Mathilde Valenchon
email: mathilde.valenchon@yahoo.fr

Jean-Louis Deneubourg

Anna P Nesterova

Odile Petit
email: odile.petit@cnrs.fr

Does a high social status confer greater levels of trust from groupmates? An experimental study of leadership in domestic horses

In collective movements, specific individuals may emerge as leaders. In this study on the domestic horse (Equus ferus caballus), we conducted experiments to establish if an individual is successfully followed due to its social status (including hierarchical rank and centrality).

We first informed one horse about a hidden food location and recorded by how many it was followed when going back to this location. In this context, all horses lead their groupmates successfully. In a second step, we tested whether group members would trust some leaders more than others by removing the food before the informed individual led the group back to the food location. In addition, two control initiators with intermediate social status for which the food was not removed were tested. The results, confirmed by simulations, demonstrated that the proportions of followers for the unreliable initiator with highest social status are greater than the ones of the unreliable initiator with lowest social status. Our results suggest an existing relationship between having a high social status and a leadership role. Indeed, the status of a leader sometimes prevail at the detriment of the accuracy of the information, because an elevated social status apparently confers a high level of trust.

Introduction

One of the most captivating social displays in the biological world is undoubtedly collective movements and concerns many species: swarms of invertebrates, shoals of fish, flocks of birds, herds of ungulates or troops of primates. This polyphyletic phenomenon reflects its highly adaptive function [START_REF] Boinski | On the Move: How and Why Animals Travel in Groups[END_REF]. However, how such coordination for moving together is achieved raises several issues. Among them, the possibility that some individuals exert a disproportionate influence on group decisions, and thus considered as leaders, is still in debate [START_REF] Strandburg-Peshkin | Inferring influence and leadership in moving animal groups[END_REF] including in humans [START_REF] Von Rueden | Leadership in small-scale societies: Some implications for theory, research, and practice[END_REF], and raises questions about its sustainability within a social group. Indeed, confidently following a leader will speed up the collective decision but it implies that the information is concentrated in this sole individual [START_REF] Couzin | Effective leadership and decision-making in animal groups on the move[END_REF][START_REF] Sumpter | Quorum responses and consensus decision making[END_REF][START_REF] Kao | Decision accuracy in complex environments is often maximized by small group sizes[END_REF][START_REF] Gavrilets | Convergence to consensus in heterogeneous groups and the emergence of informal leadership[END_REF]. In case of wrong information, the whole group will suffer the consequences of bad choices [START_REF] Conradt | Consensus decision making in animals[END_REF].

Even if all group members can initiate collective movements in nearly all species, some of these initiators are joined more successfully than others. In this context, we wonder why some individuals are elevated to leadership roles by group members, resulting in a more efficient decision-making process. These interindividual differences in leadership success can be accounted for by several factors such as age, sex, temperament, energetic state, dominance status, or centrality (see Petit and Bon 2010 for a review). Whatever the species, there is not a unique and consistent factor explaining the success of an initiator. However, it appears that several parameters underlying social status may play a preponderant role. Indeed, on one hand, we know that dominant individuals can play a key role in collective decision-making [START_REF] Radford | Vocal Coordination of Group Movement by Green Woodhoopoes (Phoeniculus purpureus)[END_REF], Sueur and Petit 2008[START_REF] Jacobs | Social Structure Affects Initiations of Group Movements but Not Recruitment Success in Japanese Macaques (Macaca fuscata)[END_REF]). On the other hand, it seems that the network of affiliative relationships amply explains joining latencies with an optimization of the process when initiators have high centrality scores (Sueur and Petit 2008[START_REF] King | A rule-of-thumb based on social affiliation explains collective movements in desert baboons[END_REF][START_REF] Sueur | Predicting leadership using nutrient requirements and dominance rank of group members[END_REF][START_REF] Briard | Personality and Affinities Play a Key Role in the Organisation of Collective Movements in a Group of Domestic Horses[END_REF]. All these factors refer to the social position of an individual among its group mates. It is thus likely that a favourable social position may lead to a higher social influence on group decisions. This social influence could reflect the trust that other group mates might place in this conspecific.

This study aims to explore the social influence of individuals in explaining successful leaders and thus focuses on followership which is generally neglected in the literature [START_REF] Van Vugt | Leadership, followership, and evolution -Some lessons from the past[END_REF]. We first predict that potential followers would be more inclined to follow an initiator proposing an activity if this initiator possess a high social status (including hierarchical rank and centrality). If this is the case, it might suggest that followers consider those individuals as more 'trustable' leaders, making them more motivated to follow.

In naturally occurring complex social systems, correlations between multiple factors result in feedback effects that make it difficult to distinguish between causes and consequences. It is therefore necessary to conduct experiments to assess the determining factors of decisionmaking. Thus, we conducted experiments simulating natural situations to establish if an individual is successfully followed due to its social status. By controlling experimentally the time and place of the group departure, we are able to narrow our focus on the influence of a leader's identity itself by discarding the influence of the pre-departure period [START_REF] Bourjade | Decision-Making in Przewalski Horses (Equus ferus przewalskii) is Driven by the Ecological Contexts of Collective Movements[END_REF], Sueur, Deneubourg et al. 2011[START_REF] Briard | How stallions influence the dynamic of collective movements in two groups of domestic horses, from departure to arrival[END_REF][START_REF] Briard | Group behaviours and individual spatial sorting before departure predict the dynamics of collective movements in horses[END_REF]). In the first step of this experiment, we informed one individual about a hidden food location containing highly preferred food and recorded how much and how rapidly it was followed in comparison to other group members. As a result, the whole group had access to a great amount of highly preferred food. We tested individuals with different hierarchical status and centrality scores and compared their performances in terms of proportion of the group members that followed the initiator. In the second step of the study, we tested whether group members would trust some leaders more than others with an extinction procedure. To investigate this extent of trust, we experimentally tested its perseverance for an initiator for which the reliability is challenged. In this second step, we informed a single individual about a food location as usual, but food was removed before the informed individual led the group back to this location. We then measured how this new unreliability impacted the followers' decision throughout several testing sessions. If group members no longer trust the informed individuals whatever their identity, we should observe a decrease of the following rates over the course of the experiment. Conversely, if congeners continue to follow specific individuals, those with high social status for example, it would provide the best demonstration that a form of trust from the followers that does not solely depend on an assessment of a leader's reliability as a keystone factor to determine group decision-making.

We conducted this work in the domestic horse (Equus ferus caballus) which is an ideal model for this experimental study as they are easily trainable. The domestic horse forms cohesive social groups of both sexes [START_REF] Cameron | Social bonds between unrelated females increase reproductive success in feral horses[END_REF]Cameron 2009), and shares physiological attributes, cognitive and social characteristics with many social primates that have been extensively studied when exploring the influence of social status on decision-making processes.

Methods

Experimental horses

Three groups of 6 females were studied (two in 2014: groups A and B, and one in 2015: group C). Subjects were Haflinger, Dartmoor and Shetland ponies aging 7.2 ± 0.6 years (mean ± SE), housed at the "Elevage du Haut-barr" (Saverne, France) and used for reproduction purposes (brood mares and their daughters) (see Table 1 for a summary of individual characteristics). All horses were familiarised with each other for several years within the farm and studied groups were formed and stabilized at least 6 months before the beginning of the experiment. Four of the 12 horses studied in 2014 were part of the 2015 group C (ARM, OTO, SER, and TAK). All horses lived in outdoor pasture all year long with periodical fodder supplementation and water available ad libitum. Each morning when an experiment took place, all horses were equipped with halters that were removed after the last test.

The three groups (A, B, C) were tested for the individual initiation tests (Experiment 1) and only the group C was finally tested for the extinction tests (Experiment 2).

Experiment 1: individual initiations tests

We provoked movement initiations with a protocol based of food hiding in order to assess individual levels of success [START_REF] Gérard | How does the expressiveness of leaders affect followership in domestic horses (Equus ferus caballus)?[END_REF]. The three groups were tested.

The experiment consisted of 73 valid tests.

At each test, we first gathered the whole group into a temporary pen (approximately 5 m x 5 m) at the departure area. Then, an experimenter concealed the food reward (19 carrots) out of sight at 30 m from the departure area. The tested horse (only one for each test) was then brought to this hidden food location and was allowed to eat only one carrot so it became motivated to come back. Then, it was brought back to the group and as soon as it entered the departure zone, the whole group was released so the informed horse could initiate a movement to the food location and being eventually followed by others. The 19 carrots were divided in a pile of 10 carrots at the centre plus 9 carrots randomly distributed at a distance of 2-3m around the centre so all horses can have access to food even when the whole group is present (i.e. non-monopolisable resource).

Three experimenters were present to build and rapidly remove the temporary pen made of 6 plastic posts and 20mm-large plastic tape. Three cameras were arranged in order to cover the departure area, the route taken by the group and the arrival area.

The number of followers per initiation was recorded (from direct observations and videos).

Any horse seen moving away (at least 20 steps) from the departure in the same direction as the initiator was identified as a follower [START_REF] Briard | Personality and Affinities Play a Key Role in the Organisation of Collective Movements in a Group of Domestic Horses[END_REF]. Only the trials when the informed horse was the first to move to go directly from the departure zone to the food reward were analysed. Three trials a day were conducted and we had at least two hours break between trials.

Experiment 2: extinction tests

Extinction tests aimed to assess the perseverance of a group to follow an initiator despite this initiator not leading them to a food reward. This experiment was conducted on group C only.

The protocol was the same as for individual initiations tests except the reward (with the exception of two carrots) was removed by an experimenter (discreetly) after the initiator was informed and before it came back followed by the group. The two carrots were left so the initiator had still access to a food reward as it arrived first (to keep it motivated to go back to the reward location), but the rest of the group was not rewarded. Two "unreliable initiators" (i.e. no reward anymore) and two control "reliable initiators" (i.e. intact reward) were tested randomly. The two misinformed initiators were: the horse with the highest social status (HS) and the one with the lowest social status (LS). The two informed initiators were two horses with intermediate social status (see paragraph (d) for social status determination).

During the first phase, we completed five rewarded initiation tests for each initiator, meaning the whole reward remained present for the group. Then, in the second phase, we conducted seven other trials per initiator. These seven trials were extinction trials for the two unreliable initiators (food reward was removed for the followers) and rewarded trials for the two reliable initiators (food reward remained present for all). For each initiator, we compared the amount of followership between the six first trials (five control trials + the 1 st trial of the second phase when the group discovered the absence of food reward for misinformed initiators) and the next six trials (6 rewarded trials for reliable initiators or 6 extinction trials for unreliable initiators).

2.4.Assessing social status

Each group was observed and filmed continuously for 6 h daily, distributed between 8a.m. and 7 p.m., during the four weeks preceding the beginning of experiment 1, for a total of 50 hours.

Agonistic interactions were recorded continuously. For each interaction, we recorded the identity of the emitter of each approach, threat or aggression, and both the identity and the response of the receiver (offensive behaviour, avoidance or no reaction). Unidirectional interactions -when only one horse was approaching or emitting offensive behaviour and the other clearly avoided them -were used to build a dominance hierarchy. We carried out an analysis of dominance hierarchy using SOCPROG 2.4 [START_REF] Whitehead | SOCPROG programs: analysing animal social structures[END_REF]) to obtain David's scores for each rank and then checked for hierarchy linearity (de Vries 1995, de Vries 1998, [START_REF] De Vries | Measuring and testing the steepness of dominance hierarchies[END_REF].

Spatial proximities between horses were recorded every 15min using instantaneous scan sampling ( [START_REF] Altmann | Observational Study of Behavior: Sampling Methods[END_REF], 200 scans total). Two individuals were considered to be close when they were <1m apart. Based on these scans, we built a proximity matrix based on the number of scans where two individuals were observed close to each other, for each possible dyad. We carried out a social network analysis to obtain a Centrality score for each horse (Eigen vector) index (Jacobs and Petit 2011)). Dominance and centrality scores have been attributed to each individual (Table 1). For the group C used for the experiment 2, the dominance hierarchy was significantly linear (de Vries test for linearity, h'=0.89, P=0.04) and dominance scores and centrality scores were positively correlated (Spearman correlation test, r=1, P=0.003). Therefore, the horse (SER) with the highest David's score and Centrality index was chosen to be the unreliable initiator with the highest social status (HS), and the horse (TAK) with the lowest David's score and Centrality index was chosen to be the unreliable initiator with the lowest social status (LS). Horses ARM and RAT were chosen as control reliable initiators 1 and 2 due to their intermediate status.

Simulation study

A general probability to follow has been calculated from the first experiment and was the same for all initiators (0.9). This probability was considered to be degraded by the failures (i.e. the unrewarded trials) and we made the assumption that this decrease occurred as soon as the first failure occurred. The probability that an individual will not follow will then be related to the number of failures by the following relationship:

Where P S is the probability to follow an initiator. This probability decreases with the number of failures. P 0 the basic probability to follow an initiator , P m the maximal probability to follow a reliable initiator, E the number of failures (number of trials with no food) and S the threshold for tolerating a failure (depending on the social status in our assumption), n the sensitivity to the number of failures. The extreme is an all-or-none response of the potential followers: P 0 and S depend on the identity or the status of the initiator: a highly potentially reliable initiator shows a high value of P 0 and S . This formula allowed us to calculate the probability to follow each unreliable initiator for each trial as well as the probability to follow the reliable initiators. Finally, we obtained a threshold for tolerating failure for the two unreliable initiators.

Eventually, we conducted simulations on the number of followers according to the respective threshold we obtained for both unreliable initiators to confirm our experimental results.

Statistical analyses

Social status analyses (Davids score, linearity tests and Eigenvector centrality scores) have been performed using SOCPROG 2.4. Statistical analyses were performed using R software (R Core Team 2019). The level of statistical significance was set at P<0.05, and the level of tendency at P<0.10. In the text, we report median and interquartile ranges in the form ''M= (1st interquartile-3 rd interquartile)".

For experiment 1, inter-individual comparisons have not been statistically investigated due to a lack of inter-individual variability in the number of followers. For experiment 2, we used generalised linear model (GLM) with binomial distribution to test how the testing phase, individual's identity, and trial number influenced the proportion of the group members that followed the informed individual. We used car package [START_REF] Fox | An Companion to Applied Regression[END_REF] for this analysis. We started with the full model that included the testing phase, individual identity, interaction between these two terms, and trial number as predictor variables.

To perform model selection, we relied on the second-order Akaike's Information Criterion (AICc) and Relative Variable Importance (RVI) as was suggested by [START_REF] Burnham | Model Selection and Inference: A Practical Information-Theoretic Approach[END_REF] and [START_REF] Symonds | A brief guide to model selection, multimodel inference and model averaging in behavioural ecology using Akaike's information criterion[END_REF]. As best-fitting models, we considered the model with the lowest AICc and any model within two AICc units. We then used model.avg function of the MuMIn package [START_REF] Barton | MuMIn: Multi-Model Inference[END_REF] to calculate RVI for our predictor variables. Our bestfitting model contained variables with an RVI of at least 0.7 and it had the lowest AICc score.

It included testing phase and individual identity as predictor variables. However, the interaction term between these two variables and trial number were not retained. To compare performance of different individuals we performed post-hoc analysis on the individual identity variable in the GLM. We used emmeans function of the emmeans package [START_REF] Lenth | Effects of observational learning on acquisition of an operant response in horses[END_REF] to perform multiple pairwise comparisons with a Tukey correction.

Ethics statement

The experimental protocols followed EU Directive 2010/63/EU guidelines for animal experiments and were approved by the Ethical Committee (CREMEAS) under agreement number AL/01/10/07/11.

Results

Experiment 1

A total of 73 trials have been analysed (i.e. when the informed individual was the first to go directly to the food location) and resulted in 3 or more trials per initiator for a total of 15 initiators (Table 2). We found no variation in the number of followers according the initiator's identity since a large majority of the trials was followed by the whole group (Group A: 96.15%, Group B: 100%, Group C: 78.57 %). Three initiators failed to validate the criteria to be retained in the analysis (DLI in Group A, CHE in Group B and OTO in Group C).

Experiment 2

During the first six trials (5 trials of phase 1-rewarded -plus the 6 th trial when followers discovered the food was missing for the first time, see protocol details above), 4 or 5 followers always followed each one of the four initiators. During the 6 following trials (phase 2), the number of followers ranged from 5 to 1 for the reliable initiators (rewarded trials).

Concerning the unreliable initiators, this number ranged from 4 to 1 for the initiator with highest social status (extinction trials), and from 2 to 1 for the initiator with the lowest social status (extinction trials, Fig. 1). The best fitting model was explained by horse's identity and testing condition (Supplementary Table XXX). There was a significant effect of Phase 2: the proportion of group members that followed decreased during the unrewarded (second) phase of the test (GLM, N =4 8, β = -3.075, SE=0.525, Z = -5.86, P < 0.001). The proportion of group members that followed also depended on the identity of the initiator (Table 3).

Specifically, the ability of the two reliable initiators (ARM, Reliable 1 and RAT, Reliable 2) to recruit the followers was 7 and 10 times respectively greater than the ability of the unreliable initiator with lowest social status (TAK, Unreliable 2) (Ration value Reliable1/Unreliable2 =7, z = 3.6, P<0.0001; Ration value Reliable2/Unreliable2 = 10.1, z = 4.00, P < 0.0001, Table 2). On the contrary, the differences between the unreliable initiator with highest social status (SER, Unreliable 1) and the other individuals were not significant (Ration value Reliable1/Unreliable1 = 2.9, z = 2.00, P = 0.19; Ration value Reliable2/Unreliable1 = 4.1, z = 2.500, P = 0.06; Ration value Unreliable1/Unreliable2 = 2.4, z = 1.800, P = 0.25, Table 3). Finally, no difference was found between two reliable initiators (Ration value Reliable1/Reliable2 =0.7, z = -0.6, P = 0.93).

Simulation study

The probability to follow the unreliable initiator with highest social status (SER) was less affected by the discovery that food was no longer present than for the unreliable initiator with lowest social status (TAK) (Fig. 2).

In our case, after a first failure, for both unreliable initiators (regardless their social status), the mean number of followers decreases but then remains constant from trial to trial (linear regression between the number of followers as a function of the successive trials, null hypothesis the slope is equal to zero, P > 0.05). However, the mean number of followers per trial is greater for the initiator with highest social status than for the other unreliable initiator (respectively SER: 2.66 and TAK: 1.17). These results suggest that equation 2b is a good approximation of the individual response to an initiator, the two thresholds (number of failures) for decreasing the probability to the initiator (S, equation 2) are equal to 1 and confirm our assumption that an individual with a high social status continues to be trusted even if it provides false information. On the contrary, an individual with a low social status is poorly followed immediately after the discovery of the food absence. The probability that each potential follower (P 0 in equation 1) follows the initiator with a low social status is lower than the P 0 for the initiator with a high social status (respectively P 0 = 1.17 / 5 = 0.23 for TAK and P 0 = 2.66 / 5 = 0.5 for SER). This result also means that the success of the reliable initiators was not influenced by the failures experienced by group members in trials with unreliable initiators.

We performed 10000 simulations of 6 successive trials in which we applied the P 0 of each unreliable initiator, P 0 being the probability that each individual follows the initiator. We obtained that the proportions of followers for the unreliable initiator with highest social status (SER) and the unreliable initiator with lowest social status (TAK) are statistically different (P < 0.05) (Fig. 2).

Discussion

In the first experiment of this study, we provided optimal conditions of departure by giving the same value to all initiators in terms of food access for the whole group. As a consequence, we did not observe any difference between the different initiators in the amount of followership. Such positive influence of possessing right information on followers has already been found in pigeons [START_REF] Watts | Misinformed leaders lose influence over pigeon flocks[END_REF]) and horses [START_REF] Andrieu | Informed horses are influential in group movements, but they may avoid leading[END_REF]. This absence of inter-individual variability contrasts with what is observed in spontaneous collective movements, where initiators were found to be of unequal value for followers which join them less quickly or not at all (Leca, Gunst et al. 2003, Sueur andPetit 2008). It suggests that phenomena that are non-specific to an initiator identity are also at play in the followers' decisions and emphasises the importance of the pre-departure period. Indeed, we know that each group member can notify its motivation to move, negotiate or vote during this period and thus contributes to the collective decision at this early step or that group dispersion and activity are decisive for moving collectively [START_REF] Bourjade | Decision-Making in Przewalski Horses (Equus ferus przewalskii) is Driven by the Ecological Contexts of Collective Movements[END_REF][START_REF] Ramseyer | Decision making in group departures of sheep is a continuous process[END_REF][START_REF] Sueur | Sequence of quorums during collective decision making in macaques[END_REF], Sueur, Deneubourg et al. 2011[START_REF] Briard | Group behaviours and individual spatial sorting before departure predict the dynamics of collective movements in horses[END_REF].

Since in optimal experimental conditions each horse appears to be able to lead successfully, in the second experiment, we placed potential followers in a more challenging environment which mimicked natural cases where an initiator could lead the group to a bad location (in terms of food expectation). These conditions simulated the crucial impact of a leader on collective choices. With this experiment, we aimed to reveal the strength of the trust the followers put into a leader, by asking directly the followers. In such a context, followers successfully displayed flexibility in their willingness to follow a leader according to its reliability. Indeed, an immediate drop in the followership was observed within the unreliable initiator with the lowest social status, while followers kept following both reliable initiators. It demonstrates that horses are able to assess the reliability of an initiator. This could be mediated by associative learning mechanisms, especially since it involves the presence or absence of food reward. Such influence of recent experience on former choice on subsequent decision has been found in human and non-human primates [START_REF] Steelandt | Children Base Their Investment on Calculated Pay-Off[END_REF][START_REF] Pele | To bet or not to bet? Decision-making under risk in non-human primates[END_REF]. Being able to discriminate social information based on the reliability of their emitting source has also been shown in other social contexts, such as for alarm calls for instance (vervet monkeys: [START_REF] Cheney | Assessment of meaning and the detection of unreliable signals by vervet monkeys[END_REF], yellow-bellied marmots: Blumstein, Verneyre et al. 2004, carrion crows: Wascher, Hillemann et al. 2015). Such ability is undeniably crucial to optimise the benefit in the use of social information. Moreover, our results suggest an even more subtle phenomenon: not only the followers have been able to adapt their choice to follow -or not-based on the reliability of the initiator, but the identity of the unreliable initiator played a role in how fast this initiator lost their trust. Indeed, whereas the number of followers dropped drastically and immediately for the unreliable initiator with the lowest social status, this was not the case for the unreliable initiator with the highest social status where the decrease was delayed and more gradual. Our results, obtained on a limited number of individuals, were confirmed by our simulations which demonstrated that the proportions of followers for the unreliable initiator with highest social status differed from the one of the unreliable initiator with lowest social status. These calculated probabilities may represent a threshold for trusting a group mate, and this threshold may depend on the leader's social status since our two unreliable initiators occupied the two most extreme social positions within their group (i.e. the most dominant and central horse vs.

the most subordinate and peripheral one). The fact that the extinction pattern is delayed and more gradual with an unreliable initiator with the highest social status suggest that even if followers were able to detect its unreliability, as shown before, there was something left to maintain their motivation to follow and it strongly suggests an existing relationship between having a high social status and a leadership role. It is often assumed that animals with a high social status, especially high-ranked individuals (i.e. in dominance hierarchy), are more successful as a leader because they are usually older and more experienced, and should therefore possess accurate information about their environment [START_REF] Mccomb | Leadership in elephants: the adaptive value of age[END_REF]. In many species, age and hierarchical ranks are indeed correlated (e.g. in horses [START_REF] Sigurjónsdóttir | Social Relationships in a Group of Horses without a Mature Stallion[END_REF][START_REF] Briard | Personality and Affinities Play a Key Role in the Organisation of Collective Movements in a Group of Domestic Horses[END_REF]), and leaders are also often high-ranking and/or older group members (Sueur andPetit 2008, Sueur andPetit 2008). As a consequence, from observational studies, it is impossible to disentangle the intrinsic abilities that contribute to lead groups successfully. On the contrary, our experimental results suggest that groupmates' decision to follow depends on the social status of the initiator, and not age since neither centrality or David's scores followed the age order. Moreover, actual signs of unreliability did not prevent the group to keep following the high-status initiator, suggesting that a high level of leadership, once acquired, does not depend directly on how well-informed the leader is, or, at least, that a highly influential leader will benefit from a margin of errors that a low-status initiator would not benefit from. Altogether, it really emphasises that the influence of a leader's social status cannot be solely explained by the actual and/or current reliability of the information it possesses. We now need to explore whether highly influential individuals are also leaders in other domains like conflict management (Petit andThierry 1994, Widdig, Streich et al. 2006), group protection and defence [START_REF] Meunier | Participation in group defence: proximate factors affecting male behaviour in wild white-faced capuchins[END_REF][START_REF] Strandburg-Peshkin | Burrow usage patterns and decision-making in meerkat groups[END_REF] or observational learning [START_REF] Sage Frith | Mechanisms of Social Cognition[END_REF].

What component of social status explain highest trust in followers and what is their causal link remain open questions. It is important to notice that in our study, we considered both the affiliative and agonistic dimensions of the social status, that were correlated in our studied group, but it is not necessarily the case in horses [START_REF] Cameron | Social bonds between unrelated females increase reproductive success in feral horses[END_REF]. There is a general tendency to oversimplify the social position of a leader to its dominance status can lead to miss the point. [START_REF] Bonanni | Effect of affiliative and agonistic relationships on leadership behaviour in free-ranging dogs[END_REF] indeed demonstrated in free-ranging dogs that leadership is influenced by both affiliative and agonistic relationships. Affiliative relationships appear to be especially important for trust, as shown by chimpanzees that trust their friends more than their non-friends in a modified version of the trust game [START_REF] Berg | TRUST, RECIPROCITY, AND SOCIAL-HISTORY[END_REF][START_REF] Tooby | Friendship and the banker's paradox: Other pathways to the evolution of adaptations for altruism[END_REF][START_REF] Engelmann | Chimpanzees Trust Their Friends[END_REF]. In the same way, Fruteau et al's study suggests that social centrality is more determinant than dominance's rank to gain the trust of other group members [START_REF] Fruteau | Supply and demand determine the market value of food providers in wild vervet monkeys[END_REF].

Indeed, in their experiment, the authors found than subordinate vervet monkeys were more groomed after providing food access to group mates, whereas no change in troop hierarchical order was reported. These results also suggest that a feedback loop between social status and trust (and thus tolerance to error) should exist even though it is difficult to determine its direction. In order to understand this causal relationship between social status and trust, we will need to manipulate again the level of social credibility of leaders by increasing the one of lower social status individuals.

It is important to note that, in our study, we only conducted six tests for each initiator and we focused on a temporary loss of reliability in a specific food context. This protocol was indeed designed to prevent any habituation or counterstrategies from group members since they might have decided to search for food reward on their own. Such process could be seen during the last trials when followership started to be disrupted even with control initiators. However, even with those limitations, we believe such extinction paradigm should be tested further, in more animals and more groups. Studying the contrast in extinction patterns is a good way to start quantifying this part of trust that animals put into a conspecific and that cannot be solely explained by trial and error mechanisms.

Finally, the flexibility that followers displayed in our experiments also raises interesting questions for the comprehension of equine social cognition. Indeed, followers appeared to adapt adequately their responses to contexts and initiators identities, suggesting flexibility when deciding which individual to follow and when. Such flexibility evokes the concept of social learning strategies, the "flexible rules that specify or bias when or how individuals should use social information, under various circumstances, to meet functional goals" [START_REF] Kendal | Social Learning Strategies: Bridge-Building between Fields[END_REF]. Social learning strategies likely involve both associative and social learning [START_REF] Kendal | Social Learning Strategies: Bridge-Building between Fields[END_REF], and eventually sophisticated social cognition in some cases. The ability of horses to be selective and flexible in what, when, and from which individual they should get and use social information could explain why many authors struggled in demonstrating (observational) social learning in this species, despite the strong assumption they should possess this capacity (classic learning a simple operant: (Lindberg, [START_REF] Lenth | Effects of observational learning on acquisition of an operant response in horses[END_REF], discrimination: [START_REF] Baer | Observation effects on learning in horses[END_REF][START_REF] Baker | Observational learning in horses[END_REF][START_REF] Clarke | Effects of observational learning on food selection in horses[END_REF], instrumental: [START_REF] Ahrendt | The ability of horses to learn an instrumental task through social observation[END_REF], or detour spatial task: [START_REF] Rørvang | Horses fail to use social learning when solving spatial detour tasks[END_REF][START_REF] Burla | Human Demonstration Does Not Facilitate the Performance of Horses (Equus caballus) in a Spatial Problem-Solving Task[END_REF]. Interestingly, the only two studies that possibly showed premises of copying mechanisms suggest that a demonstrator's influence depends on its social status (Krueger andHeinze 2008, Schuetz, Farmer et al. 2017).

The absence of clear experimental evidence of copying abilities contrasts with the knowledge we have of the horses' non-social cognitive abilities (reviews: [START_REF] Hausberger | Mutual interactions between cognition and welfare: The horse as an animal model[END_REF])), and the existence of flexible and complex social phenomena that have been studied in more naturalistic conditions, such as in the context oflike in the present study -collective movement [START_REF] Briard | Personality and Affinities Play a Key Role in the Organisation of Collective Movements in a Group of Domestic Horses[END_REF][START_REF] Briard | How stallions influence the dynamic of collective movements in two groups of domestic horses, from departure to arrival[END_REF][START_REF] Gérard | How does the expressiveness of leaders affect followership in domestic horses (Equus ferus caballus)?[END_REF]) -or third-party interactions and reconciliation (domestic horses [START_REF] Cozzi | Post-conflict friendly reunion in a permanent group of horses (Equus caballus)[END_REF], feral horses: [START_REF] Schneider | Third-party interventions keep social partners from exchanging affiliative interactions with others[END_REF], and Przewalskii horses: [START_REF] Krueger | Context-dependent third-party intervention in agonistic encounters of male Przewalski horses[END_REF]). Therefore, our study highlights the importance of including the flexibility and selectivity factors when considering social phenomena, and that the study of leadership using experimental paradigms constitutes a great framework to explore this topic.

Table 1 . Individual characteristics

 1

	657
		ID	Group	Year of	Age	Breed	David's score Centrality index
				study	(yrs)
		CAN	A	2014	2	Dartmoor	-5.46	0.52
		DAY	A	2014	1	Shetland	-5.42	0.32
		DLI	A	2014	23	Dartmoor	13	0.48
		HAV	A	2014	19	Dartmoor	5	0.54
		NOR	A	2014	18	Shetland	-1.78	0.17
		TAK* A	2014	7	Shetland	-5.33	0.28
		ARM* B	2014	4	Dartmoor	-11.57	0.55
		CAL	B	2014	2	Dartmoor	-8.63	0.28
		CHE	B	2014	2	Dartmoor	6	0.51
		CYB	B	2014	2	Dartmoor	-2.71	0.36
		OTO* B	2014	12	Dartmoor	2.33	0.26
		SER* B	2014	8	Haflinger	14.57	0.41
		ARM* C	2015	5	Dartmoor	0.58	0.23
		BEL	C	2015	4	Shetland	-9.00	0.09
		OTO* C	2015	13	Dartmoor	4.29	0.67
		RAT	C	2015	10	Shetland	0.52	0.15
		SER* C	2015	9	Haflinger	15.00	0.68
		TAK* C	2015	8	Shetland	-15.00	0.07
	658	* studied in both 2014 and 2015
	659
	660

Table 2 . Number of followers per initiation according to initiator's identity for each

 2

Table 3 . Comparison of individual performances during the testing phase.

 3 For each combination of two initiators, the ration of the proportion of individuals that followed one individual over the proportion of individuals that followed another individual has been tested.

	Individual A / Individual B	ratio	SE	z.ratio	P Value
	Reliable1 / Reliable2	0.7	0.4	-0.600	0.93
	Reliable1 / Unreliable1	2.9	1.5	2.000	0.19
	Reliable1 / Unreliable2	7.0	3.8	3.600	<0.0001
	Reliable2 / Unreliable1	4.1	2.3	2.500	0.06
	Reliable2 / Unreliable2	10.1	5.8	4.000	<0.0001
	Unreliable1 / Unreliable2	2.4	1.2	1.800	0.25

Acknowledgements

We warmly thank Thierry Bourgeois, Nans Burgarella, Rose Delacroix, Océane Liehrmann, Luc Martin, Virginie Nierat, Edouard Paumier, Louise Prévost, Julie Renard, Pierre Uhlrich, and Lucile Villot for helping with the experimental task, Cristian Pasquaretta for his guidance in statistics, and Rachel Annan for proofreading the manuscript. The project was funded by the University of Strasbourg Institute for Advanced Study and the Fondation des Treilles (Young Researcher Prize) that we thank for their support.

Animal welfare note

The experimental protocols followed the EU Directive 2010/63/EU guidelines for animal experiments and were approved by the Ethical Committee (CREMEAS) under agreement number AL/01/10/07/11.


Romain Lhermerout
email: romain.lhermerout@univ-grenoble-alpes.fr

The Influence of Mechanical Deformations on Surface Force Measurements

Keywords: contact mechanics, surface interactions, surface force balance 15

Surface Force Balance (SFB) experiments have been performed in dry atmosphere and 1 across an ionic liquid, combining the analysis of the surface interactions and deformations, and 2 illustrate that the mechanical deformations of the surfaces have important consequences for the 3 force measurements. First, we find that the variation of the contact radius with the force across 4 the ionic liquid is well described only by the Derjaguin-Muller-Toporov (DMT) model, in contrast 5 with the usual consideration that SFB experiments are always in the Johnson-Kendall-Roberts 6 (JKR) regime. Secondly, we observe that mica does not only bend but can also experiences a 7 compression, of order 1 nm with 7 µm mica. We present a modified procedure to calibrate the mica 8 thickness in dry atmosphere, and we show that the structural forces measured across the ionic 9 liquid cannot be described by the usual exponentially decaying harmonic oscillation, but should 10 be considered as a convolution of the surface forces across the liquid and the mechanical response 11 of the confining solids. The measured structural force profile is fitted with a heuristic formulation 12 supposing that mica compression is dominant over liquid compression, and a scaling criterion is 13 proposed to distinguish situations where the solid deformation is negligible or dominant. 14

Introduction 16

Understanding the behavior of liquids in nano-confinement is crucial for a range 17 of applications including energy storage (electrolyte in contact with porous electrodes), 18 lubrication (lubricant in between rough sliding surfaces) and filtration (like sea water 19 through a membrane), as well as in biological systems (ion crossing the cell membrane in 20 a nano-channel, etc.). Very often, model experiments are performed with elementary sys-21 tems of simple geometry: two surfaces facing each other in force measurements, a single 22 channel in flow measurements. This allows for a simpler mathematical description of the 23 system, while putting aside the complex collective effects happening in the presence of 24 multiple asperities or pores. The ultimate goal is to understand the underlying physics 25 at a fundamental level, independently of a particular geometry. For example, the Surface 26 Force Balance/Apparatus (SFB/SFA), and the Atomic Force Microscope (AFM) are force 27 measurements techniques using surfaces of different shapes and radii of curvatures R: 28 crossed cylinders of radii ∼ 1 cm in SFB, sphere of radius ∼ 10 nm -10 µm and a plane 29 in AFM. The normal interaction force profiles F(D) obtained can, in certain conditions, 30 be directly compared by rescaling the measured force F by the radius of curvature R.

31

Indeed, Derjaguin showed that F/R is proportional to the energy density between 32 equivalent planar and parallel surfaces, provided that (i) D R, (ii) the interactions 33 involved are additive and (iii) the surfaces are not deformed [START_REF] Derjaguin | Untersuchungen über die Reibung und Adhäsion, IV[END_REF]2]. However, real solids here, we also show that the mechanical deformations of the solid surfaces strongly affect 73 the shape of the structural force profile, with the commonly used exponentially decaying 74 harmonic oscillation being convoluted with the compression of the mica. Finally, we [START_REF] Horn | Direct measurement of structural forces between two surfaces in a nonpolar liquid[END_REF] propose a heuristic formulation to describe such convoluted structural force profile when 76 the solid compression dominates the liquid compression, as well as a general scaling 77 criterion to determine for any surface force experiment its regime of convolution (i.e.

78

solid compression negligible or dominant compared to liquid compression).

Materials and Methods

80

The Surface Force Balance (SFB) has been explained in details in previous publi-

81
cations [2,[START_REF] Perkin | Forces Between Mica Surfaces, Prepared in Different Ways, Across Aqueous and Nonaqueous Liquids Confined to Molecularly Thin Films[END_REF][START_REF] Lhermerout | Nanoconfined Ionic Liquids: Disentangling Electrostatic and Viscous Forces[END_REF]. Here we briefly recall the principle of the instrument, illustrated in ) [START_REF] Israelachvili | Thin Film Studies Using Multiple-Beam Interferometry[END_REF]. We developed an automatic 106 procedure in order to extract the apical distance D and the contact radius a from this 107 profile. The procedure is described in details in the supplementary materials, together 108 with a brief presentation of the models of contact mechanics that will be used in the next 109 section to analyze the data. Finally, the apical distance D is measured with a precision 110 of 0.02 nm given by the standard deviation of the signal, and an accuracy of 1 nm due to 111 light disalignment when changing the contact spot [START_REF] Schwenzfeier | Optimizing multiple beam interferometry in the surface forces apparatus: Novel optics, reflection mode modeling, metal layer thicknesses, birefringence, and rotation of anisotropic layers[END_REF]. The contact radius a is measured 112 with a precision of 0.03 µm given by the standard deviation of the signal, and an accuracy 113 of 1 µm due to the uncertainty on the value of the radius of curvature R.

114

The top surface can be moved normally with a stepper motor (large displacement A parabolic fit at small scale close to the apex (in green) allows to measure the apical distance D, while a parabolic fit at large scale (in blue) is used to extract the contact radius a. In both cases, the FECO images were recorded with a black and white camera, then recolored using the calibration of the wavelength axis with a mercury lamp. The two particular cases shown here correspond to the points of maximum load reached in Figure 3. top surface, using a procedure that takes into account the presence of the viscous force 126 at large separations (detailed in [START_REF] Lhermerout | Nanoconfined Ionic Liquids: Disentangling Electrostatic and Viscous Forces[END_REF]). In comparison to previous studies we performed, 127 the normal spring used in the present measurements is about 20 times stiffer; this was 128 chosen in order to apply larger load (at fixed displacement range). Note that this choice 129 of a stiff spring increase the explored ranges of load and deformation as well as the slope 130 of the spring instability, which is convenient for the investigation of contact mechanics, 131 but does not limit the generality of this study. This comes with a price in terms of 132 sensitivity limit, ∼ 10 -2 mN, which doesn't allow to detect the anomalously long-range 133 electrostatic force that has been observed with concentrated electrolytes [START_REF] Gebbie | Long-range electrostatic screening in ionic liquids[END_REF][START_REF] Smith | The Electrostatic Screening Length in Concentrated Electrolytes Increases with Concentration[END_REF]. on the mica and glue thicknesses [START_REF] Horn | Measurement of the Deformation and Adhesion of Solids in Contact[END_REF]. Note that the curves measured on approach and 165 retraction are not superimposed, a phenomenon known as the adhesion hysteresis and 166 due to non-elastic processes at play. For this system, it is generally assumed that adhesion 167 hysteresis is mainly caused by viscoelasticity and plasticity in the glue layer [START_REF] Horn | Measurement of the Deformation and Adhesion of Solids in Contact[END_REF][START_REF] Mcguiggan | Contact mechanics of layered elastic materials: experiment and theory[END_REF].

168

The variation of the force F with the distance D is plotted in Figure 3 profile in dry atmosphere is not measured, but the surfaces are slowly approached until 218 they jump-in to contact and it is at this point that the "mica thickness" is calibrated. In 219 our experiment, the surfaces are already compressed by 1.4 nm just after the jump-in.

220

The usual procedure therefore leads to an underestimation of the mica thickness by case of an infinitely thin elastic layer between two rigid solids [START_REF] Reedy | Thin-coating contact mechanics with adhesion[END_REF][START_REF] Borodich | Adhesive contact problems for a thin elastic layer: Asymptotic analysis and the JKR theory[END_REF] (applicable for 272 a e mica ), but not for the intermediate case present here (a ∼ e mica ).

273

To summarize, the JKR model reasonably fits the variation of the contact radius 274 with the force -considering the whole mica/glue/glass system as an effective homo-275 geneous material-, but completely fails to describe the variation of the force with the 276 distance -because only the indentation of the mica is measured, which forms a layer that 277 cannot be considered as semi-infinite-. In order to simultaneously fit the a(F) and F(D) 278 relationships with a coherent set of parameters, complex models that explicitly include 279 the description of all the layers are needed. So far, only a few remarkable approaches 280 have attempted to quantitatively fit the a(F) relationship with such description, using 281 approximated analytical formulations [START_REF] Math | A general contact mechanical formulation of multilayered structures and its application to deconvolute thickness/mechanical properties of glue used in surface force apparatus[END_REF] or finite element methods [START_REF] Mcguiggan | Contact mechanics of layered elastic materials: experiment and theory[END_REF][START_REF] Sridhar | Adhesion mechanics of the surface force apparatus[END_REF]. In particular, 282 they described how the effective elastic modulus K depends on the properties of the 283 different layers, and showed that these multi-layers aspects have to be taken into account 284 to obtain a correct value of adhesion energy W from the jump-out force F s [START_REF] Israelachvili | On the adhesion force between deformable solids[END_REF]. Similar 285 approaches could be followed in the future to quantitatively fit the F(D) relationship 286 while taking into account the compression of mica, that was neglected so far. another repulsive wall at ∼ 0.5 nm (thereafter reported as layer i = 1); thereafter the 296 applied load is increased continuously and very large positive forces are reached. At 297 some point (F = 17.80 mN), the direction of the motion is reversed, the applied load is 298 decreased continuously and small negative forces are reached until the spring instability 299 at F s = -0.47 mN leads to a jump-out to large distances. In this subsection, we focus 300 on the contact mechanics of the system when the liquid is composed of a single layer of 301 ions (layer i = 1). The influence of the mechanical deformations on the structural force 302 profile will be detailed in the next subsection.

303

The variation of the contact radius a with the force F is plotted in Figure 3(d).

304

In contrast with the measurements across N 2 , the jump-out clearly happens at a zero In this subsection, we now focus on the detailed shape of the structural force profile.

380

Because of the spring instability, the surfaces experience a jump-in on approach every 381 time a layer is squeezed-out, an a jump-out on retraction from a given layer. Therefore

F(D) = F 0 exp - D -D 0 ζ cos 2π D -D 0 λ , ( 2
)
where the 4 fitting parameters are the period of the oscillations λ, the decay length ζ, and

406

The two curves exhibit a good linearity, and are fitted with the relations:

407 D max i = D max 0 + iλ max D min i = D min 0 + iλ min , (3)
where the slopes λ max , λ min represent the mean layer thickness and the intersects D max 0 , with M the molar mass 426 of the ionic liquid, ρ its bulk density and N A the Avogadro's number [START_REF] Horn | Double-layer and solvation forces measured in a molten salt and its mixtures with water[END_REF]), perhaps 427 suggesting a denser packing of ions in confinement than in the bulk. However, our 428 value is also smaller than the previous measurements performed with the same ionic 429 liquid, reporting a mean layer thickness of 0.80 ± 0.04 nm between two mica surfaces 430 with a SFB [START_REF] Smith | Switching the Structural Force in Ionic Liquid-Solvent Mixtures by Varying Composition[END_REF] and 0.79 nm between a mica surface and a Si 3 N 4 tip with an AFM [START_REF] Hayes | Pronounced Structure in Confined Aprotic Room-Temperature Ionic Liquids[END_REF].

431

A possible explanation for this difference is the inherent contribution from viscosity to 432 the force profile, in particular in vicinity to the jump-in and jump-out instability. For the 433 method using the positions of the minima, viscosity tends to move the point of minimum 434 force towards larger distances even more than adhesion is larger; for the method using 435 the jump-in distances, viscosity tends to reduce the jump-in distances. In both cases, 436 viscous effects possibly lead to an underestimation of the mean layer thickness. Previous 437 studies may be less affected by viscosity, as retractions were performed by slow steps in 438 the SFB study [START_REF] Smith | Switching the Structural Force in Ionic Liquid-Solvent Mixtures by Varying Composition[END_REF] (with similar radius of curvature), and the radius of curvature was six 439 orders of magnitude smaller in the AFM study [START_REF] Hayes | Pronounced Structure in Confined Aprotic Room-Temperature Ionic Liquids[END_REF] (with a velocity less than an order of 440 magnitude larger).

441

444          F max i = F max 0 exp -i λ max ζ max F min i = F min 0 exp -i λ min ζ min , ( 4
)
where the slopes give access to the ratios the envelope of the force profile with the horizontal axis. Such asymmetry cannot be due to the van der Waals contribution which is always attractive for symmetric systems.

450

It cannot be explained by the anomalously long-range electrostatic force that has been 451 observed with concentrated electrolytes [START_REF] Gebbie | Long-range electrostatic screening in ionic liquids[END_REF][START_REF] Smith | The Electrostatic Screening Length in Concentrated Electrolytes Increases with Concentration[END_REF], because exponentially fitting the mean

452 amplitude F max i + F min i /2
gives a decay of ∼ 1 nm and an amplitude of ∼ 4 mN, 453 respectively one order of magnitude smaller and two orders of magnitude larger than for 454 long-range electrostatic force reported for this system [START_REF] Lhermerout | Nanoconfined Ionic Liquids: Disentangling Electrostatic and Viscous Forces[END_REF][START_REF] Smith | The Electrostatic Screening Length in Concentrated Electrolytes Increases with Concentration[END_REF]. In fact, the exponentially 455 decaying harmonic oscillation given by equation 2 can be predicted theoretically in the 456 asymptotic limit of large distances (far-field term), and an additional (non-oscillating)

457
exponentially decay has been proposed as a correction at small distances (short-field 458 term, with 2 additional fitting parameters) [START_REF] Hoth | Force microscopy of layering and friction in an ionic liquid[END_REF][START_REF] Moazzami-Gudarzi | Interplay between Depletion and Double-Layer Forces Acting between Charged Particles in Solutions of Like-Charged Polyelectrolytes[END_REF][START_REF] Schön | A simple extension of the commonly used fitting equation for oscillatory structural forces in case of silica nanoparticle suspensions[END_REF]. This second term, that is intrinsic to that the next squeeze-out event would take place at a force F = 1.3 ± 0.6 mN, while we 486 do not observe any additional jump-in for a force up to F = 17.80 mN (see Figure 3(c)).

487 Thus, we think that the layer seen at the maximum load is indeed composed of a single 488 layer of cations, and we identify it as i = 1.

489

In the SFA literature, the structural force profile is usually fitted with an exponen-490 tially decaying harmonic oscillation (equation 2, with 4 fitting parameters), supposing

491
that there is no surface deformations. In general, such description seems reasonable 492 because the eye measurements produce more scattered points and/or the retraction 493 branches are not explored. However, in a few studies it has been pointed out that the 494 oscillation is not rigorously sinusoidal, notably in the first paper reporting a structural 495 force for a simple liquid [START_REF] Horn | Direct measurement of structural forces between two surfaces in a nonpolar liquid[END_REF], and even more clearly in a recent study with an extended 496 Surface Force Apparatus [START_REF] Zachariah | Stepwise collapse of highly overlapping electrical double layers[END_REF]. In Figure S3, we have plotted two exponentially decaying ] between mica surfaces when approaching or retracting the top surface with the piezoelectric tube at v = 0.5 nm/s, showing structuring with 5 distinguishable layers labeled by i. For clarity, only the full approach is shown (in red), together with retractions from layers i = 1 (in blue), i = 2 (in green), i = 3 (in orange), i = 4 (in purple), i = 5 (in yellow). The black lines are the fit with equation 5. Inset: proposed structure of alternating layers of cations and anions. (b) Distances D i measured at the points of maximum and minimum force (respectively in red and blue), as a function of the layer index i. Straight lines are the corresponding linear fits (equations 3). (c) Forces |F i | measured at the points of maximum and minimum force (respectively in red and blue), as a function of the layer index i, in log-lin representation. Straight lines are the corresponding exponential fits (equations 4). middle position between the surrounding maxima, and one value of D corresponds to a 504 single value of F. In the experiment, the stable branches are always convex, the minima considered as semi-infinite in these conditions [START_REF] Mcguiggan | Contact mechanics of layered elastic materials: experiment and theory[END_REF][START_REF] Sridhar | Adhesion mechanics of the surface force apparatus[END_REF]. At larger loads, the system enters 530 in the regime described in the two previous subsections: the finite size of the mica layers 531 cuts off the range of the elastic deformations, leading to an apparent stiffening of the 532 solids compared to their bulk counterparts. To summarize, in contrast with previous 533 considerations that neglected mica compression [START_REF] Christenson | Surface Deformations in Direct Force Measurements[END_REF], we find for our system that the 534 surface deformations have a strong influence on the force profile even at low loads.

535

In particular, we expect this effect to be more important and independent of the mica 536 thickness at low loads; and smaller and reduced for thinner mica at large loads.

537

For our specific ionic liquid/mica system measured with SFB, we have shown that

572          F osci (D) = -2πRW 0 exp - D -D 0 ζ cos 2π D -D 0 λ F K (D) = KR 1/2 D 3/2 F k (D) = k[D -(D(t = 0) -vt)] . (6)
The pure liquid response is measured without spring instability if the gradient of the

         N k = dF osci dD dF k dD ∼ 4π 2 R k W 0 λ N K = dF osci dD dF K dD ∼ 8π 2 3 R 1/2 K W 0 λ 3/2 . ( 7
)
The different parameters have analogous roles, even if the exact exponents and numerical

34 are

 34 not perfectly rigid, and significant deformations of the surfaces can occur depending 35 on the strength and range of the interaction. For example, when measuring friction 36 in the boundary lubrication regime and under applied load, a strong repulsion across 37 the molecularly-thick boundary film typically leads to a substantial flattening of the 38 surfaces [3]. Suchl deformations are not limited to situations of nanometric proximity 39 of the confining solids, as they can be mediated by hydrodynamic interaction across

Figure 1 .

 1 Figure 1. (a) Chemical structure and sizes of [C 4 C 1 Pyrr][NTf 2 ]. Ion sizes are estimated from geometry, bond lengths and covalent radii, associated with the most stable configuration found by energy minimisation (Chem3D 16.0, PerkinElmer Informatics). (b) Schematic of the SFB experiment to measure the surface interactions and to characterize in-situ the geometry, when a liquid is confined between two mica surfaces. (c) Composition of the layers forming each solid surface, with associated thickness e, Young's modulus E, Poisson's ratio ν and elastic modulus K = 2

79

 79

Figure 1 (

 1 Figure 1(b), and the details particular to the present experiments.

83

  Muscovite mica is cleaved to produce atomically-smooth facets of micrometric thick-84 ness and millimetric extension, that are backsilvered and glued onto glass (fused silica) 85 cylindrical (radius R ∼ 1cm) lenses with an epoxy resin (EPON 1004, Shell Chemicals). 86 Two surfaces are made with mica of the same thickness, form a stack of different layers 87 (represented in Figure 1(c), together with their thicknesses, Young's moduli, Poisson's ra-88 tios and elastic moduli), and are arranged in a cross-cylinder geometry. First, calibrations 89 are done in a dry atmosphere, which is achieved by inserting P 2 O 5 , phosphorus pentox-90 ide (Sigma-Aldrich, 99%), in the chamber and purging the chamber with N 2 , nitrogen, 91 during about one hour prior to the calibrations. Secondly, measurements are performed 92 with an ionic liquid, because these liquids have proved to resist squeeze-out even under 93 very large loads [21], a regime in which significant mechanical deformations of the sur-94 faces are expected. The liquid used is [C 4 C 1 Pyrr][NTf 2 ], 1-butyl-1-methylpyrrolidinium 95 bis[(trifluoromethane)sulfonyl]imide (Iolitec, 99%), which chemical structure and sizes 96 are indicated in Figure 1(a) (molar mass M = 422.41 g/mol, density ρ = 1.405g/mL, 97 refractive index n = 1.422 and dynamic viscosity η = 74 mPa.s at 25 • C [22]). It is dried 98 in a Schlenck line at 60 • C and 5 × 10 -3 mbar for ∼ 10 hours and inserted in the chamber 99 just after, which contains P 2 O 5 and is purged again with N 2 during about one hour prior 100 to the measurements. 101 White light is passed through the confined medium, interferes in this optical res-102 onator, and is then directed towards a spectrometer and collected by a CCD camera 103 (QImaging Retiga R6, resolution 2688 × 2200 px 2 ). The analysis of the Fringes of Equal 104 Chromatic Order (FECO, shown in Figures 2(a), 2(c)) then allows to calculate the profile 105 of the distance between the surfaces (Figures 2(b), 2(d))

Figure 2 .

 2 Figure 2. (a) Picture of the FECO when the two solid surfaces are in contact across N 2 , observed in (wavelength λ)-(lateral distance x) space. (b) Corresponding profile of the distance z between the surfaces along the lateral coordinate x (in red). A parabolic fit at small scale close to the apex (in green) allows to measure the apical distance D, while a fit with the JKR profile (equation 8 of the supplementary materials) at all measured scales (in blue) is used to extract the contact radius a. (c) Picture of the FECO when the two solid surfaces are in contact across [C 4 C 1 Pyrr][NTf 2 ], observed in (wavelength λ)-(lateral distance x) space. (d) Corresponding profile of the distance z between the surfaces along the lateral coordinate x (in red).A parabolic fit at small scale close to the apex (in green) allows to measure the apical distance D, while a parabolic fit at large scale (in blue) is used to extract the contact radius a. In both cases, the FECO images were recorded with a black and white camera, then recolored using the calibration of the wavelength axis with a mercury lamp. The two particular cases shown here correspond to the points of maximum load reached in Figure3.

134 3 . 3 . 1 .

 331 Results and Discussion135 Calibrations in dry atmosphere 136 In Figures3(a) and 3(b) are shown the force F, the distance D and the contact 137 radius a measured when approaching (in red) then retracting (in blue) the top surface 138 with the stepper motor at 13.2 nm/s across N 2 . Initially separated by N 2 , the surfaces are 139 brought closer until they experience a strong van der Waals attraction, which together 140 with the spring instability causes a jump-in to mica-mica contact and a slightly negative 141 force (F = -0.20 mN); thereafter the applied load is increased continuously and very 142 large positive forces are reached. At some point (F = 27.91 mN), the direction of the 143 motion is reversed, the applied load is decreased continuously and large negative forces are reached until the spring instability at F s = -5.71 mN leads to a jump-out to large 145 distances. 146 The variation of the contact radius a with the force F, plotted in Figure 3(b), is totally 147 consistent with the work of Horn, Israelachvili and Fribac [6]. Clearly, the jump-out 148 happens at a non-zero value for the contact radius (a s = 10.23 µm), which is a typical 149 feature of the JKR model (see the corresponding FECO image and separation profile 150 in Figure S1(a) and Figure S1(b)). The measured a(F) relationship is compared to the 151 JKR prediction (equations 4 of the supplementary materials) in the following manner. 152 First, the force F s = -5.71 mN reached just before the jump-out is used to compute the 153 adhesion energy W = -2F s 3πR = 131.79 mN/m. Then, a fitting procedure provides the 154 elastic modulus K = 47.0 ± 0.

202

  which is not sensitive to the refractive index considered. Consequently, the variations of 203 the distance D observed here have to be interpreted as changes of N 2 thickness when 204 the surfaces are out of contact, and changes of mica thickness when the surfaces are in 205 contact. 206 In these conditions, an important question is how can we define the mechanical 207 origin, i.e. the point at which D = 0? A sensible choice is to select the point where the 208 indentation of the surfaces is zero, i.e. δ = 0. Following the JKR model, δ is positive 209 at the maximum load (compression) and negative at the jump-out (dilatation). The 210 point of zero indentation is located in between, and does not correspond to the point 211 of zero force, i.e F = 0. Considering that the measured a(F) relationship is reasonably 212 fitted by the JKR model, we have taken the point δ = 0 from the fit as the mechanical 213 zero for the measured D (as indicated in Figures 3(a) and 3(b)), corresponding to a 214 mica thickness e mica,0 = 7.431 µm. Such definition of mechanical origin then leads to a 215 distance D that can be negative when the mica layers are compressed. This calibration 216 procedure is significantly different from what is done usually. In general, the force 217

221∼ 1 "

 1 nm for this ∼ 7 µm-thick mica, and an equivalent outward shift of the force profile 222 measured after injecting the liquid (adhesion is generally much smaller across a liquid 223 than dry atmosphere). If the mica is thinner or if the spring is stiffer, the shift is expected 224 to decrease and there is less ambiguity on the definition of the mechanical origin[START_REF] Israelachvili | Measurement of forces between two mica surfaces in aqueous electrolyte solutions in the range 0-100 nm[END_REF].225But in the general case this effect is present, and taking it into account is particularly 226 important when investigating aspects at the molecular scale. In the following, we present 227 three examples of situations where this may matter.228•Since the conception of the SFA, negative distances from -0.2 nm to -1.3 nm have229been reported when two mica surfaces separated by water jump-in to contact[11, 230 28,38-42]. This is due to the washing of gas molecules and organic contaminants 231 (carbon compounds) that are spontaneously deposited on the mica surfaces in 232 air[START_REF] Poppa | The surface composition of Mica substrates[END_REF], and to the dissolution in water of the potassium ions initially present 233 on the mica surfaces. As adhesion is typically 10 times smaller in water than 234 in dry atmosphere, the mica is expected to be less compressed after the jump-in 235 across water than during the calibration after the jump-in across dry atmosphere.236Because compression of mica was not considered in these studies, the thickness 237 of the contaminant layer is maybe underestimated, albeit not by more than a few 238 angstroms given the much thinner mica used. Nevertheless, the dependence of this 239 effect on the mica thickness and spring constant may explain -at least in part-the 240 strong variability on the reported values. 241 • In the case of molecular liquid giving rise to a structural force profile under confine-242 ment, a good accuracy on the distance D is needed in order to identify the absolute 243 number of ordered layers composing the film (as illustrated in subsection titled 244 Influence of surface deformations on structural force profile"). 245 • Dynamic measurements can be performed with the SFA to determine the slip 246 length associated to a flow of liquid in vicinity of a solid surface. By definition, the 247 slip length is the distance between the hydrodynamic origin and the mechanical 248 origin. Being able to measure nanometric slip lengths therefore requires to have a 249 sub-nanometric resolution on the mechanical zero [20,44-48]. 250 Finally, one can ask whether the JKR model can also describe the F(D) relationship. 251 If we use equations 4 of the supplementary materials and a reference D ref = δ + D = 0 (i.e. the origin D = 0 corresponds to δ = 0), with the values W = 131.79 mN/m 253 and K = 47.0 GPa coming from the fit of the a(F) relationship, the model does not 254 fit at all and predicts an indentation that varies much more than in the experiment 255 (by ∼ 40 nm instead of ∼ 5 nm in the explored range of force, see green curve in Figure 256 3(a)). Qualitatively, this is because the effective parameters coming from the fit of the a(F) 257 relationship correspond to the indentation of the whole mica/glue/glass system, while 258 here we measure the indentation of the mica only. If we approximate the different layers 259 composing the solids as springs in series, we expect that the ratio of the indentation of 260 mica relative to the indentation of glue is of the order of the ratio of the Young's modulus 261 of mica relative to the Young's modulus of glue, i.e. 3.4 GPa/62 GPa = 5.5%, which is 262 indeed of the same order of magnitude than the measured ratio of 5 nm/40 nm = 12%. 263 If we now relax the parameter K to fit the F(D) relationship (black curve in Figure 264 3(a)), it does not fit the a(F) relationship (for the same reason given just before, see black 265 curve in Figure 3(b)) and it provides a value K = 900 ± 200 GPa one order of magnitude 266 larger than the elastic modulus of mica. This is because we probe a contact zone of 267 size a ∼ e mica , while the validity of the JKR model would require a e mica to consider 268 the mica as a semi-infinite solid. Qualitatively, the finite size of the mica layers cuts 269 off the range of the elastic deformations, leading to an apparent stiffening of the solids 270 compared to their bulk counterparts. Some analytical formulations exist for the opposite 271

287 3 . 2 .

 32 Contact mechanics across an ionic liquid 288 In Figures 3(c) and 3(d) are shown the force F, the distance D and the contact 289 radius a measured when approaching (in red) then retracting (in blue) the top surface 290 with the stepper motor at 10.5 nm/s across [C 4 C 1 Pyrr][NTf 2 ]. When the ionic liquid is 291 confined at the nanoscale, a structural force profile is observed, due to the organization 292 of the ions in ordered layers. Initially far away, the surfaces are brought closer until 293 they experience a repulsive wall at D ∼ 1.5 nm (thereafter reported as layer i = 2) for 294 a load up to F = 0.39 mN, then a layer is squeezed-out and the surfaces jump-in to 295

Figure 3 .2K 1 / 3 = 6 .Fribac [ 6 ] 344 • 1 ) 3 . 3 .

 31366344133 Figure 3. (a) Force F as a function of distance D and (b) contact radius a as a function of force F when approaching (in red) and retracting (in blue) the top surface across N 2 with the stepper motor at v = 13.2 nm/s. The green curve is a JKR fit of a(F) (equations 4 of the supplementary materials) with K = 47.0 ± 0.5 GPa, W = 131.79 mN/m and D ref = δ + D = 0; the black curve is a JKR fit of F(δ) with K = 900 ± 200 GPa, W = 131.79 mN/m and D ref = δ + D = 0. (c) Force F as a function of distance D and (d) contact radius a as a function of force F when approaching (in red) and retracting (in blue) the top surface across [C 4 C 1 Pyrr][NTf 2 ] with the stepper motor at v = 10.5 nm/s. The green curve is a DMT fit of a(F) (equations 3 of the supplementary materials) with K = 16.7 ± 0.5 GPa, W = 8.18 mN/m and D ref = δ + D = 1.4 nm; the black curve is a DMT fit of F(δ) with K = 600 ± 200 GPa, W = 8.18 mN/m and D ref = δ + D = 1.4 ± 0.4 nm.

382

  FigureS2shows the force profiles measured when approaching the top surface up to a 384

401 403 F

 403 the position D 0 and amplitude F 0 of the first layer. To characterize the structural force 402 profile, we have measured for each layer i the distances D max i , D min i and the forces F max i , min i at the points of maximum and minimum force (i.e. respectively just before the 404 jump-in and jump-out, as indicated in Figure 4(a) for i = 2).

Figure 4 (

 4 Figure 4(b) shows the variations of the distances D max i , D min i with the layer index i.

  position of the extrapolated layer i = 0 (fitted values indicated 409 in the figure). The mean layer thickness obtained from the maxima λ max = 1.0 ± 0.3 nm 410 is significantly larger than the mean layer thickness obtained from the minima λ min = 411 0.64 ± 0.05 nm. This observation can be interpreted as a consequence of the mica 412 compression. According to the DMT model for a given layer, the indentation of the 413 solids is zero at the minimum force (jump-out point) and continuously increases up to 414 the maximum force (jump-in point). When going from layer i = 1 to i = 5, the range 415 of force explored decreases (because of repulsive maxima and adhesive minima that 416 come closer to F = 0), as well as the amplitude of mica compression. The variation 417 of D max i with i thus includes a systematic decrease of the mica compression, leading to 418 an overestimation of the mean layer thickness. On the contrary, the variation of D min i 419 with i does not include any influence of the mica compression, and provides the true 420 mean layer thickness 0.64 ± 0.05 nm. Another method to determine the mean layer 421 thickness consists in measuring the average jump-in distance, supposing an unchanged 422 mica compression and a fast viscous relaxation during the squeeze-out events. This 423 provides a consistent value of 0.64 ± 0.01 nm, confirming our interpretation of the effect 424 of the mica compression on the mean layer thickness. Interestingly, this value is smaller 425 than the mean ion pair diameter of 0.79 nm (given by M

Figure 4 (

 4 Figure 4(c) shows the variations of the forces F max i

  λ max ζ max , λ min ζ min of the period of the oscillation on 445 the decay length, and the intersects correspond to the amplitudes F max 0 , F min 0 of the 446 extrapolated layer i = 0 (fitted values indicated in the figure). The curves obtained with 447 the maxima and the minima are clearly distinct, as a consequence of the asymmetry of 448

459 1 = 2 . 2 ±

 122 the liquid for infinitely stiff surfaces, could contribute to the asymmetry of the measured 460 force profile. The significant deformations of the surfaces could also contribute to the 461 asymmetry, in the following manner. According to the DMT model for a given layer, 462 the surfaces are not deformed at the minimum force (jump-out point), and flattened 463 at the maximum force (jump-in point). Compared to a situation without deformation 464 of the solid surfaces, the squeezing-out of the liquid layer intuitively requires a larger 465 force, leading to a maximum that is larger than the minimum in absolute value. When 466 going from layer i = 1 to i = 5, the range of force explored decreases, as well as the 467 amplitude of flattening, and the points of maximum and minimum forces are more and 468 more symmetric around the horizontal axis.469An important aspect to interpret structural force profiles is to identify the composi-470 tion of the layers. As the period is similar to the mean ion pair diameter, it is usually 471 considered that one squeeze-out event corresponds to the squeeze-out of an electroneu-472 tral "slab" of one cation layer and one anion layer. In the case of negatively charged 473 surfaces, the first layer (i = 1) is then assumed to be composed of a monolayer of cations 474 (as sketched in inset of Figure 4(a)). Direct solid-solid contact is never reached, because 475 of the strong electrostatic attraction between the cations and the negatively charged 476 surfaces. For our experiment, one can ask whether we really reach this single layer of 477 cations within the explored range of loads. The position of the first layer at the point 478 of minimum force (including no indentation of the surfaces) is D min 1.0 nm, a 479 bit larger than the cation sizes (given in Figure 1(a)). However, the measurement of the 480 absolute distance D depends on many delicate steps (alignment of the optics, calibration 481 of the mica thickness in dry atmosphere, choice of a particular run to shift the force 482 profiles laterally), and deducing the thickness of the monolayer from the cation sizes 483 requires to know their conformations, making accurate comparisons difficult. If we 484 extrapolate the exponential fit of F max i to the layer i = 1 (see Figure 4(c)), it predicts 485

497 harmonic oscillation (equation 2 )Figure 4 .

 24 Figure 4. (a) Force profile measured with [C 4 C 1 Pyrr][NTf 2] between mica surfaces when approaching or retracting the top surface with the piezoelectric tube at v = 0.5 nm/s, showing structuring with 5 distinguishable layers labeled by i. For clarity, only the full approach is shown (in red), together with retractions from layers i = 1 (in blue), i = 2 (in green), i = 3 (in orange), i = 4 (in purple), i = 5 (in yellow). The black lines are the fit with equation 5. Inset: proposed structure of alternating layers of cations and anions. (b) Distances D i measured at the points of maximum and minimum force (respectively in red and blue), as a function of the layer index i. Straight lines are the corresponding linear fits (equations 3). (c) Forces |F i | measured at the points of maximum and minimum force (respectively in red and blue), as a function of the layer index i, in log-lin representation. Straight lines are the corresponding exponential fits (equations 4).

505++ iλ min -D 3 / 2 -

 32 and maxima are not regularly spaced, and one value of D can correspond to several 506 values of F (we can have D min i > D max i+1 ). So it is clear that an exponentially decaying 507 harmonic function is not appropriate to describe the detailed shape of the measured 508 structural force profile. We propose a heuristic description, assuming on the contrary 509 that the mica compression dominates the liquid compression, i.e. that the change of D 510 within each layer comes only from the indentation of mica (elastic modulus K). For each 511 layer, we suppose that the mica indentation is given by the DMT model (equations 3 512 of the supplementary materials) with a reference D i,ref = δ i + D = D min 0 iλ min and 513 an adhesion W i = -F min 0 2πR exp -i λ min ζ min = W 0 exp -i λ min ζ min , up to a maximum force 514 F max 0 exp -i λ max ζ max . Finally, we obtain the following expression: 515 F i (D) =KR 1/2 D min 0 2πRW 0 exp -i λ min ζ min for D ≤ D min 0 + iλ min and F i ≤ F liquid controls the positions (distance at the point of minimum force) 516 and the strengths (forces at the points of minimum and maximum distances) of the 517 layers, while the mica controls the shape of the profile within the layers. Excluding 518 the radius of curvature R that is measured independently, the force profile is described 519 by 7 fitting parameters, which is the exact number of parameters required to describe 520 an asymmetric structural force profile with deformable surfaces characterized by a 521 single elastic modulus. To compare with our measurements, we have used the values 522 D min 0 , λ min coming from the linear fit of D min i , the values F min 0 , ζ min coming from the 523 exponential fit of F min i , the values F max 0 , ζ max coming from the exponential fit of F max i , 524 and we have fitted the single remaining parameter K. The fit shown in Figure 4(a) is very 525 good at low loads, with K = 40 ± 5 GPa remarkably close to the elastic modulus of bulk 526 mica (value given in Figure 1(c)). This is because a contact zone of size a 3 µm e mica 527 is probed when the force is close enough to the adhesion minimum, i.e. |F i -2πRW i | 528 0.1 mN. The elastic deformations are affecting only the top mica layers, which can be 529

538

  the detailed shape of the structural force profile is strongly affected by the mechanical 539 deformations of the surfaces, with a mica compression that is dominant compared to 540 the ionic liquid compression. A convolution of the surface forces across the liquid and 541 the mechanical response of the confining solids may be present for other solid/liquid 542 systems and force measurement techniques. In a typical AFM experiment (see for 543 example[START_REF] Hoth | Force microscopy of layering and friction in an ionic liquid[END_REF]), the deflection ∆ of a cantilever is measured as a function of the approach 544 position z imposed to the base of this cantilever. The force is deduced by multiplying the 545 deflection by the calibrated spring constant, and the distance between the tip and the 546 substrate is calculated as the difference between the cantilever deflection and the linear fit 547 of the ∆(z) relationship when the surfaces are in contact. In fact, this standard procedure 548 supposes ideal surfaces that are infinitely stiff in the fitting region. In practice real solids 549 are compliant, and this method leads to subtracting the indentation of the surfaces, 550 linearized in the fitting region. As the relationship between the force and the indentation is generally not linear, the calculated distance is not exactly equal to the distance between 552 the tip and the substrate. In addition, the amplitude of the force depends on the geometry 553 of the surfaces, and is therefore affected by their deformation. For these two reasons, 554 the mechanical deformations of the confining surfaces are expected to influence the 555 measured force profile also in the case of AFM. Irrespective of the particular system or 556 technique used, it is very important to know the degree of convolution, i.e. whether the 557 solids compression is negligible or dominant compared to the liquid compression, in 558 order to interpret properly the structural force profile regarding the compressibility of 559 the liquid layers[START_REF] Jurado | Irreversible structural change of a dry ionic liquid under nanoconfinement[END_REF][START_REF] Garcia | Nanomechanics of ionic liquids at dielectric and metallic interfaces[END_REF][START_REF] Horn | Direct measurement of structural forces between two surfaces in a nonpolar liquid[END_REF][START_REF] Smith | Monolayer and bilayer structures in ionic liquids and their mixtures confined to nano-films[END_REF][START_REF] Griffin | Direct measurements of ionic liquid layering at a single mica-liquid interface and in nano-films between two mica-liquid interfaces[END_REF][START_REF] Cheng | Effect of Concentration on the Interfacial and Bulk Structure of Ionic Liquids in Aqueous Solution[END_REF][START_REF] Lhermerout | Are Buckminsterfullerenes Molecular Ball Bearings?[END_REF]. Indeed, this question of the "elasticity" of a thin liquid 560 film is connected to a strong debate in the community, to understand how a liquid can 561 exhibit a solid-like behaviour in nanoconfinement[START_REF] Klein | Confinement-Induced Phase Transitions in Simple Liquids[END_REF][START_REF] Khan | Dynamic Solidification in Nanoconfined Water Films[END_REF][START_REF] Seddon | Conservative and Dissipative Interactions of Ionic Liquids in Nanoconfinement[END_REF][START_REF] Khan | Young's modulus of nanoconfined liquids?[END_REF][START_REF] Comtet | Nanoscale capillary freezing of ionic liquids confined between metallic interfaces and the role of electronic screening[END_REF]. Our study suggests that the 562 finite slope of the structural force profile in each layer is not necessarily due to a change 563 with load of packing fraction or structure of the molecules in the structured film, but can 564 be attributed, in some cases and at least in part, to the indentation of the confining solids.565For these reasons we propose a general criterion to distinguish the two opposite regimes 566 of convolution, for a generic system exhibiting a structural force profile. We consider the 567 mechanical response of a liquid confined between infinitely stiff solid surfaces (taken 568 as an exponentially decaying harmonic oscillation F osci (D)), the mechanical response 569 of two deformable solid surfaces in direct contact (assumed to be a simple Hertz force 570 F K (D)), the two responses being measured with an external spring (imposing a restoring 571 spring force F k (D)):

616

  By simultaneously measuring interactions and characterizing in-situ the geometry 617 of the contact, we have shown that the mechanical deformations of the confining solids 618 can have a strong influence on surface force measurements. Although this paper focus 619 on the analysis of specific SFB experiments with dry atmosphere and an ionic liquid, we 620 think some of the conclusions listed below may be of general interest for the community 621 of surface force measurements. 622 • SFA experiments are not always in the JKR regime but can be in the DMT regime, 623 typically for situations of moderate adhesion over a range of a few nanometers, 624 as we have seen in the case of the ionic liquid. Using the correct model of contact 625 mechanics is crucial, notably for quantitative investigations of adhesion or friction. 626 The two regimes of contact are usually distinguished by calculating the value of the 627 Maugis parameter from an estimate of the range of the attractive forces; it is in fact 628 more accurate to look at the value of the contact radius before jump-out. 629 • In classical SFA experiments using mica sheets glued on glass lenses, the mica 630 does not only bend but can also experience a compression, that we observed for 631 relatively thick mica.

632 •

 632 This compression has to be taken into account for a proper calibration of the 633 undeformed mica thickness in dry atmosphere; for this purpose we presented a 634 method based on the fitting of the relation between the contact radius and the 635 force with the JKR model. The usual procedure, that consists in taking the jump-in 636 point as a reference, can lead to an underestimation of the mica thickness and an 637 equivalent outward shift of the force profile measured after injecting the liquid. We 638 found that this effect amounts to ∼ 1 nm for a ∼ 7 µm-thick mica, and we expect it 639 to decrease with the mica thickness. 640 • For any system showing a structural force profile with SFA or AFM, the exponen-641 tially decaying harmonic oscillation due to local variations of liquid density may 642 be convoluted with the mechanical response of the confining solids. A correct 643 interpretation of the detailed shape of the structural force profile is necessary to 644 understand the behaviour of liquids in nanoconfinement.

  [START_REF] Wang | Out-of-Contact Elastohydrodynamic Deformation due to Lubrication Forces[END_REF] GPa. As the range of a van der Waals attractive force

	155
	156	across a gas is d ∼ 0.3 nm [6], a Maugis parameter Ma ∼ 19	1 is obtained (using
	157	equation 6 of the supplementary materials), confirming the suitability of the JKR model
	158	for this system [27]. The data are reasonably fitted by the model (see the curve in green),
	159	and the extracted W and K are consistent with the values reported in previous studies
	160	for similar systems [6,7]. As the solids used in this experiment are complex stack of
	161	layers associated with different thicknesses, Young's moduli and Poisson's ratios (values
	162	from the literature given in Figure 1(c)), the value of K extracted from the fit should be
	163	interpreted as an effective quantity. Therefore, it is expected that this value is between
	164	the Young's modulus of mica or glass and the Young's modulus of the glue, and depends

  single mica layer of variable thickness 2e mica = 2e mica,0 -2δe mica (δe mica

	169	(a). We observe
	170	a clear change of D when the surfaces are in contact, of ∼ 5 nm between the points of
	171	maximum load and of jump-out, or ∼ 2 nm between jump-in and jump-out. Potentially,
	172	imperfections of the double cantilever spring may induce a tiny rotation of the solids
	173	when varying the load, leading to a progressive dealignment of the light and so to a
	174	shift of the fringe positions and of the extracted distance. However, we ruled out this
	175	possible artefact by checking the tuning of the optics at regular intervals during the
	176	loading-unloading cycle. Note that the curves measured on approach and retraction are
	177	not perfectly superimposed, mainly because of mechanical imperfections of the set-up,
	178	i.e. non-linearity of the stepper motor and long-term thermal drifts, leading to errors in
	179	the determination of the force. In addition, we have made sure that this phenomenon
	180	is not due to a potential contamination of a particular experiment, by systematically
	181	observing that mica undergoes a significant indentation in many separate experiments
	182	with different mica sheets. As the mica is the single material separating the two silver
	183	mirrors, such variation can only be explained by a compression of the two mica layers
	184	in contact. In early SFA/SFB studies, FECO were observed directly by eye or recorded
	185	with analogical photography, that is why very thin mica was used (typically of 2 µm
	186	thickness) to maximize the resolution of the measurement, and it was estimated that the
	187	mica experiences only negligible compression, of order ∼ 0.1 nm [6,28,29]. However
	188	in recent years, as high resolution CCD cameras have been used in combination with
	189	fast spectral correlation and multilayer matrix method to analyze the FECO [30], there
	190	has been a tendency towards working with thicker mica. One expects in this case a
	191	significant compression of mica, which indeed has been repeatedly mentioned in studies
	192	using a so-called extended Surface Force Apparatus [31-37], but not studied specifically.
		Such compression of the mica spacer layers raises a technical difficulty for analyzing

193 the FECO. Like what is done usually, we have indeed supposed that the interferometer 194 is composed of two mica layers of same constant thickness e mica,0 separating a vacuum 195 layer of thickness D (with known and constant refractive indexes), which is clearly wrong 196 here when the surfaces are in contact. To be more rigorous, we have also re-analyzed 197 the FECO when the surfaces are in contact, now supposing that the interferometer is composed of a 199 defined positive for compression and negative for dilatation). It turns out that the two 200 methods provide the same amplitude of compression; qualitatively this is because in 201 both cases we are looking at a small wavelength variation around a given reference point,

  For this SFB study with an ionic liquid (R ∼ 1 cm, k ∼ 3000 N/m, K ∼ 50 GPa, 587 W 0 ∼ 1 mN/m, λ ∼ 0.6 nm), we get N k ∼ 2 • 10 2 and N K ∼ 4, in agreement 50 GPa, W 0 ∼ 50 mN/m, λ ∼ 0.8 nm), we get N k ∼ 5 • 10 2 and N K ∼ 2 • 10 -1 , in agreement with the fact that they have spring instabilities but a little effect of the 593 surface deformations on the structural force profile. For a previous SFA study with liquid crystals[START_REF] Cross | Rheological properties of a highly confined film of a lyotropic lamellar phase[END_REF] (R ∼ 0.3 cm, k ∼ 2000 N/m, 595 K ∼ 50 GPa, W 0 ∼ 0.1 mN/m, λ ∼ 6 nm), we get N k ∼ 1 and N K ∼ 6 • 10 -3 , in 596agreement with the fact that they have no spring instabilities and a little effect of 597 the surface deformations on the structural force profile. For a previous AFM study with polyelectrolytes[START_REF] Moazzami-Gudarzi | Interplay between Depletion and Double-Layer Forces Acting between Charged Particles in Solutions of Like-Charged Polyelectrolytes[END_REF] (R ∼ 2 µm, k ∼ 0.3 N/m, 599 K ∼ 50 GPa, W 0 ∼ 0.02 mN/m, λ ∼ 50 nm), we get N k ∼ 10 -1 and N K ∼ 10 -6 .Also for a previous AFM study with colloidal suspensions[START_REF] Schön | A simple extension of the commonly used fitting equation for oscillatory structural forces in case of silica nanoparticle suspensions[END_REF] (R ∼ 7 µm, k ∼ 601 0.03 N/m, K ∼ 50 GPa, W 0 ∼ 0.005 mN/m, λ ∼ 70 nm), we get N k ∼ 7 • 10 -1 and 602 N K ∼ 4 • 10 -7 . This is in agreement with the fact that both studies have no spring 603 instabilities and a little effect of the surface deformations on the structural force If the surface deformations have a little effect on the structural force profile (N K 1, 606 i.e. solids compression negligible compared to liquid compression), a fit with a semi-607 empirical exponentially decaying harmonic oscillation (equation 2) or a variation from 608 it can be attempted. On the contrary, if the surface deformations have a strong effect 609 on the structural force profile (N K 1, i.e. solids compression dominant compared 610 to liquid compression), a fit with our heuristic formulation (equation 5) or a variation 611 from is preferable. For an intermediate situation where the surface deformations have 612 a moderate effect on the structural force profile (N K ∼ 1, i.e. solids compression of the

	604
		profile.
	580
	581	factors are not the same. There is no spring instability (resp. small influence of surfaces
	582	deformations) when N k < 1 (resp. N K	1), which is fulfilled for systems with small
	583	adhesion W 0 and large period λ, measured with a spring of large stiffness k (resp. with
	584	solids of large elastic modulus K) and -less intuitively-with surfaces of small radius of
	585	curvature R. In the following, we test these criteria with studies in the literature that use
		different systems and instruments.

586

• 588 with the fact that we have spring instabilities and a strong effect of the surface 589 deformations on the structural force profile. 590 • For previous AFM studies with ionic liquids [63,68,72] (R ∼ 20 nm, k ∼ 0.1 N/m, 591 K ∼ 594 • 598 • 600 605 613 same order than liquid compression), more sophisticated models are required, like the 614 energy minimization approach proposed by Hoth et al. [68].

Version June 8, 2021 submitted to Lubricants https://www.mdpi.com/journal/lubricants

compressibilities of the mica layers and of the liquid film, i.e. the materials separating

•

We have proposed a simple scaling criterion to estimate the degree of convolution.

646

Typically, compression in the solids is dominant over compression in the liquid for 647 simple liquids (large energies, small length-scales) and easily deformable solids 648 (small elastic modulus, large radius of curvature). For SFA experiments with mica 649 sheets glued on glass lenses, the influence of mica compression is interpreted to be 650 more important and independent of the mica thickness at low loads; and smaller 651 and reduced for thinner mica at large loads. This effect is expected to be even 652 more important at all loads when mica is replaced by a softer layer (like EPON 653 glue) [START_REF] Britton | A Graphene Surface Force Balance[END_REF][START_REF] Van Engers | Direct Measurement of the Surface Energy of Graphene[END_REF][START_REF] Van Engers | A 3-mirror surface force balance for the investigation of fluids confined to nanoscale films between two ultra-smooth polarizable electrodes[END_REF], or at high loads if the distance measurement includes the indentation 654 of the whole solid bodies (not only the top layers) [START_REF] Garcia | Nanomechanics of ionic liquids at dielectric and metallic interfaces[END_REF].