Aïsha Sahaï
email: aisha.sahai@onera.fr

Andrea Desantis

Ouriel Grynszpan

Elisabeth Pacherie

Bruno Berberian

Action co-representation and the sense of agency during a joint Simon task: comparing human and machine co-agents

Keywords: joint Simon task, action co-representation, sense of agency, cooperation, humancomputer interaction

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Introduction

The sense of agency refers to the experience of controlling one's own actions, and through them, events in the outside world (Haggard & Chambon, 2012). This form of selfawareness is the basis of the legal and ethical structure of modern societies and forms the basis on which humans experience responsibility [START_REF] Moretto | Experience of agency and sense of responsibility[END_REF][START_REF] Frith | Action, agency and responsibility[END_REF][START_REF] Caspar | Coercion changes the sense of agency in the human brain[END_REF]. Interestingly, studies have shown the emergence of a form a shared control when engaged in a cooperative task: we develop an experience of agency for actions and outcomes generated by our partner, a phenomenon called we-agency. This experience of agency could support our engagement in the cooperative task. Given that engagement with automated systems is essential in the context of human-machine cooperative activities such as piloting an airplane, understanding the challenges involved in developing a sense of shared agency with automata becomes a critical concern. The present study aims at addressing this issue and in particular at investigating the factors contributing to our experiences of vicarious agency during joint tasks with humans and artificial agents.

The sense of agency in joint actions

During the last decades, the sense of agency has largely been investigated using both explicit and implicit measures (for a recent review see [START_REF] Haggard | Sense of agency in the human brain[END_REF]. For instance, one of the implicit measures most widely used to quantify people's sense of agency is the Intentional Binding phenomenon (IB; see [START_REF] Moore | Intentional binding and the sense of agency: a review[END_REF]: when a voluntary movement generates a sensory outcome, people experience an illusory temporal compression between the movement and the outcome. Importantly, this temporal binding seems to reliably occur in situations in which the participant is the agent, but not with passive movements [START_REF] Haggard | Voluntary action and conscious awareness[END_REF].

A large body of studies have used IB to shed light on how the sense of agency emerges during joint actions [START_REF] Capozzi | Temporal perception in joint action: this is MY action[END_REF]Obhi & Hall, 2011a;[START_REF] Strother | Subjective agency and awareness of shared actions[END_REF]. On the one hand, it has been shown that people can exhibit IB for actions and outcomes generated by another human agent when they are engaged in a cooperative task (Obhi & Hall, 2011a;[START_REF] Strother | Subjective agency and awareness of shared actions[END_REF]. For instance, in Strother and colleagues' (2010) study, pairs of participants were seated in front of a monitor and each member of the pair had his/her right finger on one side of the spacebar. The fingers of the participants were hidden so that no visual information about who was pressing the key was available. Participants alternately pressed the spacebar. During the operant trials, the key press triggered a tone 200 ms later. Participants had to estimate the perceived moment of the key press or the tone occurrence. By computing the action estimate and the tone estimate into one interval, the authors observed intentional binding for the other-generated shared action in a nearly similar fashion as for the self-generated action [START_REF] Strother | Subjective agency and awareness of shared actions[END_REF]. In addition, in Obhi & Hall's (2011a) study, participants were divided in pairs and were asked to press a spacebar. In a first experiment, the two participants could initiate the press at the time of their own choosing, but whenever one participant pressed the spacebar first, the other was to join in and press the spacebar as soon as possible. In a second experiment, the roles of the participants were pre-defined: one of the participants was to press the spacebar first while the second was always following her/his move. In both experiments, participants had to report their feeling of causal responsibility (i.e., their experience of agency) using a percentage scale and also to judge the onset time of the first key press or the onset of the tone. Interestingly, in both experiments, although only the initiator reported a reliable feeling of causal responsibility, both individuals showed IB (Obhi & Hall, 2011a). Notably, participants experienced IB for their own actions but also for the actions executed by the partner, suggesting that during joint tasks individuals can develop a vicarious sense of agency. It must be emphasized that it is the mere belief that another human agent caused an outcome (and not the fact of seeing or not the human-generated action) that allows individuals to develop a vicarious sense of agency.

On the other hand, there are studies that showed that intentional binding was not socially shared during cooperative tasks [START_REF] Capozzi | Temporal perception in joint action: this is MY action[END_REF][START_REF] Pfister | Action and perception in social contexts: Intentional binding for social action effects[END_REF]. For example, Capozzi and colleagues (2016) asked participants to perform a key press that triggered after 259 ms (T1) an auditory tone that served as a start signal for a (non-naïve) co-agent key press. This co-agent key press triggered a different auditory tone which occurred at a fixed delay of 629 ms after T1 (T2). In the cooperative condition, they were told to coordinate their actions as if they wanted to create a melody with the two tones. In the competitive condition, the participants were told that the coagent had to perform his key press as fast as possible as if he wanted to "wipe out" T1.

Participants had to verbally report their perceived duration of either T1 or T2. The authors found that during sequential joint actions with a co-agent, individuals demonstrated IB for their self-generated actions but not for their partner's action, independently of the context of the performed action [START_REF] Capozzi | Temporal perception in joint action: this is MY action[END_REF]. In addition, in [START_REF] Pfister | Action and perception in social contexts: Intentional binding for social action effects[END_REF]'s study, participants were paired. One assumed a leader role and the other a follower role. The leader had to press a key at the moment of their own choice and the key press triggered a tone after a certain delay (interval 1). This tone served as a go-signal for the follower to press his or her own key (interval 2). The follower's key press could trigger a tone after a random delay (interval 3) or no tone at all. Both the leader and the follower had to verbally judge the interval lengths. The results showed that the leader's interval estimations were always shorter than the follower's interval estimations, meaning that the leaders always made more IB than the followers. In addition, as the initiator of the action, the leader made IB while the follower, as the observer, did not (interval 1). For the interval 2 estimation, the leader experienced IB for the follower's action but the follower did not. Hence, leaders' sense of agency does not only concern their own actions and their adjacent effect but also predictable actions of other agents. Finally, when the follower's key press generated a sensory consequence, the temporal interval estimations (interval 3) were similar for the leader and the follower: none of them exhibited IB. That is, the follower never experienced a sense of agency whether it be over their own action and its effect or over those of the leader [START_REF] Pfister | Action and perception in social contexts: Intentional binding for social action effects[END_REF].

Interestingly, with regard to automata, there are studies that have consistently found that individuals' sense of self-agency and vicarious agency were degraded when interacting with a non-human-like artificial agent (e.g. a computer). For example, [START_REF] Obhi | Sense of agency in joint action: influence of human and computer co-actors[END_REF] showed that in a joint task participants exhibited IB for their own actions and those generated by another human co-agent, whereas, when acting jointly with a machine, IB for their own actions and those generated by the machine disappeared. In a similar vein, Berberian and colleagues [START_REF] Berberian | Automation technology and sense of control: a window on human agency[END_REF] showed that human operators experienced a very strong decrease of their sense of agency when interacting with highly automated autopilot interfaces. Moreover, they reported a lack of control over the outcomes generated by the automated system. Conversely, it has been shown that observing an action performed by a human-like automaton (e.g. an anthropomorphic hand with servo-actuated tendons rotating each finger joint) induced a temporal binding effect in a similar way as during the observation of another person performing the same action [START_REF] Khalighinejad | Social transmission of experience of agency: An experimental study[END_REF].

Taken together these studies suggest that the ability to build a sense of vicarious agency during a joint task with another agent is highly influenced by the nature of the agent one interacts with. During joint tasks with non-human-like automated machines people fail not only to experience the agency of the artificial co-actor but also their own agency. This decrease of agency during human-machine interactions might lead in turn to a reduction of involvement with the automaton since a felt lack of control would directly impact the operator's engagement and responsibility in the task [START_REF] Moretto | Experience of agency and sense of responsibility[END_REF][START_REF] Frith | Action, agency and responsibility[END_REF][START_REF] Caspar | Coercion changes the sense of agency in the human brain[END_REF].

This study aims at exploring this decrease in individuals' sense of agency during their interactions with automated agents compared to human-human interactions. Past research suggested that the experience of agency for the actions performed by another individual is based on the ability to simulate or co-represent that individual's motor plans and intentions [START_REF] Obhi | Sense of agency in joint action: influence of human and computer co-actors[END_REF][START_REF] Wohlschläger | Intentionality as a constituting condition for the own self-and other selves[END_REF]; see also [START_REF] Kunde | Sociomotor action control[END_REF]. Based on this hypothesis, people would not experience a sense of vicarious agency when interacting with computers because they fail to simulate computer-generated actions [START_REF] Obhi | Sense of agency in joint action: influence of human and computer co-actors[END_REF]. In line with this assumption, the present study investigated the link between action co-representation and individuals' experience of agency during actions performed by another human versus an automated artificial agent in a joint task. The ability to co-represent the actions of the other agent was investigated with the Social Simon task [START_REF] Sebanz | Representing others' actions: just like one's own?[END_REF]

Social Simon effect: action co-representation in joint actions

Action co-representation of human-generated actions has been classically demonstrated by the Social Simon Effect (SSE; [START_REF] Sebanz | Representing others' actions: just like one's own?[END_REF]. In the classical Simon effect people's reaction time in response to the presentation of a target stimulus decreases when the target is presented in the same relative spatial location as the response. For instance, in one version of this paradigm participants are required to execute a left hand action as soon as a red target is presented and a right hand action as soon as a green target it presented.

Red/green targets are displayed in either the same relative location as the participant's response (e.g., the target is presented to the left of a central fixation and participants have to perform a left key-press) or in the opposite location (e.g., participants have to press the right key but the target is presented on the left). It has been shown that people are faster and more accurate when the target is on the same side as the response compared to when it is presented on the opposite side. [START_REF] Sebanz | Representing others' actions: just like one's own?[END_REF] showed with a go no-go task, that when participants had to respond to only one target (e.g., respond only to the green target with the right hand) the interference disappeared. This go no-go task suggested that the Stimulus-Response congruency effect observed in the classical Simon task derives from the cognitive interference caused by two different action representations that are concurrently activated [START_REF] Simon | Choice reaction time as a function of angular stimulusresponse correspondence and age[END_REF]. However, interestingly, when the participant performs the same go no-go task with another agent (joint Simon task), and each of them is required to respond to only one target (i.e., the participant responds only to the green target, and the partner only to the red target), the interference effect for the incongruent Stimulus-Response key mapping reappeared.

Hence, it has been proposed that during a joint task, individuals integrate the behavior of their partner into their own motor scheme even though it is not task relevant [START_REF] Sebanz | Representing others' actions: just like one's own?[END_REF], but see the reference coding hypothesis [START_REF] Dolk | How "social" is the social Simon effect?[END_REF][START_REF] Dolk | The (not so) social simon effect: A referential coding account[END_REF] or the task representation hypothesis [START_REF] Yamaguchi | Sharing tasks or sharing actions? Evidence from the joint Simon task[END_REF].

Consistently, neuroimaging data showed that when someone observes or imagines a conspecific performing an action, their premotor cortex is activated -though to a lesser extent than during action execution -allowing action understanding and prediction with the help of the observer's own motor expertise (Calvo-Merino, Glaser, Grèzes, Passingham, & Haggard, 2005;[START_REF] Kilner | Predictive coding: an account of the mirror neuron system[END_REF][START_REF] Pineda | The effects of self-movement, observation, and imagination on mu rhythms and readiness potentials (RP's): toward a braincomputer interface (BCI)[END_REF]. Moreover, it has been shown that during a joint Simon task, stimuli that referred to the partner's action evoked a similar frontal cerebral signal as the stimuli that were addressed to the participants themselves [START_REF] Sebanz | Twin peaks: and ERP study of action planning and control in coacting individuals[END_REF]. Some recent studies have started investigating whether action co-representation also occurs during cooperative tasks with artificial automated systems. For example, a study showed a congruency effect during a joint Simon task when participants thought they were interacting with an unseen computer [START_REF] Wen | Neuroimaging of the joint Simon effect with believed biological and non-biological co-actors[END_REF]. However, other studies that used a similar paradigm found no congruency effect, whether at the behavioral or at the cerebral level, during a joint Simon task with an unseen computer (Tsai, Kuo, Kung, & Tzeng, 2008) or with a robot with high level of human-like appearance (i.e. a full humanoid robot with a torso, an anthropomorphic head, eyes, and two arms with fingers) when it was described as an unintentional robot [START_REF] Stenzel | When humanoid robots become human-like interaction partners: corepresentation of robotic actions[END_REF]. Similarly, cerebral motor resonance has been shown not to arise during the observation of non-human-like machine-generated actions (Mann, Williams, Ward, & Jannelle, 2007;[START_REF] Perani | Different brain correlates for watching real and virtual hand actions[END_REF]but see [START_REF] Gazzola | The anthropomorphic brain: the mirror neuron system responds to human and robotic actions[END_REF][START_REF] Kuz | Mirror neuronsand human-robot interaction in assembly cells[END_REF] while action recognition mechanisms has been shown to be sensitive to artificial humanoid avatars' actions [START_REF] Ferstl | Action recognition is sensitive to the identity of the actor[END_REF].

In sum, these studies suggest that during joint actions people tend to spontaneously simulate the action of another human partner but not the actions of an artificial agent (excepted when people believe that the artificial partner is an intentional human-like agent).

Moreover as we mentioned above, a separate line of evidence indicates that people can experience agency for other human's action but fail to experience it for actions performed by artificial systems. In line with these elements of evidence, the present study sought to investigate whether the ability to experience agency for the actions performed by another agent is linked to the ability to co-represent the agent's actions. To address this issue we ran a behavioral study combining a Social Simon reaction time task and an intentional binding task.

The reaction times observed in the Simon task served as an index of action co-representation while IB was used as an implicit measure of the sense of agency. More specifically, participants performed the Simon task alone (standard Simon condition), or with another agent (joint go-nogo conditions) that could be another human or a computer (we used a desktop computer considering that it is the artificial system that people most commonly work with). Participants were required to estimate the time interval between their action or the action of the co-agent and a subsequent auditory outcome. This allowed us to investigate two issues. Firstly, it permitted us to assess whether participants co-represent the actions performed by a human and an automated artificial co-actor. For instance, if people do not corepresent the action performed by an automated system, we should observe the classical stimulus-response congruency effect (see description of the Simon task above) when participants perform a joint task with another human partner but not when they perform it with an artificial agent. Secondly, it allowed us to evaluate whether participants' experience of self-agency and vicarious agency during joint actions is linked to their ability to corepresent the action of their partner.

Method

Ethic statement

This study was approved by the CERES (recommendation n°201726, institutional ethical research committee of the Paris Descartes University, France). The investigation was carried out in accordance with the Declaration of Helsinki and each participant provided his or her written informed consent before starting the experiment. All participants were assigned a number in order to ensure the anonymity of the data.

Participants

Sixteen healthy adults volunteered to take part in the experiment (6 women, mean age 24.75 years, SD of age 3.15 years). One participant was excluded from the sample due to his poor time discrimination performances observed during the training session. All participants were right-handed and had normal or corrected-to-normal vision. None of them had prior knowledge about the purpose of the experiment.

Material & Stimuli

Participants were sitting 46 cm from a computer screen with a refresh rate of 120 Hz.

Stimulus presentation was controlled using E-prime software (2.0 version). Visual stimuli consisted of two dots of 0.5 cm diameter, one green and the other red. An auditory tone (1000 Hz, 200 ms duration), presented via a speaker placed behind the participant, was used during the experiment as the effect of the participant/partner's button press for measuring intentional binding.

Procedure

Participants were sitting in front of a screen and had to detect, as quickly and as accurately as possible, colored dots that appeared either to the left or to the right side of a central fixation cross. This task could be performed alone, jointly with another human (the experimenter), or jointly with a computer.

In the standard Simon condition, participants were sitting in front of the center of the screen. They were instructed to perform as quickly as possible a right key-press with their right index finger when a green dot was displayed on the screen, regardless of whether it appeared on the left or right side of the screen. When a red dot was displayed they were required to execute a left key-press with their left index finger, regardless of whether it appeared on the left or right side of the screen (Figure 1a).

In the individual go-nogo condition, participants were sitting in front of the right side of the screen and an empty chair was placed in front of the left side of the screen. They were asked to perform a right key-press with their right index finger when presented with a green dot regardless of whether it appeared on the left or right side of the screen, and to not react when presented with a red dot (Figure 1b).

In the joint conditions, participants were sitting in front of the right side of the screen.

They executed a right key press with their right index finger when presented with a green dot, independently of its location on the screen. When presented with a red dot they were asked to not react. Importantly, the task could be performed jointly with another human agent (joint HH condition, Figure 1c) or with an automated artificial system, i.e. a computer (joint HM condition, Figure 1d). When the partner was a human, she was sitting on the left side of the screen and performed left key-presses with her left index finger. When the partner was the computer, an empty chair was placed in front of the left side of the screen.

In the passive observation conditions, participants were sitting in front of the right side of the screen and they had to observe another agent (another human agent in the passive observation H condition and the computer in the passive observation M condition) performing the red dot detection (Figure 1e &1d). Both the human and the automated system responded to the presentation of the red dot. When a green dot was displayed no response was required by the partner. Each trial started with a fixation cross that appeared at the center of the screen during 500 ms. Thereafter, the target immediately appeared on the screen and participants had at most 1800 ms to press their response key otherwise an error message appeared and the trial was canceled.

Participants were informed of the onset of their own action and the action of their coagent (human or computer) by the presentation of an empty square displayed around the target for a duration of 200 ms. Participants were required to fixate the computer screen throughout the experiment and to not look at the actions performed by the human agent. When the human co-agent performed the target detection task, the square onset times corresponded to the coagent's real reaction times. When the computer performed the target detection task, the square onset times were taken from a normal distribution calculated from the mean and standard deviation of the human co-agent's response times computed during a pre-test session. More specifically, square onset times were selected randomly within two standard deviations of this distribution.

Correct target detections were followed by an auditory tone presented after the response at one of three possible Stimulus Onset Asynchronies (SOA) of 400, 900 or 1400 ms. The SOA for a given trial was selected randomly. After the presentation of the sound, participants had 4 s maximum to verbally report the perceived duration between the onset of the target detection (indicated by the square appearing around the target) and the onset of the auditory tone (see Figure 2 for a summary). This time interval estimates served as an implicit measure of participants' sense of agency. Note that participants were not asked to perform the temporal estimations in the standard Simon condition as they made estimations for selfgenerated actions in the individual go-nogo condition.

Participants were trained at the beginning of each experimental condition block to estimate and report their perceived duration of the action-tone intervals. During this training, they were presented with an empty square that flashed, followed by an auditory tone with a random delay between 200 ms and 2000 ms. They had to verbally report the perceived duration of this interval in milliseconds. Then, they were given the correct delay with a visual feedback in order to accurately recalibrate their internal clock. This training session consisted in 20 trials. Thereafter, participants performed 20 trials of the given experimental condition as training. The goal was to familiarize them with the task so that they would associate their key press with the following auditory tones.

The experimental conditions were tested using a within-subject design and the order of conditions that each participant completed was selected randomly. Trials were coded as congruent when the target appeared on the side of the participant's response key, and as incongruent when the target appeared on the opposite side of the participant's response key.

Data analyses

Our dependent measures were the mean target detection Response Times (RTs) and the mean perceived action-tone temporal interval. Statistical analyses were performed with R software (3.3.1 version). Extreme values (the values that were below or above 2 standard deviations from the mean) of the participants' response times and perceived intervals were excluded from further analyses in order to eliminate outliers and allow for robust statistical analyses. The significance level was set at α = .05.

To distinguish participants' trials from the partner's trials in the joint conditions, participants' trials were labeled joint HH self and joint HM self, and the partners' trials were labeled joint HH other and joint HM other.

Response times (RTs) analysis

This analysis was based exclusively on the data gathered in the conditions in which participants performed an action (standard Simon, individual go-nogo, joint HH self, and the joint HM self). The analysis aimed at quantifying the Social Simon Effect (SSE) when participants interacted with a human co-agent (joint go-nogo HH self) and with the computer (joint go-nogo HM self). To investigate the effect of task order on participants' RTs we introduced the Order (1st, 2nd, 3rd, 4th, 5th, 6th) as a covariate in a linear mixed model (𝑚𝑚 𝑅𝑅𝑅𝑅 � ), with Task and Congruency as fixed factors and Subject as random effect. The factor Order indicates when a given task (e.g. standard Simon) was presented during the experiment.

For instance, if the participant performed firstly the standard Simon task, we would attribute the Order "1 st " to this task. Instead, if the standard Simon task was performed second, we would attribute the order "2 nd " to this task and so on for the other Order values. This model(𝑚𝑚 𝑅𝑅𝑅𝑅 � ) was then compared, using a likelihood ratio test, to a simpler model that included the same fixed and random effects but without order as a covariate (𝑚𝑚 𝑅𝑅𝑅𝑅 ): We found no significant difference between 𝑚𝑚 𝑅𝑅𝑅𝑅 and 𝑚𝑚 𝑅𝑅𝑅𝑅 � (respectively logLik = -697.94 and logLik = -695.97, χ 2 (5) = 3.94, p = .56). This indicates that including Order as a covariate did not improve our model of the observed data. Thus, Order was excluded from further analyses. We conducted a within-subjects analysis of variance (ANOVA) on reaction times with Task (standard Simon, individual go-nogo, joint HH self, joint HM self) and Congruency (Congruent, Incongruent) as factors. The Target (Green dot, Red dot) factor was not included in the ANOVA because the SSE does not rely on the target identity but rather on the congruency between the location of the target and the location of the response key. The Delay (400, 900 and 1400) factor was irrelevant for the analysis as the auditory tone was produced after participants' response and therefore could not influence their RTs.

Temporal estimation accuracy checking

To verify whether participants correctly perceived the different action-tone temporal intervals, we conducted a prior within-subjects one-way ANOVA on the raw temporal estimations Delay (400, 900, 1400) as factor. This control analysis allowed us to ensure that participants paid attention to the action-sound intervals.

Intentional binding analysis

This analysis aimed at investigate the influence of the co-agent on participants' agency experience. To characterize the IB phenomenon, we considered the mean perceived actiontone temporal interval in the passive observation M condition as our baseline, i.e. when no agency was involved at all (see [START_REF] Poonian | Implicit agency in observed actions: evidence for N1 suppression of tones caused by self-made and observed actions[END_REF][START_REF] Wohlschläger | Intentionality as a constituting condition for the own self-and other selves[END_REF]. We measured the IB by subtracting the mean perceived action-tone temporal interval in the passive observation M condition from the mean perceived action-tone temporal interval in all other experimental conditions (negative values indicate temporal underestimations in these conditions compared to the baseline). Hence, the passive observation M condition was not included as a factor in the IB analyses.

Note that for the intentional binding analyses (see below) we did not include the action -tone Delay (400, 900 and 1400 ms) as a separate factor. Indeed, in the present experiment we were interested in the way the social context influences IB in general, rather than its influence on IB for different temporal intervals.

Given that the order of the experimental conditions was fully counterbalanced between participants, we introduced the Order (1st, 2nd, 3rd, 4th, 5th, 6th) factor as a covariate in a linear mixed model (𝑚𝑚 𝐼𝐼𝐼𝐼 � ) with Action Context (Human individual action, Human-human joint action, Human-computer joint action), Agent (Self, Other) and Congruency (Congruent, Incongruent) as fixed factors and Subject as random effect. The Action Context referred to the social context in which the Simon task was performed, i.e. whether the agent was acting alone (Human individual action), co-acting with another human (Human-human joint action), or with a computer (Human-computer joint action). Accordingly, the Human individual action context included the passive observation H for the actions that were performed by the human partner and the individual go-nogo condition for the actions performed by the participant him/herself. Note that there was no temporal estimation measurement in the standard Simon condition. That is why this condition was not included in the analyses of the IB phenomenon.

The Agent (Self, Other) factor referred to the author of the action, i.e. the participants themselves or their partner respectively (see Table 1). Using a likelihood ratio test, we then The analysis showed that the 𝑚𝑚 𝐼𝐼𝐼𝐼 � model including Order as a covariate fitted our data better than the 𝑚𝑚 𝐼𝐼𝐼𝐼 model (respectively logLik = -1135.63 and logLik = -1154.96, χ 2 (5) = 38.66, p < .001). Hence, we conducted a within-subjects ANOVA on IB with Action Context (Human individual action, Human-human joint action, Human-computer joint action), Agent (Self, Other) and Congruency (Congruent, Incongruent) as fixed factors and Order (1st,2nd,3rd,4th,5th,6th)

Social Simon Effect

We first examined the SSE during joint-actions with another human and with a desktop computer (Figure 3). We assessed the normality of the RTs distributions of the differences between the congruent trials and the incongruent trials in the standard Simon, individual go-nogo, joint HH self and joint HM self conditions using the Shapiro-Wilk test.

The analyses showed that none of the RTs distribution deviated from normality (all W > .90 and all p > .10). We then computed a within-subjects 4 x 2 analysis of variance (ANOVA) Error bars represent standard errors.

Temporal estimation accuracy checking

We verified whether participants correctly perceived the different action-tone temporal intervals that were used in the task by assessing whether they adjusted their temporal estimations accordingly. We investigated the effect of Delay on the mean perceived actiontone temporal interval. We computed a within-subjects one-way ANOVA with the factor Delay (400, 900, 1400). We found a significant main effect of Delay on the perceived actiontone temporal interval estimations (F(2,28) = 316.71, p < .0001). Post-hoc comparisons using Tukey's HSD test revealed that individuals reported significantly higher temporal intervals when the Delay between the action and the tone was 1400 ms compared to when the delay was 900 ms (respectively 1133.04 ms and 797.83 ms, p < .0001) and when the delay was 400 ms (433.57 ms, p < .0001). In addition, individuals reported higher temporal intervals with a interest (indeed, we only introduced various delays in order to avoid the predictability bias), this effect demonstrated that participants were paying attention to the temporal delays and correctly discriminated the different delays.

Intentional Binding

Firstly, we calculated the amount of IB by subtracting the mean perceived action-tone temporal interval in the passive observation M condition from the mean perceived action-tone temporal interval in all the other conditions. We computed a within-subjects 3 x 2 x 2 ANOVA Action Context (Human individual action, Human-human joint action, Humancomputer joint action), Agent (Self, Other) and Congruency (Congruent, Incongruent) as fixed factors and Order (1st, 2nd, 3rd, 4th, 5th, 6th) as a covariate.

We found neither a main effect of Agent (F(1,14) = .11 p = .74, ns) nor a main effect of Congruency (F(1,14) = 0.38, p = .54, ns). However, we found a significant main effect of Action Context (F(2,28) = 10.78, p < .001). Post-hoc comparisons using Tukey's HSD test investigated this main effect and revealed that participants exhibited greater IB for human individual actions context than for human-machine joint actions context (p < .001). In addition, there was no difference on IB between the human individual actions context and the human-human joint actions context (p = .24, ns). Finally, participants exhibited greater IB for human-human actions context than for human-machine joint actions context (p < .001).

Furthermore, we found a significant Action Context x Agent interaction (F(2,28) = 11.24, p < .001) on IB (Figure 4). Considering that Congruency did not interact with the other factors, we averaged congruent and incongruent trials for further analyses.

Post-hoc comparisons using Tukey's HSD test revealed that IB for self-generated actions was greater when participants were acting alone (individual go-nogo condition) compared to when they were jointly acting with a machine (p < .001). However, IB for selfgenerated actions did not differ when participants were performing the task alone (individual go-nogo condition) compared to when they were interacting jointly with another human agent (p = .34). Similarly, IB did not differ when participants acted with another human agent compared to when they interacted with a machine, although there was a trend (p = .06, ns).

Accordingly, this indicates that IB observed for self-generated actions did not differ whether participants performed the task alone or interacted with another human agent but tended to strongly decrease when participants interacted with a machine.

Additional post-hoc comparisons using Tukey's HSD test on IB for other-generated actions revealed greater IB when participants performed the task with another human agent compared to when they were observing the same agent completing the task alone (passive observation H condition) (p < .001). In addition, we observed greater IB for other-generated actions when participants performed the task jointly with a human agent compared to when they performed it with the computer (p < .001). However, there was no difference in IB for other-generated actions when participants observed a human agent performing the task alone (passive observation H condition) compared to when they interacted with the computer (p = .81, ns). Accordingly, this indicates that the IB observed for other-generated actions strongly increased only when participants performed the task together with a human agent compared to the mere passive observation of the same action. This enhancement was not observed during joint actions with a machine.

Further post-hoc comparisons using Tukey's HSD test showed greater IB for selfgenerated actions compared to other-generated actions when participants and the other human agent were performing the task alone (human individual action context) (p = .005). During human-human joint actions, IB was weaker for self-generated actions compared to othergenerated action (p = .03). There was no difference on IB between self-and other-generated actions when participants interacted with a machine (p = 1.00, ns). In addition, no difference was observed between the IB for self-generated actions performed alone and the IB for othergenerated actions in the context of human-human joint action (p = .95, ns). In sum, participants reported stronger IB when they performed the task alone compared to when they passively observed another person performing the same action. However, during humanhuman joint action, this trend was reversed. Notably, they reported stronger IB for the action of their partner than for their own action.

Discussion

In this study, our aim was to investigate the sense of agency in joint action with another human versus a desktop computer. To assess action co-representation, we used a typical Social Simon task where participants had to detect a target that could appear on the same side as their response key or on the opposite side. Accurate target detection triggered an auditory tone after a randomized delay. Participants had to estimate the temporal delay between the target detection and the onset of the tone. This estimate served as an implicit measure of participants' sense of agency [START_REF] Haggard | Voluntary action and conscious awareness[END_REF].

We found the classical Simon effect with longer response times in incongruent trials when participants performed a double target detection task (standard Simon condition) but not when they performed a single target detection task (individual go-nogo (self) condition).

Moreover, we found a Social Simon Effect (SSE) when the participants performed a single target detection task jointly with another human agent performing the complementary action (joint go-nogo HH (self) condition). This suggests that, during human-human interactions, participants experienced the action performed by a human partner as if it was their own action. However, this effect disappeared when a computer program replaced the human partner. These findings are consistent with previous results supporting the claim that the social identity of the partner critically influences individuals' action co-representation abilities during a joint task [START_REF] Stenzel | When humanoid robots become human-like interaction partners: corepresentation of robotic actions[END_REF][START_REF] Wen | Neuroimaging of the joint Simon effect with believed biological and non-biological co-actors[END_REF]. One possible explanation is that the artificial agent was not considered as an agent either because it was not perceived as acting intentionally or because the causal relationship between the agent and the action effect was opaque or both. Indeed, a previous study investigating the SSE during interactions with a humanoid robot failed to observe this effect when the robot was described as an unintentional device that passively executed a deterministic command whereas the SSE was present when participants were induced to believe that the robot was an intentional active and intelligent human-like agent [START_REF] Stenzel | When humanoid robots become human-like interaction partners: corepresentation of robotic actions[END_REF]. In addition, it has been proposed that the perceived coagent's agency plays an important role in the emergence of SSE [START_REF] Stenzel | The joint Simon effect depends on perceived agency, but not intentionality, of the alternative action[END_REF]. In their study, Stenzel and colleagues (2014) asked participants to perform a joint Simon task next to a co-agent who either intentionally performed the response key presses (agency+/intentionality+ condition), or passively placed their finger on the response key that automatically moved up and down (agency-/intentionality-condition), or who was thought to controlled the response key with a braincomputer interface while placing their finger besides the response key (agency-/intentionality+ condition). The authors found that the SSE was manifested only in the agency+/intentionality+ condition, that is to say, when the causality between the co-agent and the action effect was not disrupted. Hence, perceived agency may play an important role in the SSE [START_REF] Stenzel | The joint Simon effect depends on perceived agency, but not intentionality, of the alternative action[END_REF].

Note that the "social" component of the SSE has been recently criticized [START_REF] Dolk | How "social" is the social Simon effect?[END_REF][START_REF] Dolk | The (not so) social simon effect: A referential coding account[END_REF], for a review see [START_REF] Dolk | The Joint Simon Effekt: A review and theoretical integration[END_REF]. Instead, a referential-coding approach has been proposed according to which salient events, such as a physical presence for example -whatever social or not -, serve as a spatial referential for action coding. When considering the referential-coding approach, one can argue that the human partner in our experiment may have been more salient than the computer partner and could thus have grabbed the attention of the participant to a higher degree, influencing participants' response times. Our study was not designed to and cannot provide an answer to this debate. However, we believe that the referential-coding approach cannot explain the influence of social factors on the SSE. For example, it has been shown that individuals showed less or even no SSE when they were in a competitive context compared to a cooperative context [START_REF] Iani | The role of group membership on the modulation of joint action[END_REF][START_REF] Ruissen | Competitive Game Play Attenuates Self-Other Integration during Joint Task Performance[END_REF] or after a negative affect induction compared to a positive affect induction between the two co-agents [START_REF] Hommel | How Social Are Task Representations?[END_REF][START_REF] Kuhbandner | The role of positive and negative affect in the "mirroring" of other persons' actions[END_REF] although the participants performed the very same task. This theoretical account also hardly explains the difference found between intentional and non-intentional artificial agents [START_REF] Stenzel | When humanoid robots become human-like interaction partners: corepresentation of robotic actions[END_REF]. We believe that this issue could be settled by further studies combining brain imaging and behavioral techniques.

Interestingly, we observed that IB phenomenon for self-generated actions remained stable regardless of whether participants performed the task alone or with another human agent. However, IB decreased strongly when participants interacted with a machine. This pattern of results indicates that the social context played a pivotal role in implicit self-agency attribution as measured by the intentional binding phenomenon. Even though participants in the present experiment had to perform the same action leading to the same sensory effect in all conditions, they tended to bind their own action and its effect differently according to the social context. The intentional binding for their action was maximal when they were performing the task alone. It decreased drastically as soon as they were collaborating with a machine. Thus, sharing a task with an artificial agent seems to decrease the sense of selfagency. This result corroborate previous findings showing a reduction of self-agency during a human-human cooperative task compared to an individual task, even when there was no ambiguity about who had caused the outcome [START_REF] Beyer | Beyond self-serving bias: diffusion of responsibility reduces sense of agency and outcome monitoring[END_REF].

Our study provides however original evidence that this process of diffusion of responsibility does mainly occur during human-computer interaction, suggesting that when interacting with a machine our sense of responsibility strongly decreases.

More importantly, the vicarious sense of agency for the partner's action disappeared when the partner was the computer, as did the congruency effect on response times. Hence, the IB phenomenon seemed to parallel the SSE effect when comparing the human-human condition with the human-machine condition. This lends support to the view that vicarious sense of agency and SSE could potentially share common underlying mechanisms, but this needs to be confirmed by neuroimaging studies. As mentioned earlier, there are currently two main hypotheses explaining the SSE. The first more classical account posits that we automatically activate our own sensorimotor representations when partnered with a human coagent [START_REF] Sebanz | Representing others' actions: just like one's own?[END_REF]. The second theory accounting for the SSE suggests that action coding and thus decision is influenced by salient environmental cues [START_REF] Dolk | How "social" is the social Simon effect?[END_REF][START_REF] Dolk | The (not so) social simon effect: A referential coding account[END_REF]. However, a third explanation combining the two accounts could also be taken into consideration, explaining both the RTs and the IB results. In the case of individual selfgenerated actions, individual's attention was focused on the ongoing task and their sensorimotor network was activated, leading to a greater sense of agency. During the joint tasks, the need of participants to coordinate their actions with their partner induced an attentional shift toward the other-generated actions, which in turn engaged the participant's sensorimotor network when the partner was a human being, leading to a vicarious sense of agency in the specific context of the human-human joint task. Finally, during the humanmachine joint task, the absence of an intentional partner brought the participants to pay less attention to the actions performed by the machine. As a consequence, participants' sensorimotor network was not activated in this case, leading also to a decrease of agency for the actions generated by the machine. However, further studies are needed to investigate which hypothesis provides the best account for the outcomes reported here. Taken together these results suggest that participants experienced vicarious agency during joint actions with other humans but not with machines. As for now, the findings might be considered to provide further evidence in favor of the view that the human representational system is biologically tuned (Tsai & Brass, 2007).

Other interesting results were observed in our study. First, IB for the actions generated by a human partner was stronger during joint actions compared to mere passive observation of the same action. These results might suggest that being involved in a joint task leads to a stronger representation of the agency of the co-actor. Second, we found that during the human-human collaborative condition, IB for the co-agent's action was stronger compared to IB for the participant's own actions. This outcome contrasts with the experiment reported by Obhi & Hall (2011a), in which intentional binding was the same for the participant's and the partner's action when they were performing a joint task. This difference might depend on the fact that Obhi & Hall (2011a) used a very different design compared to our study. In their experiment, the two participants were required to press the same button jointly, whereas in our experiment the participant and their partner pressed different buttons at different times in response to different stimuli. The way agency is allocated among partners may depend on the type of collaboration required by the task. When a participant and a co-actor are required to perform an action together (as in Obhi & Hall, 2011a), the IB observed might reflect a summation of both self-and other-agency. However, in our case since the participant's and the partner's action were separate in time, we believe that the IB we observed for the actions generated by the co-actor reflect a sense of vicarious agency. Finally, we did not observe any effect of Congruency on participants' sense of agency. This is in contrast with past studies

showing that the fluency of action selection modulates the sense of agency (Chambon & Haggard, 2012;[START_REF] Sidarus | Difficult action decisions reduce the sense of agency: a study using the eriksen flanker task[END_REF]. Notably, past research showed that when action selection is facilitated by the presentation of a cue, individuals report higher judgement of control compared to when action selection is not facilitated. In our study we did not find any effect of action facilitation on participants' sense of agency. However, unlike previous experiment on the fluency of action, we did not use an explicit rating scale but an implicit measure of agency. This might suggest that intentional binding and explicit judgments of agency do not share the same processes [START_REF] Dewey | Do implicit and explicit measures of the sense of agency measure the same thing?[END_REF].

Conclusion

We raised the question of the relationship between action co-representation and the sense of we-agency. Given the convergence of the SSE and the vicarious sense of agency, we hypothesized that as soon as individuals are able to use their own motor system to simulate their partner's action into their own motor system, they can understand their partner's intention with the help of their previous experiences, supporting the construction of a vicarious sense of agency. During human-machine interactions, as individuals do not consider the artificial system as intentional, they do not use their sensori-motor system to simulate and try to understand the machine-generated action and consequently are unable to develop a vicarious sense of agency. This leads to the question of how we can design artificial automated systems that we can understand and thus guarantee a meaningful human-machine interaction. To address this issue an increasing number of studies are now investigating whether action co-representation is facilitated by humanized automated artificial systems (for a review, see [START_REF] Sahaï | Predictive mechanisms are not involved the same way during human-human vs. human-machine interactions: a review[END_REF]. Human-like automata could more easily be considered as intentional agents and may facilitate action co-representation and in turn ensure optimal human-machine interactions. In line with this notion, it has been shown that humanized machines could enhance the individuals' sense of agency compared to traditional automata (Wohlschläger, Haggard, et al., 2003).

Figure 1 .

 1 Figure 1. Experimental setup for the standard Simon condition (a), the human-human joint condition (b), the human-machine joint condition (c), the individual go-nogo condition (d), the human observation condition (e), and the machine observation condition (f). See text for more details.

  Participants completed a total of 3600 trials, (6 Tasks (standard Simon, individual go-nogo, joint HH, joint HM, passive observation H, and passive observation M) x 2 Targets (Green dot, Red dot) × 2 Congruency levels (Congruent, Incongruent) × 3 Delays (400, 900, 1400) × 50 trials).

Figure 2 .

 2 Figure 2. Trial timeline. A fixation cross appeared for 500 ms. Then, the target appeared and the agent (the participant or the partner) had to detect it before 1800 ms. All target detections were signaled by an empty square around the target. An auditory tone was generated at a delay of 400, 900 or 1400 ms after target detection. The participant had to report the temporal delay between the onset of the target detection and the onset of the tone.

  𝑚𝑚 𝑅𝑅𝑅𝑅 : RT ~ Task * Congruency, random = Subject 𝑚𝑚 𝑅𝑅𝑅𝑅 � : RT ~ Task * Congruency + Order, random = Subject

  compared this model (𝑚𝑚 𝐼𝐼𝐼𝐼 � ) with another mixed model (𝑚𝑚 𝐼𝐼𝐼𝐼 ) with only Action context, Agent, and Congruency as fixed factors and Subject as random effect: 𝑚𝑚 𝐼𝐼𝐼𝐼 : IB ~ Action Context * Agent * Congruency, random = Subject 𝑚𝑚 𝐼𝐼𝐼𝐼 � : IB ~ Action Context * Agent * Congruency + Order, random = Subject

  with the factors Task (standardSimon, individual go-nogo, joint HH self, joint HM self) andCongruency (Congruent, Incongruent). We found a significant main effect of Congruency indicating longer mean RTs on Incongruent trials compared to Congruent trials (F(1,14) = 5.47, p = .03) but no significant main effect of Task (F(3,42) = .80, p = .50, ns). We found a significant interaction between Congruency and Task (F(3,42) = 9.99, p < .001). Post-hoc comparisons using Tukey's HSD test revealed that mean RTs on Incongruent trials was significantly longer than the mean RTs on Congruent trials in the standard Simon condition (respectively 426.19 ms and 409.25 ms, p = .001) and in the joint HH self condition (respectively 390.23 ms and 403.51 ms, p = .02) but not in the individual go-nogo condition (respectively 414.32 ms and 416.09 ms, p = .99, ns) nor in the joint HM self condition (respectively 434.78 ms and 444.16 ms, p = .23, ns). Hence, we observed the classical Simon Effect in the standard Simon and when participants performed the task with another human agent (joint HH self). No Simon effect was observed when participants interacted a machine.

Figure 3 .

 3 Figure 3. Congruency effect for the standard Simon, individual go-nogo, and joint conditions.

Figure 4 .

 4 Figure 4. Intentional binding for self-and other-generated actions in the action contexts of human individual action, human-human joint action, and human-machine joint action. The vertical bars represent the standard error values.

Table 1 .

 1 as a covariate. Factorial design for the intentional binding analysis

				Factor: Action Context
			Human individual	Human-human joint	Human-computer
			action	action	joint action
		Self	Individual go-nogo Joint HH self	Joint HM self
	Factor : Agent	Other Passive	Joint HH other	Joint HM other
			observation H
	4 Results

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Funding source

This work was funded by a French National Research Agency grant (Young researcher program -ANR-15-CE26-0010-01). Elisabeth Pacherie's work was supported by ANR-10-LABX-0087 IEC and ANR-10-IDEX-0001-02 PSL*.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Dalla Amaury

Florence Monta

Jean-Benoit Razan

Amaury Dalla Monta

Florence Razan

Jean-Benoît Le Cam

Grégory Chagnon

Using thickness-shear mode quartz resonator for characterizing the viscoelastic properties of PDMS during cross-linking, from the liquid to the solid state and at different temperatures

Keywords: Quartz Crystal Microbalance, liquid and solid states, cross-linking, viscoelasticity, PDMS

Using thickness-shear mode quartz resonator for characterizing the viscoelastic properties of PDMS during cross-linking, from the liquid to the solid state and at different temperatures

Introduction

The Quartz Crystal Microbalance (QCM) is a versatile tool, first described by Sauerbrey in his seminal work [START_REF] Sauerbrey | Verwendung von schwingquarzen zur wägung dünner schichten und zur mikrowägung[END_REF]. Due to its simplicity, this type of sensor has been widely used in various domains, enabling us to determine with accuracy measurands [START_REF] Cheeke | Acoustic wave gas sensors[END_REF] such as mass density [START_REF] Stockbridge | Resonance frequency versus mass added to quartz crystals[END_REF], viscosity [START_REF] Kanazawa | The oscillation frequency of a quartz resonator in contact with liquid[END_REF] and pressure (Stockbridge, 1966a), in a continuous and non-destructive manner, with sample as small as a microliter. As illustrated in Figure 1, it consists in a circular thin disk of piezoelectric ATcut quartz, with metallic electrodes on both sides. The application of a voltage between them generates a shear deformation of the crystal, which can then be excited into resonance when its thickness is near an odd multiple of half the acoustic wavelength. The deposit of a material sample on the crystal changes the resonance properties, which are fundamentally dependent on the characteristics, either mechanical or electrical, of the sample. For all its applications, the QCM needs an electrical interface able to apply a sinusoidal voltage between its electrodes and to measure the resonance conditions. Among all the methods of read-out, the most used is the impedance analysis [START_REF] Arnau | A review of interface electronic systems for AT-cut quartz crystal microbalance applications in liquids[END_REF], which maps the electrical admittance of the QCM as a function of the frequency, giving access to a greater number of parameters than simpler methods such as the use of an oscillator circuitry.

In biology, where the QCM is increasingly used [START_REF] Becker | A survey of the 2006-2009 quartz crystal microbalance biosensor literature[END_REF], the functionalization of the QCM surface with a definite substance enables us to measure active species absorption or deposition, and then to recognize specific pathologies like schistosomiasis [START_REF] Wang | Rapid, simple, and sensitive immunoagglutination assay with SiO2 particles and quartz crystal microbalance for quantifying Schistosoma japonicum antibodies[END_REF] or Ebola fever [START_REF] Yu | Detection of Ebola virus envelope using monoclonal and polyclonal antibodies in ELISA, surface plasmon resonance and a quartz crystal microbalance immunosensor[END_REF]. In chemistry, with a similar method, such a sensor can detect presence of harmful molecules in the air, acting as an electronic nose [START_REF] Serfaty | Polymer coated quartz crystal microbalance sensors for detection of volatile organic compounds in gas mixtures[END_REF]. In mechanics, finally, it is used for measuring complex shear modulus of polymers [START_REF] Lucklum | Gravimetric and nongravimetric chemical quartz crystal resonators[END_REF][START_REF] Holt | Measurement of polymer shear modulus using thickness shear acoustic waves[END_REF][START_REF] Serfaty | Polymer coated quartz crystal microbalance sensors for detection of volatile organic compounds in gas mixtures[END_REF] or for following its evolution during a specific process, such as dissolution [START_REF] Hinsberg | Measurement of Thin-Film Dissolution Kinetics Using a Quartz Crystal Microbalance[END_REF].

To our best knowledge, the QCM has never been used to characterize the viscoelastic properties of a rubber at different stages of the crosslinking, while it should provide additional information to better describe and understand crosslinking kinetics. This study aims at performing such measurements with a polydimethylsiloxane, from the liquid to the solid state, at different temperatures.

The paper is organized as follows. Section 2 briefly describes the theory governing the behavior of the QCM and the measurement principle. Section 3 presents the application to the characterization of viscoelasticity of a polydimethylsiloxane: the experimental device, the measurement methodology and the material. Results obtained are analyzed in light of the aforementioned objectives. Concluding remarks close the paper.

Theoretical framework and measurement principle

On a fundamental point of view, the QCM is simply a transducer, linking its load impedance with its electrical impedance. The fundamental relations governing its behavior are briefly recalled hereafter. The reader can refer to [START_REF] Johannsmann | The Quartz Crystal Microbalance in Soft Matter Research: Fundamentals and Modeling[END_REF] for further information.

Acoustic parameterization

In a first approximation, when its diameter is large compared to its thickness, the QCM can mainly be seen as a unidimensional device, a succession of homogeneous layers in which acoustic shear waves propagate along the z axis. Inside a layer, the amplitude u(z, t) of the displacement due to the wave, z and t being respectively the z coordinate and the time, is described by the well-known wave equation, which is:

∂ 2 u ∂t 2 = c 2 ∂ 2 u ∂z 2 (1)
This can be expressed in the frequency domain by:

-ω 2 û(z) = c2 d 2 û dz 2 (2)
Here, û is the complex amplitude of the displacement, c = ρ/ G is the speed of sound, ρ the density and G the shear modulus. Introducing the wavenumber k = ω/c and the imaginary unit i, the solutions of the wave equation are in the form of:

û(z) = û+ exp +i kz + û-exp -i kz (3)
û+ and û-are the amplitude of a wave traveling respectively to the left and to the right. It is therefore possible to express the velocity v(z) and the shear stress σ(z) :

v(z) = iωû(z) (4) = iωû + exp +i kz + iωû -exp -i kz σ(z) = Gdû dz (5) = i k Gû + exp +i kz -i k Gû -exp -i kz = iω Z û+ exp +i kz -iω Z û-exp -i kz
It is important to notice that the acoustic wave impedance Z = ρc is a mechanical property of the layer, and should not be confused with the acoustic impedance, the ratio between shear stress and velocity Zac = σ(z)/v(z), which is characteristic of the excitation.

Mason circuit

The acoustic shear wave is in nature very similar to an electromagnetic wave. By virtue of the electromechanical analogy, it is therefore possible to represent a layer of the QCM as a distributed-element network. Such a representation is perfectly adapted to this device, because our goal is to link a mechanical quantity with an electrical quantity. To go further, since only the quantities at the interfaces between two layers are of interest, the QCM can even be represented as a two-port network, one for each interface (see Figure 2(a)). This paragraph shows that the Mason circuit is a compatible representation [START_REF] Mason | Piezoelectric crystals and their applications to ultrasonics[END_REF].

Here, A is the area of the device in the plane orthogonal to the direction of propagation, and h is half the thickness of the considered layer. Using the Kirchhoff rules, it appears that forces and velocities are related by the following expressions that should be validated:

F1 = iA Z tan kh v1 - iA Z sin 2 kh (v 1 + v2 ) (6) F2 = iA Z sin 2 kh v1 -iA Z tan kh (v 1 + v2 ) (7)
The displacement and the shear stress can be written as:

û(z) = ûα sin kz + ûβ cos kz (8) σ(z) = ω Z ûα sin kz + ûβ cos kz (9)
Hence, being careful about the sign of the relations:

F1 = -Aσ(-h) v1 = +iωû(-h) (10) F2 = -Aσ(h) v2 = -iωû(h) (11)
This yields to Equation ( 6) and Equation ( 7) after some trigonometric manipulations, confirming the correctness of the circuit used. However, as such, the circuit is still incomplete: the quartz being piezoelectric, a third port electrical in nature should be added. Across a transformer, an electrical source connected to this port will be able to generate an acoustic wave and to be influenced by the mechanical properties of the device.

It can be shown that this three-port network satisfies the constitutive relations of piezoelectricity ; by introducing the piezoelectric stress coefficient ẽ 26 and the dielectric constant at constant strain ε of the quartz, these link the electric displacement D and the electric field Ê to the mechanical characteristics of the wave:

σ = Gdû dz - ẽ 26 εε 0 D (12) Ê = - ẽ 26 εε 0 dû dz - 1 εε 0 D (13)
Finally, in practice and on one side, the QCM is in contact with the air, which has an acoustic wave impedance negligible compared with the one of the quartz. Therefore F1 = 0, and the equivalent circuit is short-circuited on the left. On the other side, the QCM is in contact with the material to characterise, with an acoustic impedance at the interface ZL . Therefore F2 /v 2 = A ZL , and the equivalent circuit is closed with a resistance having this value. It remains in the circuit only the electrical port, which corresponds to the fact that the QCM can only be interrogated electrically through its electrodes, and not by any acoustic or mechanical mean.

Resonance condition

We can now use the Mason circuit to calculate the equivalent impedance of the circuit after the transformer, the motional impedance Zmot . Although it does not have a clear physical significance, it is remarkable because it vanishes, by definition, if a resonance occurred; and in practice, the resonant frequency can be measured with a great precision. Using the Kirchhoff rules, the subscript q indicating properties of the quartz, we have:

Zmot = - φ 2 iωC 0 - i Zq sin 2 kq h q + i Zq tan kq h q -1 + i Zq tan kq h q + ZL -1 -1 (14)
This relation can be simplified if we assume that the wavenumber kq is close to the ideal open-circuit wavenumber kq,OC = nπ/2h q , solution of the equation Zmot = 0 when the QCM is fully immerged in the air without influence of the piezoelectric effect. Using a Taylor expansion of the previous relation yields:

Zmot ≈ 1 4 - 4φ 2 iωC 0 + inπ Zq ω -ωOC ωOC + ZL (15)
Applying this relation once in a reference state (the resonant angular frequency being called ωsample ), with ZL = 0, and once in charge, the frequency shift obeys the following equation:

∆ f f 0 = ωref -ωsample ωOC n = i π Zq ZL (16)
This is a direct relation between frequencies measurement and the load impedance of the tested sample. To go further, some assumptions on the sample are required.

If it is assumed, for instance, that it is semi-infinite, that is to say there is no right-traveling wave (û -= 0), Equations ( 4) and ( 5) yield:

σ(z) v(z) = iω Zsample û+ exp +i ksample z iωû + exp +i ksample z = Zsample (17)
Hence:

ZL = σ(h q ) v(h q ) = Zsample = ρ sample Gsample (18)
Given that the density of the sample is already known, the shear modulus can be deduced. Of course, the quantities characterizing the quartz in the previous relation ( Zq and f 0 ) are not known a priori. However, they can be evaluated by making an additional measurement with a sample in a well-known material, for instance water.

Extraction of the resonance properties

The impedance analyzer used in these experiments measures the electrical admittance Ỹel,measure as a function of the frequency. This paragraph explains how the complex resonance frequency can be deduced from this measurement. It can be shown that the linearization around the open-circuit frequency used previously allows getting a simpler representation of the equivalent circuit, using only standard electrical elements (resistance, inductance and conductance), which corresponds to the Butterworth-Van-Dyke model [START_REF] Butterworth | On electrically-maintained vibrations[END_REF][START_REF] Van Dyke | The piezo-electric resonator and its equivalent network[END_REF], shown in Figure 2(d).

The Kirchhoff rules yield the electrical admittance of this circuit:

Ỹel,measure = iωC 0 + iω L1 + 1 iω C1 + R1 -1 (19)
It is possible to fit the measurement Ỹel,measure (ω) with this theoretical Ỹel (ω) and to deduce the complex resonance frequency from the values of the four elements, but it is helpful to use directly the following expanded Lorentzian functions which reduce the scatter in the fit parameters:

Ỹel = G el,max f f r Γ (f r -f ) 2 + Γ 2 cos(θ) + f r -f (f r -f ) 2 + Γ 2 sin(θ) + G el,of f (20) Ỹel = B el,max f f r Γ (f r -f ) 2 + Γ 2 cos(θ) - f r -f (f r -f ) 2 + Γ 2 sin(θ) + B el,of f (21)
Here, the fit parameters are the resonance frequency f r , the half-bandwidth at half-height Γ, the maximal conductance G el,max and susceptance B el,max , the conductance G el,of f and susceptance B el,of f offset, and the tilt angle θ. The complex resonance frequency is then simply deduced from the first two parameters:

fr = f r + iΓ (22)
3 Application to the characterization of viscoelasticity of a polydimethylsiloxane

Materials and methods

The quartz crystal and its holder used in the experiments are the commercially available QCM200 (Stanford Research Systems, CA, USA). The crystal has a resonance frequency near 5 MHz and a diameter of 2.54 cm. It is covered with circular electrodes of titanium and gold. It is physically maintained with one Oring on both side and connected with the electrical interface via BNC connectors, which are then adapted to an SMA connection. The portable network analyzer is the miniVNA PRO (mini Radio Solutions, Germany), shown in Figure 3. It can be used in a wide range of frequencies, between 100 kHz and 200 MHz in step of 1 Hz and measure impedance between 1 Ω and 1 kΩ.

It is connected to a personal computer with a USB connection and controlled by a custom program, measuring continuously the admittance of the TSM resonator in frequencies centered on the resonance frequency f r with a span of 6 • Γ, the resonance half-bandwidth, both parameters being evaluated by a fit using the trust-region reflective method with a sufficiently small termination tolerance to ensure a coefficient of variation smaller than 3 % [START_REF] Byrd | A trust region method based on interior point techniques for nonlinear programming[END_REF]. The whole setup is presented in Figure 4.

To minimize the variation of external parameters, the QCM is placed inside a thermostatic chamber at a given controlled temperature. The thermostatic chamber is located in a clean room which relative humidity is kept constant at approximately 50%. The network analyzer is initially calibrated by the short-open-load method. The quartz crystal is washed with acetone, rinsed with water and dried with nitrogen before being placed inside its holder. Once the resonance frequency is stabilized, the crystal is loaded with a volume of 900 µL of distilled water as the reference material, creating a layer thick enough to be considered as a semiinfinite medium. Finally, after stabilization of the resonance frequency, the water is removed and replaced by the same volume of 900 µL of PDMS rubber RTV615 (Momentive Performance Materials Inc., NY, USA) in the liquid state, i.e. non cross-linked. Following the manufacturer's recommendations, it was obtained by weighting out ten parts of the uncured rubber with one part of a curing agent. The components were then thoroughly mixed together, and degassed with exposition to partial vacuum for about ten minutes, in order to remove the entrapped air and guarantee an homogenous cured product.

Thus, the evolution of the viscoelastic properties was measured continuously from the liquid (uncross-linked) state to the solid (cross-linked) state. For information, Table 1 provides some mechanical properties obtained once the material is cured. The temperature of the PDMS sample surface was measured by using a CT-SF15-C3 Micro-Epsilon pyrometer, 15:1 in optical resolution, 8-14 µm in spectral range, 150 ms in response time and 0.1 K in thermal resolution. First, the analysis focuses on evolution of the viscoelastic properties at the lowest temperature. At 25 • C, the shear modulus evolution is strongly nonlinear from 4 MPa for the liquid state (only slightly cross-linked) to the stabilized value of 12 MPa for the solid state. Several regimes can be distinguished to describe the shear modulus evolution, as illustrated by the dashed line in Fig. 5(a). During the first ten hours, G' slowly increases, then strongly increases (from 10 to 17 hr), slowly increases (from 17 to 27 hr), before being almost stabilized at the 35 th hr. The loss factor decreases and evolves in a same nonlinear way from 1.6 to 0.8 (see Fig. 5(b)). The results show that in the megahertz domain the elastic modulus and the loss factor are high compared to values obtained with classical dynamic mechanical analysis (DMA), see for instance the work by [START_REF] Rey | Influence of the temperature on the mechanical behaviour of filled and unfilled silicone rubbers[END_REF] on PDMS under tension loading condition. This is expected as the frequency is increased. This measurement at 25 • C is consistent with the data provided by the material supplier, who states that the material is sufficiently cured in 24 hours to permit handling.

Results and discussion

This evolution in the viscoelastic properties is directly linked with the cross-linking kinetics. In lower frequency domains, modeling approaches have been developed

to predict the shear equilibrium modulus, most of them being based on the phenomenological models by [START_REF] Langley | Elastically effective strand density in polymer networks[END_REF] and [START_REF] Dossin | Rubber elasticity of well-characterized polybutadiene networks[END_REF]. In these models, the contribution of chemical crosslinks is distinguished from that of trapped physical entanglements. At much higher frequencies, leading to very low displacement amplitudes, the topological contribution to the shear modulus seems difficult to evaluate and the subsequent prediction of the shear equilibrium modulus is therefore delicate. Further investigations are therefore required to model the effects of the kinetics of cross-linking on the viscoelastic properties in the megahertz domain.

When the temperature is increased, the curves obtained at 50 • C and 80 • C highlight that the higher the temperature, the faster the evolution of viscoelastic properties and their stabilization. This result is expected since increasing temperature accelerates cross-linking. Furthermore, the first regime described at the lowest temperature is not observed anymore at the highest temperature at 80 • C and is reduced to the first hour at 50 • C.

In Figure 6, the time at viscoelastic properties stabilization for the three crosslinking temperatures have been compared to the times provided by the supplier (see the data-sheet curve in blue). The times measured and provided are very close.

Finally, temperature measurements were carried out at the material disc surface during the test. The results obtained at 25 , 50 and 80 • C are presented in Figure 7 in blue color, in red color and in orange color, respectively. Before analyzing the sample temperature evolution, it is necessary to recall that the QCM is placed inside the thermostatic chamber at the desired controlled temperature. The thermostatic chamber door has to be opened to place the quartz crystal inside its holder. After stabilization of the resonance frequency, the water used to calibrate the measurement is removed and replaced by the same volume of 900 µL of PDMS rubber RTV615 (the door is re-opened). Therefore, once the door is closed, a certain time is needed to achieve the thermal stabilization. At 25 • C, as the temperature is close to the ambient temperature of the laboratory, the temperature of the sample (see the blue curve in Fig. 7) is quite constant. At the highest temperatures, the door opening induces a significant effect: 40 minutes are needed for stabilizing the sample's temperature.

In a first approximation, the temperature change due to heat diffusion should follow a Fourier law, meaning that it should evolve as an exponential function of time. Nevertheless, this is not the case at 50 • C and even more at 80 • C. This is the signature of cross-linking kinetics, cross-linking being exothermal. The heat produced by cross-linking increases, becomes detectable by the pyrometer and the temperature curve shape changes.

At this stage, an important question arises. Indeed, the measurements have shown that temperature influences the cross-linking kinetics. Due to viscosity, the higher the material temperature, the lower (higher) the elastic (viscous) modulus (see the higher the cross-linking temperature, the higher the shear modulus and the lower the loss factor, meaning that the cross-linking kinetics has clearly an effect on the final viscoelastic properties.

Conclusion

The present paper aimed at presenting the QCM as a relevant tool for measuring the viscoelastic properties of rubbers in the megahertz domain. The advantage of such a technique is that the measurement can be carried out for liquids and for solids as well, i.e. from the uncross-linked to the cross-linked state. The volume of the material needed is small compared to classical characterizations. This technique is well adapted to measurements in an oven, i.e. at different temperatures. It has been used to successfully characterized the viscoelastic properties of a polydimethylsiloxane and their evolution during cross-linking at three different temperatures : 25, 50 and 80 • C. Results have shown that the final viscoelastic properties depend on the temperature during cross-linking. The higher the crosslinking temperature, the higher (lower) the final elastic (viscous) modulus at 25

• C. Therefore, increasing the temperature accelerates the cross-linking kinetics, but also influences the final viscoelastic properties (the material becomes stiffer and the loss factor decreases).

Figure 5

 5 Figure 5 gives the evolution of the viscoelastic properties in terms of shear modulus G' (Fig. 5(a)) and loss factor tan(δ) (Fig. 5(b)) versus time. Results obtained at 25 • C, 50 • C and80• C appears in red, blue and orange colors, respectively. First, the analysis focuses on evolution of the viscoelastic properties at the lowest temperature. At 25 • C, the shear modulus evolution is strongly nonlinear from 4 MPa for the liquid state (only slightly cross-linked) to the stabilized value of 12 MPa for the solid state. Several regimes can be distinguished to describe the shear modulus evolution, as illustrated by the dashed line in Fig.5(a). During the first ten hours, G' slowly increases, then strongly increases (from 10 to 17 hr), slowly increases (from 17 to 27 hr), before being almost stabilized at the 35 th hr. The loss factor decreases and evolves in a same nonlinear way from 1.6 to 0.8 (see Fig.5(b)). The results show that in the megahertz domain the elastic modulus and the loss factor are high compared to values obtained with classical dynamic mechanical analysis (DMA), see for instance the work by[START_REF] Rey | Influence of the temperature on the mechanical behaviour of filled and unfilled silicone rubbers[END_REF] on PDMS under tension loading condition. This is expected as the frequency is increased. This measurement at 25 • C is consistent with the data provided by the material supplier, who states that the material is sufficiently cured in 24 hours to permit handling.

  Figs 5(a) and 5(b)). But are the final viscoelastic properties the same? In other words, does the change in the kinetics affect the final properties? To answer this question, the viscoelastic properties of samples cross-linked at 50 and 80 • C have been measured at 25 • C.

Figure 1 .

 1 Figure 1. Two-port network representation of a layer in which propagate a shear-wave.

Figure 2 .

 2 Figure 2. Schematic diagrams of networks and circuits

Figure 4 .

 4 Figure 4. Photographs of the experimental setup.

Figure 5 .

 5 Figure 5. Evolution of the viscoelastic properties at different ambient temperatures, respectively 25 • C (in blue color), 50 • C (in red color) and 80 • C (in orange color).

Figure 6 .Figure 7 .

 67 Figure 6. Time at viscoelastic properties stabilization measured and provided by the supplier.

  Table 2 summarizes results obtained. Results show that

Corresponding author jean-benoit.lecam@univ-rennes1.fr Fax : (+33)

236 111


A Manchon Spintec

N Ryzhanova

A Vedyayev

B Dieny

Spin-dependent diffraction at ferromagnetic/spin spiral interface

Keywords: numbers: 72.25.-b, 73.43.Qt, 75.60.Ch, 75.50.Ee

Spin-dependent transport is investigated in ballistic regime through the interface between a ferromagnet and a spin spiral. We show that spin-dependent interferences lead to a new type of diffraction called "spin-diffraction". It is shown that this spin-diffraction leads to local spin and electrical currents along the interface. This study also shows that in highly non homogeneous magnetic configuration (non adiabatic limit), the contribution of the diffracted electrons is crucial to describe spin transport in such structures.

The recent observations of current-induced domain wall motion [1] and the investigation of this phenomenon by micromagnetic simulations [2] have underlined the question of spin transport in non homogeneous systems. Although proposed very early [3], current-induced domain wall motion (DWM) has attracted much attention because of its great application potential but also because of the deep fundamental question of the role of electron spin motion in temporally and spacially varying magnetic structures. Zhang et al. [4] proposed an analytical formula for spin torque in slowly varying structures (see eq. 9 in Ref. 4):

T = a ∂M ∂t + bM × ∂M ∂t -c 1 M × [M × (j e .∇)M ] -c 2 M × (j e .∇)M ( 1
)
where M is the magnetization unit vector, a is the renormalization factor of the gyromagnetic ratio and b is the renormalization factor of the damping parameter in the Landau-Lifshitz-Gilbert equation. The last two terms represent the spin transfer torque in spacially varying magnetic structure. The prefactors c 1 and c 2 are respectively proportionnal to the adiabatic and non adiabatic contribution. For smooth enough non-homogeneities of the magnetic structure, the adiabatic approximation is usually assumed : the electron spin follows the local magnetization producing a small torque proportional to the spatial derivative of the magnetization [5,6]. In this case, it is usually accepted that the non adiabatic term c 2 is small but cannot be neglected in domain wall experiments [7,8]. Furthermore, in spacially non homogeneous systems, c 1 and c 2 are non local coefficients [5,6,9]: at each point of the structure, one has to consider the contribution of all the electrons flowing through the structure.

In this article, we propose to study the non adiabatic regime in an "academic" system.

We consider two adjacent layers: the left one (z¡0, F) with a homogenous magnetization P = P z and the right one (z¿0, SS -for Spin Spiral) with a 2D helical magnetization contained in the (x,z) plane M = M(sin θ(x)x + cos θ(x)z), where θ(x) = θ 0 + Qx. The interface lies in the (x,y) plane and z is perpendicular to the interface. We consider that the regions are semi-infinite and respectively connected to a ferromagnetic and a spin spiral reservoir. The bias voltage V is applied across the interface.

Such helical spin structures exist in some compounds such as MnSi [10], oxide materials such as SrFeO 3 , NaCuO [11], rare-earth based compounds [12] or γ-iron [13]. This helical structure can be also a simplified picture of narrow stripe domains with domain wall width comparable to domain width.

In this case, the adiabatic approximation is no more valid because an electron moving through the interface keeps the memory of its spin state for some distance. Spin-polarized electrons moving from F into SS undergo "spin diffraction" as represented on Fig. 1: an impinging electron with an in-plane incident wavevector κ gives rise to transmitted (reflected)

waves with in-plane wavevectors κ + nQ/2 (κ + nQ), 2πQ -1 being the wavelength of the spin spiral. From the interference of all these waves, one can expect original electrical properties like non-zero torque acting on the magnetization of SS, local longitudinal spin current and even charge current (Hall effect and spin Hall effect [14]) along the F/SS interface.

To model the spin-dependent transport, we use Keldysh out-of-equilibrium technique [15,16] which expresses the lesser Keldysh Green functions G -+ σσ ′ (rr') as a function of the basis of wavefunctions Ψ l,rσ (r) for an electron moving from the left (right) to the right (left) reservoir:

G -+ σσ ′ (rr') = f l (µ l )Ψ * lσ ′ (r')Ψ lσ (r) + f r (µ r )Ψ * rσ ′ (r')Ψ rσ (r) (2)
where f l(r) (µ l(r) ) are the Fermi distribution functions in the left and right electrodes and µ l(r) are the chemical potentials in these electrodes so that V = (µ l -µ r )/e. The electrical current density J e , spin density m and torque T (exerted on SS) are given by the usual local definitions:

J e = ℑ[(∇ r' -∇ r )(G -+ ↑↑ (rr') + G -+ ↓↓ (rr'))] (3)
m x + im y =< σ + >= 2G -+ ↑↓ (rr') (4)
m z =< σ z >= G -+ ↑↑ (rr') -G -+ ↓↓ (rr') (5)
T = γJ sd hµ B M × m ( 6
)
where γ is the gyromagnetic ratio, µ B is the Bohr magneton and J sd is the sd exchange coupling. The torque T possesses two components: the usual spin transfer torque term (STT, adiabatic torque propotionnal to m y and lying in the (x, z) plane), and the field-like term (IEC, non adiabatic torque perpendicular to the (x, z) plane). The Hamiltonian of the system is then:

H = p 2 2m + U + J sd (σ.S) ( 7
)
where U is the potential profile, σ is the vector of Pauli matrices and S is the magnetization of the layer (S = M in SS and S = P in F). The wavefunctions are the solutions of the Schrödinger HΨ = EΨ. To solve these equation, we used the procedure developped by Calvo [17] in a spin spiral. The boundary conditions of these wavefunctions and their derivatives lead to recurrent relations between the coefficients of these wavefunctions [18].

For example, the wavefunctions of an initially majority electron originating from the left reservoir with in-plane wavevector κ are:

Ψ ↑(↑) lSS = n [c 3(κ+Q(n-1 2 )) (cos θ(x) 2 -iφ κ+Q(n-1 2 ) sin θ(x) 2 )e ik 3(κ+Q(n-1 2 )) z + c 4(κ+Q(n-1 2 )) (φ κ+Q(n-1 2 ) cos θ(x) 2 + i sin θ(x) 2 )e ik 4(κ+Q(n-1 2 )) z ]e i(qy+(κ+Q(n-1 2 ))x) (8)
Ψ

↓(↑) lSS = n [c 3(κ+Q(n-1 2 )) (sin θ(x) 2 + iφ κ+Q(n-1 2 ) cos θ(x) 2 )e ik 3(κ+Q(n-1 2
)) z + c 4(κ+Q(n-1 2 )) (-i cos θ(x) 2 + φ κ+Q(n-1 2 ) sin θ(x) 2 )e ik 4(κ+Q(n-1 2
)) z ]e i(qy+(κ+Q(n-1 2 ))x) (9) where c 3(4) are complex diffraction coefficients and k 3(4) are wavevectors for majority (minority) spin projection in SS. This procedure will be developped in a forthcoming article [18].

For the numerical simulations, we took parameters corresponding to spin transport in Co: the Fermi wavevectors for majority and minority spins are respectively k ↑ F = 1.1 Å-1 , k ↓ F = 0.6 Å-1 ; the inverse wavelength of the spin spiral is Q -1 = (2π) -1 Å-1 (highly non homogeneous magnetic system). The Fermi wavevectors of SS and F are the same and we consider the linear approximation : for a small enough bias voltage, only the electrons originating from the left reservoir with an energy located between µ l and µ r significantly contribute to the charge and spin transport. We set µ l -µ r = 38 meV.

Figure 2 displays the longitudinal charge current J e x in the (x, z) plane. Close to the interface, J e x oscillates with SS magnetization, but rapidly decreases and vanishes to zero within 5 Å. We observe the same behaviour for the perpendicular current J e z except it reaches an averaged value within 5 Å. We observe the same features for STT and IEC (not shown here). This rapid decay is due to interferences between diffracted electron waves. Close to the interface, the averaging effect is smaller than in the bulk SS, so the oscillation amplitude of J e

x (z) and STT, IEC is always higher near the interface than in the bulk. These damped oscillations strongly depend on the SS wavelength 2πQ -1 [18]. Fig. 3 displays the z-dependence of the longitudinal current J e x [Fig. 3 When Q increases, the oscillation amplitude of J e

x is reduced so that J e x vanishes more rapidly to zero: the increase of the non adiabaticity induces a more important averaging effect due to interference between multiple diffracted waves. STT and IEC have an opposite behaviour when varying Q. The oscillation amplitude of STT decreases when Q increases (similarly to J e x ), whereas the amplitude of IEC increases. This illustrates the different nature of STT and IEC: STT is the adiabatic torque (vanishes in highly non homogeneous structure) and IEC is the non adiabatic torque (vanishes in adiabatic systems).

This study demonstrates that spin diffraction gives rise to complex characteristic in spin torque and electrical currents and is of seminal importance in non adiabatic magnetic systems.

This work was partially supported within the European MRTN SPINSWITCH CT-2006-035327 and the Russian Fundings for Basic Research 07-02-00918-a.

FIGURE CAPTIONS

  (a)], STT [Fig. 3(b)] and IEC [Fig. 3(c)] for different values of SS wavelenght Q -1 .

Figure 1 :

 1 Figure 1: Cartoon of the bilayered structure. The left semi-infinite layer is a ferromagnet with a homogeneous magnetization and the right semi-infinite layer is a spin spiral with wavelength 2πQ -1 . The spin-polarized electrons undergo spin-diffraction at the interface.

Figure 2 :Figure 3 :

 23 Figure 2: Map of the longitudinal electrical current density in the (x, z) plane.