of phosphoric acid in N2 determined by drift tube ion mobility and their use in travelling wave ion mobility[END_REF] and 0.61 in helium (this work).

De Pauw and co-workers [START_REF] Haler | Predicting Ion Mobility-Mass Spectrometry trends of polymers using the concept of apparent densities[END_REF][START_REF] Hanozin | Two-Parameter Power Formalism for Structural Screening of Ion Mobility Trends: Applied Study on Artificial Molecular Switches[END_REF] have also observed such power function growth trends for polymer-cation complexes of different charge states (poly(ethoxyphosphate), poly(ethyleneoxide) and poly(caprolactone)) and oligorotaxane foldamers, with exponent values ranging from 0.50 to 0.93. They interpret the constant A as an apparent charge density or charge solvation capacity of the monomer unit. For polymers, the A value depends on the chemical nature of the monomer, but also on the charge state. Several multiply charged polymers were observed to transition from one power function fit to another depending on the degree of polymerization, reflecting a structural rearrangement. Here, the A value was the same whatever the charge state, which thus indicated a constant charge solvation capacity of phosphoric acid. Also, there was virtually no overlap with several charge states for a given aggregation number. We thus conclude that, for a given charge state when the aggregation number decreased and the charge density was too high, the clusters dissociated, and the smaller clusters should have a similar shape as the larger ones. We hypothesize that the cluster ions are roughly spherical, similarly to a nanodroplet of liquid phosphoric acid.

The fact that the exponential factors differed in helium and nitrogen gas could indicate that interactions with the drift gas also plays a role in the CCS values and trends. The exponent in helium is the closest to 2/3, but not equal. The interpretation of both A and exponential factor values should therefore not only focus on the geometry or charge solvation capacity of the analyte ions, but also on its interaction with the drift gas. Our study illustrates another interest of recording CCS trends in helium and compare them with nitrogen values on the same analytes, having defined shapes and apparent densities [START_REF] Bleiholder | Molecular Structures and Ion Mobility Cross Sections: Analysis of the Effects of He and N2 Buffer Gas[END_REF].

In figure 6b, the comparison of the DT CCSHe values of phosphoric acid cluster anions with that of the tuning mix ions [START_REF] Marchand | Drift Tube Ion Mobility: How to Reconstruct Collision Cross Section Distributions from Arrival Time Distributions?[END_REF] highlights that both sets cover a similar range of m/z but the phosphoric acid cluster anions cover a broader range of charge states and CCS. Covering several charge states is an advantage as it is currently deemed safer to match the charge state of the analyte and the calibrant [START_REF] Gelb | A study of calibrant selection in measurement of carbohydrate and peptide ion-neutral collision cross sections by traveling wave ion mobility spectrometry[END_REF][START_REF] Lavanant | Collision cross sections of negative cluster ions of phosphoric acid in N2 determined by drift tube ion mobility and their use in travelling wave ion mobility[END_REF]. The different charge states follow different mobilitym/z trends (Figure 6c).

In TWIMS, it is also recommended to match molecular classes [START_REF] Gabelica | Recommendations for reporting ion mobility Mass Spectrometry measurements[END_REF]. This latter recommendation is not always complied with and would require further investigation. In fact, the polymer poly-(ethylene oxide) monomethyl ether (PEO) have been shown to be good calibrants for peptides, in the positive ion mode, for tryptic peptides of BSA [START_REF] Haler | Comprehensive Ion Mobility Calibration: Poly(ethylene oxide) Polymer Calibrants and General Strategies[END_REF]. One condition to obtain a reliable calibration is to ascertain that the ion mobilities of the ions are comparable in different ion mobility setups, which was shown for PEO polymers [START_REF] Haler | Comparison of Different Ion Mobility Setups Using Poly (Ethylene Oxide) PEO Polymers: Drift Tube, TIMS, and T-Wave[END_REF].

A comparison of the reduced mobilities (Figure 6c) shows that the singly charged tuning mix ions alone cover a wide interval ranging from 1.82 to 4.55 cm 2 V -1 s -1 and that, except for the reduced mobility of the P#4 -(at 5.65 cm 2 V -1 s -1 ), the differently charged phosphoric acid cluster ions fall collectively in an analogous range between 2.74 and 4.36 cm 2 V -1 s -1 . Reduced ion mobilities K0 are more often used for calibration of trapped ion mobility devices (TIMS) [START_REF] Silveira | High Resolution Trapped Ion Mobility Spectrometery of Peptides[END_REF][START_REF] Hernandez | Ion dynamics in a trapped ion mobility spectrometer[END_REF]. Here, the advantage of phosphoric acid cluster anions would be to provide many values in a given ion mobility range. Table 1. DT CCSHe (Å 2 ) and K0 (cm 2 V -1 s -1 ) for phosphoric acid cluster anions with charge state up to 4-with type A statistical analysis (σK0 is the standard deviation of the reduced mobilities, u is the experimental standard deviation of the mean (calculated as σ/√𝑛) , σ is the standard deviation on DT CCSHe and n is the number of replicates).

(cm 2 V -1 s - 1 ) (cm2 V-1 s- 1) (Å 2 ) (Å 2 ) (cm 2 V -1 s -1 ) (cm 2 V -1 s - 1 ) (Å 2 ) (Å 2 )

Conclusions

Phosphoric acid is a very affordable compound that forms cluster ions by direct infusion of its solution into an electrospray ionization source. In the present paper, we assessed the influence of activating parameters before and after the ion mobility cell, on their DT CCSHe values. Different experimental conditions operated in the area before the ion mobility device expectedly produced different proportions of singly and multiply charged species, while different post-IMS conditions affected the profile of the extracted ion mobility spectra and the ion mobility peak signals. Indeed, dissociation of the phosphoric cluster anions which correspond to cleavage of hydrogen bonds after the ion mobility device could be observed for cluster ions with low charge states and aggregation number n.

After a careful identification of the peaks corresponding to the intact clusters, we observed that DT CCSHe values determined by classic multi-field measurements did not vary with pre-IMS or post-IMS experimental conditions. The relative standard deviation of DT CCSHe from three to ten replicates was less than 1.7%. The propagation of the errors on DT CCSHe from the physical and instrumental parameters led to a relative combined uncertainty of 2%. This collection of seventy-five DT CCSHe values for phosphoric cluster anions come as a first set of data for which the influence of different instrumental settings was assessed. This extensive set of values which range four different charges states for negative ions (1-to 4-) and DT CCSHe values from 95.1 to 641.7 Å 2 (K0 values from 5.861 to 2.741 cm 2 V -1 s -1 ), are all associated with one single and very accessible compound. As they are regularly spaced and close in value, these DT CCSHe values could be used in the calibration of non-uniform electric field drift cells. Using these values for calibration of TWIMS and TIMS will be the object of future work.

Electronic supplementary information

Detailed experimental conditions and related voltages used on the Agilent 6560 IMS-QTOF and Synapt G2 instruments; linear regression and evaluation on uncertainty for slope and intercept; full mass spectra of singly and multiply phosphoric acid cluster ions, ammonia and sodium adducts; CID spectrum obtained by ion mobility tandem mass spectrometry for a sample triply charged ammonia adduct; experimental DT CCSHe values (Å 2 ) of tuning mix and sample H3PO4 cluster anions; correlation between the DT CCS(He) with DT CCS(N2) values. .287 s V m -1 and intercept 0.005 m V -1 with standard errors of 0.019 s V m -1 and 3 10 -5 m V -1 respectively . R 2 was higher than 0.9999. The zoomed region shows the 95% confidence bands.

For Table of Contents Only

Figure 1 .

 1 Figure 1. ESI DTIMS in negative ion mode of a 5 mM solution of phosphoric acid: colour maps of m/z vs drift time (ms) obtained a) in soft pre-IMS and soft post-IMS conditions and b) in harsh pre-IMS conditions and soft post-IMS conditions at 10 V/cm in the drift tube. The inset on the left shows an enlarged area where two populations of triply charged ions are seen framing quadruply charged ions. We named these two populations metastable ions (grey) and main phosphoric acid cluster ions (blue). The extracted mass spectra of the metastable and main populations around 21 and 24 ms respectively showed the same m/z values, corresponding to the isotopic distribution of [(H3PO4)48 -3H] 3-(inset on the right in (a)).

Figure 2 .

 2 Figure 2. Absolute intensities of metastable ions (grey) multiplied 10 times and absolute intensities of the main stable (blue) phosphoric acid cluster ions present in soft pre-IMS/ soft post-IMS conditions as a function of the reduced electric field (Td) applied in the ion mobility cell during the multi-field measurements. Black lines are only a guide for the eyes.

  2-), m/z 978.76 (blue) (P#10 -and P#20 2-) and m/z 1027.75 (magenta) (P#21 2-) under the four experimental conditions corresponding to (a) harsh pre-IMS/ soft post-IMS, (b) harsh pre-IMS/harsh post-IMS, (c) soft pre-IMS/ soft post-IMS and (d) soft pre-IMS/harsh post-IMS. Comparing panel (a) with (c) and panel (b) with (d) in Figure

Figure 3 .

 3 Figure 3. Ion mobility spectra of m/z 782.80 (black) (P#8 -and P#16 2-), m/z 880.78 (red) (P#9 - and P#18 -), m/z 978.76 (blue) (P#10 -and P#20 2-) and m/z 1027.75 (magenta) (P#21 2-) extracted at four different combinations of experimental conditions applied in the pre-IMS and post-IMS: (a) harsh pre-IMS/ soft post-IMS, (b) harsh pre-IMS /harsh post-IMS, (c) soft pre-IMS / soft post-IMS and (d) soft pre-IMS/harsh post-IMS tunings are shown. The EIM spectra presented above were recorded at a drift field of 5 V cm -1 in the ion mobility region. The omitted y axes correspond to the signal intensity in arbitrary units normalized to the highest peak on display.

Figure 4 .

 4 Figure 4. DT CCSHe values (Å 2 ) at 296.6 K for tuning mix anions m/z 602 (a), 1034 (b), 1334 (c), 1634 (d), 1934 (e), and 2234 (f). Green diagonal crosses are DT CCSHe values determined from soft pre-IMS parameters and harsh pre-IMS parameters from tuning mix solutions. Blue crosses are DT CCSHe values determined from harsh pre-IMS parameters and harsh pre-IMS parameters from an Agilent tuning mix solution in mixture with phosphoric acid. Blue circles are DT CCSHe values determined from harsh pre-and post-IMS parameters from a pure Agilent tuning mix solution. The mean experimental value is represented as a thick black horizontal line with error bars representing the standard deviation σ. The Anal Chem 2017 [36] values at 296 K with their standard deviations σ are represented on the right with a dark horizontal mark.

Figure 5 .

 5 Figure 5. DT CCSHe values (Å 2 ) at 296.6 K for eight sample H3PO4 cluster anions with charge state between 1-and 4-: (a) P#8 -m/z 782.8, (b) P#10 -m/z 978.8, (c) P#20 2-m/z 978.8 (d) P#25 2- m/z 1223.7, (e) P#42 3-m/z 1370.6, (f) P#52 3-m/z 1397.3 (g) P#67 4-m/z 1640.1 (h) P#70 4- m/z 1713.6. Green: DT CCSHe values determined using soft pre-IMS tuning. Blue: DT CCSHe values determined using harsh pre-IMS. Open symbols (green squares and blue circles): DT CCSHe values determined from experiments with a soft post-IMS gradient. All other DT CCSHe values were determined from harsh post-IMS conditions. Green diagonal crosses and blue crosses: DT CCSHe values obtained from a mixture of phosphoric acid and tuning mix. All other DT CCSHe values were obtained from a pure phosphoric acid solution. The mean experimental value is represented as a thick black horizontal line with error bars representing the standard deviation σ (n = 10).

Figure 6 .

 6 Figure 6. a) DT CCSHe of phosphoric acid cluster anions vs their aggregation number n b) DT CCSHe vs m/z c) reduced ion mobilities K0 vs m/z for phosphoric acid cluster anions and tuning mix ions. The different charge states for H3PO4 cluster anions are shown by green crosses (z = 1 -), orange circles (z = 2 -), blue triangles (z = 3 -) and magenta diamonds (z = 4 -). Singly charged tuning mix anions (m/z 602 to 1934) are represented by black full squares. Error bars in panel (a) represent the 2% relative standard uncertainty. Error bars with the observed relative standard deviation from all replicates would have nearly the same size as the symbols.

Figure S1 .

 S1 Figure S1. Linear regression of P#8 -phosphoric acid cluster anion ([(H3PO4)8 -H] -). The slope was 10.287 s V m -1 and intercept 0.005 m V -1 with standard errors of 0.019 s V m -1 and 3 10 -5 m V -1 respectively . R 2 was higher than 0.9999. The zoomed region shows the 95% confidence bands.

Figure S2 .

 S2 Figure S2. Phosphoric acid cluster anions mass spectrum extracted for charge state up to 4obtained in soft pre-IMS and soft post-IMS condition using an electric field of 5.0 V cm -1 in helium. The insets show doubly charged (orange) and triply charged (blue) ammonia and sodium adducts.

Figure S3 . 3 Figure S4 .

 S33S4 Figure S3. Extracted ion mobility spectra of m/z 880.78 (P#9 -and P#18 -and their precursor ions). Ion mobility with soft pre-IMS conditions: green traces; Ion mobility with harsh pre-IMS conditions: purple and light blue traces. Harsh post IMS gradient: full lines (green and purple). Soft post IMS gradient: dashed lines (light green and light blue). The omitted y axes correspond to the signal intensity in arbitrary units normalized to the highest peak on display; the intensity corresponding to the signal of the singly charged cluster ions were multiplied by 3, similarly to figure3.

Figure S5 .

 S5 Figure S5. Correlation between the DT CCS(He) values in this study with DT CCS(N2) values determined in our previous work [ref 19]. Green crosses are singly charged phosphoric cluster anions, orange circles are doubly charged charged phosphoric cluster anions, and blue squares are triply charged charged phosphoric cluster anions.

Table S1 (

 S1 a). List of the tuning parameters applied in the pre-IMS region

			Soft or harsh PRE-IMS	Soft or Harsh PRE-IMS
	Experimental conditions	Soft POST-IMS	Harsh POST-IMS
		Source: gas temperature	220°C	220°C
		Source: drying gas	1.5 L/min	1.5 L/min
		Source: Nebulizer pressure	9 psig	9 psig
		Source: capillary	-3500V	-3500V
		Optics 1: FRAGMENTOR	soft: -300 V	soft: -300 V
			harsh: -600 V	harsh: -600 V
		IM front funnel: high pressure	-110 V	-110 V
		funnel delta
		IM front funnel: high pressure	100-180 Vp-p	100-180 Vp-p
	Pre-IMS Tuning	funnel RF IM front funnel: trap funnel delta IM front funnel: trap funnel RF IM front funnel: trap funnel exit IM trap: trap entrance grid low	-140 V 160 Vp-p -10 V -70 V	-140 V 160 Vp-p -10 V -70 V
		IM trap: trap entrance grid delta	-2 V	-2 V
		IM trap: trap entrance	-69 V	-69 V
		IM trap: trap exit	-67 V	-67 V
		IM trap: trap exit grid 1 low	-64 V	-64 V
		IM trap: trap exit grid 1 delta	-5 V	-5 V
		IM trap: trap exit grid 2 low	-63 V	-63 V
		IM trap: trap exit grid 2 delta	-9 V	-9 V
		Acquisition: trap fill time	1000 s	1000 s
		Acquisition: trap release time	200 s	200 s
		IM drift tube: drift tube exit	-210 V	-210 V

Table S1 (

 S1 b). List of the tuning parameters applied in the post-IMS region.

		Soft or harsh PRE-IMS	Soft or Harsh PRE-IMS
	Experimental conditions	Soft POST-IMS	Harsh POST-IMS
	IM rear funnel: rear funnel	-200V	-200V
	entrance
	IM rear funnel: rear funnel RF	180 Vp-p	180 Vp-p
	IM rear funnel: rear funnel exit	-26 V	-35 V
	IM rear funnel: IM Hex Entrance	-24 V	-32 V
	IM rear funnel: IM Hex Delta
	Post-IMS Tuning

Table S2 .

 S2 List of supplementary experimental parameters used to perform ion mobility mass spectrometry measurement on the Synapt G2.

	ESI-V Resolution mode	DC potentials (V)	Traveling wave parameters
	m/z range	50 -4000 Trap Collision Energy	5	Source Wave Velocity (m/s)	300
	Capillary (kV)	2.7	Transfer Collision Energy	0	Source Wave Height (V)	0.2
	Source Temperature (°C)	90	Trap DC Entrance	3	Trap Wave Velocity (m/s)	300
	Sampling Cone (V)	60	Trap DC Bias	40	Trap Wave Height (V)	0.5
	Extraction Cone (V)	2	Trap DC	-1	IMS Wave Velocity (m/s)	350
	Source Gas Flow (mL/min)	20	Trap DC Exit	0	IMS Wave Height (V)	20 or 15
	Desolvation Temperature (°C)	250	IMS DC Entrance	25	Transfer Wave Velocity (m/s)	350
	Cone Gas Flow (L/hr)	0	Helium Cell DC	35	Transfer Wave Height (V)	5
	Desolvation Gas Flow (L/hr)	500	Helium Exit	-5
	Trap Gas Flow (mL/min)	0.4	IMS Bias	0
	Helium Cell Gas Flow	180	IMS DC Exit	0
	IMS Gas Flow (mL/min)	70	Transfer DC Entrance	defa ult
	Sample Infusion Flow Rate (µL/hr)	300	Transfer DC Exit	defa ult

Table S3 :

 S3 Experimental DT CCSHe values (Å 2 ) of figure4for tuning mix anions, in helium at 296.6 K compared to the Anal Chem 2017[START_REF] Zeleny | On the ratio of the velocities of the two ions produced in gases by Röntgen radiation; and on some related phenomena[END_REF] DT CCSHe at 296 K.

			MIXTURE WITH H3PO4 (JANUARY 2019)		TUNEMIX anions (MAY 2018)	All experiments	Anal Chem 2017 [1]
	m/z	Soft-pre-IMS (F 350V)	Harsh pre-IMS (600 V)	Harsh pre IMS (600 V)	Mean DT CCSHe (Å 2 )	σCCS ((Å 2 ))	DT CCSHe (Å 2 )	σCCS ((Å 2 ) n = 3
									n=6
	602	118.7	119.0	119.0	118.8	118.2	119.0	118.8	0.3	118.6	1.5
	1034	191.2	191.5	191.3	191.0	189.1	190.1	190.7	0.9	190.0	1.3
	1334	215.9	216.6	216.7	216.6	214.9	216.6	216.2	0.7	215.5	1.4
	1634	245.4	245.8	245.9	245.8	242.7	245.4	245.2	1.2	244.3	1.5
	1934	273.7	273.1	272.9	273.1	271.6	273.4	273.0	0.7	272.2	1.7
	2234	296.1	296.8	297.0	297.1	295.5	298.2	296.8	0.9	295.8	1.9

Table S4 .

 S4 Experimental DT CCSHe values (Å 2 ) at 296.6 K of figure5for sample H3PO4 cluster anions with charge state between 1-and 4-in different experimental conditions The last columns represent the mean CCS ( DT CCSHe) and the standard deviation of CCS (σCCS) values. .3 292.4 292.3 291.2 289.7 291.6 290.5 290.9 289.9 291.1 1.0 1370.7 42 3-401.7 401.3 400.5 400.1 401.1 400.0 400.1 400.1 399.9 397.9 400.3 1.0 1697.3 52 3-451.8 453.7 454.3 453.4 451.0 451.2 450.8 450.9 452.7 449.9 452.0 1.5 1640.1 67 4-539.6 540.8 540.4 538.8 540.7 541.3 539.1 538.2 537.8 536.6 539.3 1.5 1713.6 70 4-553.8 554.1 554.8 553.8 549.4 550.3 553.2 551.5 553.2 552.0 552.6 1.8

				Harsh post-IMS	Soft post-IMS	All experiments
	m/z P# Charge	Soft pre-IMS (F 350V) January 2019	Soft pre-IMS (F 300V) May 2018	Harsh pre-IMS (F 600V)	Soft pre-IMS (F 300V)	Harsh pre-IMS (F 600V)	Mean DT CCSHe (Å 2 )	σCSS (Å 2 ) n =10
	782.8 8	1-	149.2 149.3 152.0 149.9 150.3 152.0 151.1 150.6 152.7 152.3 150.9	1.3
	978.8 10	1-	172.4 173.0 171.7 170.5 168.9 168.0 169.1 170.3 172.3 170.1 170.6	1.7
	978.8 20	2-	255.3 254.6 255.3 254.6 256.5 255.3 254.5 254.9 255.2 255.5 255.2	0.6
	1223.7 25	2-	292.0 290

Acknowledgements

This work was financed by Région Normandie (GREENCHEM), the Labex SynOrg (ANR-11-LABX-0029) and the European Regional Development Fund (ERDF HN0001343). The DTIMS experiments were performed at the Plateforme de BioPhysicoChimie Structurale at IECB.

Supporting Information

Tel : +33235522932 helene.lavanant@univ-rouen.fr

Electronic supplementary information

Detailed experimental conditions and related voltages used on the Agilent 6560 IMS-QTOF and Synapt G2 instruments; linear regression and evaluation on uncertainty for slope and intercept; full mass spectra of singly and multiply phosphoric acid cluster ions, ammonia and sodium adducts; CID spectrum obtained by ion mobility tandem mass spectrometry for a sample triply charged ammonia adduct.


L Beauregard
email: laurent.beauregard@isae-supaero.fr

Dr A Urbano

D Colbeck

Dr S Lizy-Destrez

J Morlier

CONFERENCE FOR AERONAUTICS AND AEROSPACE SCIENCES (EUCASS) Multidisciplinary Design Optimization of a Reusable Lunar Vehicle

The Lunar Orbital Platform-Gateway will be the successor to the ISS and will be placed around the Moon. To bring crew onto the lunar surface, a lunar lander must be designed and used. This work will present a system design tool for lunar landers which utilizes OpenMDAO, a multidisciplinary design optimization library. Moreover, different mission architectures will be compared independently. As a benchmark, a design for a one-stage LH2/LOX will be produced and compared to an existing design.

Introduction

With its end of life approaching, the international community is looking for a replacement to the International Space Station (ISS). The leading candidate is a space station, called the Lunar Orbital Platform-Gateway (LOP-G) or Gateway, placed in a L2 southern Near Rectilinear Halo Orbit (NRHO). The orbit would have a periapsis altitude of 1500 km, an apoapsis altitude of nearly 70,000 km and a period of roughly 6 days and 13 hours. This station would serve as a hub for human exploration and scientific progress. The NRHO is strategically chosen to have constant Earth communication, nearly no solar eclipses, low ∆v maintenance cost, high observability of the lunar south pole, relative ease of access from the Earth and decent access to the lunar surface [START_REF] Whitley | Options for staging orbits in cislunar space[END_REF]. A dedicated vehicle will be required to bring crew and payload on the surface. However no design for this "lunar shuttle" has been chosen yet by the international community. Objectives for this vehicle are varied but often include the ability to bring 2 to 4 crew members unto the lunar surface for a duration of 2 to 14 days. The vehicle should have some level of reusability and be refueled at the LOP-G in between the missions. The present work focuses on two objectives, first to establish a list of viable mission architectures and the second is the development of a tool for the preliminary sizing of a lunar lander. To tackle the design of such complex systems, Multidisciplinary Design Optimization (MDO) has gained significant traction in the past decades and has had a number of successes in the field of aerospace [START_REF] Jasa | Design and Trajectory Optimization of a Morphing Wing Aircraft[END_REF] [START_REF] Gray | Aero-propulsive Design Optimization of a Turboelectric Boundary Layer Ingestion Propulsion System[END_REF]. Application to the space sector has also seen a recent surge of interest [START_REF] Balesdent | Multidisciplinary Design Optimization of Launch Vehicles[END_REF] [START_REF] Hosseini | Multidisciplinary design optimization of an expendable launch vehicle[END_REF]. However few research has focused on applying MDO to the design of a reusable lunar lander. In this work, a MDO software using the OpenMDAO library [START_REF] Gray | Openmdao: an opensource framework for multidisciplinary design[END_REF] is used as a preliminary design tool. In chapter 2, the concepts, nomenclature and mathematical framework behind MDO are introduced. In section 3, the different disciplines of propulsion, trajectory and structure are introduced along with the objective function. Chapter 4 will described the results obtained: first optimal trajectories will be analyzed, then mission architectures will be optimized independently and lastly a MDO will be performed for a one-stage LH2/LOx lunar lander and compared to an existing design.

Multidisciplinary analysis and optimization

At the heart of a MDO is the solving of a nonlinear optimization problem, the formulation used in this article is min x,y,z f (x, y, z)

Subject to g(x, y, z) = 0 h(x, y, z) ≥ 0 y = C(x, y, z) R(x, y, z) = 0 (1)
where x ∈ R N are called the design variables, y ∈ R D are the coupling variables and z ∈ R S are the state variables

f : U f ⊆ R N × R D × R S → R is the objective function to minimize g : U g ⊆ R N × R D × R S → R M eq are the equality constraints h : U h ⊆ R N × R D × R S → R M ineq are the inequality constraints C : U C ⊆ R N × R D × R S → R D are the disciplines R : U R ⊆ R N × R D × R S → R S are the residuals
The design variables x are independent parameters that can be modified to optimize the system. The coupling variables, y are outputs of the disciplines. The states variables z are internal parameters that are necessary in the computation of the various models.

Considering the disciplines C(x, y, z) and residuals variables R(x, y, z), the process of solving for z and y given a choice of design variables x is called a multidisciplinary analysis (MDA). Such system can be solved in a variety of ways; in general, a Newton's method or quasi-Newton's method is employed to this effect. For the discipline equations C, fixed point iteration schemes, such as Gauss-Seidel or Jacobi method can be used.

In this article, only monolithic architectures are considered, that is, only one system optimizer is present. Within monolithic architectures, several methods to optimize the system are applicable. The distinctive feature between these methods is how the variables y and z are handled by the optimizer, either solving for them explicitly at every step of the optimization or letting the optimizer handle the variables. These choices separates the architecture into 4 categories, see table 1 • Multidisciplinary Feasible -MDF

• Individual Discipline Feasible -IDF

• Simultaneous Analysis and Design -SAND

• All-At-Once -AAO Table 1: Types of monolithic optimizer y is solved y is free z is solve MDF IDF z is free SAND AAO
Whichever architecture is ultimately picked, an optimization algorithm must be chosen, those include BFGS, COBYLA, CG, Powell, Nelder-Mead and SLSQP among others. The choice of the algorithm depends on the nature of the problem, the constraints present and whether analytical derivatives are available. Refer to the documentation of the optimization toolbox of Python for more information [START_REF]The SciPy community[END_REF].

Models

To perform multidisciplinary analyses and optimization of the system, models of the subsystems must be chosen, along with an objective function. For a lunar lander system, 3 main subsystems have been identified: propulsion, trajectory and structure. These disciplines are coupled through inputs and outputs.

Propulsion

The working principle of a chemical rocket engine is that reactants are combined in a chamber generating a vast amount of energy in the form of heat which is then converted to mechanical work by a nozzle. In this work, only liquid-fuel engines are considered. The objective of the propulsion discipline is to calculate the specific impulse I sp , rocket engine mass M Engine and tank mass M t from the type of reactants, mixture ratio R m , combustion chamber pressure P c , expansion ratio and total propellant M p . The propulsion module is separated into 3 subdisciplines, the combustion analysis, the fluid expansion and the tank system.

The state of the art for combustion analysis is the NASA computer program Chemical Equilibrium with Applications (CEA). Given the fuel type, R m and P c , CEA computes the combustion temperature T c , the adiabatic index γ and the average molecular weight M w of the combustion products. Given an expansion ratio, , one can solve for the pressure at the exit,

P e = γ -1 γ + 1 2 γ + 1 1 γ-1 1 P e P c 2 γ -P e P c γ+1 γ (2)
Where R = 8.314 J K mol is the ideal gas constant. From the variables above, the speed of the exhaust v e can be found to be

v e = 2γ γ -1 RT c M w         1 - P e P c γ-1 γ         (3)
The (vaccum) specific impulse can be found to be

I sp = 1 g 0         v e + RT c M w v e P e P c γ-1 γ         (4)
In this work, the estimation of the mass of the rocket engines will be based on regression of existing engines. The choice of the model has an impact on the optimization possibilities, for example, some models includes the effects of the expansion ratio and some do not. Trying to optimize the expansion ratio with a model that does not accurately take it into account will result in nonsensical results.

M Engine [kg] = T [N] 7.81 × 10 -4 + 3.37 × 10 -5 √ + 59 (5) M Engine [t] = 0.0135 × N 0.4118 eng × T [kN] 0.471 × m propellant [t] 0.3574 (6) M Engine [kg] = 0.00514 × T [N] 0.92068 (7) M Engine [kg] = T [N] 107.4 ln(T [N]) -792 ( 8
)
where T is the thrust of the engine, N eng is the number of engines, m propellant is the mass of propellant. With equations 5, 6, 7 and 8 coming from [1][16][21] and [START_REF] Larson | Space propulsion analysis and design[END_REF] respectively. Other regressions are considered in Castellini [START_REF] Castellini | Multidisciplinary Design Optimization for Expendable Launch Vehicles[END_REF].

Given a propellant mass M p and mixture ratio R m , the mass of fuel M F and oxidizer M O are given by

M F = 1 1 + R m M p (9)
M O = R m 1 + R m M p (10)
Tank mass depends on the geometry of the vessel, usual shapes are spheres or cylinders (or shapes approximating those). The approximate relation between the tank mass, M t , the pressure of the fluid P, the volume V, the density of the tank material ρ t and the maximum allowable stress σ is given by

M t = S F cPV ρ t σ ( 11
)
where c is a constant that depends on the geometry of the vessel, c = 3/2 for a sphere, c = 2 for a cylinder and S f ∼ 2 is a safety factor. Since the volume of the fuel/oxidizer is given by the total mass of fluid M L divided by its density ρ L a near linear relationship exist between the tank mass and the fuel/oxidizer mass.

M t = S F c P σ ρ t ρ L M L = αM L (12)
This constant can be obtained from various existing tanks [START_REF] David | Mass estimating relations[END_REF] • Hydrogen, α = 0.128

• Liquid oxygen, α = 0.0107

• RP-1, α = 0.0148 • LCH4, α = 0.0287

Trajectory

Efficient ascent and descent trajectories to and from the surface of Moon are primordial for mass saving. The input/output of the trajectory discipline are summarized at the end of this section in table 3. The trajectory can be modeled in two phases as shown in figure 1.

• Phase 1: Near Rectilinear Halo Orbit (NRHO) to Low Lunar Orbit (LLO).

• Phase 2: LLO to lunar surface

¨ r = - µ E | r -r E | 3 r -r E - µ M | r -r M | 3 r -r M -ω × ω × r -2 ω × ˙ r ( 13
)
where r is the vectorial position of the spacecraft with respect to the Earth-Moon Barycenter, µ E and µ M are the standard gravitational constant of the Earth and Moon respectively and ω is the vectorial representation of the rotation of the Earth-Moon system. The thrust to weight ratio (TWR) of the lander is assumed to be greater than one because it is necessary to lift-off from the lunar surface, this allows the maneuvers to be considered impulsive as a first order estimate. The problem then becomes a boundary value problem. Values of ∆v from the NRHO to LLO varies between 730 m/s to 900 m/s depending on when during a lunar orbit the transfer is performed [START_REF] Whitley | Options for staging orbits in cislunar space[END_REF].

Phase 2's dynamics can be modeled by the following system of differential equations

ṙ = v r (14) θ = v t r (15)
vr = T m sin(α) - µ M r 2 + v 2 θ r ( 16
) vt = T m cos(α) - v t v r r ( 17
) ṁ = - T w ( 18
)
where r is the distance from the craft to the center of the Moon, θ is the angle of the craft with respect to a reference point on the surface (usually the starting point), v r is the radial component of the velocity, v θ is the tangential component of the velocity, µ M is the standard gravitational parameter of the Moon, m is the total mass of the system, T is the thrust generated by the engines, w = I sp g 0 is the effective exhaust velocity and α is the pitch angle. Since gravity losses Figure 2: Geometry of the ascent/descent trajectory variables are much more important in phase 2 than phase 1, the application of optimal control theory is of greater benefit.

The subject of optimal control is a vast and still is an ongoing area of research. The problem is usually formulated in the following manner

min u {S } (19)
where

S = φ(t f , x f ) + t f t i L (t, x, u) dt (20) subject to ẋ = f (t, x, u) (21)
h(t, x(t), u(t)) ≥ 0 (22) Φ(t f , x f ) ≥ 0 ( 23
)
where S is composed of two costs: φ, the boundary cost and L, the running cost, x are the state variables, u are the control variables, f (t, x, u) is the dynamics of the states, h(t, x, u) are path constraints and Φ are boundary constraints. Software developed to solve such problems broadly fall into two main categories: indirect vs direct methods. In this work, only direct methods will be considered. Direct methods discretizes the dynamics/variables/cost of the system.

This discretized system is then fed into a nonlinear optimizer such as SNOPT or IPOPT. A broad class of algorithms are called "Pseudospectral methods" where both the states and the control variables are discretized and approximated by orthogonal polynomials [START_REF] Herman | Direct optimization using collocation based on high-order gauss-lobatto quadrature rules[END_REF].

For the trajectory discipline, the objective function of the optimal control problem only has a boundary term: the amount of fuel necessary to achieve orbit S = M p . The system dynamics is given by equations 14 to 18. The only constraint a priori is r ≥ R Moon , that is, the craft's trajectory must not intersect the Moon's surface. The optimal control was solved with the software "Dymos: Open Source Optimization of Dynamic Multidisciplinary Systems". Transcription methods in Dymos include Gauss-Lobatto Collocation and Radau Pseudospectral Method [START_REF] Falck | Optimal control within the context of multidisciplinary design, analysis, and optimization[END_REF]. Two cases will be considered: without surface obstacles and with obstacles given by an inequality constraints on the trajectory. While there are many options to take into consideration the obstacles from the terrain [START_REF] Ma | A unified trajectory optimization framework for lunar ascent[END_REF], one simple way to implement a trajectory constraint is to set

r(θ) ≥ r constr (θ) = R + kh min (Rθ) k(Rθ) + h min ( 24
)
This function has two defining properties: the slope k and the minimum height, h min

dr constr d(Rθ) | θ=0 = k (25) lim θ→∞ r constr = R + h min ( 26
)
The geometry of the constraint is shown in figure 3.

Structure

The purpose of this discipline is to evaluate the structural mass of the lander. Several models can be employed to model structural mass, among the simplest one is a fixed proportionality between the structure mass and the total mass of the system, usually called the structural efficiency σ.

ρ B = 1 + R m 1 ρ F + R m ρ O (30)
where ρ F and ρ O are the density of the fuel and oxidizer respectively. The regressions in equations 27, 28 and 29 can be found in [START_REF] Perczynski | Mass estimating relationships for manned lunar lander and ascent vehicle concept exploration[END_REF], [START_REF] Perczynski | Mass estimating relationships for manned lunar lander and ascent vehicle concept exploration[END_REF] and [START_REF]Lunar lander conceptual design[END_REF] respectively.

Objective function

In theory, the system should be optimized for the lowest possible cost while respecting the mission constraints. Although cost models have been considered before for launch vehicles [START_REF] Koelle | The transcost-model for launch vehicle cost estimation and its application to future systems analysis[END_REF],they carry larger uncertainties than engineering models, this is why surrogate variables are usually preferred. For Earth launchers, a common optimization metric is the total mass of the system M Total also known as the gross liftoff weight (GLOW) for a fixed payload [START_REF] Depince | New mdo approaches for launch vehicle design[END_REF]. For reusable lunar landers, there seems to be no universally agreed upon metric. To identify a relevant metric, two limiting cases are considered: when the lunar lander is fully expendable, the metric should scale as the total mass of the system M Total and considering an idealized case where the lander is infinitely many times reusable, then the cost should scale as the proprellant mass used M Propellant . In general, the metric is chosen to be the average mass used per mission, which is a linear combination of M Total and M Propellant

f Objective = M Total + N Reuse × M Propellant 1 + N Reuse ( 31
)
where N Reuse is the number of times the lander can be reused.

Results

Trajectory

The two most important parameters that influences the performance of an ascent/descent trajectory are the thrust to weight ratio (in lunar gravity) and the specific impulse of the vehicle. Without any constraints, the most efficient ascent trajectory is the one that grazes the lunar surface up to orbital speed followed by a Hohmann transfer to the correct height. While unreasonable trajectories in themselves, they still provide a lower bound for the ∆v or fuel necessary. The fuel mass ratio as a function of initial thrust to weight ratio (TWR) and specific impulse is depicted in 4. When physical obstacles are taken into considerations, the ∆v to reach orbit increases. The results of the optimal control simulation in Dymos is summarized in figure 5 Figure 4: Fuel fraction necessary to perform an ascent trajectory without constraints Figure 5: Fuel fraction necessary to perform an ascent trajectory with constraints

Mission analysis

The primary goal of the vehicle is to bring a payload from the NRHO to the lunar surface and back in orbit, schematically this is represented by figure 1. Even more abstractly, this mission architecture can be represented by figure 6, where "1" refers to the first (and only) vehicle. Up to now, only one-stage landers have been considered, however mission architectures with multiple stages can reduce the total mass of propellant used. The inclusion of multiple stages implies the non-trivial choice of the best mission architecture. An example of a two stages mission architecture in shown in figure 7, where double (or triple) lines indicates that two (or three) stages are moving together with the first number indicating which vehicle is providing the thrust. The number of possible mission architecture depends on the number of allowed transfers, for example, 5 transfers mean each craft can do a maximum of 5 transfers during the mission (including being ferried). A minimum of 4 transfers is necessary to perform the mission as one craft must at least perform the following sequence of transfers:

1) NRHO → LLO 2) LLO → Surface 3) Surface → LLO 4) LLO → NRHO
The number of possible mission architectures grows very rapidly with the number of allowed transfers as shown in table 5. However, the way the enumeration of valid mission architecture is done in this work leads to a large number of equivalent architectures which only differs by rearrangements. The mission architectures should be ranked with respect to an appropriate objective function (which will be described down below). However, the sheer size of the most numerous case considered (5 transfers, 3 stages) imposes the use of a simple and fast model so as to limit the computational time. A linear model is finally chosen as described here

M = P + T + F + S (32)
T = αF (33)

F = M 1 -e -∆v/w (34) S = βM (35)
Where P, T, F, S , M are the payload mass, the fuel tank mass, the fuel mass, the structure mass and the total mass, α, β, w are the tank mass to fuel mass ratio, the structural index and the effective exhaust velocity respectively. The advantage of such a simple model is that the whole system can be solved for analytically, the total mass is

M = 1 (1 + α)e -∆v/w -α -β P (36)
As mentioned previously, to rank different mission architectures, an objective function must be introduced. The one chosen in this work is the mass necessary to bring a unit payload from the NRHO to the lunar surface and back. The mass used depends on whether a given vehicle is reusable or non-reusable, examples are shown in figure 8 and9. For reusable vehicles, the mass penalty is given by the mass of propellant used for the mission, for non-reusable vehicles, the metric is the total mass of the vehicle. While this method attributes the same "cost" to bringing any type of mass to the NRHO and a more accurate model would give different weights to the type of material that is brought to orbit, it is reasonable, at first order, to assume the cost of bringing something to the NRHO is dominated by the launch cost. For this analysis, conservative values for α, β and a methalox value for the effective exhaust velocity w were chosen α = 0.05 (37) β = 0.1 (38)

w = 3.6 km/s (39)
The ordering of the best mission architectures are shown in figure 10 and 11, with the best architecture for a 2-stages and 3-stages architecture shown explicitely in figure 12 and 13. The lowest achievable mass fraction for different vehicle numbers are given in table 6. While the ordering of mission architectures depends on the values of α, β and w,

α = β = 0) 3.482
it is expected that there is no dramatic change in the optimal ordering if these coefficients are changed. The analysis can be performed again for a different set of constants α, β, w but only considering the N best scenarios found for the original set of parameters. This process essentially allows the "weeding out" of millions of unreasonable mission architecture. A more thorough study can then be done on the set of optimal mission architecture. Since the LMT lander is planned to be used approximately 10 times, the objective function of the MDO will consider N Reuse = 10. The payload of the lander is chosen to be similar to the mass of the Orion capsule plus one ton [START_REF]Orion quick facts[END_REF]. The engine chamber pressure and expansion ratio is set to the RL-10 engines [START_REF] Castellini | Multidisciplinary Design Optimization for Expendable Launch Vehicles[END_REF]. The optimization variables are the thrust and mixture ratio. After convergence, a preliminary design is obtained. Table 7 summarizes the results and compares them with the Lockheed Martin lander.

Conclusion

A tool to obtain preliminary designs of lunar landers was developed based on techniques of multidisciplinary design optimization. The three disciplines of propulsion, trajectory and structure were discussed and several models were presented. To compare with the existing Lockheed Martin design, an optimization with respect to the thrust T and mixture ratio R m was performed on a one stage LH2/LOX lunar lander, the comparison can be seen in table 7.

An exhaustive search for the best mission architecture was done in parallel to the MDO using a simpler model to limit the computational time. Scenarios were ordered by the amount of mass necessary to perform the mission. Scenarios for 2 and 3 stages architecture using the least amount of mass were identified. While the optimality of the scenario depends on the coefficient α, β and w of the model, a more thorough analysis can be done on the best mission architecture obtained so far.

Future work will include higher fidelity models, the addition of more disciplines such as the reaction control system, fuel feed system (turbopump, pressurization system), cooling system, three dimensional ascent/descent trajectories, more accurate gravity models and more accurate vehicle body dynamics, among others. In this work, the optimization of the mission architecture and the vehicle itself were done independently, the most significant step forward will be the combining of the mission architecture with the multidisciplinary design optimization. To this end, multi-stage systems with a framework general enough to include any mission architecture will have to be implemented in OpenMDAO.

Figure 1 :

 1 Figure 1: Schematic of the mission

Figure 3 :

 3 Figure 3: Geometry of the ascent trajectory with constraints (Not to scale)

Figure 6 :Figure 7 :

 67 Figure 6: Representation of a onestage lander mission

Figure 8 :

 8 Figure 8: Non-reusable architecture Figure 9: Reusable architecture

Figure 10 :

 10 Figure 10: Best 180 mission profile for 2 stage architecture

Figure 11 :

 11 Figure 11: Best 630 mission profile for 3 stage architecture

Figure 12 :Figure 13 :

 1213 Figure 12: Best 2 stage architecture (cost: 5.096)

Table 2 :

 2 Propusion discipline Input/Output

	Inputs	Symbol Outputs	Symbol
	Type of fuel/oxidizer	-	Combustion temperature	T c
	Mixture ratio	R m	Product molecular weight	M w
	Combustion chamber pressure	P c	Adiabatic index	γ
	Expansion ratio		Specific impulse	I sp
	Maximum thrust	T	Engine mass	M Engine
	Propellant mass	M p	Propellant tank mass	M t

Table 3 :

 3 Trajectory discipline Input/Output

	Inputs	Symbol Outputs	Symbol
	Thrust	T	Propellant mass	M p
	Total mass	M Total
	Specific impulse	I sp

  Typical values for structural efficiencies of Earth based launchers are σ ∼ 10% [17]. For lunar landers, different regression models are considered M Structure [t] = 0.0684 m dry [t] 1.7851 F -0.0645

		D	+ m payload [t] 0.9062 H D	0.6921	+ 0.4528	(27)
	M Structure [t] = 0.1882 m dry [t] × F D	0.2834 + 0.1055	H D	0.5179	+ 7.0574 × 10 -4 v H [m 3 ] 1.1652 + 0.057	(28)
	M Structure [kg] = 0.04928M Total [kg] + 390					(29)
	where v H is the habitable volume, H/D is the height to diameter of the system, F D =	ρ B 1150 kg/m 3 and ρ B is the bulk
	density given by

Table 4 :

 4 Structure discipline Input/Output

	Inputs	Symbol Outputs	Symbol
	Payload mass	T	Structure mass	M S
	Engine mass	M Engine
	Tank mass	M t
	Propellant mass	M p
	Mixture ratio	R m
	Habitable volume	v H
	Geometric factors	-

Table 5 :

 5 Number of valid mission architectures

		2 stages	3 stages
	# of transfer: 4	419	240,168
	# of transfer: 5	4,539	6,749,884
	# of transfer: 6 47,110 271,575,898

Table 6 :

 6 Best mission architectures

	Normalized mass used per mission

Table 7 :

 7 Single stage, LH2/LOX lunar lander

	Type	Parameter	Symbol Value Units LMT Lander
	Objective Average mass used	f Objective 40.64	t	42
	Fixed	Payload	M Payload	12.0	t	-
	Fixed	Combustion pressure	P c	24.0	bar	-
	Fixed	Expansion ratio		40.0	-	-
	Optimized Thrust	T	230.2	kN	266.8
	Optimized Mixture ratio	R m	5.17	-	-
	State	Total mass	M Total	56.92	t	62
	State	Thrust to weight ratio	TWR	2.27	-	2.66
	State	Propellant mass	M P	39.01	t	40
	State	Structure mass	M S	3.195	t	-
	State	Tank mass	M t	1.570	t	-
	State	Engine mass	M Engine 1.148	t	-
	State	Dry mass	M Dry	17.91	t	22
	State	Velocity increment	∆v	5148	m/s	5000
	State	Specific impulse	I sp	453.9	s	450+
	State	Combustion temperature	T c	3235	K	-

DOI: 10.13009/EUCASS2019-863

Acknowledgments

This work is part of the activities of the ONERA-ISAE-ENAC joint research group. This work was supported by the Advanced Space Concept Laboratory (SACLAB) and the industrial partners of Airbus and ArianeGroup.