Axel Gomez

Zeke A Piskulich

⇤, ‡ Ward H Thompson
email: wthompson@ku.edu

Damien Laage
email: damien.laage@ens.psl.eu

†pasteur

Water Diffusion Proceeds via a Hydrogen-Bond Jump Exchange Mechanism

The self-diffusion of water molecules plays a key part in a broad range of essential processes in biochemistry, medical imaging, material science and engineering. However, its molecular mechanism and the role played by the water hydrogen-bond network rearrangements are not known. Here we combine molecular dynamics simulations and analytic modeling to determine the molecular mechanism of water diffusion. We establish a quantitative connection between the water diffusion coefficient and hydrogen-bond jump exchanges, and identify the features that determine the underlying energetic barrier.

We thus provide a unified framework to understand the coupling between translational, rotational, and hydrogen-bond dynamics in liquid water. It explains why these different dynamics do not necessarily exhibit identical temperature dependences although they all result from the same hydrogen-bond exchange events. The consequences for the understanding of water diffusion in supercooled conditions and for water transport in complex aqueous systems, including ionic, biological, and confined solutions are discussed.

range of processes and systems, including, e.g., fuell cell membranes, 1 cryopreservation of biomolecules, 2 protection against corrosion, 3 and biomedical imaging of brain stroke and cancer. 4 However, a molecular picture of the mechanism that governs how water molecules move through the liquid and of the energetic barrier that determines the diffusion coefficient has remained elusive. A distinctive feature of water is the extended hydrogen-bond (H-bond) network that permeates the liquid. 5,6 Since each water molecule is connected to its nearest neighbors by H-bonds, any displacement beyond the molecular cage is expected to require a rearrangement of the H-bond network. However, while H-bond exchanges have been shown to be responsible for water reorientation, 7,8 such a connection has been missing for translational dynamics, despite the great importance that identifying the molecular translational diffusion mechanism would have for an improved understanding of water transport properties.

Water diffusion has often been described with random walk and jump diffusion models involving elementary hops. [9][10][START_REF] Kikutsuji | Diffusion Dynamics of Supercooled Water Modeled with the Cage-Jump Motion and Hydrogen-Bond Rearrangement[END_REF] However, these models have only been used as effective descriptions, and the molecular mechanism of these elementary hops and their connection with H-bond rearrangements have not been characterized. In addition, the molecular parameters obtained from these effective models were often not consistent with other dynamical studies: for example, the inferred delays between successive hops [9][10][START_REF] Kikutsuji | Diffusion Dynamics of Supercooled Water Modeled with the Cage-Jump Motion and Hydrogen-Bond Rearrangement[END_REF] did not match any known timescale for the H-bond network dynamics. In the absence of a characterization of the elementary events that would justify a jump diffusion description, it has also been suggested 6,[START_REF] Sciortino | Supercooled Water and the Kinetic Glass Transition[END_REF] that a continuous model would be preferable.

Here we establish an explicit connection between H-bond network rearrangements and molecular translational diffusion in liquid water. We use molecular dynamics (MD) simulations to characterize the mechanism that governs the translational displacements induced by H-bond exchanges, and show how an extended jump diffusion model provides an excellent description of water diffusion dynamics. We identify the molecular contributions to the diffusion energetic barrier, and explain why translational and rotational water dynamics, ature dependences. We finally discuss the consequences for water dynamics in complex environments.

To elucidate the impact of H-bond rearrangements on water translational dynamics, our starting point lies in the characterization of elementary H-bond exchange events that govern every change in the H-bond network connectivity. An H-bond exchange occurs when a water OH group switches H-bond acceptors, 7,8 and its mechanism has been shown via simulations to proceed through large-amplitude angular jumps, which cause the H-bond donating OH group to reorient 7,8 (Fig. 1a). Jump H-bond exchanges have been found in a wide range of aqueous systems, 8 and have been characterized by nonlinear vibrational spectroscopy experiments in aqueous salt solutions. [START_REF] Moilanen | Ion-Water Hydrogen-Bond Switching Observed with 2D IR Vibrational Echo Chemical Exchange Spectroscopy[END_REF][START_REF] Ji | Large Angular Jump Mechanism Observed for Hydrogen Bond Exchange in Aqueous Perchlorate Solution[END_REF] Because their mechanism involves the elongation and contraction of the H-bonds being broken and formed, 7,8 they are expected to induce translational displacements of the water partners, and prior simulations [START_REF] Mukherjee | Microscopic Origin of Temporal Heterogeneities in Translational Dynamics of Liquid Water[END_REF][START_REF] Liu | Interplay Between Translational Diffusion and Large-Amplitude Angular Jumps of Water Molecules[END_REF] showed that molecules that experience these jumps tend to diffuse faster. However, the precise impact of these jumps on the H-bond donor and acceptors' translational dynamics has thus far not been determined.

Prior works have established that classical MD simulations are a particularly incisive tool to investigate water H-bond exchanges. 7,8 Our present study employs classical MD simulations on liquid water at 298.15 K and the equilibrium density found at 1 atm (see SI for details). In the following, we present results obtained with the TIP4P/2005 potential, [START_REF] Vega | Simulating Water with Rigid Non-Polarizable Models: A General Perspective[END_REF] which provides a very good description of water dynamics and of its temperature dependence. [START_REF] Vega | Simulating Water with Rigid Non-Polarizable Models: A General Perspective[END_REF][START_REF] Piskulich | Examining the Role of Different Molecular Interactions on Activation Energies and Activation Volumes in Liquid Water[END_REF] Complementary results obtained with the SPC/E force field [START_REF] Berendsen | The Missing Term in Effective Pair Potentials[END_REF] are also reported and show that our model is robust vis-a-vis the choice of water potential.

To determine the translational displacements associated with H-bond exchanges in our molecular dynamics trajectory, we consider all the events where a water OH group (noted S2 for fit parameters).

O ⇤ H ⇤ in
employed to focus on exchanges between stable H-bonds. For each exchange, the transition state crossing is defined to be the instant t 0 when the O ⇤ H ⇤ group crosses the bisector plane between the initial and final acceptors with the largest angular velocity. Trajectories before and after t 0 are collected as long as neither the O ⇤ H ⇤ group nor any of the parent water molecule's other H-bonds experience another stable exchange; this ensures that the impact of a single exchange is considered. Averages are then calculated over the collection of exchange events.

Figure 1b shows that the breaking of the initial H-bond with O i and the formation of the new bond with O f lead to the elongation and contraction of the O ⇤ O i and O ⇤ O f distances, respectively. H-bond exchanges thus induce translational displacements of the water molecules involved and we now determine their respective amplitudes in the laboratory frame. In contrast to OH reorientation which occurs between stable orientations defined by the initial and final H-bond acceptor positions, 7,8 there is no stable position for the water oxygen that can be easily defined, e.g., based on the surrounding hydration shell configuration. We therefore calculate the oxygen velocity time-correlation function It is interesting to note that the sum of the donor and acceptor root-mean-squared jump am-

C vv (t) = h ! v O (t 0 ) • ! v O (t 0 + t)i
plitudes p h⇢ 2 O ⇤ i+ q h⇢ 2 O i,f i '1.
7 Å is almost equal to the distance between the first ('2.8 Å) and second (' 4.5 Å) shell peak positions in the oxygen-oxygen radial distribution function.

The average jump vectors for each water partner are schematically shown in Figure 1a and given in Table S1. Their orientations are not isotropically distributed in the local molecular frame: the average O ⇤ displacement lies in the O ⇤ O i O f plane and is parallel to the O i O f axis.

The O i and O f partners move away from and toward O ⇤ , respectively. The displacements of the three partners are thus evocative of a reaction coordinate at the transition state that is an antisymmetric vibrational mode, even if the trimer cannot be considered as an isolated system because of the surrounding liquid.

H-bond jump exchange dynamics are characterized by the jump time-correlation function (tcf) 1 hp i (0)p f (t)i where p i (resp. p f ) is 1 when the O ⇤ H ⇤ group forms a stable H-bond with O i (resp. O f ) and 0 otherwise. [START_REF] Laage | On the Molecular Mechanism of Water Reorientation[END_REF] Absorbing boundary conditions in the product state ensure that only a single H-bond exchange is considered. The jump tcf exhibits a single exponential decay (see SI Appendix Fig. S1) which suggests that jumps are well described by a Poisson process at ambient conditions. The H-bond jump time (inverse rate) is defined as 20

⌧ HB = Z 1 0 dt [1 hp i (0)p f (t)i] (1)
and is calculated to be ⌧ HB =3.657±0.007 ps. [START_REF] Piskulich | Examining the Role of Different Molecular Interactions on Activation Energies and Activation Volumes in Liquid Water[END_REF] Each water molecule is involved in successive H-bond exchanges, either as an H-bond donor via one of its two hydrogen atoms, or as an acceptor via its oxygen atom. To connect these exchanges to the resulting diffusive motion, we use the continuous time random walk (CTRW) model, [START_REF] Montroll | Random Walks on Lattices[END_REF] which generalizes the discrete random walk model to a continuous distribution from which waiting times between successive hops are randomly drawn. For simplicity, we consider that successive jumps on the same water molecule are independent, and adopt the CTRW form adapted to a single Poisson process. The water diffusion coefficient due to H-bond exchanges directly involving the molecule that diffuses is therefore the sum of independent CTRW contributions from the two H-bond donating hydrogen atoms and the two H-bond accepting sites on the oxygen atom (because we consider only stable H-bond exchanges, transient under-and over-coordination events do not need to be considered),

D HB = D don + D acc = 2 h⇢ 2 O ⇤ i 6⌧ HB + 2 h⇢ 2 O i,f i 6⌧ HB = h⇢ 2 O ⇤ i + h⇢ 2 O i,f i 3 ⌧ HB , (2)
and at 298.15 K our simulations yield D HB = (1.273 ± 0.004) ⇥ 10 5 cm 2 • s 1 (see Table S3).

However, water molecules are not necessarily immobile while they keep their H-bonds with their nearest neighbors intact. For reorientation, it has been shown 7,8 that an intact H-bonded pair of water molecules can tumble in the liquid, thus reorienting the local frame between successive H-bond jump exchanges and bringing a (minor) contribution to the overall reorientation. In a similar fashion, we determine the translational diffusion coefficient for a water molecule surrounded by the same four nearest neighbors. To circumvent the difficulty caused by the very short lifetime of such an intact cluster, we apply soft restraints between the central water molecule and each of its four neighbors (see SI Appendix), so that these H-bonds can transiently break but no stable exchange can occur. These restraints -which do not significantly perturb the local structure (see SI Appendix S3). This frame diffusion component that takes place between successive H-bond exchanges is thus small but non-negligible, and since frame diffusion and H-bond exchanges are statistically independent translational displacement sources, the overall diffusion coefficient results from the sum of the respective diffusion coefficients.

Frame diffusion presumably must arise from H-bond exchanges taking place at the interface between the H 2 O(H 2 O) 4 complex and the rest of the liquid; a CTRW modeling of the latter would be expected to involve the same ⌧ HB jump time but now affecting a larger number of H-bonds resulting in smaller amplitude displacements of the central water molecule.

The value of the H 2 O(H 2 O) 4 msd at short times, before the diffusive regime is established, can be used to estimate the amplitude of the fast molecular position fluctuations while the hydration shell is intact. Figure 2 shows that this amplitude is ' 0.7 Å, which is thus of the same order as the amplitude of translational jumps induced by H-bond exchanges (Fig. 1d).

The overlap between fast local fluctuations and jump displacement amplitudes explains why a visual inspection of the trajectory 6 does not easily reveal sudden discrete hops that are typically expected in a jump diffusion process. Our simulations are performed in finite-size boxes which are periodically replicated, and an additional hydrodynamic correction due to the periodic boundary conditions (pbc) [START_REF] Dünweg | Molecular Dynamics Simulation of a Polymer Chain in Solution[END_REF][START_REF] Yeh | System-Size Dependence of Diffusion Coefficients and Viscosities from Molecular Dynamics Simulations with Periodic Boundary Conditions[END_REF] should be included for comparisons with experimental values,

D corr = k B T ⇣ 6⇡⌘L ( 3
)
where T is the system temperature, ⌘ the shear viscosity, L the cubic simulation box side and ⇣ '2.837297. The shear viscosity was determined from the time integral of the pressure tensor time-correlation function, calculated from a separate set of simulations performed on the same system (see SI Appendix).

We can now express the overall diffusion coefficient as the sum of the diffusion coefficients coming from i) H-bond exchanges involving the central water molecule, ii) the local frame diffusion due to H-bond exchanges further away, and iii) the pbc correction:

D model = D pbc model + D corr = ⇣ D HB + D pbc frame ⌘ + D corr (4)
The resulting D model = (2.232 ± 0.006) ⇥ 10 5 cm 2 • s 1 value is in excellent agreement with the experimental value D NMR = 2.299 ⇥ 10 5 cm 2 • s 1 measured 24,25 by NMR at 298.15 K.

Even more importantly to assess the validity of our translational jump model, the agreement is also excellent between our model prediction and the value directly determined from the simulated mean-square displacement of a single water molecule and corrected for pbc,

D MD = D pbc MD + D corr = (2.37 ± 0.01) ⇥ 10 5 cm 2 • s 1 .
In addition, this very good agreement between model and simulations is obtained for both TIP4P/2005 and SPC/E force fields, as graphically summarized in Fig. 3a and detailed in Table S3. These results show that our jump model provides a quantitative description of water translational diffusion, and reveal that H-bond jump exchanges bring the major contribution to the overall translational diffusion. Because H-bond jumps go through a 5-coordinate transition-state, our results thus explain why prior simulations [START_REF] Sciortino | Effect of Defects on Molecular Mobility in Liquid Water[END_REF] had observed that overcoordinated water molecules tend to diffuse faster, but our model also stresses that such configurations are unstable.

The picture that emerges from this extended jump model for water translational dynamics is that diffusion primarily derives from H-bond exchanges involving the water molecule either as an H-bond donor or as an acceptor, with a minor but non-negligible contribution due to the local frame diffusion between successive H-bond exchanges.

We now briefly contrast this picture with prior jump descriptions of water translational diffusion. Prior studies 10,[START_REF] Kikutsuji | Diffusion Dynamics of Supercooled Water Modeled with the Cage-Jump Motion and Hydrogen-Bond Rearrangement[END_REF][START_REF] Liu | Interplay Between Translational Diffusion and Large-Amplitude Angular Jumps of Water Molecules[END_REF][START_REF] Chen | Molecular-Dynamics Study of Incoherent Quasielastic Neutron-Scattering Spectra of Supercooled Water[END_REF] considered that water structural dynamics could be separated into a fast rattling motion within a local basin, for example formed by the surrounding water molecules in the cage model, [START_REF] Chen | Molecular-Dynamics Study of Incoherent Quasielastic Neutron-Scattering Spectra of Supercooled Water[END_REF] and slower inter-basin jumps. However, our results in Fig. 2 show that water molecules continue to diffuse even when their four H-bond neighbors are unchanged, so that a separation into different mechanisms based on the displacement amplitude is not adequate. The approaches used in these prior studies have faced several limitations. For example, structural clustering based on a 0.7 Å minimum distance between local basins 10 ignored small-amplitude jumps (shown to be important in Fig. 1d) and consequently led to an overestimated '3 ps jump time at 297.1 K, 10 which is surprisingly long since it is close to the H-bond exchange time for a single OH group, while the ⇠ 4 H-bond partners of a given molecule would be expected to lead to four times more frequent jumps for the central oxygen. A cage-jump model was independently proposed, [START_REF] Kikutsuji | Diffusion Dynamics of Supercooled Water Modeled with the Cage-Jump Motion and Hydrogen-Bond Rearrangement[END_REF] where translational jumps occur only once all 4 H-bonds of a molecule break; this is in contrast to our results which show that a water molecule undergoes a translational jump as soon as any one of its 4 H-bonds is exchanged. Consequently this assumption leads to long ('10 ps) resting periods between successive translational jumps, not seen in molecular dynamics trajectories. A key advantage of our present study is that we identify the elementary jumps directly from H-bonds exchanges, and thus avoid the limitations due to coarse-graining and arbitrary thresholds used in these prior works.

Experimentally, jump models have been used to analyze water QENS spectra, 9,10,[START_REF] Teixeira | Experimental Determination of the Nature of Diffusive Motions of Water Molecules at Low Temperatures[END_REF][START_REF] Tassaing | The Dynamics of Supercritical Water: A Quasielastic Incoherent Neutron Scattering Study[END_REF][START_REF] Laage | Reinterpretation of the Liquid Water Quasi-Elastic Neutron Scattering Spectra Based on a Nondiffusive Jump Reorientation Mechanism[END_REF][START_REF] Noguere | Temperature-Dependent Dynamic Structure Factors for Liquid Water Inferred from Inelastic Neutron Scattering Measurements[END_REF] and translational jump time values close to the values obtained here have been shown to be consistent with experimental linewidths. [START_REF] Laage | Reinterpretation of the Liquid Water Quasi-Elastic Neutron Scattering Spectra Based on a Nondiffusive Jump Reorientation Mechanism[END_REF] However, QENS spectra probe displacements of water H atoms [START_REF] Teixeira | Experimental Determination of the Nature of Diffusive Motions of Water Molecules at Low Temperatures[END_REF] and, while the latter share the same diffusion coefficient as water O atoms, their motions result from the combination of larger-amplitude jumps when the H atom exchanges H-bond acceptors and smaller-amplitude jumps when the parent O atom experiences an H-bond exchange which does not involve this H. A specific translational jump model for water H atoms combining these two types of exchanges would thus provide a way to model QENS spectra without having to use the traditional but questionable translationrotation decoupling approximation. [START_REF] Laage | Reinterpretation of the Liquid Water Quasi-Elastic Neutron Scattering Spectra Based on a Nondiffusive Jump Reorientation Mechanism[END_REF] We now examine the temperature dependence predicted by our jump model for the diffusion coefficient and consider the activation energy,

E D a = k B T 2 @lnD @T (5)
Our motivation is twofold. First, this will assess whether our model can successfully describe water diffusion over a broad range of temperatures. Second, since the activation energy for a chemical reaction can be interpreted [START_REF] Tolman | Statistical Mechanics Applied to Chemical Kinetics[END_REF][START_REF] Piskulich | Activation Energies and Beyond[END_REF] as the difference in average internal energy between the reacting species and the reactants, we will use the molecular insight given by E D a to identify the origin of the underlying energetic barrier to diffusion.

The activation energy is traditionally obtained from a numerical temperature derivative calculated via an Arrhenius plot constructed from measurements (or simulations) at multiple temperatures. In contrast, here we use the recently-introduced fluctuation theory for dynamics method [START_REF] Piskulich | Activation Energies and Beyond[END_REF][START_REF] Piskulich | Removing the Barrier to the Calculation of Activation Energies: Diffusion Coefficients and Reorientation Times in Liquid Water[END_REF] that directly determines E a from the analytic temperature derivative calculated from a single simulation. This presents several advantages: it requires only simulations at a single temperature, it avoids ambiguities due to the choice of temperature range for non-Arrhenius processes analyzed with the traditional approach, and it reveals which interactions contribute to the activation energy.

The activation energy predicted by our jump model is obtained by combining eqs 4 and 5. The activation energy of the pbc-corrected diffusion coefficient is

E D model a = D HB D pbc model E D HB a + D pbc frame D pbc model E D pbc frame a + D corr D model ✓ E Dcorr a E D pbc model a ◆ (6)
The diffusion E D model a is a weighted average of the H-bond exchange, frame diffusion, and pbc correction activation energies. Using eq. 2, E D HB a can be further decomposed into terms due to the H-bond jump time and jump amplitude temperature dependence,

E D HB a = E 1/⌧ HB a + E h⇢ 2 i a = E 1/⌧ HB a + h⇢ 2 O ⇤ i h⇢ 2 O ⇤ i + h⇢ 2 O i,f i E h⇢ 2 O ⇤ i a + h⇢ 2 O i,f i h⇢ 2 O ⇤ i + h⇢ 2 O i,f i E h⇢ 2 O i,f i a (7) a b c
Figure 3: a) Diffusion coefficients from the model eq. 4, from our simulated mean-square displacement and from experiments, [START_REF] Mills | Self-Diffusion in Normal and Heavy Water in the Range 1-45[END_REF][START_REF] Holz | Temperature-Dependent Self-Diffusion Coefficients of Water and Six Selected Molecular Liquids for Calibration in Accurate 1H NMR PFG Measurements[END_REF] together with the contributions determined from the jump model, obtained from our TIP4P/2005 and SPC/E simulations (see Table S3). The respective contributions to the jump model diffusion coefficient activation energy are reported in Fig. 3b-c and Table S3 for both TIP4P/2005 and SPC/E water force fields (see SI for calculations of activation energies in eqs. 6-7). A first important result is that, for both water models, the jump model diffusion activation energy is in excellent agreement with the simulated value (Fig. 3b) which shows that the model provides an accurate description of the diffusion coefficient over a range of temperatures around ambient conditions and provides further support to the jump model validity.

These results thus suggest that the jump model can be used to gain a molecular insight into the molecular origin of the energetic barrier to diffusion. First, eq. 6 and the values in Fig. 3b-c and Table S3 show that the dominant contribution to E D model (see Table S3) because of the additional E h⇢ 2 i a term in eq. 7 coming from the increase in translational jump amplitude with increasing temperature.

In addition, the model offers an interpretation to the difference between diffusion and reciprocal viscosity activation energies. [START_REF] Mendis | Tests of the Stokes-Einstein Relation Through the Shear Viscosity Activation Energy of Water[END_REF] The latter contrasts with the expectation from the Stokes-Einstein equation and has been abundantly discussed, especially in the supercooled liquid (see, e.g., refs 36-38). While the hydrodynamic approximation involved in the Stokes-Einstein equation is questionable for the diffusion of a solute particle that has the same size as the solvent particles, one can expect H-bond jump exchanges to play an important role for both diffusion and viscosity. Eyring and coworkers [START_REF] Kincaid | The Theory of Absolute Reaction Rates and Its Application to Viscosity and Diffusion in the Liquid State[END_REF] proposed a simplified picture describing viscosity as an activated process involving displacements between equilibrium positions in the liquid, and simulations suggested [START_REF] Mendis | Tests of the Stokes-Einstein Relation Through the Shear Viscosity Activation Energy of Water[END_REF][START_REF] Stirnemann | Mechanisms of Acceleration and Retardation of Water Dynamics by Ions[END_REF] that H-bond jump exchanges could be these elementary events. Our model suggests that while the same H-bond jump exchanges are involved in diffusion and shear viscosity, any difference in activation energies between D and 1/⌘ arises from the different temperature dependences of the impact that each jump has on the molecular position and on the stress tensor.

The present results also provide new insight into the coupling between rotational and translational dynamics in water, of particular interest in the supercooled regime. [START_REF] Paschek | Simulation Study on the Diffusive Motion in Deeply Supercooled Water[END_REF][START_REF] Faraone | Model for the Translation-rotation Coupling of Molecular Motion in Water[END_REF][START_REF] Mazza | Connection of Translational and Rotational Dynamical Heterogeneities with the Breakdown of the Stokes-Einstein and Stokes-Einstein-Debye Relations in Water[END_REF][START_REF] Bove | Translational and Rotational Diffusion in Water in the Gigapascal Range[END_REF][START_REF] Manogaran | Understanding Translational-Rotational Coupling in Liquid Water Through Changes in Mass Distribution[END_REF] The role of H-bond jumps for translation elucidated here can be compared with the extended angular jump model for reorientation. 7,8 While this rotation-translation coupling is often viewed through the lens of the Debye-Stokes-Einstein equation, [START_REF] Mazza | Connection of Translational and Rotational Dynamical Heterogeneities with the Breakdown of the Stokes-Einstein and Stokes-Einstein-Debye Relations in Water[END_REF][START_REF] Bove | Translational and Rotational Diffusion in Water in the Gigapascal Range[END_REF] our model avoids the hydrodynamic approximation, which is not appropriate for a small solute, and the assumption of rotational diffusion for water reorientation, which was shown to be invalid. 7,8 H-bond jump exchanges are central to both rotational and translational dynamics (including at supercooled temperatures [START_REF] Stirnemann | Communication: On the Origin of the Non-Arrhenius Behavior in Water Reorientation Dynamics[END_REF] ), and the jump models indicate that the slight difference between the reorientation time ⌧ 2 and translational diffusion D activation energies arises from the larger temperature dependence of the translational jump amplitude compared to that of the rotational amplitude. [START_REF] Piskulich | Activation Energies and the Extended Jump Model: How Temperature Affects Reorientation and Hydrogen-Bond Exchange Dynamics in Water[END_REF] This suggests an alternative explanation for why D ⇥ ⌧ 2 increasingly deviates from its ambient temperature value upon supercooling, [START_REF] Qvist | Rotational Dynamics in Supercooled Water from Nuclear Spin Relaxation and Molecular Simulations[END_REF][START_REF] Piskulich | The Dynamics of Supercooled Water Can Be Predicted from Room Temperature Simulations[END_REF] based on differences in the temperature dependences of the translational and rotational jump amplitudes. In contrast to prior suggestions, 50 such a mechanism would thus require neither a decoupling between rotational and translational motion nor a change in diffusion mechanism.

Finally, our results also pave the way to an improved understanding of water transport properties in aqueous solutions and at interfaces. Since our model reveals the connection between the diffusion coefficient and the H-bond jump time and amplitude, the solute's or interface's impact on the latter will be of great importance. The H-bond jump time has been shown in prior studies 8 to sensitively depend on the water-solute or water-surface Hbond interaction strength. Regarding the H-bond amplitude, it is expected to depend on the relative masses of the water molecule and of its H-bond partner. This should provide a framework to understand how water diffusion is affected in electrolytes, including, e.g., the diffusion enhancement and reduction induced by different salts, [START_REF] Kim | Self-Diffusion and Viscosity in Electrolyte Solutions[END_REF] but also in materials and in biomolecular hydration shells where it should provide a connection between the H-bond jump time heterogeneity [START_REF] Laage | Water Dynamics in the Hydration Shells of Biomolecules[END_REF] and variations in the water diffusion coefficient.

In conclusion, we have described an H-bond jump model for translational dynamics of water that quantitatively describes the water diffusion coefficient and its activation energy.

We have shown that translational diffusion of a water molecule is mainly due to jumps which take place when this molecule exchanges H-bond partners, with a minor contribution from diffusion of molecules that keep their H-bonds intact. Our model thus provides a unified framework which establishes how H-bond network rearrangements are responsible for both translational and rotational dynamics in liquid water. Our analysis of the activation energies further reveals why translation and reorientation activation energies slightly differ, leading to an apparent decoupling while they proceed from the same H-bond exchanges. The molecular insight provided by the jump model for translation dynamics will be instrumental to understand how water transport properties are impacted in more complex environments.

We performed classical molecular dynamics simulations on systems of 343 water molecules, described respectively with the SPC/E and TIP4P/2005 potentials and propagated with LAMMPS. 1 Each system is first equilibrated with a Nosé-Hoover thermostat at 298.

H-bond jumps

H-bond jump times are determined from the jump tcf (Fig. S1). Jump times and their associated activation energies for the TIP4P/2005 and SPC/E water models have been calculated

previously for the same system and in the same conditions, and we use the values reported in ref.

The H-bond jump amplitude probability densities in Fig. 1d are well described by skew S2 normal distributions eq. S1 whose parameters are reported in Table S2. S1. From this Table , we see that all atoms move in the plane formed by the three oxygen atoms and the displacements are reported graphically in Fig. 1a of the main text.

4⇡⇢ 2 p(⇢) = 1 p 2⇡ exp " (⇢ ⇢ 0 ) 2 2 2 # " 1 + erf ↵ (⇢ ⇢ 0 ) p 2 !# . ( S1
Table S1: Average displacement of the atoms directly involved in the H-bond exchange during the jump (Numbers between parentheses indicate uncertainties in the trailing digit).

along ũ1 ( Å) along ũ2 ( Å) along ũ3 ( Å) O ⇤ -0.

Activation energy calculations

Following prior studies, [5][6][7] we employ the fluctuation theory for dynamics method to determine the activation energies from a single trajectory at 298.15 K.

For the di↵usion coe cient obtained from the simulated msd, the activation energy has been shown 5,6 to be

E D pbc MD a = lim t!1 h H(0) | ! r (t) ! r (0)| 2 i h| ! r (t) ! r (0)| 2 i = lim t!1 h H r 2 (t)i h r 2 (t)i , (S2)
where H(0) = H(0) hHi is the fluctuation in energy from its average value, and h H r 2 (t)i is the mean square displacement where each square displacement is weighted by the energy fluctuation.

The frame di↵usion activation energy is calculated with eq. S2. The intact

H 2 O(H 2 O) 4
cluster mean square displacement h r 2 (t)i and mean-square displacement weighted by the energy fluctuations h H r 2 (t)i are shown in Fig. S3. Linear fits of h r 2 (t)i and h H r 2 (t)i over the 10-50 ps interval are used to determine the long-time limit in eq. S2.

The pbc correction activation energy is determined from the viscosity activation energy 8

S4

(determined here from a numerical derivative, see below and Fig. S5), E Dcorr a = E 1/⌘ a + k B T .

The activation energy for the H-bond jump rate constant 1/⌧ HB is The activation energy associated with the jump amplitude temperature dependence 7 is

E 1/⌧ HB a = ⌧ HB Z 1 0 dth H(0)p i (0)p f (t)i . ( S3
E h⇢ 2 O ⇤ ,i,f i a = 1 h⇢ 2 O ⇤ ,i,f i Z 1 0 d⇢ O ⇤ ,i,f @p (⇢ O ⇤ ,i,f ) @ 4⇡⇢ 4 O ⇤ ,i,f , (S4)
and the reciprocal temperature derivative of the jump amplitude probability density is

4⇡⇢ 2 O ⇤ ,i,f @p (⇢ O ⇤ ,i,f ) @ = 1 N N X j=1 H j h ⇢ j O ⇤ ,i,f ⇢ O ⇤ ,i,f i , (S5)
where the sum runs over the N H-bond jump events collected during the simulation.

The computed activation energy values are reported in Table S3. Our model shows that the minor frame di↵usion contribution, arising from water molecules which di↵use while keeping their H-bonds intact, exhibits a larger activation energy E . For water reorientation, the activation energy for the frame tumbling contribution had also been shown 7 to be larger (by ⇠ 0.2 kcal/mol) than that for jump reorientation. Consequently, for both translation and reorientation dynamics in supercooled water, the more collective frame di↵usion should become less and less important when the temperature decreases, and dynamics is expected to be increasingly due to H-bond exchanges directly involving the molecule that di↵uses or rotates.

In addition, good agreement between the model and simulations is found when decomposing the activation energy into additive components. Following refs. 5-7, the instantaneous fluctuation in system energy is expressed as a sum of contributions arising S5 from the Lennard-Jones and Coulombic interaction energies as well as the kinetic energy:

H = KE+ V LJ + V Coul .
In accordance with prior works [START_REF] Piskulich | Examining the Role of Di↵erent Molecular Interactions on Activation Energies and Activation Volumes in Liquid Water[END_REF]6 the results reported in Table S4 indicate that the di↵usion activation energy is mostly determined by Coulombic interactions but also show that the jump model provides an excellent description of the activation energy decomposition.

Viscosity

The shear viscosity is calculated from the Green-Kubo equation involving the pressure tensor time-correlation function,

⌘ = V k B T Z 1 0 hP ↵ (0)P ↵ (t)idt , ( S6
)
where V is the simulation box volume and the hP ↵ (0)P ↵ (t)i correlation is averaged over the five correlations of the anisotropic stress tensor terms P ↵ 8 (Fig. S4).

Because of the very long simulations required to converge the viscosity activation energy with the fluctuation theory method, 8 here we determine the activation energy from an Arrhenius plot constructed from a series of six distinct simulations performed at 290, 295, 298.15, 300, 305 and 310 K using the same procedure as described above, with an initial NPT equilibration followed by a 10 ns NVT run from which 5 uncorrelated starting points are taken

Figure 1 :

 1 Figure 1: Hydrogen-bond jump exchange mechanism and translational displacement. a) Schematic representation of H-bond exchange mechanism, with average displacement vector orientations for each molecule (yellow). b) Time evolution of the average O ⇤ O i (blue) and O ⇤ O f (green) distances. c) Velocity time-correlation functions for the O ⇤ (orange), O i (blue) and O f (green) oxygen atoms. d) Jump amplitude probability densities 4⇡⇢ 2 p(⇢) for the O ⇤ (orange), O i (blue) and O f (green) oxygen atoms, and skew normal fits (dashes) (see TableS2for fit parameters).

  Fig. S2) -are key to calculating the H 2 O(H 2 O) 4 complex mean square displacement (msd) at long delays. The latter are essential to reach the diffusive regime visible in Fig. 2 and correctly determine the diffusion coefficient (without such restraints, a prior study 15 could only access the intact cluster msd at shorter delays where the msd appears sub-diffusive).Figure 2 compares the msd of a single water molecule and of a molecule which keeps its four nearest neighbors. A molecule that does not undergo any H-bond exchange with one of its four neighbors diffuses approximately four times more slowly than an average molecule (see Table

Figure 2 :

 2 Figure 2: Mean square displacement of a single molecule (orange) and of a molecule keeping the same four H-bonded neighbors (blue), with their short-time behavior shown in the inset. D pbc MD and D pbc frame are determined from the slopes of linear regressions (dashes) on the 10-50 ps interval.

  Figure3: a) Diffusion coefficients from the model eq. 4, from our simulated mean-square displacement and from experiments,[START_REF] Mills | Self-Diffusion in Normal and Heavy Water in the Range 1-45[END_REF][START_REF] Holz | Temperature-Dependent Self-Diffusion Coefficients of Water and Six Selected Molecular Liquids for Calibration in Accurate 1H NMR PFG Measurements[END_REF] together with the contributions determined from the jump model, obtained from our TIP4P/2005 and SPC/E simulations (see TableS3). b) Activation energies for the model diffusion coefficient E D model a eq. 6 (showing the weight of each contribution in the sum), for the simulated diffusion coefficient E D MD a

  a

.

a

  comes from the H-bond exchange E D HB a term because these exchanges provide the fastest diffusion pathway, while the pbc correction term is almost negligible because of the very similar values of E Dcorr a while E D model a is very close to the activation energy of the H-bond jump exchange diffusion term E D HB a , E D model a is notably greater than the H-bond jump exchange rate E 1/⌧ HB a

  15 K and a 100 fs relaxation time, and a Nosé-Hoover barostat at 1 atm and a 1000 fs relaxation time. The simulation box volume is then fixed at its average value (box side of 21.752204 Å for TIP4P/2005, and 21.725311 Å for SPC/E) and a 10 ns NVT trajectory is propagated with the velocity rescaling thermostat using a 100 fs relaxation time. A configuration is saved every 2 ns, leading to 5 independent configurations, which are then used as starting points for 5 independent 1 ns trajectories propagated in NVT with the same thermostat. These trajectories employ a 1 fs timestep, periodic boundary conditions with a 10.5 Å cuto↵ of Lennard-Jones interactions, an Ewald sum description of long-range electrostatic interactions with a 10 4 tolerance, and configurations are saved every 5 fs. Water molecules are kept rigid with the RATTLE algorithm. For the viscosity calculation, the 5 independent starting configurations are used to propagate 50 ns-long trajectories where only the pressure tensor is saved. For the calculation of activation energies, a 200 ns-long NVT trajectory is propagated, where configurations are saved every 1 ps. The resulting 200,000 independent configurations are used as starting points for 50 ps trajectories propagated in NVE.

)

  Average displacements induced by a hydrogen-bond exchangeThe local frame in which an H-bond jump occurs is defined for each jump at t 0 by the three O atoms involved: O ⇤ , O i and O f . O ⇤ is taken to be the origin of the local frame. The first frame vector ũ1 is defined along the (O i O ⇤ O f ) angle bisector. The second one ũ2 is orthogonal to ũ1 and aligned with the vector connecting O i to O f in the plane formed by these three O atoms. The last vector ũ3 is orthogonal to both ũ1 and ũ2 and oriented to form a direct reference frame {ũ 1 , ũ2 , ũ3 }. The three vectors are then normalized. The average displacements of the atoms involved in the jump between t 0 100 fs and t 0 + 100 fs are given in Table

)

  It has already been calculated by two of us and reported in ref.2 for the SPC/E and TIP4P/2005 water models, and here we use the values determined in these prior studies.

  exchange term E D HB a

to propagate 5

 5 independent 50 ns NVT trajectories. The resulting temperature dependence is shown in Fig. S5 for the TIP4P/2005 water model. The slope of the reciprocal viscosity along the reciprocal temperature is shown to be constant over this temperature range, which supports the validity of the activation energy determination from a finite di↵erence around the ambient temperature. S6

S7FigureS8Figure S2 :S9Figure S3 :S10Figure S4 :S11Figure S5 :

 S2S3S4S5 Figure S1: H-bond jump exchange time-correlation function 1 hp i (0)p f (t)i (orange) with the exponential fit (dashes) used to determine the jump time ⌧ HB determined from our TIP4P/2005 simulations.

  and use the time when it reaches its first minimum before or after t 0 as an approximate determination of the delay required for the oxygen to reach its stable position. Figure1cpresents C vv for the H-bond donor oxygen O ⇤ , and the initial O i and

	final O f H-bond acceptors, and shows that a stable position is reached for a delay t '100 fs
	after the exchange transition state is crossed. We therefore determine the translational jump
	displacement of each oxygen atom as the vector connecting the positions before and after
	the H-bond exchange ! ⇢ = ! r O (t 0 + t) ! r O (t 0	t).
	For the three water oxygen atoms involved in the H-bond exchange, jump amplitude prob-
	ability density distributions 4⇡⇢ 2 p(⇢) are reported in Fig. 1d. They are all well described by
	skew normal distributions, with a peak at '0.6-0.7 Å and a tail extending to larger ampli-
	tudes. Time reversibility requires that the O i and O f distributions be strictly identical. In ad-
	dition, our results find only minor differences between the O ⇤ O i and O ⇤ O f donor-acceptor dis-
	tributions, with the former exhibiting a slightly broader tail. The resulting average squared
	amplitudes are thus fairly similar h⇢ 2 O ⇤ i=0.675±0.003 Å 2 and h⇢ 2 O i,f i=0.722±0.002 Å 2 . (All
	reported uncertainties correspond to the Student's t-distribution 95% confidence interval.)

  Di↵usion of the H 2 O(H 2 O) 4 complexTo follow the long-time di↵usion of a water molecule in absence of H-bond exchanges involving the H-bonds that it accepts or donates, we have employed the following set of restraints in order to prevent H-bond exchanges around O ⇤ (but not transient breaks followed by the S3 reformation of the same H-bond): i) a soft half-harmonic potential on the O ⇤ -O distances to the four nearest neighbors, with a 5,000 kcal•mol 1 •nm 2 force constant that applies only beyond 3.33 Å, ii) a soft half-harmonic potential on O ⇤ -O distances to all water molecules which are not the four nearest neighbors, with a 5,000 kcal•mol 1 •nm 2 force constant that applies only below 3.33 Å, and iii) a strict half harmonic potential on the four H-bond angles involving O ⇤ , i.e. the two (HO ⇤ O) angles for the two H-bonds donated by O ⇤ and the two (HOO ⇤ ) angles for the two H-bonds accepted by O ⇤ , with a 400 kcal•mol 1 •rad 2 force constant that applies only beyond 30 . These restraints were implemented in the LAMMPS simulations with the PLUMED library.3,4 FigureS2shows that the oxygen radial distribution function around O ⇤ is only marginally a↵ected by these soft restraints.

		004(2)	0.415(2)	-0.000(2)
	O i	0.313(2)	-0.276(2)	0.0000(8)
	O f	-0.3090(9)	-0.2752(9)	0.000(2)
	H ⇤	-0.007(1)	0.936(1)	0.000(2)

Table S2 :

 S2 Parameters of the skew normal fits to the jump amplitude probability density distributions.

		⇢ 0 ( Å)	( Å)	↵
	O ⇤	0.402(3)	0.481(3)	2.16(3)
	O i	0.402(4)	0.508(4)	2.29(6)
	O f	0.402(4)	0.507(2)	2.30(4)

Table S3 :

 S3 Di↵usion coe cients determined from our model eq. 4 with their respective contributions, from the simulated msd, and from experiments, and associated activation energies (Numbers between parentheses indicate uncertainties in the trailing digit). MD (10 5 cm 2 /s) 2.37(1) 2 2.85(2) 2 D NMR (10 5 cm 2 /s) 2.299 9,10 E

							TIP4P/2005	SPC/E
				⌧ HB (ps)	3.657(7) 2	3.160(3) 2
	h⇢ 2 O ⇤ i ( Å2 ) h⇢ 2 O i,f i ( Å2 ) D HB (10 5 cm 2 /s)	0.675(3) 0.722(2) 1.273(4)	0.721(2) 0.7464(9) 1.548(3)
	D pbc frame (10 5 cm 2 /s) D corr (10 5 cm 2 /s)	0.623(3) 0.335(3)	0.749(5) 0.406(4)
	D model (10 5 cm 2 /s)	2.232(6)	2.703(6)
	D 1/⌧ HB a	(kcal/mol)	3.63(5) 2	3.09(4) 2
	E	h⇢ 2 O ⇤ i a	(kcal/mol)	0.5(2)	0.40(6)
	E	h⇢ 2 O i,f a	i	(kcal/mol)	0.55(6)	0.51(4)
	E D HB a E D pbc f rame (kcal/mol) a (kcal/mol)	4.15(7) 4.4(2)	3.55(5) 3.9(3)
	E	1/⌘ a	(kcal/mol)	3.7(4)	3.2(3)
	E Dcorr a E D model a E D pbc MD a	(kcal/mol) (kcal/mol) (kcal/mol)	4.3(4) 4.25(9) 4.10(5) 2	3.8(4) 3.68(9) 3.61(9) 2
	E D NMR a	(kcal/mol)	4.2-4.4 9,10

Table S4 :

 S4 Decomposition of activation energies into kinetic energy, Lennard-Jones interaction energy and Coulombic interaction energy terms (in kcal/mol) (Numbers between parentheses indicate uncertainties in the trailing digit).

	E a	E 3.63(5) 2 1/⌧ HB a	E 0.5(2) h⇢ 2 O ⇤ i a	E 0.55(6) h⇢ 2 O i,f i a	E D HB a 4.15(7)	E 4.4(2) D pbc f rame a	E Dcorr a 4.5(5) 8	E D model a 4.25(9)	E 4.10(5) 2 D pbc MD a
	KE	1.04(4) 2	0.33(4)	0.36(3)	1.39(5)	1.2(1)	1.0(4) 8	1.26(7)	1.16(4) 2
	LJ	-1.10(6) 2 -0.16(7) -0.24(4) -1.30(8)	-1.2(2)	-0.8(2) 8 -1.20(8) -1.24(5) 2
	Coul	3.69(8) 2	0.31(5)	0.43(7)	4.06(9)	4.5(3)	3.8(3) 8	4.1(1)	4.18(6) 2

Acknowledgement

Silvio Pipolo (now Univ. Lille, France) is gratefully acknowledged for preliminary work. This work was supported by the National Science Foundation under Grant No. CHE-2102656 (W.H.T.). Z.A.P gratefully acknowledges a National Science Foundation Graduate Research Fellowship under Grant Nos. 1540502 and 1451148 and the Graduate Research Opportunities Worldwide program. Activation energy calculations were performed at the Center for Research Computing (CRC), University of Kansas.

Supporting Information Available

The following files are available free of charge.


Karim Tabia
email: tabia@cril.univ-artois.fr

Analysis of Jeffrey's rule of conditioning in an imprecise probabilistic setting (Preprint version)

While sets of probability measures and imprecise probabilities in general, are widely accepted as a powerful and unifying framework for handling uncertain and incomplete information, updating such belief sets with new uncertain inputs has not received enough attention. In this paper, we provide an analysis of Jeffrey's rule of conditioning for updating sets of probability measures with new information, possibly uncertain and imprecise, also expressed as sets of probability measures. The paper first provides properties for updating sets of probability measures in the spirit of Jeffrey's rule, then provides and analyses extensions of Jeffrey's rule to three main imprecise probability representations: i) finite sets of probability measures and ii) convex set of probability measures specified by extreme points. The proposed extensions capture the proposed postulates and recover the standard Jeffrey's rule in case where the updated set and the new input are single probability measures.

Introduction

Imprecise probability theory [START_REF] Levi | The enterprise of knowledge : an essay on knowledge, credal probability, and chance / Isaac Levi[END_REF][START_REF] Walley | Towards a unified theory of imprecise probability[END_REF] is a unifying uncertainty theory particularly suited for encoding and reasoning with imprecise or ill-known information. This framework is often seen as a probabilistic setting with relaxed parameters and it is typically used to reason with multiple expert information [START_REF] Nau | The aggregation of imprecise probabilities[END_REF], perform sensitivity analysis [START_REF] Bock | Global sensitivity analysis for MAP inference in graphical models[END_REF], decision making with incomplete or scarce information [START_REF] Antonucci | Credal networks for operational risk measurement and management[END_REF], etc. Imprecise probabilities are often associated with a robust Bayesian interpretation [START_REF] Berger | An overview of robust bayesian analysis[END_REF] assuming that the probability measure corresponding to the actual beliefs exists and it is unique but it is unknown, that's why it is expressed in an imprecise way using the concept of sets of probability measures, credal sets [START_REF] Levi | The enterprise of knowledge : an essay on knowledge, credal probability, and chance / Isaac Levi[END_REF][START_REF] Walley | Towards a unified theory of imprecise probability[END_REF] or using other representations (such as interval-based probabilities (de Campos, Huete, and Moral 1994) and probabilistic logic programs [START_REF] Lukasiewicz | Probabilistic logic programming with conditional constraints[END_REF]). Given a set of initial uncertain beliefs, one may have new information which can be in the form of a hard evidence or in the form of uncertain or soft evidence (e.g. unreliable input) or simply new uncertain information regarding some events1 . Our focus is on updating a set of probability measures with new information expressed also as a set of probability measures. In the standard probabilistic setting, Jeffrey's rule [START_REF] Jeffrey | The logic of decision[END_REF] generalizes the standard probabilistic conditioning to the case of uncertain inputs. This conditioning rule has been adapted and studied in many uncertainty settings (for instances, see [START_REF] Dubois | A synthetic view of belief revision with uncertain inputs in the framework of possibility theory[END_REF] for the possibilistic setting, (Ma et al. 2011) for Dempster-Shafer theory). In [START_REF] Benferhat | A framework for iterated belief revision using possibilistic counterparts to jeffrey's rule[END_REF], it is claimed that this rule can successfully recover most of belief revision rules such as natural and drastic belief revision. Many works highlighted the necessity of updating probabilistic information with sets of probability measures [START_REF] Karlsson | Characterization and empirical evaluation of bayesian and credal combination operators[END_REF][START_REF] Skulj | Jeffrey's conditioning rule in neighbourhood models[END_REF][START_REF] Tang | Generalized jeffrey's rule of conditioning and evidence combining rule for a priori probabilistic knowledge in conditional evidence theory[END_REF][START_REF] Rens | On revision of partially specified convex probabilistic belief bases[END_REF]. For example, in [START_REF] Skulj | Jeffrey's conditioning rule in neighbourhood models[END_REF], the author updates a probability measure to create some neighborhood of imprecise probabilities for some events. In [START_REF] Karlsson | Characterization and empirical evaluation of bayesian and credal combination operators[END_REF], the authors study combining multiple evidences provided in the form of credal sets. There is to the best of our knowledge no study on an extension of Jeffrey's rule to sets of probability measures. The main contributions of the paper are: i) We provide natural properties that an extension of Jeffrey's rule to imprecise probabilistic settings should satisfy. ii) We provide extensions of Jeffrey's rule to sets of probability measures and convex credal sets. iii) We study the properties of the proposed extensions. Interestingly enough, the proposed extensions satisfy the defined postulates and collapse to the standard Jeffrey's rule in case where the prior belief set and the new input consist only in single probability measures.

Imprecise probabilities: Basic concepts

In the following, Ω={ω 0 , ω 1 , .., ω m } denotes the universe of discourse (all possible states of the world) and ω i denotes a given state (also called interpretation). Sets of interpretations ϕ⊆Ω, ψ⊆Ω are called events.

Sets of probability measures and credal sets

Let ∆ denote the set of all probability measures over Ω. A set of probability measures K is a subset of ∆. K denotes a finite 2 or inifinite set of classical probability measures p over Ω. In order to avoid heavy notations, a set of probability measures will be denoted K, the same notation used for credal sets. In this paper, a credal set represents all the probability measures satisfying some requirements or constraints. More precisely, a credal set is defined as follows: Definition 1 (Credal set) A credal set K is a closed convex set of probability distributions.

Credal sets are generally induced by probabilistic beliefs encoded by means of interval-based probabilities (de Campos, Huete, and Moral 1994) or probabilistic constraints as in conditional logic programs [START_REF] Lukasiewicz | Probabilistic logic programming with conditional constraints[END_REF]. Intuitively, if K is a convex set of probability measures, then linearly mixing 3 any two distributions p 1 and p 2 from K will result in a distribution p belonging to K. Given that a credal set may contain an infinite number of probability measures, there are three main commonly used ways to encode imprecise beliefs. i) Vertex-based representation where the uncertainty is encoded by a finite set of standard probability distributions representing extreme points of the convex set K. ii) Interval-based representation where every interpretation ω∈Ω is associated with upper and lower probabilities. iii) Constraint-based representation where the uncertainty is specified by means of constraints as in the comparative probabilities framework [START_REF] Miranda | Extreme points of the credal sets generated by elementary comparative probabilities[END_REF] or in probabilistic logic programs [START_REF] Lukasiewicz | Probabilistic logic programming with conditional constraints[END_REF]. In this paper, we focus only on the vertex representation.

Vertex-based representation

This representation defines a convex credal set K by a finite number of probability measures called extreme points. Such a credal set is called a finitely generated credal set. Any probability measure of K can be expressed as linear combination of extreme points.

Definition 2 (Extreme point) An extreme point (also called vertex) p of a credal set K is a probability measure such that it is impossible to find two different probability distributions p 1 ∈K and p 2 ∈K such that p=α*p 1 +(1-α)*p 2 with α∈]0, 1[. This representation is for instance used in the Jav-aBayes 4 platform for modeling and reasoning with 2 For instance, in case we have 10 experts where each expert i having his own beliefs in the form of a probability measure pi then this set is composed by the 10 probability measures p1,..,p10.

3 Mixing here means linearly combining a set of distributions p1 .. p k as follows: p= ∑ k i=1 (ai * pi) where ∑ k i=1 ai=1. 4 http://www.cs.cmu.edu/~javabayes/Home/ Bayesian and credal networks [START_REF] Cozman | Credal networks[END_REF]. In the following, ext(K) denotes the set of extreme points of the credal set K. Definition 3 (Convex hull) The convex hull of credal set K, denoted CH(K) is the closed set of probability measures whose polytope is characterized by the set of extreme points ext(K).

Example 1 In Figure 1, a convex credal set K is depicted using a barycentric representation. Here Ω={ω 0 , ω 1 , ω 2 } and K is finitely generated by three extreme points p 1 =(.45, 0, .55), p 2 =(.7, .1, .2) and p 3 =(.1, .6, .3).

w0(1, 0, 0) w1(0, 1, 0) w2(0, 0, 1)

p1 p2 p3
Figure 1: Example of extreme points p 1 , p 2 and p 3 using a barycentric representation.

Note that any closed convex set can be encoded by a finite number of extreme points [START_REF] Levi | The enterprise of knowledge : an essay on knowledge, credal probability, and chance / Isaac Levi[END_REF][START_REF] Wallner | Extreme points of coherent probabilities in finite spaces[END_REF].

Reasoning with credal sets

Reasoning tasks are performed on sets of probability measures or credal sets by exploring all the models of that credal set5 . For instance, marginalizing a credal set K(X, Y ) on two sub-sets of variables X and Y is done as follows:

K(X) = { ∑ Y p(X, Y ) : p ∈ K(X, Y )} (1)
Conditioning on an event ϕ⊆Ω is defined as follows:

K(ω i |ϕ) = {p(ω i |ϕ) : p ∈ K and p(ϕ) > 0} (2)
Note that for practical computational reasons, reasoning on K is done on ext(K) which provides an equivalent representation. Indeed, inference on a credal set K is equivalent to inference on its extremes points (de Campos, Huete, and Moral 1994). For instance, for marginalization, given a credal set K(X, Y ) on two subsets of variables X and Y . Then,

K(X) = CH({p(X) : p ∈ ext(K(X, Y )}) (3)
We assume that K is a finitely generated credal set, namely K is the convex hull of its set of extreme points ext(K). In the following, we propose extensions of Jeffrey's rule to sets of probability measures.

Extending Jeffrey's rule to sets of probability measures

This section analyzes a straightforward extension of Jeffrey's rule to sets of probability measures.

Motivating example

Let us assume we are dealing with learning probabilities from a dataset over five boolean variables X 1 , X 2 , X 3 , X 4 and X 5 . Suppose a small dataset D is collected where the values of some variables are missing. Assume also we are interested in deriving an imprecise probability distribution (here an interval-based probability distribution) from this small dataset. In this case, for each configuration x 1 ..x 5 of the five variables (X 1 , .., X 5 ), we will have a lower bound l corresponding to the frequency of x 1 ..x 5 in D and an upper bound u corresponding to the proportion of entries of D that are either x 1 ..x 5 or that can be x 1 ..x 5 (for instance, if the value of variable X 5 is missing in a given entry of D then it can be any value of X 5 ). Let P be the intervalbased probability distribution derived from D.

X1 X2 X3 X4 X5 P (X1X2X3X4X5) 0 0 0 0 0 [.001, .04] 0 0 0 0 1 [.02, .1] ... ... ... ... ... ... 1 1 1 1 1 [.005, .035]
Assume now that we have a latest and bigger dataset D ′ but only on a subset of variables X 1 ..X 2 . D ′ also contains some missing data. Let P ′ be the interval-based probability distribution computed from D ′ . It fully makes sense to revise the initial distribution P by P ′ since this latter is more recent and more representative of the problem as it concerns a large amount of data. It is important to note that the information to update is a set of probablity measues (all probability measures complying with the intervals of P ) and the new input is also a set of probability measures. This update task is fully in the spirit of Jeffrey's rule but there is to the best of our knowledge no extension of Jeffrey's rule to sets of probability measures. Of course, the need to revise sets of probabilities by new sets of probabilities can be encountered either when dealing with empirical data (typical situations are dealing with missing data and small datasets) or when dealing with subjective beliefs of agents.

Jeffrey's rule

Jeffrey's rule [START_REF] Jeffrey | The logic of decision[END_REF]) is an extension of the classical probabilistic conditioning to the case where the new observation is uncertain. It allows to update an initial probability distribution p into a posterior one p ′ given the uncertainty bearing on a set of mutually exclusive and exhaustive events λ 1 ,..,λ n . The new input is of the form (λ i , α i ), i=1..n where α i denotes the new probability of λ i . Jeffrey's rule lies on the two following principles: i) Success principle: After the update operation, the posterior probability of each event λ i must be equal to α i , namely ∀λ i , p ′ (λ i )=α i . ii) Probability kinematics principle: This constraint ensures a kind of minimal change principle. Jeffrey's method assumes that in spite of the disagreement about the events λ i in the initial distribution p and the new one p ′ , the conditional probability of any event ϕ⊆Ω given any uncertain event λ i remains the same in the original and the revised distributions. Namely,

∀λi ⊆ Ω, ∀ϕ ⊆ Ω, p(ϕ|λi) = p ′ (ϕ|λi).
(4)

Given a probability measure p encoding the initial beliefs and new inputs the form (λ i , α i ). The updated probability degree of any event ϕ⊆Ω, is done as follows:

p ′ (ϕ) = ∑ λ i αi * p(ϕ, λi) p(λi) . ( 5
)
The posterior distribution p ′ obtained using Jeffrey's rule always exists and it is unique (Chan and Darwiche 2005).

Jeffrey's rule for sets of probability measures [START_REF] Yue | Revising imprecise probabilistic beliefs in the framework of probabilistic logic programming[END_REF]. For the sake of simplicity, the input, the belief set to update is given in the form of a credal set denoted K. The new information is also given in the form of a credal set K in over a partition of Ω. This form for the inputs is general enough to capture sure observations, uncertain observations and imprecise ones. Moreover, we assume that K and K in are not empty sets. Let K ′ be a the updated set obtained by updating K with K in . Let us now see what an extension of Jeffrey's rule could aim to satisfy in an imprecise probabilistic setting.

(P1) K ′ (λ 1 ..λ n )⊆ K in (P2) ∀λ i ⊆Ω, ∀ϕ⊆Ω, K(ϕ|λ i )=K ′ (ϕ|λ i ) (P3) udp(K, K in )=
∪ p∈K,pin∈Kin udp(p, p in ). Postulate P1 corresponds to the success postulate ensuring that the new information should be accepted (the inputs are seen as constraints to be satisfied). Of course, the success postulate may be questionable in some contexts, but it may be a desired property in some applications such as in [START_REF] Skulj | Jeffrey's conditioning rule in neighbourhood models[END_REF]. In order to stay in Jeffrey's rule spirit, we just rephrase this postulate in the context of sets of probabilities. The converse inclusion K in ⊆K ′ (λ 1 ..λ n ) is strong as there may exist λ i and p in ∈K in such that p in (λ i )>0 while ∀p∈K, p(λ i )=0

preventing the application of Jeffrey's rule on an a priori impossible event as in the standard case. P2 is the statement of kinematics principle adapted to the case of sets of probability measures. This postulate aims to ensure that K ′ and K preserve the conditional credal sets on the events λ 1 ,..,λ n . P3 extends the one proposed in [START_REF] Grove | Updating sets of probabilities[END_REF] in order to capture the fact that updating a set of probability measures by another set of measures should take into account every measure in the initial set and every measure in the new input. This makes sense within a robust Bayesian interpretation of sets of probability measures.

Lemma 1 If |K|=|K in |=1 then postulates P1 and P2 recover with the success and probability kinematics principles of Jeffrey's rule respectively.

Obviously, if the credal sets K and K in are singletons (namely, each one composed of only one probability measure), then P1 will recover the success principle (the input K in is fully accepted as in Jeffrey's rule) while P2 will recover the probability kinematics principle. Consequently, the only solution satifying these properties is the one obtained using Jeffrey's rule and it always exists [START_REF] Chan | On the revision of probabilistic beliefs using uncertain evidence[END_REF].

Conditioning sets of probabiliy measures with uncertain inputs

One direct way to extend Jeffrey's rule to finite sets of probability measures is to update every member of the belief set K by every member of the new input K in as follows:

Definition 4 Let K be a set of probability measures representing the current beliefs over the universe of discourse Ω. Let the new information be K in .

K ′ = {p ′ : p ′ = Jeffrey(p, pin), p ∈ K : ∀λi, p(λi) > 0, pin ∈ Kin} (6)
where Jeffrey(p, p in ) is the update according to Jeffrey's rule given in Equation 5 of the probability measure p with the new input p in =(p in (λ 1 ), ..,p in (λ n )).

Updating using Definition 4 in straightforward in case where the belief sets K and K in consist of finite sets of probability measures. It is clear that if both K and K in contain only one probability measure then Definition 4 comes down to Jeffrey's rule in the standard probabilistic setting.

Example 2 Let us assume that Ω={a 1 b 1 , a 1 b 2 , a 2 b 1 , a 2 b 2 } and that the current beliefs about a given problem over two binary variables A and B is a set composed of three probability distributions p 1 , p 2 and p 3 . Suppose that we receive new information (for example new data) saying that the probability p in (b 1 )=.9 and p in (b 2 )=.1. Applying Jeffrey's rule to each probability measure p 1 , p 2 and p 3 will give three updated distributions p ′ 1 , p ′ 2 and p ′ 3 . Proposition 1 Let K be a finite set probability measures over Ω. Let the new information be K in which Table 2: The posterior set K ′ obtained from K of Table 1.

is a set on an exhaustive and mutually exclusive set of events λ 1 ,..λ n . Let K ′ be the results of updating K ′ with K in using Definition 4. Then K ′ satisfies postulates P 1, P 2 and P 3.

Proof 1 (Sketch) • For P1, to show that K ′ (λ 1 ..λ n )⊆K in , let p ′ ∈K ′ and show that ∀λ i , ∃p in ∈K in s.t. p ′ (λ i )=p in (λ i ).
If p ′ ∈K ′ then ∃p∈K and ∃p in ∈K in such that p ′ =Jeffrey(p, p in ). Since p ′ is obtained by updating p with p in with Jeffrey's rule, then ∀λ i , p ′ (λ i )=p in (λ i ). • For P2 and P3, the proof is also straightforward for finite sets of probability measures since by Definition 4 the update is done using Jeffrey's rule applied individually on each member of K and on each member of K in .

In practice, the credal set K to update may be finite or infinite (in case of convex sets). In the following, we extend Jeffrey's rule to closed convex credal sets.

Conditioning credal sets with uncertain inputs

In this section, the belief set to update is a closed convex set K specified by its extreme points ext(K) and the new input K in is also a closed convex set specified by its extreme points ext(K in ). One direct way to extend Jeffrey's rule is to update only extreme points of K with the ones of K in , namely update each p∈ext(K) with each p in ∈ext(K in ) using Jeffrey's rule. Definition 5 Let K be the closed convex set to update.

Let the new information be K in which is a closed convex set on set of exhaustive and mutually exclusive events λ 1 ,..,λ n .

K ′ = CH({p ′ : p ′ = jeffrey(p, pin); p ∈ ext(K) and pin ∈ ext(Kin), ∀λi, p(λi) > 0}), (7)
Given that it is impossible to update every p∈K, update of Definition 5 proceeds by updating only the set of extreme points of K by the set of extreme points of K in then recovers a convex set using the convex hull operator.

Example 3 (Example 1 continued) Let us reuse the credal set K of Example 1 where Ω={ω 0 , ω 1 , ω 2 } and K is finitely generated by three extreme points p 1 =(.45, 0, .55), p 2 =(.7, .1, .2) and p 3 =(.1, .6, .3). Assume now that new information K in regarding two events λ 1 ={ω 1 , ω 2 } and λ 2 ={ω 3 } has become available. Assume also that ext(K in ) consists of two extreme points {(.7, .3); (.6, .4)}.

w0(1, 0, 0) w1(0, 1, 0) w2(0, 0, 1) As shown in Figure 2, the number of extreme points of

p ′ 1 p ′′ 1 p ′ 2 p ′′ 2 p ′ 3 p ′′ 3
K ′ is at most to |ext(K)|*|ext(K in )|.
Proposition 2 Let K be the closed convex set to update and K in be the new input. Let K ′ be the updated set computed according to Definition 5 then K ′ satisfies postulates P1-P3.

Proof 2 (Proof sketch) For P1, in order to show that

K ′ (λ 1 ..λ n )⊆K in , let p ′ ∈K ′ and show that ∃p in ∈K in s.t. p ′ (λ 1 ..λ n )=p in (λ 1 ..λ n ). Since p ′ ∈K ′ then p ′ ∈CH({p ′ : p ′ =Jeffrey(p, p in ); p ∈ ext(K), p in ∈ ext(K in )}).
It is clear that in case where p'=Jeffrey(p, p in ) with p∈ext(K) and p in ∈ext(K in ) then p ′ (λ 1 ..λ n )∈K in since p ′ is obtained by updating p with p in using Jeffrey's rule. Now, for any p ′ ∈K ′ that is not an extreme point of K ′ , p ′ can be expressed using the extreme points ext(K ′ ) as a convex combination of extreme points of K ′ : p ′ (ω)= ∑ i,j α i,j * p ′ i,j (ω) where p ′ i,j ∈ext(K ′ ) obtained by updating the extreme point p i ∈ext(K) with the extreme point p in j ∈ext(K in ) using Jeffrey's rule, namely

p ′ i,j (ω)= pi(ω) * pin j (λ) pi(λ) . Hence, p ′ (λ)= ∑ ω∈λ ( ∑ i,j α i,j * p ′ i,j (ω)) with ∑ i,j α i,j =1.
The proof is consists in starting with expressing p ′ (λ) as a convex combination of extreme points of ext(K ′ ) and ending up with expressing p ′ (λ) as a convex combination of ext(K in ). For P2, it is enough to see that if p ′ ∈ext(K ′ ) then necessarily ∃p∈ext(K) and ∃p in ∈ext(K in ) such that p ′ =Jeffrey(p, p in ). Hence, ∀ψ⊆Ω, ∀λ i ⊆Ω, p(ϕ|λ i )=p ′ (ϕ|λ i ). Since K and K in are convex sets, then ∀p ′ ∈K ′ , ∃p∈K and ∃p in ∈K in such that p ′ =Jeffrey(p, p in ). Hence, ∀ψ⊆Ω, ∀λ i ⊆Ω, p(ϕ|λ i )=p ′ (ϕ|λ i ). For P3, the idea of the proof is based on the convexity of K ′ obtained by combining two convex sets K and K in . Indeed, K ′ is obtained by the convex hull operator on a kind of cartesian product of elements of ext(K) and ext(K in ). Let K ′ pin be the credal set obtained by updating K with only one point p in ∈K in . Then ∀p∈K, ∃p ′ in K ′ pin s.t. p ′ in =Jeffrey(p, p in ). Now, by updating K by every member of K in and taking the convex hull of the obtained points, it holds that ∀p∈K, ∀p in ∈K in , ∃p ′ ∈K ′ s.t. p ′ =Jeffrey(p, p in ). □

In the following, we study Jeffrey's rule extension in another widely used representation of imprecise probabilities, namely interval-based probability distributions.

Updating interval-based probability measures

Let us see now how to uptate interval-based probability distributions as the ones of the motivating example. Let P be an interval-based probability distribution (IPD for short) encoding the initial beliefs where each interpretation ω∈Ω is associated with a sub-interval of [0, 1].

Given an IPD P encoding the current knowledge and new information P in , there are basically two possible ways to update P with P in :

• A credal-based method: This consists in updating the credal set K underlying P (denoted K(P ) and containing all the models of P ) by the credal set K in underlying P in (denoted K in (P in )) using Definition 5. Once K ′ computed, the IPD P ′ can be computed from K ′ . • An interval-based method: The main drawback of updating at the credal level is that it manipulates extremes points of IPDs while the number of such extreme points for an IPD with m interpretations can be up to m! [START_REF] Wallner | Extreme points of coherent probabilities in finite spaces[END_REF]. The alternative then is to manipulate directly the intervals of the IPD to accommodate the input P in . This method will be addressed in future works. The credal-based update method is defined as follows: Definition 6 Let P be IPD to update and P in be the new input IPD on set of exhaustive and mutually exclusive events λ 1 ,..,λ n . Let K ′ be the updated credal set computed according to Definition 5 on K(P ) and K in (P in ). P ′ is an IPD on Ω such that ∀ω i ∈Ω,

P ′ (ωi) = [inf p ′ ∈K ′ (p ′ (ωi)), sup p ′ ∈K ′ (p ′ (ωi))]. (8)
Example 4 Let us assume in this example that the current beliefs about a given problem over two binary variables A and B are given by the IPD P (AB). In Table 3 3.

by P ′ (B) are fully accepted (see the marginal distribution P ′ (B) computed from the updated distribution P ′ (AB)).

Proposition 3 states that this updating ensures that the postulates P1-P3 are satisfied.

Proposition 3 Let P be IPD to update. Let the new information be the IPD P in on set of exhaustive and mutually exclusive events λ 1 ,..,λ n . Let K ′ be the updated credal set computed according to Definition 5 on K(P ) and K in (P in ). Let P ′ the posterior IPD computed from P and P in following Definition 6. Then P ′ satisfies P1-P3.

Proof 3 (Sketch) The proof directly follows from the fact that K(P ) is an equivalent representation of models of P and the fact that updating using Definition 5 satisfies P1-P3.

Related works and concluding remarks

This paper proposed extensions of Jeffrey's rule of conditioning to the case where the information is encoded in an imprecise probabilistic setting. More precisely, the paper rephrases the two postulates of Jeffrey's rule and added another one to enforce the update operation to take into account every member of the initial set of probability measures and every member of the new input set. The paper extends Jeffrey's rule to i) sets of probability measures and ii) convex credal sets in a vertex-based representation. These extensions are shown to satisfy the proposed postulates and collapse to standard Jeffrey's rule when the initial set and new information are singleton distributions. Updating sets of probability measures is not a new topic [START_REF] Grove | Updating sets of probabilities[END_REF] [START_REF] Levi | The enterprise of knowledge : an essay on knowledge, credal probability, and chance / Isaac Levi[END_REF]) [START_REF] Walley | Towards a unified theory of imprecise probability[END_REF]. However, all these works update sets of probability measures with hard evidence or observations while the focus of the work is updating sets of probability measures with new inputs expressed by means of a set of probability measures. The existing extensions of Jeffrey's rule are limited to special imprecise probabilistic information such as the extensions proposed in (Ma et al. 2011) for Dempster-Shafer theory or the possibilistic extension of Jeffrey's rule proposed in [START_REF] Dubois | A synthetic view of belief revision with uncertain inputs in the framework of possibility theory[END_REF]. In [START_REF] Skulj | Jeffrey's conditioning rule in neighbourhood models[END_REF], the author use Jeffrey's rule to update a single probability distribution in order to obtain the desired neighborhood of events of interest expressed only in terms of interval probabilities. In [START_REF] Yue | Revising imprecise probabilistic beliefs in the framework of probabilistic logic programming[END_REF], the authors dealt with updating imprecise knowledge in the framework of probabilistic logic programming. In case where the imprecise knowledge is compactly encoded by means of belief graphical models called credal networks, there is only one work (J. C. F. da Rocha and de Campos 2008) dealing with updating with soft evidence.

Figure 2 :

 2 Figure 2: Credal set K ′ obtained by updating K of Figure 1 with K in whose extreme points are {(.7, .3); (.6, .4)}.

Table 1 :

 1 Example of a belief set K characterized by three extreme points p 1 , p 2 and p 3 .

	A	B	p1(AB)	p2(AB)	p3(AB)
	a1	b1	.6	.65	.7
	a2	b1	.15	.1	.1
	a1	b2	.1	.1	.1
	a2	b2	.15	.15	.1
	A	B	p ′ 1 (AB)	p ′ 2 (AB)	p ′ 3 (AB)
	a1	b1	.72	.78	.79
	a2	b1	.18	.12	.11
	a1	b2	.04	.04	.05
	a2	b2	.06	.06	.05

  , we have the marginal distribution of A (namely, P (A)), the one of B (namely, P (B)) and the conditional distribution of B given A (namely, P (B|A)).Let us now assume that we have new uncertain inputs given in probability distribution P in (B) such that P in (B=b 1 )=[.7, .8] and P in (B=b 2 )=[.2, .3]. In order to update P to accommodate P in using Definition 6, we update K(P ) with K in (P

					A	P (A)
	A a1 a2 a1 a2	B b1 b1 b2 b2	P (AB) [.50, .70] [.05, .25] [.10, .10] [.15, .15]	a1 a2 B b1 b2	[.60, .80] [.20, .40] P (B) [.75, .75] [.25, .25]
			A	B	P (A|B)
			a1	b1	[.67, .93]
			a2	b1	[.07, .33]
			a1	b2	[.40, .40]
			a2	b2	[.60, .60]

in ) using Definition 5. Note that K(P )) has two extreme points p 1 =(.70, .05, .1, .15) and p 2 =(.50, .25, .1, .15) and K in (P in ) has also two extreme points, namely p in1 =(.7, .3) and p in2 =(.8, .2). p 1 will be updated into p ′ 1 =(.65, .05, .12, .18) and

Table 3 :

 3 Example of an initial IPD P and the underlying marginal and conditional distributions. The updated distribution is given by P ′ of Table4. Table3 and 4show that the input beliefs encoded

	p ′′ 1 =(.75, .05, .08, .12) and p 2 will be updated into p ′ 2 =(.47, .23, .12, .18) and p ′′ 2 =(.53, .27, .08, .12). Hence K ′ =CH({p ′ 1 , p ′′ 1 , p ′ 2 , p ′′ 2 }).
	A a1 a2 a1 a2	B b1 b1 b2 b2	P ′ (AB) [.47, .75] [.05, .27] [.08, .12] [.12, .18]	A a1 a2 B b1 b2	P ′ (A) [.59, .83] [.17, .41] P ′ (B) [.7, .8] [.2, .3]	A a1 a2 a1 a2	B b1 b1 b2 b2	P ′ (A|B) [.67, .93] [.07, .33] [.40, .40] [.60, .60]

Table 4 :

 4 Updated beliefs of the distribution given in Table

On the different meanings of hard, soft and uncertain evidence, see(Ma and Liu

2011).

Alternative approaches consist for instance in selecting the most informative model (in the sense of information entropy for example) of K to draw inferences as it is done in[START_REF] Lukasiewicz | Probabilistic logic programming with conditional constraints[END_REF]).


Daphnée Villoing

Charbel Koumeir

Arnaud Guertin

Ferid Haddad

Vincent Métivier

Freddy Poirier

Vincent Potiron

Noël Servagent

Stéphane Supiot

Grégory Delpon

Response characterization of a new radiochromic film to ultra high dose rates for light ions beams

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

RESPONSE CHARACTERIZATION OF A NEW RADIOCHROMIC FILM

TO ULTRA HIGH DOSE RATES FOR LIGHT IONS BEAMS Daphnée Villoing 1 , Charbel Koumeir 2 , Arnaud Guertin 3 , Ferid Haddad 2,3 , Vincent Métivier 3 , Freddy Poirier 2 , Vincent Potiron 1 , Noël Servagent 3 , Stéphane Supiot 1 , Grégory Delpon 1,3 , Sophie Chiavassa 1,3

1 Institut de Cancérologie de l'Ouest, Saint-Herblain, France 2 GIP ARRONAX, Saint-Herblain, France 3 IMT Atlantique, Nantes Université, CNRS/INP, Nantes, France

Particle dependency

Calibration curves of OC-1 were acquired for several type of particles, in the dose ranging from 1Gy to 100Gy (Figure 2). No dependency on the particle type was observed for photons (10MV), electrons (18MeV) and protons (68MeV) while for alpha-particles (68MeV), the color induced is less intense for the same dose and thus requires the use of a dedicated calibration curve.

Fading

The staining of OC-1 films continues to intensify several weeks after irradiation. Figure 4a illustrates the OC-1 behavior on the red channel up to 45 days post-irradiation for 20Gy. The scanner response was corrected with a test pattern. Similar result was obtained with the green channel with R² = 0.9415. Consequently, the use of different postirradiation times before scanning, even 10 days after irradiation, can significantly overestimate the dose as shown in figure 4b for red and green channel.

Objectives

The ARRONAX facility can deliver proton and alpha beams (68MeV) at ultra-high dose rates (UHDR). Our preclinical beamlines are specifically set up to easily shift between conventional and UHDR irradiation conditions. A major challenge of these irradiations is to control their dosimetry in both modes. In a previous study [START_REF] Villoing | Proton beam dosimetry at ultra-high dose rates (FLASH): Evaluation of GAFchromic (EBT3, EBT-XD) and OrthoChromic (OC-1) film performances[END_REF] in proton beams, we showed a significant increase of the net optical density (netOD) for EBT3 and EBT-XD at 7.5kGy/s and above 10Gy compared with conventional dose rate while OC-1 response was not impacted.

In this study, we first evaluate the response of OC-1 in proton beam up to 60kGy/s and in alpha beam up to 20kGy/s. Then, with the objective to use OC-1 films as dosimeter, we characterize its response in terms of fading, particle dependency, orientation and lateral scanner effect.

OC-1 dose rate independency

The dose rate independency of OC-1 is confirmed for proton beam up to 60kGy/s and alpha beam up to 20kGy/s in one pulse compared with conventional dose rates of 0.2Gy/s and 20Gy/s respectively (Figure 3). Similar results were obtained for the green channel. Significant impact is observed in netOD for EBT3 and EBT-XD in these conditions. Batch number: 12131904 (EBT3), 11062002 (EBT-XD), 2-20200131-1 (OC-1).

Online dosimetry: Faraday cup (Proton and alpha), ionization chamber PTW ROOS 34001 (electrons) and ionization chamber PTW 30013 (photons).

Lateral scanner effect

The lateral scanner effect was studied over the whole scanner area for dose ranging from 2 to 125Gy. The maximal impact in netOD was 2.6% and 3.1% in red and green channel respectively. These variations are not significant relatively to the netOD uncertainty [START_REF] Villoing | Proton beam dosimetry at ultra-high dose rates (FLASH): Evaluation of GAFchromic (EBT3, EBT-XD) and OrthoChromic (OC-1) film performances[END_REF].

Conclusions

The response of OC-1 is independent of • the particle type for protons, electron and photons, • the dose rate in proton and alpha beams up to UHDR, • the lateral scanner effect, • the orientation during the digitalization process.

The fading of OC-1 that persists for several weeks after irradiation can be very detrimental. Therefore, it should be corrected or a constant delay between irradiation and scanner should be observed.

Figure 4 .

 4 Figure 4. (a) netOD variation with respect to the time between irradiation and digitalization for the red channel and (b) its impact on the calculated dose compared to the one determined 10 days post-irradiation.

Figure 3 .

 3 Figure 3. netOD of OC-1, EBT3 and EBT-XD in red channel at conventional dose rate (0.2 and 20Gy/s) and UHDR (60 and 20kGy/s) in (a) proton and (b) alpha beams. The post-irradiation time is 4 days (a) and 2 days (b).

Figure 2 .

 2 Figure 2. Relation between netOD and dose for photons (10MeV), electrons (18MeV) , protons and alpha (68MeV), on both red and green channels. The post-irradiation time is 10 days.

Figure 5 .

 5 Figure 5. Influence of the film orientation during reading on the measured netOD for red and green channels: portrait (continuous) vs. landscape (dotted). Irradiation was made in proton beam (68MeV) and the post-irradiation time was 10 days.

Figure 1 .

 1 Figure 1. Scan of OC-1 film irradiated at 2,5,10,20 and 30Gy, respectively from left to right.