N Peschken

E Athanassoula

S A Rodionov

Forming disc galaxies in major mergers: III. The effect of angular momentum on the radial density profiles of disc galaxies

Keywords: galaxies, spiral -galaxies, structure -galaxies, kinematics and dynamics

We study the effect of angular momentum on the surface density profiles of disc galaxies, using high resolution simulations of major mergers whose remnants have downbending radial density profiles (type II). As described in the previous papers of this series, in this scenario, most of the disc mass is acquired after the collision via accretion from a hot gaseous halo. We find that the inner and outer disc scalelengths, as well as the break radius, correlate with the total angular momentum of the initial merging system, and are larger for high angular momentum systems. We follow the angular momentum redistribution in our simulated galaxies, and find that, like the mass, the disc angular momentum is acquired via accretion, i.e. to the detriment of the gaseous halo. Furthermore, high angular momentum systems give more angular momentum to their discs, which affects directly their radial density profile. Adding simulations of isolated galaxies to our sample, we find that the correlations are valid also for disc galaxies evolved in isolation. We show that the outer part of the disc at the end of the simulation is populated mainly by inside-out stellar migration, and that in galaxies with higher angular momentum, stars travel radially further out. This, however, does not mean that outer disc stars (in type II discs) were mostly born in the inner disc. Indeed, generally the break radius increases over time, and not taking this into account leads to overestimating the number of stars born in the inner disc.

INTRODUCTION

The early pioneering work of Freeman (1970) showed clearly that the radial surface density profile of disc galaxies is well fitted by an exponential. Later work (Van der Kruit 1979, and later e.g. Pohlen et al. 2002;Erwin, Beckman & Pohlen 2005;[START_REF] Gutiérrez | [END_REF] revealed the presence of a break in the profile of most galactic discs, as well as the fact that both the inner and the outer parts are well described by exponentials.

The break can be of two kinds, depending on the scalelengths of the two exponentials. If the slope of the outer part of the disc, hereafter called outer disc, is steeper than the slope of the inner part (inner disc), i.e. if the inner disc scalelength is greater than the outer disc one, the profile is called downbending, or type II (Pohlen & Trujillo 2006). This is the most common profile for disc galaxies (e.g. Pohlen & Trujillo 2006;Azzollini, Trujillo & Beckman 2008;Laine et al. Contact e-mail: npeschken@camk.edu.pl 2014). In the opposite case, the outer disc is shallower than the inner disc, and the profile is called upbending, or type III. The single exponential case, where no break is observed, is called type I.

Several mechanisms producing those different types of discs have been proposed so far. While it has been argued that a single exponential is the canonical profile for a disc [START_REF] Gunn | Astrophysical Cosomology Proceedings[END_REF]Lin & Pringle 1987;Yoshii & Sommer-Larsen 1989;Ferguson & Clarke 2001;Elmegreen & Struck 2013, 2016;Herpich, Tremaine & Rix 2016;Struck & Elmegreen 2017), many formation scenarii have been proposed for the downbending discs, sometimes related to each other. They are thought to be created by bars via the Outer Lindblad Resonance (Debattista et al. 2006;Pohlen & Trujillo 2006;Muñoz-Mateos et al. 2013;Kim et al. 2014), by star formation thresholds (Schaye 2004;Pohlen & Trujillo 2006;Elmegreen & Hunter 2006), or connected with morphological components such as rings or spirals (Laine et al. 2014). Stellar migration is also found to be important in the creation of type II profiles, being able to redistribute stars from the inner disc to the outer disc, and can be coupled with a star formation threshold (Roškar et al. 2008). On the other hand, type III profiles (upbending discs) remain poorly understood. They are sometimes associated with a spheroidal component such as a stellar halo, or with the superposition of a thin and a thick disc, and could be the result of minor mergers, or be linked to a strong bar (Erwin et al. 2005;Younger et al. 2007;Bakos & Trujillo 2012;Comerón et al. 2012;Herpich et al. 2015b).

Although the majority of the present-day spirals is thought to have experienced at least one major merger in their history (e.g. Hammer et al. 2009), so far most of the simulated galaxies used to investigate the formation of these different disc types have been evolved in isolation. Recently, however, we presented in Athanassoula et al. (2016, hereafter A16) three fiducial examples from a large sample of high resolution simulations of a major merger between two disc galaxies with a hot gaseous halo each, and showed that the remnants are good models of spirals. In this paper, we aim to study the role of angular momentum in shaping type II profiles, obtained from our large sample of major merger simulations.

The outline is as follows. In section 2, we briefly summarise the necessary parts of A16, and describe the fitting procedure for the radial density profiles. We compute the initial angular momentum and link it to the scalelengths in section 3. In section 4, we follow the angular momentum redistribution to explain how the initial angular momentum can affect the final disc properties. We discuss our results in section 5, and conclude in section 6.

TECHNICAL ASPECTS

Description of the simulations

In this section, we will briefly summarise the simulations characteristics, which have been obtained and described in some detail in A16. Our simulations start from two spherical protogalaxies consisting only of dark matter (DM) and hot gas, which we set on a given orbit. We use a total of 5.5 million particles (2 million for the baryons, 3.5 million for the DM), with softenings of 25 pc for the gas and 50 pc for the DM. Each particle's mass is 5×10 4 M for the gas and stars, and 2×10 5 M for the dark matter. By the time of the merging, a disc has formed in each of the progenitors, which is destroyed by the merging, its stars ending up mainly in a classical bulge. Gas continues to fall from the halo, and new discs, both thin and thick, are gradually formed in the remnant. Well-defined spiral arms soon develop in the thin disc, as well as a bar and a boxy/peanut bulge. Each simulation ends after 10 Gyr evolution, showing a remnant with a classical bulge-to-total ratio which is consistent with that of real spiral galaxies.

Our simulations are made using the N-body/SPH code GADGET3, including gas and its physics. The description of this code can be found in Springel & Hernquist (2002) and Springel (2005). Stars and dark matter are modeled by Nbody particles, and gas by SPH particles, with fully adaptive smoothing lengths. Gravity is computed with a hierarchy tree algorithm, and the code uses subgrid physics for the feedback, star formation and cooling, described in Springel & Hernquist (2003).

For a description of the technical aspects of the simulations we refer the reader to Rodionov, Athanassoula & Peschken (2017, hereafter R17). To avoid an excessive central concentration in our simulated galaxies, which would lead to unrealistic circular velocity curves and would delay the formation of the bar, we added AGN feedback (R17). This is based on a density threshold ρAGN and a temperature TAGN , while the physics underlying it is described in detail in A16.

The simulations have a variety of different orbits for the two merging protogalaxies. These orbits are characterised by their ellipticity, and the initial distance between the two progenitors. Each orbit leads to a different merging time, which is difficult to define precisely, but can be approximated by deriving the time beyond which the distance between the two centers of density stays below 1 kpc (see A16).

The two protogalactic haloes start with an initial spin, characterised by a spin value f representing the fraction of particles rotating with a positive sense of rotation. Thus for f = 1, all the particles rotate in the direct sense, while for f = 0.5 there is no net rotation.

The simulations used in this paper have rather early merging times, i.e. between 1.2 and 2.2 Gyr after the start of the simulation. About two thirds of our simulations have spin axes perpendicular to the orbital plane, but for the other simulations we tilt one or both protogalaxies by a chosen angle, to see the effect of the spin axis orientation. In most simulations the spin value f is the same for both protogalaxies, but in some simulations we introduced a different spin in each protogalaxy. We ran simulations of mergers with mass ratios between the two protogalaxies of 1, 1/2, 1/3, 1/4 and 1/8.

A central AGN is present in the remnant galaxy of the majority of our simulations, as described in A16 and R17, but we include 21 simulations without AGN to cover a larger part of the available parameter space. Nevertheless, the presence of our AGN affects mainly the central part of the disc (R17), and thus should not have an impact on the analysis presented in this paper.

We will first consider a subsample of 132 simulations which all have the same total mass, i.e. the same number of particles (5.5 million), and call it sample A. We also ran 67 simulations with various masses, each different from the mass of the simulations in sample A. This is done by keeping the same mass for single particles, but changing the number of particles. We add this group of 67 simulations (called sample B) to sample A to obtain a new sample of 199 simulations, sample A+B, which we will use in the discussion (section 5.2).

Fitting the radial density profiles

To derive the stellar radial density profile, we use axisymmetric concentric cylindrical annuli, and choose z lim =1 kpc as maximum height to keep only the thin disc profile. As shown in A16, the disc at the end of our three fiducial simulations is composed of an inner and a downbending outer disc (type II), separated by a break. This is also the case for the 199 simulations of the sample used in this paper (sample A+B), therefore we fit the disc part of our profiles with two exponential functions. We use a "piecewise" fit, which means For clarity, the second simulation (in black) has been shifted down by 1 logarithmic unit. Simulation 1 has a higher initial spin parameter λ than simulation 2.

that we fit the inner and the outer disc separately with an exponential:

Σin(R) = Σi exp( -R hi ), R < R break (1) Σout(R) = Σo exp( -R ho ), R > R break (2)
where Σi, Σo are normalization factors, hi and ho are the inner and outer disc scalelengths, and R break the break radius. The latter is derived by taking the intersection between the inner and the outer disc fits. The interval to fit for each part of the disc is selected manually, and we exclude, after visual inspection, every simulation for which the fit is not reliable. Two examples of fits are shown in Fig. 1. Another way to make this fit would be to use a double exponential function (or "broken-exponential") for the whole disc, which has been shown to give very similar results (differences in the scalelengths < 5 per cent, Erwin et al. 2008). We chose the present approach because of its simplicity.

The scalelengths and the break radius change with time, as shown both by observations (Perez 2004;Azzolini et al. 2008), and simulations (Roškar et al. 2008;Athanassoula, Peschken & Rodionov, in prep, hereafter Paper IV), and to be able to compare consistently the scalelengths derived for the different simulations, we need to compare all galaxies at the same evolutionary time, i.e. at the same time after the merging. We thus take a fixed evolution time of 7.8 Gyr after the merging for every simulation. Given that the merging times have values between 1.2 and 2.2 Gyr, the times at which we now study the disc are therefore between 9 and 10 Gyr, depending on the simulation. We will hereafter refer to this time as the final state.

We here aim to investigate the role of angular momentum in the formation of the new disc formed after the merging, in the secular evolution period of the galaxy. Nevertheless, some of the stars formed before or during the merging were spread all over the galaxy by the merging, and can pollute our density profiles. These stars represent between 10 and 30 per cent of the total stellar mass in the thin disc. Most of them are located in the bulge region (about 65 per cent), but they can also form a stellar halo or a thick disc, which will add particles to the thin disc, changing the value of the scalelengths by up to 20 per cent. These stars are not related to the disc formation and its dependance on the angular momentum, and we thus chose to remove them, selecting only the stars formed after the merging to derive our density profiles and the corresponding scalelengths.

THE ROLE OF ANGULAR MOMENTUM

Definition of the global, dimensionless spin parameter

Here, we use the dimensionless spin parameter λ (Peebles 1969):

λ = L|E| 1/2 GM 5/2 , ( 3
)
where L, E and M are respectively the total angular momentum, energy and mass of the system including its dark matter, gas and stars, computed with respect to the center of mass of the system, and G is the gravitational constant. The total energy is computed adding the kinetic energy, the gravitational potential energy and the gas internal energy. The total angular momentum L is a conserved quantity, but the total energy varies with time because of processes such as the cooling of the gas and the stellar feedback. We therefore have to specify the time at which we will calculate λ, and make sure it is calculated consistently for all simulations. To avoid the technical difficulties inherent in using the merging time, we use t=1 Gyr before the merging, which we call the initial state.

Angular momentum and scalelengths

How does the disc structure relate to the global spin parameter λ? From the definition of angular momentum, we expect that for two galaxies of the same total mass, the galaxy with the highest angular momentum will be more extended, as it has been shown in previous studies (e.g. Dalcanton, Spergel &Summers 1997 andKim &Lee 2013). We checked this result in our simulations by plotting the size of our galactic discs at the final state as a function of λ, for the simulations of sample A. As an estimate of the disc size, we take R95, the cylindrical radius containing 95 per cent of the total stellar mass. We can see on Fig. 2 that the size of the final galaxy increases linearly with λ, with a high correlation coefficient.

We thus find the disc to be globally larger for high λ galaxies, as expected. However, since the disc is constituted of an inner and an outer part, which part of the disc is affected most? We expect at least one of the two discs (inner or outer) to be larger at higher λ. To have a first insight of the effect of the spin parameter λ on the parameters of the radial density profiles, we plot in Fig. 1 the profiles at the final state (as defined in section 2.2) for two simulations of sample A with different λ values. We can see that the scalelengths of the simulation with a higher λ (in blue) are larger, and its break is located further out. The next step is to see whether this preliminary result is valid for all the simulations in sample A, as described below.

We examine the discs of the remnants at the final state, and plot the three parameters derived from the fits of the radial density profiles (inner, outer disc scalelength and break radius) as a function of the initial λ for each simulation in sample A (Fig. 3). As described in section 2.2, we excluded all the simulations for which there was some uncertainty in the fit of the corresponding parameter, which explains why the number of points is different for each plot. We thus have a total of 97 values for the inner disc scalelength (hereafter inner scalelength), 99 for the outer disc scalelength (outer scalelength) and 84 for the break radius (for which the fits of both the inner and the outer disc have to be reliable).

We see that the inner scalelength, the outer scalelength and the break radius all increase linearly with λ, so that larger angular momentum systems produce discs with larger inner, outer scalelengths and break radii. This confirms the results of Herpich et al. (2015a) for the inner scalelength and the break radius for cases with isolated galaxies with no mergers (see discussion in section 5.1), and adds a further argument showing that the remnant of a major merger possesses all the characteristics of a disc galaxy.

The correlations of the fits in Fig. 3 are tight, especially for the break radius. The scalelengths show a larger spread, presumably because their values can be more strongly influenced by the presence of morphological components such as bars, spirals or rings than the break radius.

It is important to note that all the simulations we have so far considered (sample A) have identical total masses (gas + stars + DM) and similar total stellar masses at their final state, so that the spread of scalelengths cannot be simply due to a mass dependence. Thus the spread should be due to the different parameters of the initial conditions in the various simulations, such as the halo spin value, and, in particular, to the different orbits of the two progenitors.

UNDERSTANDING OUR CORRELATIONS

The dimensionless spin parameter λ is computed 1 Gyr before the merging, and is dominated by the halo (hot gas and DM), because the discs of the protogalaxies are of short extent and of low mass, as the merging occurs early in our sample of simulations. Therefore, λ does not seem to be directly related to the final disc of the remnant and to its properties such as the scalelengths. To understand how it can yet influence the final disc, we will first look at the angular momentum in the final disc.

Angular momentum in the baryonic disc

We define the baryonic disc as the 2 kpc thick pill-box (|z| < 1 kpc, consistently with section 2.2) of maximum radius Rmax = 1.5 R break , and exclude the inner 3 kpc to remove the bulge part. We then compute the total angular momentum of the baryonic matter (gaseous + stellar particles) in this volume at the final state, and plot it against the scalelengths and the break radius for all the simulations of sample A in Fig. 4. We find clear correlations, showing that the properties of the disc are directly linked to its angular momentum.

We now need to relate this disc angular momentum to the initial halo-dominated spin parameter λ. We showed in A16 that the gas from the gaseous halo accretes onto the disc and thus rebuilds it after the merger, by fuelling star formation in the disc throughout the whole simulation. We suggest here that gas accretion from the halo is the main mechanism gradually transferring angular momentum from the gaseous halo to the disc, and investigate this in the following subsections.

Angular momentum redistribution

To be able to study the exchanges of angular momentum between the disc and the gaseous halo, we first made sure that the angular momentum exchanges between the baryonic and the dark matter haloes are relatively small and thus can be neglected in the context of our very simple qualitative explanation. Indeed, we found such exchanges to be of the order of few per cent.

To illustrate the angular momentum exchanges in our simulations, we will focus on two fiducial simulations of sample A, mdf 732 and mdf 780, presented in A16, which have quite different λ values, mdf 732 having a considerably higher λ than mdf 780. Note, however, that we made the same analysis with other simulations as well, and found similar results.

We define the disc as in section 4.1, and the halo as everything else. The baryonic halo is mainly constituted of gas, but a few stars are expected to be found as well. Nevertheless, we find the total angular momentum L of these stars to be a thousand times lower than the halo gas, and we thus neglect them.

We first plot in Fig. 5 the evolution over time of the total mass of the halo gas, the disc gas and the stars in the disc, separately. As expected (A16), the gaseous halo looses mass while the stellar disc gains it. In Fig. 6 we further plot the angular momentum as a function of time for these three components, and we can see that the halo gas is gradually loosing angular momentum over time, while the angular momentum of the stellar disc is increasing. The gas in the disc is loosing mass and angular momentum due to star formation in the disc. These plot thus show how the gas in the halo is accreted onto the disc, where it forms stars (see A16). Therefore, the stellar disc is gaining angular momentum by gas accretion and star formation.

Note that this simple angular momentum redistribution picture implicitly assumes that our system is relatively isolated. Otherwise, it will be able to exchange matter and/or angular momentum with other galaxies, so that the total mass and angular momentum of our system need not be conserved quantities.

Thus, the disc grows from material taken from the gaseous halo, which directly gives a fraction of its angular momentum to the disc. To find how this fraction varies for different simulations, we plot (Fig. 7) the final baryonic disc angular momentum (as defined in section 4.1) versus the initial total baryonic angular momentum of the system (which in practice we calculate at the merging time). We find a clear correlation, which shows that gaseous haloes with more angular momentum create on average discs with more angular momentum.

To conclude, the initial properties of the halo -and in particular its angular momentum -affect the final disc structure via gas accretion, which constitutes a plausible reason for the disc properties (scalelengths and break radius) to be linked to the initial spin parameter λ.

DISCUSSION

Robustness of the correlations

To test the robustness of our results with respect to the time defined as final state, we fitted the radial density profiles also 5.8 Gyr after the merging, instead of 7.8 Gyr. The values of the correlation coefficients are reported in Table 1 (2nd row). As expected due to the growth of the disc, the individual values of the scalelengths and the break radii are different, but we still find increasing linear trends with λ. Therefore, the time chosen as final state does not seem to be important for this study, provided it is consistent for all simulations and provided it is sufficiently long after the merging for the disc to have settled. Under these conditions, the increasing trend with λ should be valid regardless of the disc evolution time. We can thus keep 7.8 Gyr after the merging as the final state without being concerned about the effect of our choice.

We also changed the time at which we calculate (the initial state) and used the merging time. We found very little difference in the results, and the correlation coefficients are similar (Table 1, 3rd row). We repeated this analysis, calculating λ at various times covering the range between the start of the simulation and the merging, and conclude that the time chosen does not change the results significantly. We can also use λ at the final state instead of the initial state, to link λ and the disc structure both taken at the same time. We found again the same increasing trends, with the correlation coefficients reported in Table 1 (4th row). This points out the existence of a correlation between the initial and final λ, which we confirmed by plotting one against the other, and argues that for an analysis of this type, one can take the value of angular momentum at any time, as long as it is consistent for all simulations.

We also used λ at t=0 Gyr for all simulations (sample A), and again find correlations for the scalelengths and the break radius, although the correlation coefficients are slightly lower (Table 1, last row).

To see if the results obtained for our merger simulations are also valid for isolated galaxies, we included 12 simulations which take only one of the protogalaxies used in the merger simulations, and see how it evolves in isolation. This protogalaxy contains the same number of particles as the simulations of sample A, to keep the same total mass. To be able to compare the merger to the isolated simulations, we define the initial state as the start of the new disc formation for both, i.e. at the end of the merging for the former, and at t=0 for the latter. We take 7 Gyr after the initial state as the final state. We find that the isolated galaxies fit well with the merger simulations (see Fig. 8), which argues that the increasing linear trends of the two scalelengths and the break radius with the angular momentum seem to be the same if the disc is formed in isolation or from a major merger. Our results should therefore also be valid for isolated galaxies. Herpich et al. (2015a) did a similar analysis to ours, looking at the dependence of the scalelengths and the break radii with the angular momentum, but using a sample of 9 simulations of galaxies evolved in isolation. They computed the spin parameter λ of the halo at the start of the simulation, and also found that the inner scalelength and the break radius increase with λ. However, they observed a decreasing trend of the outer scalelength with λ, while we have an increasing one. We explain this difference by the fact that they included upbending discs in their analysis, whose outer scalelengths are naturally much higher than for downbending discs, while we only have downbending discs in our sample. Upbending profiles will be discussed in detail in a future paper (Paper IV). As a consequence, Herpich et al. (2015a) find an increasing trend with λ for the ratio of the inner scalelength to the outer scalelength, while we do not, our values showing no correlation with λ.

We mentioned in section 2.1 that in some simulations, the spin axis of one or both protogalaxies is tilted by a given angle; this concerns 34 simulations in sample A. This angle can be around the X or the Y axis, Z being the axis perpendicular to the orbital plane. We wanted to see if these simulations behave differently in our results than the simulations where both protogalactic spin axes are parallel, and perpendicular to the orbital plane. In Fig. 9 we plotted again our correlations between the scalelengths and λ, but showing the effect of spin orientation. We can see that the simulations where the spin axes are not parallel (angle = 0) fit reasonably well with the others. We can thus conclude that our results seem to be valid regardless of the spin axis orientation.

We also looked at the effect of the presence of a central AGN on our correlations, and found that the simulations without AGN also fit well with the ones having an AGN. This confirms that our AGN only affects the central part of the galaxy (R17).

The effect of the initial density distribution of gas and DM have also been tested since our sample contains simulations with different distribution parameters, such as the initial characteristic radii of the halo (see A16) or the presence of a central core for DM. Although the corresponding parameter space is very large, our relatively few trials argue that such parameters do not seem to have an effect on the correlations we found. Various other parameters, such as the merging orbit, the baryonic to total mass ratio, and the softening of the gas and DM, were also shown to have no significant impact on our results.

Introducing simulations with different total masses

Here we will analyse the effect of changing the total mass in the simulations, using sample B (see section 2.1). The problem when analysing simulations of different masses is that the evolution time-scale is also different. In our analysis we took a fixed evolution time after the merging (7.8 Gyr) to compare the properties of the disc, which introduces a small inconsistency when we compare simulations with different total masses. Nevertheless, we expect the time-scales to be similar if the total masses are not too different, which is the case of the simulations in our subsample of 67 galaxies (sample B) since they have total masses between 0.5 and 2 times the total mass of simulations in sample A. We include them in our correlations (Fig. 10) and find again high correlation coefficients; the sample B fits well with sample A. Furthermore, we added simulations of galaxies evolved in isolation as in section 5.1, but with half the number of particles, i.e. with a mass twice lower than the simulations of sample A. We found that these simulations match well in the correlations, which again suggests that our results are valid also for galaxies formed in isolation. Moreover, we included the simulations of sample B in the correlation of the final baryonic disc angular momentum as a function of the initial baryonic angular momentum (as in Fig. 7), and found again a very good match.

Since the simulations of sample B follow similar trends as sample A despite their total masses being different, in the following subsections we will include them in our analysis, using the A+B sample.

Migration and angular momentum

The formation of outer discs in downbending profile galaxies is still debated, but one of the main possible scenario relies on the presence of outwards stellar migration to build the outer dic (Roškar et al. 2008). We thus investigate where the stars in the outer disc of our remnant galaxies come from, by calculating the radius at which they were born. The stellar discs of the two protogalaxies are separate before the collision and thus the birth radius with respect to the remnant galaxy cannot be defined. We therefore use in our analysis only the stars born after the merging, as in section 2.2.

We define at any given time the stars of the outer disc as all the stellar particles located beyond the break. We plot in Fig. 11 the distribution of the radii at birth of the stars which by t=10 Gyr end up in the outer disc, for simulation mdf 730 from sample A, which was chosen because of its smooth profile, allowing us to make our points on migration clearer. Since every star is born at a different time, this distribution does not represent a real stellar distribution at any time of the simulation, but it helps visualise the fraction of stars born at a given radius throughout the whole simulation. We can see that most stars of the outer disc (78 ± 18 per cent for sample A+B) were born at a radius less than the break radius found at t=10 Gyr. This means that these stars have migrated radially outwards to form the outer disc, establishing the presence of inside-out stellar migration in the disc. This is in agreement with the results of Roškar et al. (2008) presenting simulations with a downbending disc profile induced by a star formation threshold and outward migration, and with observations e.g. in Radburn-Smith et al. ( 2012) and Zheng et al. (2015), and thus adds a further argument to those presented in A16 that the remnant of a major merger is a disc galaxy and behaves in all aspects as such.

Having established that there is inside-out stellar migration in our discs, we explore the effect of angular momentum on this phenomenon. To characterise the migration, we calculate the distances travelled by these stars migrating towards the outer disc, throughout the whole simulation. To derive the migration distances for a given simulation, we first compute the radial distance travelled by each star ending up in the outer disc between its time of birth and the final state (as defined in section 2.2):

Dmig = R f inal -R birth , (4)
where R f inal and R birth are the radii at the final state and at birth for each star. We remove again the stars born before the merging since migration only starts after the collision, or, more precisely, after the disc formation begins (t bd , A16). Since we are only interested in the outward migration towards the outer disc, we also remove the stars migrating inwards, which represent only about 5 per cent of the outer disc population. We then take the mean over all the remaining stars to get a single value for each simulation, which we plot versus λ. We repeat this for all the simulations in sample A+B and plot the result in Fig. 12. We see a clear increasing trend of the migration distances with λ, allowing us to conclude that higher initial angular momentum systems lead to discs with higher migration distances towards the outer disc. In fact, we find the same result if we consider inside-out migration in the whole disc, and do not restrict ourselves to outer disc stars. The stars travel globally further out in high angular momentum systems, and this is particularly important in the outer disc, where 94 (±5) per cent of the stars have undergone inside-out migration (compared to 46±5 per cent for the whole disc). The angular momentum thus seems to play an important role in the radial outward migration, by redistributing the stellar content in the outer disc efficiently (with large travelling distances) or not.

This result is consistent with outer discs in high λ simulations having larger scalelengths: if stars travel further out, their final distribution will be more extended, resulting in a larger outer disc.

Migration and birth radius

In section 5.3, we showed that most particles ending up in the outer disc were born inside the break radius, as calcu-lated at t=10 Gyr. However, the stars whose birth distribution has been plotted in Fig. 11 were born at all times after the merging. As will be shown for our simulations in Paper IV (see also Perez 2004 andAzzollini et al. 2008 for observations), the break radius generally increases with time, so that stars which are born inside the 10 Gyr break in Fig. 11 could in fact be born beyond the break taken at their time of birth. Therefore, we cannot claim that about 80 per cent of the outer disc's stars that were born at a radius which corresponds to the inner disc at 10 Gyr were actually born in the inner disc, since their birth radius could be in the outer disc at their birth time. Although this does not change the fact that most of these stars clearly migrated outwards, it could question the claim that they travelled from the inner disc to the outer disc.

To solve this, we took the same stars (i.e. the ones in the outer disc at 10 Gyr), and computed the fraction which was born in the inner disc defined with respect to the break at their time of birth. This fraction however depends strongly on the value taken for the break radius, since the break is often located near the end of structures such as spirals (Laine et al. 2014;Paper IV), which are high star formation areas (A16 for simulations and e.g. Silva-Villa & Larsen 2012 for observations). Besides, the break radius can be difficult to measure precisely at early times (a few Gyr after the merging). We thus defined errorbars for the estimated break position at each time step for every simulation, and excluded all the particles born closer to the break radius than these errorbars. To make sure our 10 Gyr outer disc definition is also reliable, we defined the particles of the outer disc as the ones located beyond the external errorbar of the break radius.

To exclude the effect of radial displacements due to radial oscillations of stars in their orbit, we chose to use RG -the guiding radius of each star -instead of their regular radius to derive their position with respect to the break. RG is the radius of the circular orbit associated to the angular momentum of a given star, and can be derived by solving the equation:

RG = Lz vcirc(RG) , (5)
where Lz is the vertical component of the angular momentum of the star, and vcirc is the circular velocity at the guiding radius.

The fractions of outer disc stars that are born in the inner disc derived with this method cover a wide range of values depending on the simulations, from 10 to 80 per cent. Nevertheless, we find globally lower fractions than the ones from section 5.3, as well as the ones from Roškar et al. (2008, ∼ 85 per cent), which can be explained by the fact that the break radius tends to increase with time (see paper IV), so that some stars were born inside the 10 Gyr break but outside their birth time break, and thus should be considered as being born in the outer disc. Note that these low fractions can not be explained by our choice to take the guiding radius instead of the regular radius, or to exclude the stars within the errorbars of the break radius: we performed the same analysis using the radius instead of the guiding radius, and also keeping the particles inside the errorbars, and found similar results in both cases. For each star, the inner disc is defined here with respect to the break radius derived at its time of birth, as explained in section 5.4.

To understand the spread of values derived for the fraction of stars born in the inner disc, we tried to plot these fractions again versus the spin parameter λ (taken for all the components, 1 Gyr before the merging) in Fig. 13. Although having a large scatter (probably because of the sensitivity of the values to the break location and its errorbars, and the difficulty to define them precisely at early times), this plot shows again an increasing trend of the fractions with λ. Therefore, galaxies with high angular momentum will have an outer disc which is populated by a majority of stars coming from the inner disc, whereas in low angular momentum galaxies, stars in the outer disc will mostly be born in the outer disc. It is important to note that the trend found in Fig. 13 involves the fraction of stars, i.e. relative and not absolute numbers. It thus does not imply that low angular momentum galaxies form more stars in their outer parts, since the number of stars in the outer disc is different in every simulation.

We thus have simulations which have outer discs formed mostly by radial migration, but with a significant amount of stars in the outer parts which were born in the outer disc. We conclude that not taking into account the fact that the break radius changes with time, can lead to overestimating the fraction of stars born in the inner disc in the context of radial migration. Therefore, the picture of outer disc stars mostly coming from the inner disc has to be used with caution, since it depends on the angular momentum of the galaxy. Only galaxies with high spin parameter λ (λ>0.08 in our simulations) seem to be able to have about 80 per cent of their outer disc stars born in the inner disc. In galaxies with average or yet lower angular momentum (λ<0.06 in our simulations) this fraction drops below 50 per cent, disclaiming the inner disc as the main origin of the outer disc stars.

For this series of papers we ran a sample of high resolution simulations of which three fiducial cases were described in A16, where more general information on the runs was also given. It was shown there that after a merging, a new disc forms in the remnant from gas accreting from the halo. In this paper, we used subsamples derived as described in section 2.1. We derived the stellar radial density profiles of the remnants at the end of all the simulations, and found downbending (type II) profiles. We then derived the corresponding inner and outer disc scalelengths, as well as the break radius. We used the spin parameter λ computed 1 Gyr before the merging to have a definition of the angular momentum consistent for all our simulations. Plotting for our sample the values derived from the radial density profiles fits as a function of λ, we found that the inner, the outer scalelength and the break radius increase with λ. Therefore, both the inner and the outer discs are larger for higher angular momentum systems.

To explain how the initial orbit-and halo-dominated spin parameter λ of the merging system can affect the properties of the final remnant disc (mostly composed of stars born after the merging), we looked at the angular momentum redistribution in our simulations. The scalelengths and break radius correlate with the baryonic angular momentum of the disc, suggesting a link between the latter and the initial angular momentum. To understand how the angular momentum of the disc was acquired, we investigated the transfers of the baryonic matter and angular momentum between the halo and the disc, after making sure that the total baryonic angular momentum is conserved. In the framework of disc galaxy formation from a major merger of two progenitors with extended gas-rich haloes (A16), it is easy to understand that the gaseous halo gives angular momentum to the disc by accreting its gas onto it. We showed this in two fiducial simulations with different λ values, with the halo gas gradually loosing a fraction of its mass and angular momentum to the stellar disc, by gas accretion and star formation. We further found that haloes with higher initial total angular momentum (or λ) will create final discs with higher angular momentum, and larger scalelengths and break radius.

Naturally this scenario is only valid for an isolated system with no interactions with the environment. Thus our results, as well as those of other dynamical (and thus necessarily idealised) simulations, can not be straightforwadly extended to a cosmological context, since interactions with environment could interfere with the angular momentum exchanges between the gaseous halo and the baryonic disc.

The correlations of the scalelengths and the break radius with λ are robust and do not change with the time at which we fit the radial density profiles, or the time used to compute λ. Furthermore, simulations of isolated galaxies follow the same trends, so that our results should be valid for discs formed both in isolation and in major mergers. In some simulations, the spin axes of the protogalaxies are tilted, and we showed that the spin axis orientations of the two merging protogalaxies do not seem to play a role in this analysis.

We also included in the correlations a sample of 67 merger simulations with total masses lower or higher than the simulations from the main sample. Although the evolution time-scale of these simulations is also different, we found that these simulations (which have total masses reasonably close to our main sample) fit well with the others in the correlations, and thus added them to the sample.

We analysed the outer disc origin, and found that the stars ending up in the outer disc were mostly born at smaller radii (∼95 per cent), suggesting inside-out migration as the main formation driver for the outer disc, in good agreement with previous work on galaxies formed in isolation (Roškar et al. 2008). To see the effect of the angular momentum on this migration, we computed the distance radially travelled by the stars towards the outer disc, and plotted it against the spin parameter λ for the simulations of our sample. We found a clear correlation, galaxies with higher angular momentum having larger inside-out migration distances.

We also showed that to study the origin of the outer disc, it is necessary to take into account the fact that the break location changes with time, so as to avoid overestimations of the fraction of stars born in the inner disc. While ∼80 per cent of the outer disc stars were born inside the break derived at t=10 Gyr, this fraction can take lower values (under 50 per cent) using the time dependent break, which sets doubt on the picture of the outer disc formed mainly from the inner disc. This fraction depends on the angular momentum, and is higher in high angular momentum galaxiess. Thus, in some low angular momentum systems, this fraction can drop to values even lower than 20 per cent.

We can thus conclude that the angular momentum is a key parameter in the creation of disc structures, as it affects radial migration, and can explain the large range of values observed for the inner and outer scalelengths in disc galaxies of a given mass.

Figure 1 .

 1 Figure1. Projected surface stellar density radial profiles for two simulations, together with the corresponding fits for the disc part. For clarity, the second simulation (in black) has been shifted down by 1 logarithmic unit. Simulation 1 has a higher initial spin parameter λ than simulation 2.

Figure 2 .

 2 Figure 2. Cylindrical radius containing 95 per cent of the stellar mass at the final state, as a function of the spin parameter λ computed at the initial state, for the simulations of sample A. The corresponding linear fit is plotted in black, with the equation and correlation coefficient given in the top left corner.

Figure 3 .

 3 Figure 3. From top to bottom we plot respectively the inner scalelengths, the break radii and the outer scalelengths derived from the fits at the final state, each as a function of the spin parameter λ taken at the initial state, for the simulations in sample A. The corresponding linear fits are plotted in black, with the equation and correlation coefficient given in the top left corner of each plot.

Figure 4 .

 4 Figure 4. Correlations of the inner scalelength, the break radius and the outer scalelength, with the total baryonic angular momentum of the final disc, for sample A.

Figure 5 .

 5 Figure 5. Total mass of the gaseous halo, the gaseous disc and the stellar disc as a function of time for two simulations, mdf732 (high λ, upper pannel) and mdf780 (low λ, lower pannel).

Figure 6 .

 6 Figure 6. Total angular momentum of the gaseous halo, the gaseous disc and the stellar disc as a function of time for two simulations, mdf732 (high λ, upper pannel) and mdf780 (low λ, low pannel).

Figure 7 .

 7 Figure 7. Final baryonic disc angular momentum, as a function of the total baryonic angular momentum of the system computed at the merging time, for simulations in sample A.

Figure 8 .

 8 Figure8. Same plots as in Fig.3, but adding simulations of isolated galaxies (in blue).

Figure 9 .

 9 Figure9. Same plots as in Fig.3, but showing the effect of spin orientation on our correlations, with the angle between the two protogalaxies axes.

Figure 10 .

 10 Figure10. Same plots as in Fig.3, but adding a sample of 67 simulations (sample B) with total masses different from those of the simulations in sample A.

Figure 11 .

 11 Figure 11. Radial density profile for a snapshot at t=10 Gyr (in blue), and distribution at time of birth of the stars ending up in the outer disc (in green). The fit of the profile at t=10 Gyr is plotted in red, and the corresponding break radius is indicated with a black vertical line.

Figure 12 .

 12 Figure 12. Mean inside-out migration distance of the stars ending up in the outer disc, as a function of the initial spin parameter λ, for sample A+B.

Figure 13 .

 13 Figure 13. Fraction of outer disc stars which were born in the inner disc, as a function of the spin parameter λ, for sample A+B.For each star, the inner disc is defined here with respect to the break radius derived at its time of birth, as explained in section 5.4.

Table 1 .

 1 Parameters of the linear fits for different plots of λ versus the inner scalelength, break radius and outer scalelength. t i , t f and tmerg are respectively the times of the initial state, the final state and the merging, and r 2 c is the correlation coefficient for the linear fit with equation: f (λ) = aλ + b.

		Inner Scalelength	Break Radius		Outer Scalelength
		r 2 c	a	b	r 2 c	a	b	r 2 c	a	b
	λt=t i , t f = tmerg + 7.8	0.56 63.1 1.7	0.92 303.6 -2.5 0.74	27.6 0.4
	λt=t i , t f = tmerg + 5.8	0.48 55.0 2.3	0.91 283.5 -1.9 0.74	24.8 0.4
	λt=t merg , t f = tmerg + 7.8 0.56 58.0 1.7	0.90 277.9 -2.2 0.72	24.9 0.5
	λt=t f , t f = tmerg + 7.8	0.55 54.0 1.6	0.90 259.8 -3.0 0.73	23.7 0.4
	λt=t f , t f = tmerg + 5.8	0.47 47.0 2.2	0.87 238.6 -1.8 0.71	20.9 0.4
	λ t=0 , t f = tmerg + 7.8	0.53 59.8 2.0	0.89 291.1 -1.1 0.72	26.6 0.6

MNRAS 000, 1-??(2017)

ACKNOWLEDGEMENTS

We thank Jean-Charles Lambert for computer assistance, and the referee for useful suggestions. This work was supported in part by the Polish National Science Centre under grant 2013/10/A/ST9/00023, and was granted access to the French HPC resources of [TGCC/CINES/IDRIS] under the allocations 2014-[x2014047098], 2015-[x2015047098] and 2016-[x2016047665], made by GENCI, as well as the HPC resources of Aix-Marseille Université financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program « Investissements d'Avenir » supervised by the Agence Nationale de la Recherche.


Alexandre Poirier
email: alexandre.poirier@umontpellier.fr

Amélie Banc
email: amelie.banc@umontpellier.fr

Antonio Stocco
email: antonio.stocco@umontpellier.fr

Martin In
email: martin.in@umontpellier.fr

Laurence Ramos
email: laurence.ramos@umontpellier.fr

Multistep building of a soft plant protein film at the air-water interface

Keywords: Gliadin, soft plant protein, air-water interface, tensiometry, dilatational rheology, ellipsometry, master curve, conformational change

Gliadins are edible wheat storage proteins well known for their surface active properties. In this paper, we present experimental results on the interfacial properties of acidic solutions of gliadin studied over 5 decades of concentrations, from 0.001 to 110 g/L. Dynamic pendant drop tensiometry reveals that the surface pressure  of gliadin solutions builds up in a multistep process. The series of curves of the time evolution of collected at different bulk protein concentrations C can be merged onto a single master curve when  is plotted as a function of t where t is the time elapsed since the formation of the air/water interface and  is a shift parameter that varies with C as a power law with an exponent 2. The existence of such time-concentration superposition, which we evidence for the first time, indicates that the same mechanisms govern the surface tension evolution at all concentrations and are accelerated by an increase of the bulk concentration. The scaling of  with C is consistent with a kinetic of adsorption controlled by the diffusion of the proteins in the bulk. Moreover, we show that the proteins adsorption at the air/water interface is kinetically irreversible. Correlated evolutions of the optical and elastic properties of the interfaces, as probed by ellipsometry and surface dilatational rheology respectively, provide a consistent physical picture of the building up of the protein interfacial layer. A progressive coverage of the interface by the proteins occurs at low . This stage is followed, at higher , by conformational rearrangements of the protein film, which are identified by a strong increase of the dissipative viscoelastic properties of the film concomitantly with a peculiar evolution of its optical profile that we have rationalized. In the last stage, at even higher surface pressure, the adsorption is arrested; the optical profile is not modified while the elasticity of the interfacial layer dramatically increases with the surface pressure, presumably due to the film ageing.

Introduction

Interfacial properties govern many structural and mechanical properties of bulk materials and dispersions.

Common examples include the importance of grain-boundaries in the mechanical properties of atomic, molecular and colloidal polycrystals, and of surface actives species in stabilizing foams and emulsions, thanks to their adsorption at air/liquid or liquid/liquid interfaces [START_REF] Langevin | Influence of interfacial rheology on foam and emulsion properties[END_REF] . Characterizing in details those liquid interfaces is the key to understand and possibly tune the bulk properties of foams and emulsions, which are crucial in a wide range of industrial applications from cosmetics to agrochemistry and pharmacy.

There is a wealth of surface active species (e.g. soaps, phospholipids and polymer molecules with different architectures) that spontaneously adsorb at a liquid interface. Proteins, which are amphiphilic compounds made of polar and non-polar amino acid residues, also tend to accumulate at air/water and oil/water interfaces. Understanding and controlling the interfacial adsorption of proteins is important in many industrial fields from food science, to purification and implantable medical devices, for instance.

Proteins films are usually characterized by a high viscoelasticity and can undergo a glass or gel transition due to a protein crowding at the interface even for low bulk concentrations of the subphase [START_REF] Dickinson | Adsorbed protein layers at fluid interfaces: interactions, structure and surface rheology[END_REF] . Up to now, the majority of detailed studies on interfacial properties of protein films has been conducted with animal proteins. Many works focused on bovine serum albumin, -lactoglobulin and ovalbumin, which are globular proteins considered as hard proteins, and on -casein, which is considered as a soft protein prone to conformational changes at the interface and yielding softer interfaces than hard proteins [START_REF] Bos | Interfacial rheological properties of adsorbed protein layers and surfactants: a review[END_REF] .

Interfacial properties of plant proteins have been much less investigated than their animal counterpart [START_REF] Ducel | Adsorption kinetics and rheological interfacial properties of plant proteins at the oil-water interface[END_REF] .

Among vegetable proteins, wheat storage proteins, commonly known as gluten proteins, represent an important part of the occidental people diet. Indeed, many food products traditionally contain wheat flour.

In addition, gluten is also used as a texturing agent by the food industry that exploits the unique properties of the viscoelastic network created by these proteins. The interfacial properties of these proteins were also explored to better understand their role in the gas holding capacity of dough, and to determine their potentiality for the development of new products [START_REF] Ornebro | Interfacial behaviour of wheat proteins[END_REF] . These studies were performed on either whole gluten proteins [START_REF] Tschoegl | The surface chemistry of wheat gluten II. Measurements of surface viscoelasticity[END_REF] or separately on the two main classes of gluten proteins: glutenins, the polymeric proteins, and gliadins, the monomeric ones. Gluten proteins are non-globular proteins that behave as polymers and display a persistence length similar to that of disordered proteins [START_REF] Dahesh | Polymeric Assembly of Gluten Proteins in an Aqueous Ethanol Solvent[END_REF] . Their secondary structure is labile and depends of the environment [START_REF] Georget | Effects of temperature and water content on the secondary structure of wheat gluten studied by FTIR spectroscopy[END_REF] and osmotic pressure [START_REF] Boire | Osmotic Compression of Anisotropic Proteins: Interaction Properties and Associated Structures in Wheat Gliadin Dispersions[END_REF] . It was established that gliadins are more surface active than glutenins [START_REF] Mita | Physicochemical Studies on Wheat-Protein Foams[END_REF] and display at equilibrium lower surface tension at the air-water interface than soy glycinin, casein [START_REF] Bos | Foams and surface rheological properties of betacasein, gliadin and glycinin[END_REF] or bovine serum albumin proteins [START_REF] Takeda | Emulsifying and surface properties of wheat gluten under acidic conditions[END_REF] . Furthermore, gliadins were demonstrated to be very surface active for oil-water interfaces and form highly viscoelastic films 4a . As a consequence, gliadins were studied to stabilize foams 10a, 13 and emulsions [START_REF] Takeda | Emulsifying and surface properties of wheat gluten under acidic conditions[END_REF] . However, the precise mechanism of gliadin adsorption at interface, the structure of the film formed and its rheology are still not well described.

In the present work, we investigate the adsorption of gliadins at the air-water interface from acidic solutions displaying a large range of concentration of proteins, from 1 mg/L to 110 g/L. Solutions are prepared at acidic pH, for which gliadins are the most soluble in aqueous solvent due to their neutral isoelectric point. A combination of tensiometry, dilatational viscoelasticity and ellipsometry measurements is performed as a function of time. A time-concentration superposition is evidenced whatever the subphase concentration and reveals that protein adsorption at the interface is dominated by bulk diffusion. We propose a consistent physical picture of the multistep diffusion-controlled irreversible adsorption of the gliadin proteins at an air/water interface, and discuss our main conclusive results in light of the literature. Overall all our experimental results indicate that gliadin displays a behavior typical of soft proteins.

Material and methods

Material

Gliadin extraction and characterization

Gliadins are extracted from industrial gluten (courtesy of Tereos Syral, France) according to a protocol previously described by some of us [START_REF] Dahesh | Polymeric Assembly of Gluten Proteins in an Aqueous Ethanol Solvent[END_REF] . Briefly, gluten powder is mixed with 50% (v/v) ethanol/water solvent and submitted to a continuous rotating agitation for 19 hours. After centrifugation, the supernatant is recovered and placed at 4°C to induce a liquid-liquid phase separation. The light phase, enriched in gliadins, is recovered and freeze-dried. This fraction (labelled S2 in Ref. [START_REF] Dahesh | Polymeric Assembly of Gluten Proteins in an Aqueous Ethanol Solvent[END_REF] ) is characterized by Sodium Dodecyl Sulfate PolyAcrylamide Gel Electrophoresis, SDS-Page) and size-exclusion high performance liquid chromatography (SE-HPLC). Figure 1a displays the SE-HPLC profile measured on a TSK G400 SWXL column using the experimental conditions described in Dahesh et al. [START_REF] Dahesh | Spontaneous gelation of wheat gluten proteins in a food grade solvent[END_REF] . The molecular weight (M w ) composition deduced from the profile is 78% gliadins (25 000<M w <55 000g/mol), 15% glutenins (60 000<M w <400 000g/mol) and 7% albumins-globulins (M w ≈20 000 g/mol).

Sample preparation

We use a 50mM acetic acid aqueous solvent (purity 99%) (pH=3) to fully solubilize gliadin proteins (gliadins are insoluble at neutral pH). Protein/solvent mixtures of concentration C=50 g/L and higher are put in a rotary shaker overnight at room temperature to allow sample homogenization. They are subsequently filtered with a 0.22 µm cellulose mixed ester membrane. Diluted solutions are prepared from the stock solution at C=50 g/L by dilution with 50mM acetic acid aqueous solvent to reach the target concentrations.

Bulk sample characterizations

Size of the proteins

The hydrodynamic radius of the proteins in a acetic acid solution (50mM) is characterized by dynamic light scattering (DLS) performed at a wavelength =532nm for 6 different scattering angles ranging between 40 and 120°. The number size distribution is dominated by objects with a hydrodynamic radius of 5 nm. This size is consistent with the radius of gyration of gliadins (3<Rg<5nm) measured by Thomson et al. [START_REF] Thomson | Small angle X-ray scattering of wheat seed-storage proteins: alpha-, gamma-and omega-gliadins and the high molecular weight (HMW) subunits of glutenin[END_REF] who modeled gliadins as prolate ellipsoids with an aspect ratio comprised between 4 and 6. We expect that the interfacial properties, as measured in this study, are majorly due to individual gliadins.

Dielectric constant and refractive index

The refractive index of the protein solutions in the visible range, , is measured with an Abbe refractometer, for protein concentrations C from 0 to 50 g/L. Concentrations are converted in volume fraction of dry protein,    assuming a protein relative density =1.32 [START_REF] Zhang | Solvent effects on the molecular structures of crude gliadins as revealed by density and ultrasound velocity measurements[END_REF] . For   in the range (0-0.038), the data can be very well accounted for by a linear variation of with the volume fraction of proteins   Fig. 1b) as predicted theoretically. In the Wiener effective medium approximation [START_REF] Aspnes | Optical-Properties of Thin-Films[END_REF] , the dielectric constant  = n 2 (with n the refractive index) is indeed expected to change linearly with     Hence the refractive index of the protein solutions, , reads (Eq. 1) with the index of refraction of the solvent, and the index of refraction of the proteins.

Equation 1 can be rewritten as (Eq. 2)

The best fit of the experimental data with Equation 2 yields , giving an index of refraction of the proteins, = 1.541. This value is consistent with the literature (n=1.555 was found for gliadins in 0.1M acetic acid solution) [START_REF] Robertson | On the refractive indices of solutions of certain proteins. V. Gliadin[END_REF] .

Pendant drop tensiometry and dilatational rheology

Pendant drop tensiometry

A drop profile analysis tensiometer (PAT-1, SINTERFACE Technologies, Germany) is used for surface tension and surface dilatational rheology measurements. An aqueous drop (typical volume 15 mm 3 ) is formed at the capillary tip inside an empty measuring glass cell. The images of the drop are recorded and the interfacial tension is computed from the drop shape [START_REF] Loglio | Drop and bubble shape analysis as tool for dilatational rheology of interfacial layers[END_REF] . The surface pressure is defined as

 =  solv ,
where  solv is the surface tension of the acetic acid aqueous solvent in absence of protein and  is the surface tension of the protein solution against air.

Droplet washing

To assess the reversible character of adsorption, we use the drop profile tensiometer with a coaxial capillary to form a drop and subsequently wash the inside of the drop by replacing a volume V=1mm 3 of the interior of the drop by pure solvent. This step lasts about 1s. To ensure a complete washing of the drop this procedure is repeated 450 times, so that the total volume exchanged is thirty times that of the drop (15 mm 3 ). The time evolution of the surface tension, before, during and after the washing procedure is measured.

Dilatational rheology measurements

We use the drop profile tensiometer to study the dilatational rheology of proteins at the air-water interface.

To do so, an aqueous drop is perturbed by small harmonic oscillations of the interfacial area A leading to harmonic oscillations of the surface tension . From the Fourier transform F of the response relative to the perturbation the complex visco-elastic modulus E is calculated 19 :

(Eq. 3)

where the real part, E ' , is the storage modulus that characterizes the dilatational interfacial elasticity, and the imaginary part, E " , is the loss modulus and is related to the dilatational interfacial viscosity. We impose relative deformations of 5%, respectively 10%, at a frequency of 0.025Hz, respectively 0.05Hz, for samples with bulk concentrations below, respectively above, 50 mg/ml. These parameters are chosen to optimize the quality of raw data while remaining in the linear regime of the response.

Wilhelmy plate tensiometry, ellipsometry and Brewster angle microscopy

Planar interfaces are characterized using a set-up that combines a Wilhelmy plate (KSV, Nima), a

Brewster angle microscope (Optrel, Germany) and an ellipsometer (Optrel, Germany). The time evolution of the air-water interface is probed after a glass Petri dish of 17 cm diameter has been filled with protein solutions.

On the one hand, Brewster angle images are recorded using the ellipsometric device equipped with a 10X magnification objective and an acquisition camera. The contrast of the images is controlled by rotating the polarizer angle, the analyzer angle being kept constant (vertically polarized).

On the other hand, the optical profile and surface excess concentrations of proteins at the air/water interface are measured by ellipsometry. is defined as:

(Eq. 4)

Here z is the axis normal to the interface (with the convention > 0 in air) and = 0 is the location of the Gibbs dividing surface, C is the solute concentration as a function of the distance from the interface and the solute concentration in the bulk solution [

].

An ellipsometer working with a green laser light (λ = 533 nm) is used to measure the quantities and .

Measurements are performed at the incident angle =55°. and are related to the ratio of reflection field coefficients by: (Eq. 5)

Here is the component of field reflection coefficient parallel to the reflection plane, and is the component perpendicular to reflection plane. Data are analyzed in the framework of perturbation theory, which describes as the deviation from the reflection coefficient expected for a sharp step-like (Eq.7)

Here , respectively , is the component of the dielectric constant parallel, respectively perpendicular, to the interface, and .

If the interfacial layer is locally isotropic, = = , and Equation 7 reads (Eq. 8)

In the limit of optically thin interfacial layers, I 1 is related to the surface excess concentration measured by ellipsometry , which is approximately equal to the surface excess concentration (Eq. 4):

(Eq. 9)

where gives in a good approximation the dielectric constant surface excess.

Finally, the surface excess concentration reads 21 :

(Eq. 10)

Experimental results

Surface tension.

We show in Figure 2a the time evolution of the air/water surface tension, , measured for several protein concentrations C in the aqueous phase, spanning more than 5 orders of magnitude (from 0.01 to 110 g/L), over a time scale ranging from 5 s to 10000 s, using the pendant drop tensiometry. We find that the surface tension of the solvent is constant ( solv  73 mN/m) over the whole duration of the measurements, ensuring reliable data over the full time scale. For the lowest protein concentration (0.01g/L), an induction period of about 50 s is observed during which the surface tension of the protein solution stays constant and equal to that of the solvent and then continuously decreases with time. For the other protein concentrations the induction period is too short to be measured with our experimental set-up. We find indeed that the surface tension at the shortest accessible time is systematically smaller than that of the solvent and decreases with the protein concentration. Note that several regimes of surface tension decrease with time can be identified in a semi-log scale.

Interestingly, all the data acquired for C>0.1 g/L, overlap on a single master curve once plotted as a function of a normalized time t, where t is the real time elapsed since the drop formation and  is a scale factor. We arbitrarily choose C=1 g/L as the reference concentration (=1). The scaling is shown in Figure 2b where the surface pressure , is plotted as a function of t. The master curve evidences two exponential growths of the surface pressure associated with characteristic times t 1 500 s and t 2 =19 000 s (fits not shown) followed by asymptotic logarithmic evolution. The scaling does not hold for the two lowest protein concentrations (0.01 and 0.1 g/L). For these dilute solutions, the drop (typical volume 15 mm 3 ) does not constitute an infinite reservoir of protein, and the dynamic surface tension is expected to be strongly influenced by the depletion of proteins in the bulk due to their adsorption at the interface [START_REF] Kairaliyeva | Surface Tension and Adsorption Studies by Drop Profile Analysis Tensiometry[END_REF] . To overcome this problem and complete our experimental data in the low concentration limit, we perform similar measurements with a planar geometry, for bulk protein concentrations between 0.001g/L and 10g/L. In these conditions, a regime of infinite reservoir of proteins is systematically reached. We use a Petri dish filled with the protein solution and measure the surface tension as a function of time using a Wilhelmy plate. A master curve, surface pressure versus rescaled time t, similar to that obtained in the pendant drop experiment, is also obtained. The scaling factors, , used to build master curves with the experimental data collected in the two experimental geometries are displayed in Figure 2c. We find that  increases as a power law of the concentration,  ~ C p with p=1.9. We note that this value is close to 2, the numerical value expected for a diffusive process. This point will be discussed below.

To address the reversibility of gliadin adsorption we measure the time evolution of the surface pressure after dilution of the bulk subphase. In the case of reversible adsorption, a decrease of the bulk concentration should lead to a decrease of the surface pressure [START_REF] Ferri | Desorption kinetics of surfactants at fluid interfaces by novel coaxial capillary pendant drop experiments[END_REF] . This has been checked with a soluble surfactant (dodecyltrimethylammonium chloride, DTAC), ensuring the reliability of our experimental procedure (as detailed in the Materials and methods section). By contrast, in the case of gliadin,  never decreases after washing, as shown in Figure 3. Three experiments carried out at different stages of the time evolution of the surface pressure are presented. At low surface pressures (= 11 mN/m and = 21 mN/m), dilution of the bulk does not lead to any significant decrease of the surface pressure, although it dramatically reduces the rate of increase of the surface pressure. At high pressure (= 30 mN/m) dilution of the bulk turns out to have no effect on the increase rate of the surface pressure. Thus, our findings indicate that, at low pressure the evolution of is determined by protein transfer from the bulk to the interface, while at high pressure the further evolution of the pressure is independent of mass transfer to the interface. Moreover, since no decrease of the surface pressure could be evidenced, the desorption of the protein from interface is certainly extremely slow if ever it occurs. Adsorption of gliadins at interface can thus be safely considered as irreversible.

Dilatational viscoelasticity of the protein interfacial layer

We measure the dilatational viscoelasticity of the interfacial layer at different times after the drop formation and for various protein concentrations in the subphase. We find that the viscoelastic measurements only depend on the surface pressure of the interface and that data collected at different protein concentrations (from 6.5 mg/L to 800 mg/L) reasonably fall onto a single master curve (Fig. 4). In the whole range of surface pressure investigated (5 mN/m <  < 40 mN/m), the interfacial protein layer is essentially elastic with an elastic modulus, E', always larger than the loss modulus, E". The elastic modulus varies non-monotonically with the surface pressure, and three regimes can be defined for the (E' vs ) master curve. In regime I,  <  c1  19 mN/m, and regime III,  >  c2  31 mN/m, E' continuously increases with surface pressure, whereas in the intermediate regime

( c1 <  <  c2 ), E'
decreases and reaches a minimum with . Interestingly the minimum of E' with the surface pressure is accompanied by a large increase of the loss modulus (Fig. 4b): the ratio E"/E' reaches about 0.4 in regime II, whereas this ratio is of the order of (0.1-0.15) in regimes I and III. Furthermore, it is worth mentioning that, in the intermediate regime, the collapse of the data collected at different protein concentrations is not as good as in the two other regimes, but instead the maximum of E"/E' seems to increase with C. The limits of these three regimes are also reported on the tensiometry plot that displays washing test in each regime (Figs.

3).

Optical properties of the protein interfacial layer.

The structure of the interfacial protein films prepared in Petri dishes is observed by Brewster angle microscopy. Figure 5b displays characteristic images obtained for the 3 different regimes of surface pressure and viscoelasticity defined above. In all cases, the films appear very homogeneous at the microscopic scale. To get pictures with a good contrast it is necessary to adjust the polarizer angle from 0° to 15°. The increase of the brightness with surface pressure indicates an increase of surface reflectivity that we associate to an increase of the amount of proteins at the interface.

In order to quantitatively characterize the optical properties of the interfacial layer we use ellipsometry.

Similarly to our findings for the viscoelastic properties, we measure that the structure of the interfacial layer only depends on the surface pressure and that data acquired at different protein concentrations for the subphase (in the range 0.001 to 5 g/L) collapse on a single curve (Fig. 5a). The ellipsometric signal, I 1 , varies non-monotonically with , which allows distinguishing the same three regimes as before with the same cross-over pressures  c1 and  c2 . In regime I, I 1 increases with , whereas I 1 decreases with  in regime II, and is more or less constant in regime III. In addition, we also observe a preliminary regime in which I 1 increases in time whereas the surface pressure is maintained at zero.

Discussion

We have investigated the interfacial properties of acidic solutions of gliadin studied over 5 decades of concentrations from 0.001 to 110 g/L. Gliadins adsorb irreversibly at the air-water interface in accordance with other studies on proteins [START_REF] Freer | Interfacial rheology of globular and flexible proteins at the hexadecane/water interface: Comparison of shear and dilatation deformation[END_REF] . The structural and viscoelastic properties of the interfacial gliadin film build up in a multistep process, the evolution of which is completely characterized by the surface pressure.

The master curves obtained for the surface pressure, the viscoelasticity and the optical properties of the interfacial layer indicate that the properties at the air/water interface are determined by the concentration of the subphase and the time elapsed since the formation of the interface.

Adsorption kinetics

For low surface pressures,  <  c1 , the optical profile is assumed to be locally isotropic (and equations 8-10 hold) because of the weak interaction between proteins, which are randomly oriented at the interface.

We determine the surface excess concentration  according to Equation 10. Figure 6a displays the evolution of  as a function of time for three different concentrations, and for two sets of data for each concentration. We measure that the higher is the bulk concentration, the faster is the adsorption.

Experimental data can be compared to an analytical model of diffusion-controlled adsorption kinetics 25 :

(Eq. 11)

With D the bulk protein diffusion coefficient and the maximum surface excess concentration. Equation 11takes into account the decreasing difference of chemical potential of the proteins between bulk and interface along the adsorption process [START_REF] Douglas | A Simple Kinetic-Model of Polymer Adsorption and Desorption[END_REF] . Hence at longer time and higher concentration, a steady state is reached with this model that takes into account saturation. Far from interface saturation on the other hand, when the chemical potential at interface can reasonably be considered as being constant, hence at short time and low concentration [START_REF] Langmuir | The effect of dissolved salts on insoluble monolayers[END_REF] , Equation 11 can be approximated as (Eq. 12)

and Note that equation 12 is the short time limit of the Ward and Tordai model [START_REF] Ward | Time-Dependence of Boundary Tensions of Solutions .1. The Role of Diffusion in Time-Effects[END_REF] that is classically used to model interfacial adsorption of surface active components. However, we do not use this model because it considers an instantaneous equilibrium of adsorbing species between the bulk and the interface and allows thus for back-diffusion which we have demonstrated to be negligible for gliadin at the air-water interface.

The theoretical curves presented in the figure 6a are calculated using the bulk diffusion coefficient measured by dynamic light scattering and the maximum surface excess concentration measured at the end of regime I (D=4.41 10 -11 m²/s and 3.1 mg/m²) and quantitatively compared to the experimental data.

Overall a good agreement is found between the theoretical expectations and the experimental results (Fig. 6a). Equation 12account nicely for the experimental data measured at C=0.001 g/L. Experimental data with C=0.01 g/L and C=0.1 g/L are instead better described taking into account a saturation (Eq. 11).

Surface pressure master curve: Time-concentration superposition

In line with those results, a time-concentration superposition is clearly evidenced, for the first time, to the best of our knowledge, for proteins at a liquid interface. It indicates that interfacial measurements performed with solutions at high protein concentration can predict the long ageing time behavior of interfaces formed from solutions at lower concentrations. The existence of a master curve furthermore indicates that the surface pressure obtained with a bulk concentration and an elapsed time t can also be obtained with the reference concentration, , with an elapsed time :

  (Eq. 13) with (Eq. 14)

With the assumption that only one surface concentration,  is associated to a given surface pressure, , one can also write (Eq. 15)

According to the time evolution of the surface excess concentration in regime I, we have shown that the protein adsorption is controlled by diffusion. Combining the diffusion-controlled models previously introduced (Eqs. 11 and 12) with Equations 14 and 15, one derives the value of the scaling factor :

(Eq. 16)

Figure 2c shows that the theoretical prediction (Eq. 16) with = 1 g/L fits very nicely the experimental data over a 4 orders of magnitude for the protein bulk concentration (from 0.001 to 10 g/L). These concentrations cover regimes I and II, for which we have demonstrated, thanks to the droplet washing procedure, that the evolution of the interfacial pressure is mainly associated to protein adsorption. We mention that in the literature [START_REF] Ybert | Study of protein adsorption by dynamic surface tension measurements: Diffusive regime[END_REF] , the diffusion-controlled models were restricted to dilute regimes. Ybert and di Meglio 29a previously performed a time-concentration rescaling of tensiometry data for bovine serum albumin at an air/water interface but were puzzled by the correspondence between curves at high surface pressure as they expected dynamics to be governed by adsorption mechanisms. In the case of gliadins the adsorption of additional proteins to an already relatively dense interface appears not to be the limiting step, even when the protein concentration at the interface is relatively large (excess surface concentration >3mg/m²). We think that the diffusive protein mass transfer remains possible thanks to the creation of free space at the interface with the reorganization of proteins previously adsorbed due to the protein crowding.

So, the propensity of gliadins to easily reorganize at interfaces could explain this behavior. In line with our physical picture, Miura et al [START_REF] Miura | Effects of surface affinity on the ordering dynamics of self-assembled monolayers of chain molecules: Transition from a parallel to a perpendicular structure[END_REF] showed by simulations that the overall adsorption kinetics of chain molecules on a surface is not strongly affected by their internal reorganization. They rationalized their result using a theoretical model that considers bulk diffusion, Langmuir adsorption and rearrangement of anisotropic molecules through a second order reaction.

On the other hand, for higher concentrations, we measure that  is smaller than theoretically expected for a diffusive process, showing that the increase of bulk protein concentration does not speed up the time evolution of the surface pressure as much as expected for a purely diffusion-controlled process. For these concentrations, the shift factor is determined according to surface tension profiles dominated by the regime III that is very rapidly reached. In this regime, the drop washing test showed that adsorption is not involved, explaining the discrepancy between  determined at high concentration and the model. Overall, we show that the surface tension dynamics is controlled by bulk diffusion of gliadin toward the air-water interface at short times (low surface pressures), during regime I, but also during regime II.

Surface excess concentration and protein conformational changes by ellipsometry

Thanks to the concomitant measurement of the surface pressure and surface excess concentration, the surface pressure isotherm of gliadin proteins at the air-water interface,  versus , can also be plotted for  c1 (Fig. 6b). In the early regime, below   0.5 mg/m² (which corresponds to a surface of 116 nm² occupied on average by each protein), no evolution of the surface pressure could be measured due to the high molecular weight of the surface active species. According to the ideal gas law a very weak dependence of the surface pressure with the surface excess concentration is predicted in this concentration range: =RT/M w , with R the ideal gas constant, T the temperature and M w the molecular weight of the species adsorbed at the interface. For objects with M w =35 000 g/mol (Fig. 1), one predicts =0.07i.e.

a surface pressure of 3.5 10 -2 mN/m for   0.5mg/m². Such low value is not measurable with our technique (see line of the ideal gas law in Fig. 6).

For   , the surface pressure increases faster with the surface concentration as a result of proteins interactions. The pressure increases linearly with up to a value of  max =3.1 mg/m² that we considered as the saturation. Modeling the protein layer as a monolayer of hard spheres of constant thickness h, at maximal packing, the protein volume fraction at interface is =0.6. Using  at saturation, with the protein relative density , we find that the thickness of the film is =4 nm. This value is consistent with the size of gliadins, as measured by dynamic light scattering. The preliminary regime, characterized by a negligible surface pressure, corresponds to a gas regime in which proteins adsorb on a nearly "free interface" whereas in the regime I proteins adsorb on an interface already covered by interacting proteins.

In regime II, the ellipsometric signal I 1 decreases while the surface pressure increases. In this regime, equations 8-10 do not hold anymore because the surface concentration cannot decrease if the surface pressure increases. In the absence of desorption, such behavior can only be accounted for by a conformational change of the interfacial layer. Hypotheses of formation of chain loops or multilayer in the bulk can be safely discarded as such processes would lead to constant or increasing I 1 according to equation 8. Partial protein dehydration or the development of an anisotropic refractive index in the interfacial layer is instead consistent with the decrease of I 1 as detailed below.

In the hypothesis of a progressive dehydration and protrusion of proteins in air, equation 9 cannot be used.

A two-step dielectric constant profile for the protein film with a constant thickness h is assumed. We define as the fraction of the thickness of the interfacial protein film in air. The protein film is therefore decomposed into the sum of a layer of thickness h in air, and a layer of thickness h in the solvent. Equation 8thus becomes (Eq. 17 Here, is the index of refraction of air, is the index of refraction of the subphase, as measured experimentally (Eqs. 1&2).

, respectively y, is the index of refraction of the protein layer in air, respectively solvent. Following Equation 2, (Eq. 20) and (Eq. 21)

with the volume fraction of proteins in the interfacial layer. Whatever , As a consequence, the decrease of I 1 observed in regime II could be accounted by an increase of and associated to protein dehydration. Considering (the numerical values found at the end of regime I), we estimate that continuously increases in regime II and at the end of this regime reaches . Here we assume that the amount of protein at the interface is constant and equal to its value at the transition from regime I to regime II, while further protein adsorption should contribute in this regime according to drop washing test. is thus presumably underestimated.

However, this value can be compared to the proportion of hydrophobic residues in gliadin sequences that is about one third.

An alternative model is to suppose that the refractive index of the protein interfacial film becomes anisotropic, due to the alignment of the proteins in the film. Such alignment seems realistic considering that gliadins are usually modeled as prolate ellipsoids [START_REF] Thomson | Small angle X-ray scattering of wheat seed-storage proteins: alpha-, gamma-and omega-gliadins and the high molecular weight (HMW) subunits of glutenin[END_REF] . This would induce a decrease of I 1 with the development of anisotropy of the refractive index of the protein layer. According to Equation 7, for one slab dielectric constant profile of the protein film of thickness h, equation 7 reads:

(Eq. 22)

Here , where (resp. ) is the index of refraction of the protein layer orthogonal to (resp. in the plane of) the film. Considering (the numerical values found at the end of regime I), and a constant index of refraction perpendicular to the film equal to the index of refraction of gliadin solution at ( ), we estimate that continuously decreases in regime II and reaches a value , or equivalently at the end of regime II. Assuming constant in this regime, instead of increasing, may underestimate the latter value. However, the order of magnitude appears reasonable in comparison with measurements obtained for flagellin proteins on a hydrophobic surface by optical waveguide light mode spectroscopy (n=-0.015) [START_REF] Kovacs | Optical Anisotropy of Flagellin Layers: In Situ and Label-Free Measurement of Adsorbed Protein Orientation Using OWLS[END_REF] . Finally, in regime III, the value of I 1 is roughly constant indicating that the interfacial optical profile is not modified by the increase of interfacial pressure.

Interfacial regimes

To summarize, the dynamical properties distinguish three main regimes (hereafter referred to as regimes I, II and III) while optical properties reveal a preliminary regime in addition.

In the preliminary regime (regime 0) ellipsometry data show that proteins start to adsorb. However, a null surface pressure is measured. The negligible surface pressure indicates that proteins are in a very dilute regime at the air-water interface. This regime occurs during the so-called induction time of tensiometry kinetics [START_REF] Miller | Dynamic surface tension and adsorption kinetics of beta-casein at the solution air interface[END_REF] and can only be observed with the lowest bulk concentration (0.001g/L).

In regime I, 0<<19mN/m, proteins continue to adsorb but progressively start to interact with each other at the interface as probed by the linear increase of surface concentration and dilatational elasticity with the surface pressure. Dilatational viscoelastic master curves with a similar regime were previously described in different studies 3b, [START_REF] Benjamins | Viscoelastic properties of triacylglycerol/water interfaces covered by proteins[END_REF] . Interestingly, the behavior of gliadin displays similarities with that of soft proteins like -casein. Indeed, the maximum value of E' at the transition between regimes I and II is relatively weak (26 mN/m), and comparable to that measured for -casein at an air/water interface (20-30mN/m) [START_REF] Cicuta | Studies of a weak polyampholyte at the air-buffer interface: The effect of varying pH and ionic strength[END_REF] , whereas larger values (typically 70-80mN/m) were reported for several globular/hard proteins 3b . In addition, the proportionality constant between E' and  (i.e. the slope of the E' vs  plot) is quite small (approximately 1). According to polymer scaling laws, this low value would be associated to the weak compactness of proteins at interface 3c, 34 . This weak compactness would induce soft rearrangements of proteins at the interface that are in line with the model of Lucassen-Reynders et al 3b . In this model, proteins are assumed to be able to adopt smaller molecular area with increasing surface pressure, a wider range of conformations being adopted by soft proteins that are more compressible because of their flexible structures. In addition, it was previously demonstrated by a polarization modulation infrared reflection absorption spectroscopy (PM-IRRAS) study that, at the air-water interface, gliadins, which are initially weakly folded and flatly oriented at low surface pressure, progressively fold and orientate their secondary structures as surface pressure increases [START_REF] Banc | Structure and orientation changes of omega-and gamma-gliadins at the air-water interface: A PM-IRRAS Spectroscopy and Brewster angle microscopy study[END_REF] .

In regime II, 19 <<31mN/m, a master curve can be plotted with viscoelastic data, even if the maximum of E"/E' slightly increases with the bulk concentration. This behavior, analogous to the one observed for -lactoglobulin 33c,f , can be attributed to the fast interfacial adsorption of proteins with high bulk concentration that would result in more disordered films, more prompt to relax with the oscillatory dilatational stress. The general increase of dilatational dissipative viscoelastic properties associated with an anomalous pressure dependence of the optical profile suggests a significant change of protein conformation. In the literature 33c, 34, 36 , the decrease of E' is associated to relaxation processes. Different processes are proposed: reorientation, internal reconformation (formation of loops in the bulk or in air, are specific cases of reconformation), phase transition, formation of 3D structures at the interface, collapse.

However, it is very difficult to experimentally prove the exact physical processes involved in the relaxation. In our work, thanks to the combination of different techniques we are able to discard the formation of multilayer and loops in the bulk subphase and their collapse as possible processes. We propose two models to rationalize our ellipsometric data, assuming either a progressive dehydration of the proteins as water is not a good solvent for gliadin or the development of anisotropy for the protein refractive index resulting from the anisotropic shape of gliadin protein [START_REF] Thomson | Small angle X-ray scattering of wheat seed-storage proteins: alpha-, gamma-and omega-gliadins and the high molecular weight (HMW) subunits of glutenin[END_REF][START_REF] Ang | Structure and heterogeneity of gliadin: a hydrodynamic evaluation[END_REF] . Both processes were previously identified for several proteins at the air water interface by neutron or X-ray reflectivity [START_REF] Lu | Structural conformation of lysozyme layers at the air/water interface studied by neutron reflection[END_REF] . Even if the numerical values obtained with the reorientation model seem more realistic that those obtained with the partial dehydration model, we expect that a more complex behavior involving both phenomena might occur. Overall, our experimental data demonstrate that significant conformational changes occur in regime II, induced by the surface pressure, due to thermodynamics [START_REF] Joos | The Principle of Braun-Le Chatelier at Surfaces[END_REF] .

Both regimes I and II are related to a protein mass transfer from the bulk to the interface that is controlled by diffusion. By contrast, in regime III, >31mN/m, no protein adsorption is required to induce the surface pressure evolution. In addition, a drastic increase of the elastic modulus E' with is measured, signing the increasing incompressibility of the interfacial filmwhile a constant optical profile is observed.

We attribute this regime to ageing of the protein film at the air-water interface. A modification of interactions inside the protein film with time can be at the origin of the behavior observed. Gelation phenomena could occur but interfacial shear rheology would be required to fully characterize it.

The sequence of three regimes, adsorption-reorganization-gelation, was previously proposed for different proteins at interfaces 40 . However, generally the shape of dynamic interfacial tension displays only one decay step. The induction time (during which surface pressure  ≈ 0) was associated to protein adsorption, the steep increase of surface pressure with time being associated to protein relaxation and the final slow increase of surface pressure being associated to gelation. This interpretation was only based on surface tension measurements and quite speculative while the combination of techniques used in our study gives more credit to our interpretation. Note finally that three comparable regimes were previously identified for gliadins in the bulk [START_REF] Boire | Osmotic Compression of Anisotropic Proteins: Interaction Properties and Associated Structures in Wheat Gliadin Dispersions[END_REF] . Upon osmotic compression, after a dilute regime of repulsive colloids, a highly compressible regime with secondary structure changes was evidenced, and followed by a regime of strongly interacting proteins.

Conclusion

We have studied the adsorption of gliadin proteins at the air-water interface from acidic protein solutions over a broad range of concentrations. Combination of dilatational viscoelastic measurements and optical profile measurements provides a consistent physical picture of the three steps process, as summarized in Figure 7. In the preliminary regime (regime 0, not represented), the surface pressure is equal to zero and proteins start to adsorb on a very dilute interface. Then, in the first regime, at small surface pressure  (regime I), the ellipsometric signal I 1 and the elastic modulus E' increase with the surface pressure, while the loss modulus E" remains small. This regime corresponds to a progressive coverage of the interface by individual weakly interacting flexible proteins, up to a critical surface pressure  c1 . The second regime ( c1 < <  c2 ) results from significant conformational rearrangements of the protein film induced by further protein adsorption, which are identified by a strong increase of the dissipative viscoelastic properties of the film concomitantly with an anomalous evolution of its optical profile. Finally, in regime III ( >  c2 ), the ellipsometric signal does not evolve with the surface pressure, suggesting that the structure of the interfacial layer does not change significantly. In this regime, on the other hand, a dramatic increase of the elastic modulus is measured while no further adsorption occurs. The two experimental observations can be consistently interpreted as resulting from ageing of the interfacial layer.

The softness of gliadins, which we have evidenced at the air/water interface, is presumably linked to its partially disordered secondary structure due to the abundance of proline and glycin residues in the repetitive domains of their primary sequence. This behavior could be associated to the biological function of these proteins that is to store amino-acid for the future development of the plant in the wheat grain. An efficient accumulation of proteins in a minimum volume should be reached thanks to their capacity to reorganize with an increasing osmotic pressure. Furthermore, from an industrial point of view, the gliadin softness is a clear asset as it should limit emulsion drop breakup in shear flow as demonstrated for a model flexible protein by Erni et al [START_REF] Erni | Emulsion Drops with Complex Interfaces: Globular Versus Flexible Proteins[END_REF] .

  index of air and the bulk liquid phase; and and are the angles related by the Snell's law: . The term I 1 is related to the interfacial optical profile as 20 :

Figure 1 : 2 Figure 2 :

 122 Figure 1: Characterization of the protein extract. (a) Size-exclusion high performance liquid chromatography (SE-HPLC) profile of the gliadin extract. The dashed lines delineate the molecular weight (M w ) limits for albumin (alb) and globulin (glob) (M w <23000 g/mol), for gliadin (23000 g/mol< M w <60000 g/mol) and for glutenin (M w >60000 g/mol). (b) Variation of the square of the refractive index of gliadin solutions as a function of the protein volume fraction. The line is a linear fit of the experimental data points.

Figure 3 :

 3 Figure 3: Surface pressure as a function of the normalized time, t. The gray data points correspond to the master curve shown in Fig. 2b. The colored symbols correspond to experiments where the interior of the drop has been washed at different surface pressures as indicated in the caption. The gray horizontal lines display the limits ( c1 and  c2 ) of the different regimes identified by dilatational viscoelasticity and ellipsometry experiments.

Figure 4 :

 4 Figure 4: Dilatational elastic modulus E' (a) and ratio between the viscous and elastic moduli, E"/E' (b) as a function of the surface pressure , for different protein concentrations as indicated in the legend.Thick gray lines delineate 3 regimes with cross-overs at  c1 and  c2 ..

Figure 5 :

 5 Figure 5: (a) Typical Brewster angle microscopy images of the planar interface taken in the three regimes. From left to right, the polarizer angle is 5°, 10° and 15°. (b) Ellipsometric response, I 1 , as a function of the surface pressure, , for different protein concentrations as indicated in the legend. Thick gray lines delineate the 3 regimes with cross-overs at  c1 and  c2 .

Figure 6 :

 6 Figure 6: (a) Surface excess concentration  as a function of time t. Symbols correspond to experimental data obtained by ellipsometry for different protein bulk concentrations as indicated in the caption. Dashed and continuous lines display diffusive models expressed in Equations 11 and 12 respectively. (b) Equation of state, surface pressure as a function of the surface excess concentration, in regime I.

Figure 7 :

 7 Figure 7: Scheme of the three interfacial regimes identified thanks to structural and rheological measurements of the gliadin air/water interface. The three regimes are defined by the interfacial pressure.

Acknowledgments

This work was performed, in partnership with the SAS PIVERT, within the frame of the French Institute for the Energy Transition (ITE P.I.V.E.R.T., www.institut-pivert.com) selected as an Investment for the Future by the French Government under the reference ANR-001-01.We thank Marie-Hélène Morel for the development of gliadin extraction and the HPLC measurements. We also acknowledge the Laboratory of Excellence NUMEV (ANR-10-LAB-20) for partial funding of the Brewster Angle Microscope.