Jordan Navarro
email: jordan.navarro@univ-lyon2.fr

Emanuelle Reynaud
email: emanuelle.reynaud@univ-lyon2.fr

Catherine Gabaude
email: catherine.gabaude@ifsttar.fr

EYE-MOVEMENT ANALYSIS IN DYNAMIC SCENES: PRESENTATION AND APPLICATION OF DIFFERENT METHODS IN BEND TAKING DURING CAR DRIVING. Short title: EYE-MOVEMENT ANALYSIS IN DYNAMIC SCENES

Keywords: Eye movement, evaluation methodology, moving images, car driving, cornering behaviours

HAL is

Figures equivalent signs count

L'oculométrie est une technique qui offre l'opportunité d'investiguer, au travers des mouvements des yeux, ce qui se trouve derrière : notre cerveau. Comme de nombreux autres domaines, l'oculométrie moderne bénéfice à la recherche en psychologie ergonomique. Parmi la variété des situations étudiées en psychologie ergonomique, l'activité de conduite automobile a fait l'objet de nombreuses études dont un bon nombre font appel à l'oculométrie. Les analyses des données oculométriques ont progressé à mesure du développement de cette technique, et sont aujourd'hui nombreuses et abouties en ce qui concerne l'analyse des parcours oculaires sur des images statiques. En revanche, dans le cadre de l'étude de la conduite automobile, comme des activités dynamiques en général, les images présentées au conducteur sont dépendantes à la fois de l'environnement de conduite, mais aussi de ses propres actions sur le véhicule via le volant et les pédales, ce qui rend les techniques d'analyse habituelles des données oculométriques peu pertinentes. Face à cette situation, les chercheurs ayant un intérêt pour l'étude de la conduite automobile ont développé plusieurs techniques d'analyses des mouvements oculaires qui peuvent être regroupées en quatre classes de méthodes. Les quatre méthodes se définissent par rapport à l'analyse des positions du regard (1) dans un référentiel écran (sans indication sur ce qui est visible dessus) en deux dimensions, (2) selon des zones d'intérêts fixes définies dans ce même référentiel écran, (3) selon des zones d'intérêts dynamiques définies dans un référentiel relatif à la tâche de conduite (un élément de la scène visuelle qui se déplace dans celle-ci) et (4) dans un référentiel défini sur un point dynamique de la scène visuelle (un point de la scène visuelle qui se déplace dans celle-ci). Chacune de ces quatre classes de méthode a été présentée et appliquée ici à la prise de virage en conduite automobile simulée en condition de conduite classique et en conduite hautement automatisée. En condition hautement automatisée, l'assistance maintenait le véhicule dans sa voie en agissant directement sur le volant sans aucune intervention de la part du conducteur. Dix-huit conducteurs ont pris part à une étude sur simulateur de conduite où les positions du regard ont été enregistrées. Selon la méthode d'analyse considérée, les résultats obtenus varient considérablement. A l'appui des résultats d'analyse collectés via les quatre classes de techniques d'analyse, les analyses relatives à un point ou une zone dynamique dans la scène visuelle sont apparues les mieux adaptées à l'analyse des données oculaires en contexte de conduite automobile. Ce sont les seules méthodes qui ont permis de mettre en avant et de spécifier une différence dans les prises d'informations visuelles en présence de l'assistance et comparativement à la condition non-assistée. La discussion présente les avantages et les limites de chacune des quatre classes de méthode et l'impact du choix de la méthode d'analyse des données oculaires sur l'analyse ergonomique des situations considérées et les enjeux sécuritaires associés.

Mots clés : Mouvements oculaires, méthodologie d'analyse, images animées, conduite automobile, prise de virages.

I INTRODUCTION

"Les yeux sont les fenêtres de l'âme" or "Eyes are the windows to the soul" for a translation, Georges Rodenbach. This citation illustrates an old idea that eyes might offer a special access to human cognition. Indeed, it is known for long that the eyes and the mind are tightly linked. Consequently, eye movements have always been of interest so as to investigate what is behind the eyes: our brain. This idea has been put into practice numerous times in a variety of tasks such as reading, music reading, typing, visual search, and scene perception. Based on a literature review on these different topics, with a major emphasis on reading, researchers concluded that "eye movement data reflect moment-to-moment cognitive processes in the various tasks examined" [START_REF] Rayner | Eye movements in reading and information processing: 20 years of research[END_REF]. Other researchers also found anatomical relationships between eye movements and cognitive processes. Eye movements and attention share anatomical bases in the brain for instance [START_REF] Corbetta | A common network of functional areas for attention and eye movements[END_REF], reinforcing the idea that eye movements might allow to understand the brain quite directly.

Even if modern eye-trackers have been expanding rapidly for about two decades, eye-tracking techniques started in the early 1900s. The educational psychologist Edmund Burke Huey is known to have built what is often referred to as the first eye-tracker in 1908. This eye-tracker was designed to study reading and was very intrusive, as readers had to wear contact lenses with holes for the pupils. The lenses were attached to a pointer which followed eye movements, revealing them to the experimenter [START_REF] Huey | The psychology and pedagogy of reading[END_REF]. Since that very first idea, eye-tracking techniques have improved with the sophistication of the tools available to record eye movements and particularly the development of video cameras. Nowadays, eye-tracking techniques are very accurate and relatively easy to implement in real life activities. Depending on the activity considered, people do not have to wear any apparatus directly, the video cameras used to record eye movements can be located inside a computer screen or in front of the person of interest. Other techniques sometimes require people to wear a helmet linked to a computer or even more recently glasses linked to a smartphone for instance.

As many other domains, including cognitive and educational psychology or marketing among others, cognitive ergonomics benefit from those techniques improvements. Among the variety of activities studied by cognitive ergonomics, an activity that appeared after Huey's first eye tracker received particular attention regarding visual exploration analysis: car driving. Because most of the driving activity is based on visuomotor coordination, researchers focusing on car driving had a direct interest to understand visual explorations of the driving environment. This led early car driving researchers to suggest that eye movements analysis may be the best method to assess driving behaviours [START_REF] Walraven | Perception research and human engineering[END_REF]. The first eye-tracking experiment carried out on car driving studied the impact of route familiarity on visual patterns, showing an increase of gaze positions compactness with the increase of route familiarity [START_REF] Mourant | Mapping eye movement patterns to the visual scene in driving: An experimental study[END_REF]. Caution must be taken when interpreting eye-tracking data, because those devices only focus on central vision and no information on peripheral vision is collected. This limitation of eye-tracking method has been pointed out since the first eye-tracking experiment on car driving [START_REF] Mourant | Mapping eye movement patterns to the visual scene in driving: An experimental study[END_REF][START_REF] Rockwell | Eye movement analysis of visual information acquisition in driving: An overview[END_REF]. This is particularity important in the context of car driving because it is known for long that peripheral vision participates to car steering by monitoring the lane edge markers for instance [START_REF] Mourant | Mapping eye movement patterns to the visual scene in driving: An experimental study[END_REF][START_REF] Summala | Maintaining lane position with peripheral vision during in-vehicle tasks[END_REF].

Driving is composed of three main sub-tasks: control, navigation and hazard identification [START_REF] Stanton | Automating the driver's control tasks[END_REF]. Control refers to car steering and speed adjustment, navigation refers to planning the journey and hazard identification to the detection of potential hazard. All these sub-tasks rely on visual analysis of the driving scene. Therefore, these three driving sub-tasks can beneficiate from visual exploration analysis. Among these three sub-tasks, eye-tracking offered an important contribution to the understanding of car control, in particular since the seminal work on drivers eyes positions while steering the car in bends [START_REF] Land | Where do we look when we steer[END_REF]. These authors observed that during bend taking, drivers spent about 75 percent of the time looking around a particular region of the driving environment. This particular region was referred as the Tangent Point (TP), a dynamic point changing according to the driver position but that remained located "on the inside of each bend, where the edge of the road reverse direction" [START_REF] Land | Where do we look when we steer[END_REF]. This result was replicated several times and drivers indeed spend between 50% and 75% looking in the TP area while negotiating bends [START_REF] Kandil | Driving is smoother and more stable when using the tangent point[END_REF][START_REF] Mars | Steering and gaze control modifications from directed or spontaneous use of a visual augmentation cue[END_REF][START_REF] Panerai | Eye movement patterns in nighttime driving simulation: conventional and swivelling headlights[END_REF][START_REF] Wilson | Eye movements coordinated with steering benefit performance even when vision is denied[END_REF]. The TP area might be looked at by drivers because (1) this particular point of the road might indicate directly to the driver the curvature of the bend [START_REF] Land | Does steering a car involve perception of the velocity flow field?[END_REF], (2) this particular point of the road corresponds to the vehicle future trajectory and is therefore a point followed by drivers [START_REF] Wann | Why you should look where you are going[END_REF][START_REF] Wilkie | Eye-movements aid the control of locomotion[END_REF].

Analysis of car driving, and particularity steering control has beneficiated from the use of modern eye-tracking techniques. But eye-trackers have been initially designed to record and analyse gaze positions on still images. These devices are therefore perfectly adapted for cognitive activities such as reading. For instance, in a reading task, a page composed of text and/or images is displayed for a period of time and the reader's visual exploration of this page is analysed. However, cognitive activities relying on moving images are also of interest. For instance, and as introduced previously, a large community in the cognitive ergonomics field is working on car driving, involving moving images situations. For car driving, as in any dynamic environment, the images (i.e. visual driving environment) are not only changing repeatedly in a defined order as in a video but also depend on the drivers' actions. The driving scene available to the driver at any time depends upon the environment itself (like in the reading example) but also upon the driver's actions on the vehicle controllers (steering wheel and pedals). Depending on the driven vehicle position on the road, the visible driving scene is different. As soon as the driver turns the steering wheel for instance, the visible driving scene is changing. To illustrate this point, when driving in straight lines, the road visible to the driver is mostly in front of him/her but when negotiating bends, the road visible is mostly on the right of the driver in a right bend and on the left of the driver in a left bend. These modifications of the driving environment cannot be predicted prior the drive itself because they depend on the driver behaviours at all times.

The softwares developed by eye-trackers manufacturers to analyse gaze data were designed to analyse precisely activities relying on still images. These softwares are not adapted to dynamic environments such as car driving because the dynamic nature of images displayed is not taken into account. When considering still images, some of the most popular gaze positions analysis are gaze plots, scanpaths, areas of interests and heat maps [START_REF] Gitelman | ILAB : A prog ram for postexperimental eye movement analysis[END_REF][START_REF] Holmqvist | Eye tracking: A comprehensive guide to methods and measures[END_REF]. Gaze plots are a representation of the eyes positions when scanning a visual scene.

Scanpaths can be defined as eye positions trajectory when scanning a visual scene, here the chronology of the eyes positions is considered (Noton & Stark, 1971a[START_REF] Noton | Scanpaths in saccadic eye movements while viewing and recognizing patterns[END_REF]. Areas of interest are areas defined on the visual scene where eye movements data are analysed. Heat maps offer a representation of gaze positions by separating the different levels of observation intensity [START_REF] Špakov | Visualization of eye gaze data using heat maps[END_REF]. In the case of a dynamic environment, these softwares and associated techniques can still be used to analyse gaze positions. Nevertheless, in this particular case, the moving images display (real or virtual) is only used as a system of reference and the images themselves are not taken into account. This is a serious limitation when it comes to data interpretation, because looking to the right of the driving scene for instance has a different meaning depending of the visual environment available (e.g. looking at pedestrians about to cross the road or looking at advertising billboards). These gaze position analyses remain valid even in dynamic environments if researchers only focus on visual parts of the driving scene that do not change over time such as the speedometer or other dashboard elements (e.g. [START_REF] Victor | Sensitivity of eye-movement measures to in-vehicle task difficulty[END_REF].

Because eye-trackers softwares are not designed to take into account moving images especially when the images displayed are dependent of the subject actions (e.g. simulated or real car driving situation where the visual environment depend on drivers' actions on the steering wheel and pedals), an actual challenge for researchers dealing with dynamic situations such as car driving is to analyse eye movement data taking into account the dynamics of the situation.

Through the years, several options have been imagined to extract the meaningful information from the collected data in dynamic environments. Here the focus is set on car driving, but the methods used are relative to gaze position analysis, and therefore potentially transferable to all other dynamic situations.

As a mean to escape from the difficulty of dynamic environment and to collect accurate eye movements data, some researchers made successive pictures of the driving situations of interest and then recorded eye movements when drivers were confronted to those pictures. Using this methodology, drivers were found to gaze at cyclists when facing at a T-junction before any other relevant area [START_REF] Walker | Drivers' gaze fixations during judgements about a bicyclist's intentions[END_REF]. Even if this idea is a clever attempt to explore visual scanning in a dynamic environment, it obviously limits the interactions of the driver with his/her environment to the few selected images.

Apart from that methodology, four main eye movements analysis techniques have been identified from the driving literature:

(1) Gaze plots are the representation of gaze positions over a period of time [START_REF] Konstantopoulos | Driver's visual attention as a function of driving experience and visibility. Using a driving simulator to explore drivers' eye movements in day, night and rain driving[END_REF][START_REF] Wilson | Eye movements coordinated with steering benefit performance even when vision is denied[END_REF]. This type of analysis is rough because it does not take into account the visual elements available on the driving scene but simply the driving scene as a frame of reference.

(2) Areas of interest, visual areas defined by an expert as relevant for performing a goaloriented task. Within the context of car driving, AOIs are visual areas, defined on the driving scene, that researchers judged as relevant for performing the driving task and worthwhile analysing regarding eye movements. These Areas Of Interest (AOI) are selected by the experimenters according to the objective pursued. Researchers may focus on a single AOI defined on the visual scene, such as the proportion of time fixating the road ahead [START_REF] Edquist | Effects of advertising billboards during simulated driving[END_REF]. The focus can also be put on several AOIs, for instance a "relevant" area for driving and another area used to display experimental cues [START_REF] Megías | Modulation of attention and urgent decisions by affect-laden roadside advertisement in risky driving scenarios[END_REF]. The entire driving scene may also be considered and divided into five AOIs for instance [START_REF] Carsten | Control Task Substitution in Semiautomated Driving: Does It Matter What Aspects Are Automated? Human Factors[END_REF][START_REF] Hughes | The effect of attentional demand in eye movement behaviour when driving[END_REF]. These researchers defined their five regions as (a) a circular 6° central region on the screen, (b) a left region covering a variety of elements such as a DVD screen, radio controls, the left side mirror, door window and traffic in adjacent lane, (c) a right region covering gaze positions to the right side mirror, door window, traffic in the adjacent lane and in the opposite carriageway, (d) a bottom region covering mainly the dashboard with a variety of gauges including the speedometer, (e) and a top region covering mainly the sky [START_REF] Carsten | Control Task Substitution in Semiautomated Driving: Does It Matter What Aspects Are Automated? Human Factors[END_REF]. Finally, another option is not to define AOIs based on researchers' expectations but rather to divide the entire driving scene in equal AOIs using a grid [START_REF] Di Stasi | Behavioral and eye-movement measures to track improvements in driving skills of vulnerable road users: First-time motorcycle riders[END_REF] or to use density plots of gaze positions on the visual screen [START_REF] Chapman | Visual search of driving situations: Danger and experience[END_REF][START_REF] Crundall | Attraction and distraction of attention with roadside advertisements[END_REF].

(3) Dynamic AOIs allow accurate eye movements analysis within the context of dynamic environments. A dynamic AOI is an area that has different locations on the visual scene but follows the movements of an element of the visual environment. In that case, gaze positions are considered relatively to an element of the visual scene. Taking into account the dynamics of the situation. Dynamic AOIs may be relative to visual objects of the driving environment [START_REF] Salvucci | Control and monitoring during lane changes[END_REF][START_REF] Salvucci | The time course of a lane change: Driver control and eyemovement behavior[END_REF] or to invisible elements of the driving scene such as the tangent point [START_REF] Land | Where do we look when we steer[END_REF]. Results may be represented on density maps [START_REF] Land | Where do we look when we steer[END_REF].

(4) Finally the fourth category, dynamic point is derived from the previous method, offering an analysis of gaze positions relative to a dynamic point on the visual scene but with extra precision and a decomposition of gaze positions in horizontal and vertical components [START_REF] Mars | Where we look when we drive with or without active steering wheel control[END_REF].

Within the actual context of highly automated driving and associated human-machine cooperation issues [START_REF] Hoc | Cooperation between drivers and automation: implications for safety[END_REF][START_REF] Navarro | Lateral control assistance in car driving: classification, review and future prospects[END_REF], an important question that cognitive ergonomics researchers have to address is how automation interfere with driving behaviours. Modifying the way drivers steer their cars may also have an impact on how they explore their environment. The focus was set on the influence of a highly automated vehicle (here the automation steered the car without any human intervention) on eye movements during bend taking. To do so, drivers gaze positions have been recorded in two experimental conditions: manual driving and automated driving, offering the opportunity to compare gaze positions distributions between the two conditions.

Here, the four main techniques used to analyse eye movements data while driving have been applied to gaze positions analysis while negotiating bends, under two conditions: manual and automated driving. Our objective was to evaluate the impact of the technique used on the assessment of gaze positions changes between manual (No Assistance -NA-) and automated driving (Assistance -A-). The underpinning objective is to help future researches to select the most appropriate technique to analyse eye movements in dynamic environments.

II METHOD

II.1 PARTICIPANTS

Eighteen drivers (16 men) with normal vision participated in this experiment. Their age ranged from 21 to 49 years old with an average of 27 years. None experienced simulator sickness.

II.2 DRIVING SIMULATION

The experiment used a fixed-base simulator (SIM 2 , developed by IFSTTAR-MSIS).

Participants took place in a driving seat and steered the vehicle through force feedback steering wheel and pedals (for more details, see [START_REF] Espié | A high performance/low-cost mini driving simulator alternative for human factor studies[END_REF]. The simulated driving environment represented a two-lane road similar to a typical French rural road composed of 13 bends linked the one to the other by straight lines. It took drivers about 3-5 minutes to complete the simulated circuit. No traffic was present during the experiment.

The driving scene was projected on a screen of 2.4 m width and 1.91 m height viewed by drivers at a distance of about 2 m. Thus the screen offered a 62° width by 51° height of field of view.

A speedometer was displayed on the bottom left part of the screen.

II.3 EYE TRACKER

A head-mounted IviewX system video based/corneal reflexion tracker (Sensomotoric Instruments) equipped with a magnetic tracker (Polhemus) were used to track drivers' gaze positions taking into account their head movements. Gaze positions were recorded at a frequency of 50 Hz, after a 13-points calibration procedure was performed.

II.4 AUTOMATED DRIVING

Under automated driving, participants were provided a steering assistance that took over the lateral control task. In other words, the steering wheel was automatically turned on by the assistance in order to keep the vehicle on its driving lane without any action of the driver. The automation was designed by IFSTTAR-LIVIC on real cars [START_REF] Chaib | H∞, adaptive, PID and fuzzy control: A comparison of controllers for vehicle lane keeping[END_REF][START_REF] Netto | Different modes on shared lateral control[END_REF] and implemented on the driving simulator by IFSTTAR-MSIS. Drivers were instructed to keep both hands on the steering wheel, to get haptic feedbacks from automated actions through the steering wheel displacements. Participants kept manual control of the vehicle speed and adjusted it to negotiate the different bends of the virtual environment.

II.5 PROCEDURE

Participants were instructed that they had to drive as they usually do (manual driving) and with a perfectly reliable automation that will steer the vehicle (automated driving). After a familiarisation with the driving simulator by performing the two first drives, the 8 experimental drives started. 4 consecutive drives were performed under manual driving and 4 under automated driving. The presentation order was counterbalanced between participants. All things considered, the experiment lasted about two hours, including eye tracker calibration, driving practice, experimental driving and breaks.

II.6 DATA ANALYSIS

Eye movement data were analysed for two different bends, of similar curvature radius, of the experimental drives: a left bend with a curvature radius of 73 m, and a right bend with a curvature radius of 80 m. As the method used to analyse eye movement was different for each technique considered, specific data analysis material is available in the results sub-sections.

III RESULTS

III.1 GAZE PLOT

Gaze plot is the most straightforward eye movements analysis methodology, and is regularly used including in car driving studies (e.g. [START_REF] Konstantopoulos | Driver's visual attention as a function of driving experience and visibility. Using a driving simulator to explore drivers' eye movements in day, night and rain driving[END_REF][START_REF] Wilson | Eye movements coordinated with steering benefit performance even when vision is denied[END_REF] In order to complement this type of representations with statistical analysis, horizontal and vertical variability of gaze positions can be analysed [START_REF] Konstantopoulos | Driver's visual attention as a function of driving experience and visibility. Using a driving simulator to explore drivers' eye movements in day, night and rain driving[END_REF][START_REF] Underwood | Driving simulator validation with hazard perception[END_REF]. In the current experiment, the average standard deviation of gaze positions for all drives in the NA and A conditions were computed (see Figure 2). Very different eye movements variability was collected in the considered left (below 10°) and right bend (above 15°). This indicates that visual exploration of the driving scene is largely dependent on the driving context. For the left bend, no significant difference between the assistance and the no assistance conditions was observed in the X-axis variability (NA-A; d=7-6.5= 0.5°; t(17)=.61; p=.55) and a small but significant difference was found for the Y-axis (NA-A; d=4.1-5.1= -1°; t(17)=3.07; p<.007). For the right bend, both the X-axis and Y-axis variabilities appeared to be significantly different between the assistance and the non assistance conditions (X-axis: NA-A; d=31.9-18.8= 13.1°; t( 17

III.2 AREAS OF INTEREST (AOI) III.2.A On static parts of the environment

All the main eye-trackers and associated softwares available on the market offer the possibility to define easily AOI on the screen, although AOI definition can be a tricky exercise that requires researchers' expertise. As for gaze plots, this type of analysis is not particularly well suited to dynamic environments as the areas defined on the screen did not take into account the different images displayed on the screen. In other words, looking at a particular area on the screen is different from looking at a particular area on the images displayed (here the driving environment). It is more relevant, however, to analyse the AOI that are static in the driving environment such as the mirrors (e.g. [START_REF] Recarte | Mental workload while driving: effects on visual search, discrimination, and decision making[END_REF][START_REF] Taoka | Duration of drivers' glances at mirrors and displays[END_REF] or in-vehicle signs for instance [START_REF] Caird | Do in-vehicle advanced signs enhance older and younger drivers' intersection performance? Driving simulation and eye movement results[END_REF]. Such an analysis was not the purpose of the current experiment because the focus was not set on the static parts of the environment. However, the speedometer was displayed at the bottom left of the screen at all times during the experiment. A brief descriptive analysis indicated that drivers gazed more often at the speedometer in the assistance condition compared to the no assistance condition (increase of 137% in gaze proportion on the left bend and of 37% in the right bend).

III.2.B On dynamic parts of the environment

Some researchers use AOIs defined on the visual screen and based on their expectations to analyse eye movements while driving. Even if inaccurate by nature, this method might be acceptable under very restricted driving conditions such as driving in a straight lane or with very gentle bends and using roughly defined AOI to analyse eye movements [START_REF] Carsten | Control Task Substitution in Semiautomated Driving: Does It Matter What Aspects Are Automated? Human Factors[END_REF]. However, a better option is not to consider any AOI prior to the experiment, but to explore eye movements based on the results collected [START_REF] Di Stasi | Behavioral and eye-movement measures to track improvements in driving skills of vulnerable road users: First-time motorcycle riders[END_REF]. Following this rationale the AOIs will emerge from the eye movement data, offering the possibility to compare different conditions.

Figure 3 presents four heat maps, each based on the normalized amount of gaze positions in the visual screen. The warmer the colour the more gaze positions were recorded at that particular location on the screen. The heat maps were computed by dividing virtually the screen in 400 potential AOIs (using a 20 x 20 grid). The number of gaze positions recorded in each of the 400 potential AOIs were added up over the total number of participants. The AOIs with the biggest gaze positions numbers were then coloured in the warmest colour (intense red) of the heat map, the AOIs with the smallest gaze positions numbers were coloured in the coldest colour (intense blue). From a general perspective each AOI was associated to a colour depending of the relative number of gaze positions received. In order to define the AOIs allowing a direct comparison between the colours represented in the four different heat maps, the number of gaze positions collected for each potential AOI has been normalized between 0 and 1, 1 corresponding to the AOI that received the maximum gaze positions. As for gaze plots analysis, and even if the representation here is much richer because it includes all participants, it is hard to notice differences between NA and A conditions. Drivers looked more to the left of the screen in the left bend and more to the right of the screen in the right bend, confirming the observations made on gaze plots. (en condition non assistée à gauche et avec un véhicule hautement automatisé à droite). Les deux graphiques situés en bas représentent le virage à droite (en condition non assistée à gauche et avec un véhicule hautement automatisé à droite). Les positions du regard sont exprimées en degrés dans un repère centré sur le centre de l'écran de projection de l'environnement de conduite virtuel. 0 sur l'axe des abscisses correspond au centre de l'écran horizontalement, une valeur positive indique un regard sur la partie droite de l'écran tandis qu'une valeur négative indique un regard sur la partie gauche de l'écran. 0 sur l'axe des ordonnées correspond au centre de l'écran verticalement, une valeur positive indique un regard dans la partie haute de l'écran tandis qu'une valeur négative indique un regard dans la partie basse de l'écran.

In order to complement this type of representations with statistical inferences, the matrices of values collected with and without assistance were compared. A Wilcoxon signed-ranks test indicated that the assistance and the no assistance conditions significantly differed the one from the other for both the left and the right bend (Left bend: Z = 2.32; p<.03; Right bend: Z = 3.65; p<.0003).

III.3 DYNAMIC AREA OF INTEREST (AOI)

Introduced by the famous work of [START_REF] Land | Where do we look when we steer[END_REF] showing the importance of the Tangent Point (TP) on steering control, the dynamic area of interest technique offers an analysis of eye movements relative to a dynamic reference on the driving scene rather than on the display screen. This method is time consuming because it is relying on an image per image analysis of the gaze position collected relatively to a dynamic reference on the driving scene. Gaze positions analysis relative to the TP is particularity well suited when focusing on vehicle control. Indeed the TP is known to be particularly useful to steer the car [START_REF] Land | Where do we look when we steer[END_REF].

Dynamic area of interest is not limited to the TP and can be defined in relation to a lead vehicle or any visible or invisible object present in the visual scene. Such analysis was performed in order to identify road sections during lane change manoeuvres for instance [START_REF] Salvucci | Control and monitoring during lane changes[END_REF][START_REF] Salvucci | The time course of a lane change: Driver control and eyemovement behavior[END_REF]. This technique is of particular interest because it offers an analysis of eyes positions relative to the driving environment and is not based on the visual screen as described with the previous methods (gaze plots and AOIs). Following [START_REF] Land | Where do we look when we steer[END_REF] work, Figure 4 represents four contour plots showing the distribution of gaze positions relative to the tangent point (located at 0 in X-axis and Y-axis) for the left bend (the two top graphs) and the right bend (the two bottom graphs) in the no assistance condition (the two graphs on the left) and in the assistance condition (the two graphs on the right). Four levels of contours are displayed (0.2, 0.4, 0.6, 0.8). The contours are lines relying the points that have the same gaze positions density levels. The outer contour (0.2) for instance is a line relying points with the same proportion of density of gaze positions relative to the maximum density (here 0.2). Descriptively, the areas delimited by the contours appeared to be centred around the TP. These areas also appeared larger in the assistance condition compared with the no assistance condition, with more gaze positions density on the left and beyond the TP in the left bend, and more gaze positions density on the right and beyond the TP in the right bend (Figure 4). In order to complement this type of representations with statistical inferences, the percentage of time spent by drivers looking at a region of 5° around the tangent point was computed. This analysis is directly inspired from Land and Lee's work because contour plots are not offering the possibility to perform detailed statistics [START_REF] Land | Where do we look when we steer[END_REF]. The percentage of time drivers spent looking at a region of 5° around the tangent point revealed a significant difference between the two conditions for both the left and the right bend (Left bend: NA-A; d=58.4%-31.7%=26.7%; t(17)=5.97; p<.0001; Right bend: NA-A; d=61.4%-37.6%=23.8%; t(17)=5.82; p<.0001; see Fig. 5). With assistance, drivers spent less time looking at the TP area compared to the no assistance condition.

III.4 DYNAMIC POINT

The last method that can be used is in line with the previous dynamic AOIs analysis, relative to a dynamic point of the visual scene relevant for explaining the participants' behaviours. Here the tangent point (TP) has been used because of the importance of this particular point while negotiating bends. Gaze positions proportions are decomposed on X-axis and Y-axis dimensions and by degree of visual angle. The distribution of gaze positions was divided in intervals of 1° of angular deviation from the TP. This procedure was repeated for both vertical (Y-Axis) and horizontal (X-Axis) gaze position components for all participants. All gaze positions that deviated more than 15° from the TP in one direction or the other were grouped in two extreme intervals (gaze positions >15° and <-15°). All gaze positions collected in all drives and for all participants were decomposed in vertical and horizontal components and are presented in the different conditions in Figure 6. The four graphs of Figure 6 are illustrating gaze positions distributions in the assistance and no assistance conditions for the left and right bends on the X-axis and Y-axis dimensions. In order to complement this type of representations with statistical inferences, two-ways repeated measures analyses of variance (ANOVA), with assistance presence (assistance vs. no assistance) and the angular deviation from the tangent point (32 levels) as independent variables, were performed on both vertical and horizontal gaze position data. This analysis was run in order to investigate the interaction between the assistance condition and the angular deviation from the tangent point. The results were as follows:

For the left bend proportion of gaze positions on the X-axis: a significant interaction was found between the assistance condition and the horizontal angular deviation from the tangent point (F(31,544)=8.98; p<.0001). Bonferroni post-hoc analysis indicated a significant difference between the NA and A conditions at seven different angular deviations (p<0.05, starred in Figure 6). Compared to NA, drivers gazed less around the TP with assistance redirecting their gaze further left of the TP. For the left bend proportion of gaze positions on the Y-axis: a significant interaction was found between the assistance condition and the horizontal angular deviation from the tangent point (F(31,544)=7.37; p<.0001). Bonferroni post-hoc analysis indicated a significant difference between the NA and A conditions at three different angular deviations (p<0.05, starred in Figure 6). Again, compared to NA, drivers gazed less around the TP with assistance. Gaze positions were insignificantly more numerous further beyond the tangent point with assistance compared to NA.

For the right bend proportion of gaze positions on the X-axis: a significant interaction was found between the assistance condition and the horizontal angular deviation from the tangent point (F(31,544)=4.53; p<.0001). But post-hoc analysis did not reveal any significant differences between NA and A conditions. Descriptively, all gaze positions were slightly deviated to the right with assistance compared to NA. For the right bend proportion of gaze positions on the Y-axis: a significant interaction was found between the assistance condition and the horizontal angular deviation from the tangent point (F(31,544)=12.46; p<.0001). Posthoc analysis indicated a significant difference between the NA and A conditions at four different angular deviations (p<0.05, starred in Figure 6). As observed for the left bend, compared to the NA condition, drivers gazed less around the TP with assistance. Gaze positions were more numerous further beyond the tangent point with assistance, this was significant for gaze positions over 15°.

IV DISCUSSION

First of all, it is important to underline the variety and the number of methods used to analyse gaze positions during car driving. This might be due to (1) the relative novelty of eye-trackers in research laboratories, resulting in the lack of a standard methodology to assess drivers' visual scanning and (2) the variability and complexity of eye movements collected during car driving.

Four main gaze positions analysis techniques used in the context of a dynamic car driving environment were identified in the literature. The underlying objective was to compare these techniques in order to identify the most adapted to qualify drivers' visual explorations. Many other techniques and measures were not considered here (see [START_REF] Holmqvist | Eye tracking: A comprehensive guide to methods and measures[END_REF] for a comprehensive description of all methods). They were not considered because not adapted and therefore not used in the context of dynamic environments. The analysis carried out here through the four selected techniques showed an important variability of results depending on the technique considered. Based on the results collected while negotiating bends with a highly automated vehicle or a regular vehicle, it appears that the analysis techniques relative to a dynamic area or point (namely dynamic AOIs and dynamic point) in the visual scene were the best suited for the car driving dynamic environment considered. Therefore, these two methods may be considered as the gold standard methods in the context of driving environments. In the following lines, advantages and drawbacks of each analysis technique will be considered and the impact on safety and human factors assessment discussed.

Considering gaze plots analysis, descriptive conclusions were not clear, even if the gaze positions recorded were not completely similar between the NA and A conditions. For the left bend, gaze position in the A condition seemed located upper on the screen compared to the NA condition. But this observation was not so clear for the right bend. For both bends, more gaze positions were observed on the speedometer area, located at the bottom left of the screen (Figure 1). Statistics revealed a smaller gaze positions variability both horizontally and vertically in the NA condition compared to the A condition, as if drivers concentrated their visual explorations when driving with assistance. Even if significant in the left bend for vertical variability, very little differences between NA and A (within 1° of visual angle) gaze positions variability were observed in the left bend (Figure 2). This difference between bends, although unexpected, may be due to the difference between curvature radii (slightly more important in the right bend), the road section ahead of the bend and the general driving context (e.g. the presence of a wall on the right of the road during a part of the left bend limiting visual explorations) as well. If more precise, AOIs based on the results analysis did not bring more information than gaze plots.

Participants looked more on the left of the screen during left bends and more on the right during right bends. Heat maps representing the density of gaze positions did not bring forward any notable differences between NA and A conditions whatever the bend considered, apart from denser gaze positions found in the speedometer area with A for both bends (Figure 3). If global significant differences were found for both bends between NA and A conditions, those differences cannot be clearly located.

Results gained using the dynamic AOI technique are much different and more instructive.

Based on contour plots, it is notable that drivers looked at the vicinity of the TP most of the time while negotiating bends (Figure 4). This result replicates previous findings [START_REF] Kandil | Driving is smoother and more stable when using the tangent point[END_REF][START_REF] Land | Where do we look when we steer[END_REF][START_REF] Mars | Steering and gaze control modifications from directed or spontaneous use of a visual augmentation cue[END_REF][START_REF] Wilson | Eye movements coordinated with steering benefit performance even when vision is denied[END_REF]. The shape of contours indicates higher gaze positions density to the left of the TP for the left bend and to the right of the TP for the right bend. This trend was present without assistance but is reinforced with highly automated driving. Assistance also expends gaze density behind the TP, as if drivers, relieved from steering control, engaged in a more anticipative visual scanning. This suggests that drivers tend to anticipate future parts of the road by exploring the driving scene even further ahead with highly automated driving. Statistics confirmed that drivers spent significantly less time looking at the TP area in A than in NA condition. These results are confirmed and detailed using the dynamic point technique. For the left bend, assistance introduced a shift in gaze positions behind and to the right of the TP. Whereas in the right bend, assistance introduced a shift in gaze positions behind and to the left of the TP (Figure 6). All things considered, it appears that the results and their interpretation can be strongly influenced by the analysis technique selected.

Dynamic area and point in the visual scene techniques appear to be the best suited for the dynamic car driving situation considered.

In the following paragraphs, the advantages and drawbacks of each technique used are discussed. Gaze plots offer a nice and easy preview of eye movements data within the visual screen considered as the frame of reference. These representations are highly understandable because of the familiarity of two-dimensions plots. The understandability of the gaze plots is even improved when a background image is added in order to offer an image of reference.

Nevertheless, gaze plots in the context of dynamic car driving come with several drawbacks.

The most important is due to the fact that despite the background image selected to illustrate the driving situation, no real indications on what was actually projected on the screen are available. In addition, in experiments with several participants (here eighteen participants) and with several repetitions (here four drives were run per condition) adding all gaze positions on the same representation is not possible due to the visual overload of the considered representation. Data interpretation might also be misguided by the background image selected.

Considering the bottom illustrations of Figure 1 may lead to interpret gaze positions as very close in driving environment, although this is simply induced by the image not corresponding to the actual driving scene. In relatively sharp bends, as those used in the current experiment, most of the virtual road was closer to the driven vehicle than it appears on the selected background image. Statistical inferences are also limited to general analysis such as gaze positions variability because of the lack of references in the dynamic environment.

Areas Of Interest (AOI) offered an additional value to the previously described gaze plots.

Using this technique, researchers may define areas of the screen that are of particular interest for them. Elements that remain located at the same position on the screen at all times, as the speedometer in our case, benefit the most of such an analysis. It is easily and accurately possible to analyse the density of visual scanning on the speedometer between in NA and A conditions.

Even if this is a specific situation that was not the focus of the current work, it could be noted that drivers looked more at the speedometer with assistance compared to the unassisted condition. Concerning gaze positions in all other parts of the driving scene, this methodology offers the possibility to select or to make emerge AOIs based on the screen as the frame of reference. Both selection and emergence approaches can be useful depending on the study objective. If AOIs are based on the results and not selected by researchers, the analysis is undoubtedly neutral and results are formally more accurate because of the much greater number of AOIs considered. This second option was therefore adopted. In addition to gaze plots, all participants and repetitions are taken into account in the heat maps illustrations. As for gaze plot analysis, the main drawback of this technique is that no real indications on what was actually projected on the screen are available, because the screen and not the driving environment is taken as the frame of reference. Despite the improvement compared to gaze plots, data interpretation remains difficult due to the frame of reference used and the lack of references in the dynamic environment. This technique might be more successful when comparing very different driving situations. Statistical inferences are also limited because carried out on the entire matrix of AOIs.

Dynamic AOIs method best suits the dynamic environment. Descriptively the contour plots centred on a dynamic area offer very convincing results. Here the TP region was selected as the dynamic AOI because of the particular importance of that road section when negotiating bends.

The AOI was defined as the 5° of visual angle around the TP. But any object present (i.e. lead vehicle, lane edges, billboards…) or virtual (on simulated environments such as the TP here) on the driving scene could be used. Contour lines link points with the same gaze positions density. Repeating the procedure for various levels of density led to plot areas where drivers gazed more. Inferential statistics are possible based on the amount of gaze positions located in the AOI considered in the different conditions of the experiment. Here the AOI (i.e. 5° of visual angle around the TP) was gazed less in the A condition than in the NA condition. Using this technique, information on gaze position location on the screen is lost, but without any negative consequences. However, this technique suffers from two drawbacks. The first is that researchers need to define the AOI(s) for the considered experiment. If the AOI was relatively obvious in the current work, this may not always be the case. The second is related to technical difficulties to record a dynamic AOI (if possible) or the time required to analyse image per image the eyetracking videos relatively to a dynamic AOI. The dynamic point technique is directly in line with the dynamic AOI one and presents the same advantages and drawbacks, apart that it offers a more detailed description of gaze positions in the driving scene, because it is relative to a point, and not to an area.

Many studies tend to avoid the difficulty of analysing eye movements during car driving by limiting the eye-tracker use to visual areas that remain at the same location on the screen at all times such as in-vehicle information systems. Total glance duration, frequency and single glance duration are the most common measures used to do so (ISO 15007-2. Road vehicles, n.d.). This approach is fully effective as far as the information displayed on that part of the visual scene is not changing over time, such as a speedometer that constantly indicates the vehicle speed. The approach is only partly effective when the areas considered are fixed on the screen but displaying dynamic moving images, for example the vehicle mirrors. In that case it is of interest to count the number of glances to the mirrors for instance but the problem of not knowing what drivers are looking at in the mirrors remains. In sum, the issue of eye movements analysis in dynamic environments may be partly solved focusing only on fixed areas, but using dynamic AOIs or dynamic points offers a more complete analysis alternative.

As stressed previously, the results appeared to be different depending on the analysis technique used. Differences in gaze positions between the NA and A conditions were not shown accurately using gaze plots or AOI analysis based on the visual screen. Only the dynamic AOI and the dynamic point analysis pointed out visual scanning differences between the NA and A conditions. Therefore, the selection of the most appropriate technique is a key element in eye movements interpretation. This is of particular importance when it comes to safety in car driving. In the current example, drivers' visual disengagement from the TP region is critical because looking at that region is a key contributing factor to steer the vehicle efficiently [START_REF] Wilson | Eye movements coordinated with steering benefit performance even when vision is denied[END_REF]. Not being able to stress these changes in visual scanning behaviours could lead to ineffective assistance design, considering for instance that gaze behaviours are not critically impacted while driving a highly automated vehicle. Difficulties were observed when drivers had to return to manual control while driving with highly automated vehicle [START_REF] Hoc | Human-machine cooperation in car driving for lateral safety: delegation and mutual control[END_REF]. Not using a visual scanning analysis technique based on dynamic area or point would lead to consider that difficulties drivers face when returning to manual control are not due, at least partly, to changes in visual scanning of the driving environment Although dynamic AOI and dynamic point techniques are allowing an accurate interpretation of eye movement data in dynamic environments, some limitations must be considered in the interpretation. The driving context is a key element that clearly impacts the visual scanning of the driving scene. Drivers' behaviours are different with traffic and without for instance.

Generally speaking, complex driving environments offer more opportunities for drivers to explore the driving scene. Eye movements were recorded here in relatively simple and flat road edges environments but the results might be affected by more complex driving backgrounds. In addition, even if it is possible to multiply dynamic AOIs and dynamic points, the results might be harder to understand with several dynamic elements of the driving scene. Another limitation is directly linked to the use of eye-tracking. Only central vision is considered and no information on peripheral vision is recorded, although it is known that drivers use peripheral vision [START_REF] Mourant | Mapping eye movement patterns to the visual scene in driving: An experimental study[END_REF][START_REF] Summala | Maintaining lane position with peripheral vision during in-vehicle tasks[END_REF]. As a consequence, even if parts of the driving scene are not gazed directly, they may be of importance to complete the driving task and not taken into account based on gaze positions analysis. Despite these general limitations, any dynamic situation, including all virtual reality environments, could benefit from analysis based on a dynamic reference on the visual scene. These techniques could particularly benefit to cognitive ergonomics where dynamic situations are often under the scope of analysis [START_REF] Hoc | Adaptation et gestion des risques en situation dynamique[END_REF].

  Figures equivalent signs count: 10600 (3 x 1/3 page & 2 x 1/3 page) Total: 80930

  figure but with a background image applied to the gaze plots. As any image used for that purpose, the illustration is not relevant because it is not possible to represent the visual scene accurately with a single image, due to the fact that the image is changing during the acquisition of eye movements. To illustrate that point, a background image representing the very beginning of the left and the right bend has been selected. It appears clearly that, apart for the glances at the fixed-position speedometer, the drivers' gaze positions were distributed over a part of the background image drivers had no reasons to look at. Therefore, it appears non relevant to represent them over a single, still image. With this type of representation, the interpretation of the results displayed is mostly dependent of the image selected to illustrate the results. For the considered examples it is hard to conclude on differences between NA and A driving. It seems that drivers looked more to the left of the screen in the left bend and more to the right of the screen in the right bend.

Figure 1 :

 1 Figure 1: Example of gaze positions for a left bend (the two gaze plots on the left) and for a right bend (the two gaze plots on the right) for one participant. The blue circles represent the gaze positions collected in the No Assistance condition (NA) and the red diamonds the gaze positions collected for the Assisted condition (A). The top and bottom gaze plots are the same, an "unrepresentative" background image was simply added to pinpoint the lack of accuracy of this representation. All gaze plots are represented in degrees of visual angle relative to the screen where the driving scene was projected. The origin of the coordinate system is the centre of the screen. Positive values refer to the top and the right parts of the screen, negative values to the bottom and the left parts.

Figure 1 :

 1 Figure 1 : Exemple de distribution des positions du regard dans la scène visuelle pour un virage à gauche (les deux graphiques situés à gauche) et pour un virage à droite (les deux graphiques situés à droite) pour un participant. Les cercles bleus représentent les positions du regard dans la condition Non Assistée (NA) et les losanges rouges les positions du regard dans la condition Assistée (A) où les participants se déplaçaient à l'aide d'un véhicule hautement automatisé prenant complétement en charge le guidage latéral de la voiture. Les deux graphiques situés en haut sont identiques à ceux situés en bas de la figure. Une image "non représentative" de la scène de conduite a simplement été ajoutée en arrière plan sur les graphiques du bas pour illustrer le manque de précision de ce type de représentations. Les positions du regard sont

  )=3.79; p<.002; Y-axis: NA-A; d=38.7-28.7= 10°; t(17)=2.96; p<.009). Both horizontal and vertical variabilities were decreased in the assistance condition compared to the assistance condition.

Figure 2 :

 2 Figure 2: Gaze positions horizontal (X) and vertical (Y) variability in the assistance (A) and the no assistance (NA) conditions for left and right bends. The error bars represent one standard error.

Figure 2 :

 2 Figure 2 : Variabilité horizontale (X) et verticale (Y) de la position du regard dans la scène visuelle avec assistance (A) et sans assistance (NA) pour le virage à gauche et le virage à droite considéré. Les barres d'erreur représentent l'erreur type.

Figure 3 :

 3 Figure 3: Heat maps representing all gaze positions collected for all participants. The two top graphs represent the left bend (NA on the left and A on the right). The two bottom graphs represent the right bend (NA on the left and A on the right). All gaze positions are represented in degrees of visual angle relative to the screen where the driving scene was projected. The origin of the coordinate system is the centre of the screen, top and right are positive values, left and bottom are represented by negative values. A normalized colour scale was set for the four heat maps. The warmer the colour the greater the number of gaze position received in that particular location of the screen.

Figure 3 :

 3 Figure 3 : Cartes de chaleur représentant toutes les positions du regard recueillies pour l'ensemble des participants. Les deux graphiques situés en haut représentent le virage à gauche

Figure 4 :

 4 Figure 4: Contour plots representing all gaze positions collected for all participants relative to a dynamic point on the visual screen (i.e. the tangent point). The two top graphs represent the left bend (NA on the left and A on the right). The two bottom graphs represent the right bend (NA on the left and A on the right). All gaze positions are represented in degrees of visual angle relative to the tangent point for the considered bend. 0 on the X axis represents the tangent point on the horizontal dimension, a positive value on the X axis indicates a gaze located on the right of the tangent point and a negative value a gaze on the left of the tangent point. 0 on the Y axis represents the tangent point on the vertical dimension, a positive value on the Y axis indicates a gaze located below the tangent point and a negative value a gaze behind the tangent point.Four levels of contour are represented (0.2, 0.4, 0.6, 0.8).

Figure 3 :

 3 Figure 3 : Cartes en iso-lignes représentant toutes les positions du regard recueillies pour l'ensemble des participants relativement à un point dynamique de la scène visuelle (i.e. le point tangent). Les deux graphiques situés en haut représentent le virage à gauche (en condition non assistée à gauche et avec un véhicule hautement automatisé à droite). Les deux graphiques

Figure 5 :

 5 Figure 5: Percentage of gaze positions within 5° around the tangent point in the assistance (A) and no assistance (NA) conditions for left and right bends. The error bars represent one standard error.

Figure 2 :

 2 Figure 2 : Pourcentage de positions du regard situées dans une zone de 5° autour du point tangent avec assistance (A) et sans assistance (NA) pour le virage à gauche et le virage à droite considéré. Les barres d'erreur représentent l'erreur type.

Figure 6 :

 6 Figure 6: Proportion of gaze positions collected for all participants relative to the tangent point, degree of visual angle by degree of visual angle. The two top graphs represent the left bend. The two bottom graphs represent the right bend. The two graphs on the left represent the Xaxis dimension of gaze positions and the two graphs on the the Y-axis dimension of gaze positions. All gaze positions are represented in degrees of visual angle relative to the tangent point for the considered bend. 0 on the X axis represents the tangent point on the horizontal dimension, a positive value on the X axis indicates a gaze located on the right of the tangent

Figure 6 :

 6 Figure 6 : Proportions de positions du regard pour l'ensemble des participants dans un repère ayant pour origine le point tangent et avec un échelonnage degré d'angle visuel par degré d'angle visuel. Les deux graphiques situés en haut représentent le virage à gauche et les deux situés en bas représentent le virage à droite. Les deux graphiques situés à gauche représentent les coordonnées en abscisse des positions du regard et les deux graphiques situés à droite les coordonnées en ordonnées de ces mêmes positions du regard. Les positions du regard sont exprimées en degrés dans un repère centré sur le point tangent. 0 sur l'axe des abscisses correspond au point tangent horizontalement, une valeur positive indique un regard à droite du point tangent tandis qu'une valeur négative indique un regard à gauche du point tangent. 0 sur l'axe des ordonnées correspond au point tangent verticalement, une valeur positive indique un regard au delà du point tangent tandis qu'une valeur négative indique un regard en deçà du point tangent. Les étoiles indiquent les degrés d'angle visuel pour lesquels la différence entre les conditions sans assistance et assistée est significative.

Acknowledgments

This work was performed within the framework of the LABEX CORTEX (ANR-11-LABX-0042) of Université de Lyon, within the program ''Investissements d'Avenir'' (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).


Edward Brightman
email: edward.brightman@enocell.com

David Pasquier

Measurement and adjustment of proton activity in solid polymer electrolytes

Keywords: Ionomers, pH, proton activity, polymer electrolyte membrane, reference electrode

Technological progress in electrochemical energy conversion devices requires new solid polymer electrolyte membrane (PEM) materials with particular properties. The strongly acidic nature of Nafion ® is not always desirable; for instance, CO2 electroreduction requires a low proton activity to avoid excess hydrogen evolution. This communication presents a novel measurement technique for determining the acidity of a PEM, using a pH-sensitive electrode half-cell attached to the membrane sample, connected to a reference electrode via a salt bridge. A dynamic hydrogen electrode on platinised-platinum surface (DHE) was found to give repeatable results within 5% uncertainty. Several membranes based on Nafion ® impregnated with various basic species were tested, as well as an anion-exchange membrane and a poly-phosphonic acid membrane.

The technique is expected to have wider reaching applications in PEM fuel cell and redox flow battery development, as well as in the electrolysis and electrodialysis industries.

Introduction

Proton-conducting polymer electrolyte membrane (PEM) materials are widely used in electrochemical devices. The benchmark PEM, Nafion ® , contains sulfonate moieties on fluorinated side chains imparting strongly acidic properties to the polymer in its protonated form [1,[START_REF] Kreuer | On the development of proton conducting polymer membranes for hydrogen and methanol fuel cells[END_REF]. This acidity is not always desirable for the application; for instance, gas phase electroreduction of CO 2 requires a low proton activity to avoid excess hydrogen evolution [START_REF] Aeshala | Effect of cationic and anionic solid polymer electrolyte on direct electrochemical reduction of gaseous CO2 to fuel[END_REF][START_REF] Delacourt | Design of an Electrochemical Cell Making Syngas (CO + H2) from CO2 and H2O Reduction at Room Temperature[END_REF]. Therefore, a large effort is under way to develop new membranes with tailored properties specific to the application. There is, however, no standard measurement technique of the acidity of such membranes.

The concept of pH in a solid polymer electrolyte is challenging to define. Conventionally, pH is defined as 𝑝𝐻 = -log 10 (𝑎 𝐻 + ), where the proton activity, 𝑎 𝐻 + , is related to volumetric concentration by the activity coefficient, γ, to account for non-ideal behaviour. Thus, in dilute solutions, pH is interpreted in terms of concentration, but this cannot apply in a solid polymer electrolyte. There is nevertheless a need to measure the acidity of such materials. For example, in selecting metallic bipolar plate materials for PEM fuel cells, the local pH between the membrane and the plate determines the rate of corrosion [START_REF] Antunes | Corrosion of metal bipolar plates for PEM fuel cells: A review[END_REF][START_REF] Mele | Localised corrosion processes of austenitic stainless steel bipolar plates for polymer electrolyte membrane fuel cells[END_REF][START_REF] Hinds | Towards more representative test methods for corrosion resistance of PEMFC metallic bipolar plates[END_REF].

The internal structure of Nafion ® has been described [START_REF] Kreuer | On the development of proton conducting polymer membranes for hydrogen and methanol fuel cells[END_REF][START_REF] Kreuer | Transport in Proton Conductors for Fuel-Cell Applications: Simulations, Elementary Reactions, and Phenomenology[END_REF][START_REF] Mauritz | State of Understanding of Nafion[END_REF], as a network of water-filled channels connecting micellar domains lined with hydrophilic sulfonic acid groups, while the hydrophobic polymer backbone forms a supporting matrix. With increasing membrane humidification, the structure approaches a continuous water phase, which solvates the acidic protons and can conceivably have a measurable pH in the conventional sense.

Seger et al. [START_REF] Seger | Proton Activity in Nafion Films: Probing Exchangeable Protons with Methylene Blue[END_REF] attempted to measure the internal pH of Nafion ® , using a colorimetric technique by impregnating membranes with methylene blue indicator. They found that the internal pH reflected the presence both of fixed protons, originating from the Nafion ® itself, and "labile" protons from the surrounding solution when immersed in solutions of varying pH. In other words, in contafct with an aqueous solution, Nafion ® is transparent to pH. However, this paper found the intrinsic proton activity of Nafion ® , i.e. the value obtained in contact with ultra-pure water, is equivalent to 1.2 M H 2 SO 4 [START_REF] Seger | Proton Activity in Nafion Films: Probing Exchangeable Protons with Methylene Blue[END_REF].

A separate study using an electrochemical method was performed by Umeda et al. [START_REF] Umeda | Proton activity of Nafion 117 membrane measured from potential difference of hydrogen electrodes[END_REF] who measured the potential difference between a reversible hydrogen electrode (RHE), made from platinum-black on the surface of a Nafion ® membrane, and a Normal Hydrogen Electrode (NHE), connected via a glass capillary salt bridge touching the membrane. Their results concurred with those of Seger et al., although the authors did not attempt to validate their method with any calibration.

Here we present a simple technique for measuring the proton activity of a PEM. This was developed for screening candidate membranes for a CO 2 electrolysis cell, but the technique will also be useful for other applications. The potential of a pH-sensitive electrode, in contact with the membrane of interest, is measured versus a reference electrode that is insensitive to pH. To demonstrate the measurement technique, a range of membranes was prepared based on Nafion ® modified by neutralising the protons with bases such as imidazole and triethylamine. Additionally, hydrocarbon-based cation and anion exchange membranes, and phosphonic acid-containing membranes were studied.

A conventional pH electrode consists of an electrochemical half-cell whose electrode potential depends on pH, measured against a pH-invariant reference electrode. This principle is widely used and understood for aqueous-phase measurements, but is equally valid for solid electrolytes provided a suitable half-cell and reference electrode is used. One option would be to immerse the working electrode/membrane assembly directly in a saline solution containing the reference electrode. However, according to Seger et al. [START_REF] Seger | Proton Activity in Nafion Films: Probing Exchangeable Protons with Methylene Blue[END_REF] this setup would be expected to measure the pH of the chosen solution.

A salt bridge reference electrode design has recently been developed for PEM fuel cells and electrolysers at NPL, UK [START_REF] Brightman | In situ characterisation of PEM water electrolysers using a novel reference electrode[END_REF][START_REF] Brightman | In situ mapping of potential transients during start-up and shut-down of a polymer electrolyte membrane fuel cell[END_REF][START_REF] Hinds | In situ mapping of electrode potential in a PEM fuel cell[END_REF], and we have adapted this concept for the present work. The salt bridge consists of a fine tube made of Nafion ® and filled with water to ensure maximum conductivity. The reference electrode is placed in a saline solution, into which is immersed one end of the Nafion ® tube. The other end is pressed against the membrane of interest, which is cast or compressed onto a dynamic hydrogen electrode (DHE) acting as the pH sensitive electrode.

Materials and methods

pH electrodes

The measurement apparatus is illustrated in Figure 1(a). The membrane sample was placed in contact with a modified commercial screen-printed electrode (Dropsens, Metrohm, France) and mounted in an adapted Flow-Cell apparatus (Dropsens, Metrohm, France) which held the salt bridge in contact with the membrane. The Flow-Cell is marketed as a wall-jet flow cell apparatus, but for the present study the vertical fluid port was adapted to accommodate a Nafion ® tube salt bridge (TT-030, Perma Pure, USA) which made contact with the membrane under test. The geometry of the Flow-Cell holds the membrane flat to the surface of the working electrode, with the salt bridge attached to the top side of the membrane in the centre of the electrode area. The other end of the salt bridge sits in a vessel containing 0.1 M K 2 SO 4 electrolyte and a saturated calomel reference electrode (SCE). The junction potential of the salt bridge was tested with several buffer solutions and found to be invariant with pH. The screen-printed electrodes (Dropsens, Metrohm, France) consisted of a gold working electrode (4 mm diameter), platinum counter electrode and silver reference electrode. In this work, since a separate reference electrode was used, the screen-printed silver reference electrode was isolated with acrylic varnish.

The working electrode was platinised according to the method recommended by Feltham and Spiro [START_REF] Feltham | Platinized platinum electrodes[END_REF], after first cleaning the gold surface by cyclic voltammetry in 0.1 M H 2 SO 4 at 100 mV/s between -0.40 V and +1.36 V (SCE) for 10 cycles. These electrodes were used as DHEs, i.e. generating hydrogen in situ by applying a small current between the platinized working electrode and the screen-printed counter electrode. The current density was controlled at -1 mA cm -2 (60 s), -0.2 mA cm -2 (60 s) and at open circuit (60 s) in order to account for ohmic and activation overpotentials.

Selection of membranes for testing

Nafion ® N115 was used as a benchmark material for this study, as its proton conductivity and acidity are well characterised. Several studies have been published documenting composite membranes formed by doping Nafion ® with molecules containing a basic moiety [START_REF] Li | Immobilization of imidazole moieties in polymer electrolyte composite membrane for elevated temperature fuel cells[END_REF][START_REF] Ainla | Nafion®-polybenzimidazole (PBI) composite membranes for DMFC applications[END_REF][START_REF] Fu | Nafion-Imidazole-H3PO4 Composite Membranes for Proton Exchange Membrane Fuel Cells[END_REF][START_REF] Guan | Proton conducting membranes based on semiinterpenetrating polymer network of Nafion® and polybenzimidazole[END_REF][START_REF] Hongsirikarn | Influence of ammonia on the conductivity of Nafion membranes[END_REF][START_REF] Lin | Amino-silica modified Nafion membrane for vanadium redox flow battery[END_REF][START_REF] Martinez | Proton-conducting ionic liquid-based Proton Exchange Membrane Fuel Cell membranes: The key role of ionomer-ionic liquid interaction[END_REF][START_REF] Meredith | Hydrophobic Salt-modified Nafion for Enzyme Immobilization and Stabilization[END_REF][START_REF] Neves | Methanol and gas crossover through modified Nafion membranes by incorporation of ionic liquid cations[END_REF][START_REF] Romero | Modification of Nafion Membranes by IL-Cation Exchange: Chemical Surface, Electrical and Interfacial Study[END_REF][START_REF] Schauer | Ion-conductive polymer membranes containing 1-butyl-3-methylimidazolium trifluoromethanesulfonate and 1-ethylimidazolium trifluoromethanesulfonate[END_REF][START_REF] Sen | Nafion/poly(1-vinyl-1,2,4-triazole) blends as proton conducting membranes for polymer electrolyte membrane fuel cells[END_REF][START_REF] Sen | Anhydrous proton conducting membranes for PEM fuel cells based on Nafion/Azole composites[END_REF][START_REF] Sood | Proton Conducting Ionic Liquid Doped Nafion Membranes: Nano-Structuration, Transport Properties and Water Sorption[END_REF][START_REF] Wycisk | Direct methanol fuel cell membranes from Nafion-polybenzimidazole blends[END_REF][START_REF] Gleason | Proton Exchange Membranes and Membrane Electrode Assemblies for Enhanced Direct Methanol Fuel Cell Performance[END_REF]. These were for a diverse range of applications requiring modification of the intrinsic properties of Nafion ® , for example as electrolyte membranes in direct methanol fuel cells (DMFC), high temperature PEM fuel cells, and redox flow batteries, as well as for biological applications. Such modification of Nafion ® by cation exchange is expected to affect the acidity of the polymer, but this aspect was not considered in the cited studies themselves. Here we have reproduced some of these materials as well as some unrelated ion conducting polymers, as listed in Table 1. All Nafion ®based membranes were prepared from Nafion ® N115 (PaxiTech, France) that was first treated by successively boiling for 1 h in 5 wt % H 2 O 2 ; H 2 O; 0.5 M H 2 SO 4 and again in H 2 O, followed by drying at 90 °C in a vacuum oven overnight. The subsequent preparation steps are given in the table. Membranes were fully humidified by soaking in liquid water and dried with tissue paper prior to testing.

The pH response of the electrodes was characterised with calibrated phosphate buffer solutions verified with a conventional pH electrode (Mettler Toledo). After initial testing in bulk solution (not shown here), the electrodes were calibrated in the FlowCell/salt-bridge setup using porous polyamide felt battery separator material soaked in the calibrated buffer solutions. The junction potential of the salt bridge was also assessed and found to be invariant with pH.

Table 1. Membranes used in this study and summary of results

Results and discussion

The potential of a DHE vs SCE during a typical experiment is shown in Figure 1(b) (shown for pH 2.13 phosphate buffer), illustrating the initial cathodic reduction of the Pt surface, followed by a steady-state region where hydrogen evolution takes place at a constant rate. The potential becomes less negative when the current density decreases, due to a smaller iR overpotential. Finally, after stopping the current, the open circuit potential (OCP) slowly drifts positive as oxygen diffuses to the electrode surface from the atmosphere. Figure 1(c) shows the calibration curve, with the OCP values giving a straight line within 1% of the theoretical Nernstian gradient. The most notable feature from the calibration is the offset of the point at pH 7 under -0.2 mA cm -2 , compared to the value at OCP, indicating a significant polarisation and/or ohmic overpotential for this buffer solution. This could be due to local depletion of H + close to the platinum surface skewing the local pH to higher values, which is a known phenomenon for hydrogen evolution in solutions of neutral pH.

The results of membrane pH measurements are shown in Figure 2. Measurements for each membrane were carried out at least three times to determine the uncertainty of the measurement, and the averaged curve is shown. Each membrane shows the characteristic curve approaching steady state from a positive potential during the -1 mA cm -2 stage, changing to a less negative steady state potential during -0.2 mA cm -2 , and finally a gradual irregular increase in potential at open circuit. For some membranes (e.g. PS-co-Styphos) the open circuit potential immediately jumps to a less negative value, followed by a more gradual increase. This indicates a high iR overpotential, suggesting a low membrane conductivity. An improved OCP-only technique using a palladium-hydrogen electrode may be possible but palladium electrodes were not available for this work. Nevertheless, repeatable steady state results were obtained for most membranes, with the exception of Nafion ® 982 (cathode siderepeatable data could not be obtained) which may be due to the textured woven support structure on the cathode side affecting contact with the electrode; the anode side of Nafion ® 982 gave very similar results to H+-form Nafion ® 115 (within 11 mVnot shown). The poly(phosphonic acid) membranes (Figure 2b) also displayed erratic results, possibly due to poor lateral conductivity between the working and counter electrodes across the membrane surface.

From these data it can be immediately observed that the hydrogen evolution potential is significantly more negative for the modified membranes. The electrode potentials at OCP for each membrane are summarised in Table 1. These values were obtained by averaging the first 30 s at OCP for each test, and then averaging that value for the three repeat tests. Uncertainty is given as two times standard deviation.

A first simple comparison can be made between the structurally similar polystyrene-based membranes AMI7001 and CMI7000, which are functionalised with a quaternary amine and a sulfonate respectively. Here an alkaline pH is observed for the anion exchange membrane and an acidic pH is observed for the cation exchange membrane. Interestingly, however, the anion exchange membrane does not give as high a pH as might be expected for a pure OH --conducting electrolyte, possibly due to reaction of atmospheric CO 2 with OH -to form carbonate.

The results for the modified Nafion ® membranes show that imidazole and butylimidazole both have a similar buffering effect on the acidity of Nafion ® , with a slightly stronger effect from triethylamine and diethylamine.

The difference is likely to arise from the higher pKa of protonated amines (~11) compared to imidazoles (~7). These membranes would be interesting candidates for a CO2 electroreduction cell, which was the main motivation for this study.

Conclusions

For the first time, the proton activities of a range of polymer electrolyte membranes have been measured and compared using a straightforward and inexpensive technique. A salt bridge connects the top surface of the membrane under test to a reference electrode, while the bottom surface of the membrane is in contact with a pH-sensitive electrode.

 A platinized-platinum DHE gave good repeatability for ranking the acidity of a range of PEM membranes.

 Membranes based on Nafion ® modified by exchanging protons for protonated bases such as butylimidazole or triethylamine were found to have a near-neutral acidity. Further characterisation of these membranes is required to measure the conductivity and the rate of leaching of the molecular species during prolonged operation.

 The novel technique has enabled rapid screening of membranes for use in a PEM electrochemical cell for reduction of CO 2 , where near-neutral pH is desirable. The technique will also provide unique

Figure 1 .

 1 Figure 1. (a) Illustration of Dropsens Flow-Cell configuration. (b) Typical potential of DHE under

Figure 2 .

 2 Figure 2. Raw data for DHE measurement of several membranes.

Figure 1 .

 1 Figure 1. (a) Illustration of Dropsens Flow-Cell configuration. (b) Typical potential of DHE under

Figure 2 .

 2 Figure 2. Raw data for DHE measurement of several membranes.

Table 1 . Membranes used in this study and summary of results

 1 Membranes International, Inc., USA) was soaked in 1 M NaOH for 24 h at ambient temperature to exchange Cl -ions for OH -. Membranes International, Inc., USA) was soaked in 0.5 M H 2 SO 4 for 24 h at ambient temperature to exchange Na + ions for H + .

	Membrane	Preparation method	Ref	Measured	Calculated	Absolute
	name			E vs. SCE	pH	uncertainty
				(V)		in pH
	Nafion ® (H + )	N/A		-0.295	1.4	± 0.20
	Nafion ® (Na + )	Heat to 80 °C for 1 h in 1 M NaOH (aq), followed by		-0.849	10.7	± 0.67
		24 h at room temperature. Then rinse in two successive
		baths of H 2 O at 80 °C for 1 h
	Nafion ®	Soak in imidazole-methanol solutions of (a) 0.1 g in 10	[18]	-0.637	7.1	± 0.40
	(Imidazole)	mL solution; (b) 2.0 g in 10 mL solution at 55 °C for 2
		h, followed by drying in a vacuum oven at 90 °C
		overnight.
	Nafion ® (Butyl-	Soak in 1-(n-butyl)imidazole-methanol solutions of (a)	Adapted	-0.634	7.1	± 0.36
	imidazole)	0.18 g in 10 mL solution; (b) 3.6 g in 10 mL solution at	from
		55 °C for 2 h, followed by drying in a vacuum oven at	[18]
		90 °C overnight.
	Nafion ®	Soak in 1 M triethylamine aqueous solution for 24 h at		-0.656	7.5	± 0.31
	(Triethylamine)	ambient temperature.
	Nafion ®	Soak in 1 M Diethylamine aqueous solution for 24 h at		-0.663	7.6	± 0.36
	(Diethylamine)	ambient temperature.
	AMI7001(OH -)	AMI7001 ([3]	-0.734	8.8	± 0.22
	CMI7000(H + )	CMI7000 ([3]	-0.325	1.9	± 0.13
	PMMA-co-	Poly(methylmethacrylate-co-styrenephosphonic acid)		-0.352	2.3	± 1.45
	Styphos	random copolymer, 9:1 ratio (Specific Polymers,
		France) was dissolved in DMAc solvent and cast on a
		porous polyolefin battery separator support.
	PS-co-Styphos	Poly(styrene-co-styrenephosphonic acid) random		-0.822	10.2	± 0.35
		copolymer, 9:1 ratio (Specific Polymers, France) was
		dissolved in DMAc solvent and cast on a porous
		polyolefin battery separator support.
	Nafion N982	Boil for 2 h in 1 M HNO 3 , then rinse in deionized	[31]	(Anode	1.6	± 0.46
		water.		side)
				-0.306
					(Cathode side)
				Unable to produce repeatable result

Acknowledgements

The authors thank IFP Energies Nouvelles for financial support and Aurélie Courbon for her helpful contribution to the experiments.

insight into novel membranes for many other electrochemical devices, as well as for understanding corrosion of metallic bipolar plates.


Pierre-Damien Thizy

SHARP QUANTIZATION FOR LANE-EMDEN PROBLEMS IN DIMENSION TWO

Keywords: Mathematics Subject Classification. 35B05, 35B06, 35J91. 1

In this short note, we prove a sharp quantization for positive solutions of Lane-Emden problems in a bounded planar domain. This result has been conjectured by

Introduction

Let Ω be an open, non-empty, connected and bounded subset of R 2 with smooth boundary ∂Ω and let ∆ = -(∂ xx + ∂ yy ) be the (positive) laplacian. In this paper, we are interested in the asymptotic behavior as p → +∞ of a sequence (u p ) p of smooth functions, positive in Ω, and satisfying the so-called Lane-Emden problem

∆u p = |u p | p-1 u p in Ω , u p = 0 on ∂Ω , (0.1)
together with the bounded-energy type assumption

p Ω |∇u p | 2 ds = O(1) , (0.2)
for all p. Up to now, the most general results on this problem were obtained by De Marchis, Ianni and Pacella [START_REF]Asymptotic profile of positive solutions of Lane-Emden problems in dimension two[END_REF]. In particular, for such a given (u p ) p satisfying (0.1)-(0.2), it is proved in [START_REF]Asymptotic profile of positive solutions of Lane-Emden problems in dimension two[END_REF] that, up to a subsequence, there exists an integer n ≥ 1 and a subset B = {x 1 , ..., x n } of Ω such that the following quantization

lim p→+∞ p Ω |∇u p | 2 ds = 8π n j=1 m 2 j (0.3)
holds true, where the m j 's can be obtained through

m j = lim β→0 + lim p→+∞ u p C 0 (Bx j (β)) , (0.4)
for j ∈ {1, ..., n}, where B xj (β) is the ball of center x j and radius β. Observe that, in particular, the x j 's are not in ∂Ω. It is also proved in [START_REF]Asymptotic profile of positive solutions of Lane-Emden problems in dimension two[END_REF] that we necessarily have that m j ≥ √ e , (0.5) for all j ∈ {1, ..., n} and that

lim p→+∞ u p = 0 in C 2 loc ( Ω\B) . (0.6)
In [START_REF]Asymptotic profile of positive solutions of Lane-Emden problems in dimension two[END_REF]Remark 1.2], it is conjectured that we must have equality in (0.5), so that, in some sense, the constant 8πe plays here the same role as the Sobolev constant in dimensions greater than 2 (see Struwe [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF]). This is the point in the following theorem.

Theorem 0.1. Let Ω be a smooth bounded domain of R 2 . Let (u p ) p be a sequence of smooth functions positive in Ω, and satisfying (0.1) and (0.2). Then, up to a subsequence, there exists an integer n ≥ 1 such that

lim p→+∞ p Ω |∇u p | 2 ds = (8πe) × n . (0.7)
Moreover, there exists a subset {x 1 , ..., x n } of Ω such that

m j = √ e , (0.8)
where m j is given by (0.4), for all j ∈ {1, ..., n}.

In addition, by [START_REF]Asymptotic profile of positive solutions of Lane-Emden problems in dimension two[END_REF], we get from (0.8) that

lim p→+∞ pu p = 8π √ e n j=1 G xj in C 2 loc ( Ω\B) , (0.9)
and that

∇ xj   H xj (x j ) + i =j G xi (x j )   = 0 , (0.10)
for all j ∈ {1, ..., n}, where G is the Green's function of ∆ with Dirichlet boundary conditions and where H is its regular part, which is smooth in Ω 2 and given by

G x (y) = 1 2π log 1 |x -y| + H x (y) ,
for all x = y.

Concerning the previous works, Ren and Wei [START_REF] Ren | On a two-dimensional elliptic problem with large exponent in nonlinearity[END_REF] and [START_REF]Single-point condensation and least-energy solutions[END_REF] where able to prove that (0.7) with n = 1 holds true if the u p 's are minimizers, i.e. if we assume in addition that u p is proportional to a solution of the problem

min {v∈H 1 0 s.t. Ω v p ds=1} Ω |∇v| 2 ds.
Answering to a former question, Adimurthi and Grossi [START_REF] Adimurthi | Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity[END_REF] were able to prove that

lim p→+∞ u p C 0 (Ω) = √ e , (0.11)
in the case of minimizers, while they discovered the way to perform the first rescaling for the u p 's as p → +∞, and the key link with the Liouville equation. Observe that (0.4), (0.6) and (0.8) clearly imply (0.11) in general case. Now in the radial case where Ω is a disk, observe that the u p 's are necessarily minimizers, since (0.1) admits only one solution (see Gidas-Ni-Niremberg [START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF] and the nice survey by Pacella [START_REF] Pacella | Uniqueness of positive solutions of semilinear elliptic equations and related eigenvalue problems[END_REF]); according to the previous discussion, we necessarily then have that n = 1 in (0.7). In contrast, if Ω is not simply connected, Esposito, Musso and Pistoia [START_REF] Esposito | Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent[END_REF] were able to prove that, for all given integer n ≥ 1, there exists a sequence of positive functions (u p ) p satisfying (0.1)-(0.2) such that (0.7) holds true, together with (0.8)-(0.11). Thus, in some sense, Theorem 0.1 is sharp. We mention that very interesting complementary results were obtained recently by Kamburov and Sirakov [START_REF] Kamburov | Uniform a priori estimates for positive solutions of the Lane-Emden equation in the plane[END_REF]. At last, we also mention that, even if the situation is far from being as well understood in the nodal case, where we no longer assume that the u p 's are positive, some asymptotic-analysis [START_REF] De Marchis | Asymptotic analysis and signchanging bubble towers for Lane-Emden problems[END_REF][START_REF] Grossi | Lane-Emden problems: asymptotic behavior of low energy nodal solutions[END_REF], as well as some constructive [START_REF]Morse index and sign-changing bubble towers for Lane-Emden problems[END_REF][START_REF]On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity[END_REF][START_REF] Gladiali | Quasi-radial nodal solutions for the Lane-Emden problem in the ball[END_REF] results were obtained.

To conclude, as explained in De Marchis, Ianni and Pacella [START_REF]Asymptotic analysis for the Lane-Emden problem in dimension two[END_REF], the techniques to get the quantization result in [START_REF]Asymptotic profile of positive solutions of Lane-Emden problems in dimension two[END_REF] are not without similarity with the ones developed by Druet [START_REF] Druet | Multibumps analysis in dimension 2: quantification of blow-up levels[END_REF] to get the analogue quantization for 2D Moser-Trudinger critical problems. Both results [START_REF]Asymptotic profile of positive solutions of Lane-Emden problems in dimension two[END_REF][START_REF] Druet | Multibumps analysis in dimension 2: quantification of blow-up levels[END_REF] can be improved by showing that all the blow-up points necessarily carry the minimal energy. It is done here in the Lane-Emden case and in Druet and Thizy [START_REF] Druet | Multi-bumps analysis for Trudinger-Moser nonlinearities I-Quantification and location of concentration points[END_REF] in the Moser-Trudinger case. Unfortunately, the authors of [START_REF] Druet | Multi-bumps analysis for Trudinger-Moser nonlinearities I-Quantification and location of concentration points[END_REF] were not able to find an as easy argument as here, in the more tricky Moser-Trudinger critical case.

1. Proof of Theorem 0.1 Let (u p ) p be a sequence of smooth functions, positive in Ω and satisfying (0.1)-(0.2). Then by [START_REF]Asymptotic profile of positive solutions of Lane-Emden problems in dimension two[END_REF], (0.3)-(0.6) hold true. Thus, the proof of (0.7)-(0.8), i.e. that of Theorem 0.1, reduces to the proof of

lim p→+∞ u p C 0 (Ω) ≤ √ e . (1.1)
Here and in the sequel, we argue up to a subsequence. Now, let (y p ) p be a sequence in Ω such that u p (y p ) = u p C 0 (Ω) , for all p. By (0.4)-(0.6), we have that

lim p→+∞ d(y p , ∂Ω) := 2δ 0 > 0 , (1.2)
where d(y, ∂Ω) denotes the distance from y to ∂Ω. Now, let µ p > 0 be given by

µ 2 p p u p (y p ) p-1 = 8 . (1.3)
By (0.4) and (0.5), we get from (1.3) that

log 1 µ 2 p = p log u p (y p ) 1 + O log p p , (1.4)
and in particular, that µ p → 0 as p → +∞. Let τ p be given by

u p (y p + µ p y) = u p (y p ) 1 - 2τ p (y) p , so that τ p ≥ 0 and τ p (0) = 0 , (1.5)
by definition of (y p ) p . By (0.1) and (1.3), we have that

∆(-τ p ) = 4 1 - 2τ p p p in Ω p := Ω -y p µ p , (1.6)
so that, by (1.5), positivity of u p and concavity of the log function, we get that

0 < ∆(-τ p ) ≤ 4 . (1.7)
By (1.2), (1.5)-(1.7) and standard elliptic theory, including the Harnack principle, we get that there exists a function

τ ∞ ∈ C 2 (R 2 ) such that lim p→+∞ τ p = τ ∞ in C 2 loc (R 2 ) , (1.8)
and then that

∆(-τ ∞ ) = 4 exp(-2τ ∞ ) in R 2 , τ ∞ (0) = 0 , ∇τ ∞ (0) = 0 , (1.9)
using also that ∇τ p (0) = 0, by definition of (y p ) p . Let R > 0 be given. Integrating by parts, using (1.3)

(1 + r 2 ) 2 + o (r p /µ p ) 2 1 + (r p /µ p ) 2 , (1.13)
using that the laplacian commutes with the average in spheres. Then, using the fundamental theorem of calculus and ūp (0) = u p (y p ), we easily get from (1.

14 ) 2 p+ log µ 2 p

 1422 [START_REF] Grossi | Lane-Emden problems: asymptotic behavior of low energy nodal solutions[END_REF]) that ūp (r p ) ≤ u p (y p ) 1 -2 tp (r p )Picking now r p = δ 0 for all p, according to (1.2), we get from (1.14) thatūp (δ 0 ) ≤ u p (y p ) 1 -2 log u p (y p ) (1 + o(1)) + O 1 p ,(1.15)by(1.4), writing merely log 1 µ + δ 2 0 = tp (δ 0 ). Since ūp ≥ 0 and since u p (y p ) = u p C 0 (Ω) , we easily get from (1.15) that (1.1) holds true. As explained at the beginning of the proof, this concludes the proof of Theorem 0.1.

  , (1.8), (0.2), ∆u p ≥ 0 and u p ≥ 0, we get that By using that u p (y p ) 2 ≥ (1 + o(1))e , and by observing that the above RHS does not depend on R > 0, which can be arbitrarily large, we get that

	lim inf p→+∞	(8u p (y p ) 2 )	B0(R)	4 exp(-2τ ∞ )dy ≤ lim p→+∞	p	By p (Rµp)	(∆u p )u p dy ,	(1.10)
									≤ lim p→+∞	p	Ω	|∇u p | 2 dy < +∞ .
									exp(-2τ ∞ )dy < +∞ .	(1.11)
								R 2
	By Chen and Li [2], (1.9) and (1.11) imply that
									τ ∞ = log(1 + | • | 2 ) .	(1.12)
	Then, we let t p be given by
								t p (y) = log 1 +	|y -y p | 2 µ 2
			f (r) =	1 2πr ∂By p (r)
									0.1), (1.3),
	(1.8), (1.12) and Fatou's lemma, we get that
	-2πr p	dū p dr	(r p )
			=					(∆u p ) 2πrdr ,
						By p (rp)
			≥	p 2u p (y p )	0	rp/µp	8πr dr

p

.

From now on, if f is a given continuous function in Ω, we let f be the unique continuous function in [0, d(y p , ∂Ω)) given by f dσ , for all r ∈ (0, , d(y p , ∂Ω)) .

Let (r p ) p be any sequence such that r p ∈ [0, d(y p , ∂Ω)) for all p. By (