Hélène Maire
email: helene.maire@univ-lorraine.fr

Rawan Charafeddine

Jean-Baptiste Van Der Henst

Interaction and its failures: an approach through embarrassment and shame

Keywords: social norm, transgression, embarrassment, shame, power situations, mixed-gender interactions, children

The present study is a theoretical and methodological proposal rooted in the field of social developmental psychology and describes three objectives. First, it aims to show how transgressive situations are conducive to study of social norms governing interactions. Second, the similarities and differences between two emotions, namely shame and embarrassment, are outlined in order to better understand how their respective measures can highlight social norms in interactions. Third, we illustrate our proposal to study social norms through emotional reactions by presenting an ongoing experiment, that examines how children may react to a power situation.

Studying the contours of interactions through transgressions of social norms

Social norms: definitions and functions

The term interaction refers to "reciprocal influence that participants have on each other's actions when they are in physical presence" [START_REF] Goffman | Interaction ritual: Essays on face to face behavior[END_REF]. Far from being random or arbitrary, an interaction is guided by social norms or conventions, namely, rules of socially valued behaviors or attitudes a given context (e.g., ways of speaking, dressing, using artifacts, etc.). Social norms play not only a descriptive role (i.e., as mere reflections of what is done, of what is taken for granted) but also a prescriptive one, as "compass for action", by encouraging certain socially desirable behaviors and attitudes while discouraging others. Social norms invite group members to coordinate and to display their commitment to the group, and thus they aim to reduce interpersonal conflicts and facilitate a cooperative functioning [START_REF] Rakoczy | The early ontogeny of social norms[END_REF][START_REF] Rakoczy | Young children understand the normative force of standards of equal resource distribution[END_REF].

The ability to assimilate and enforce social norms emerges at an early age, not only when norms are perceived as legitimate and emanating from an already established social group, but also when they are perceived as new and arbitrary. For instance, [START_REF] Abrams | The development of subjective group dynamics[END_REF] observed that children aged 9 to 11 judged peers (i.e., ingroup members) who deviate from group norms more negatively than outgroup deviants, and preferred outgroup children who condemn deviance over ingroup members who do it ("black sheep effect"). Moreover, when 3-year-olds were taught the rules of a new game and then watched a puppet playing the game without following the rules, they taught the puppet the right way to play [START_REF] Rakoczy | The sources of normativity: young children's awareness of the normative structure of games[END_REF][START_REF] Rakoczy | Young children's understanding of the contextrelativity of normative rules in conventional games[END_REF][START_REF] Schmidt | Young children attribute normativity to novel actions without pedagogy or normative language[END_REF][START_REF] Schmidt | Young children enforce social norms selectively on the violator's group affiliation[END_REF]. Thus, even very young children are not merely passive targets of social norms, but they actively contribute to enforcing them.

Sanctions for transgressions

In everyday interactions, the behavioral guidance of norms is performed by social sanctions emanating from the other interactants. Sanctioning a deviant behavior allows one to: 1) show adherence to norms, in order to build a positive public image, and 2) reinforce the predictability and the controllability of conducts, thus stabilizing coordination and promoting cooperation [START_REF] Rakoczy | The early ontogeny of social norms[END_REF]. These other interactants sanctions take the form of verbal and/or non-verbal behaviors: some sanctions are positive and approving, rewarding the exemplary compliance of normative behaviors (e.g., smiling, explicit or tacit acquiescence, etc.), while other sanctions are negative and disavowing, punishing the infraction in response to deviant behavior (e.g., admonishment, angry looks, etc.) [START_REF] Goffman | Interaction ritual: Essays on face to face behavior[END_REF].

However, these two types of sanctions are not equally salient. For instance, in a Western country, a woman wearing a skirt and mascara is unlikely to elicit any particular reaction and is perceived as normal, while a man dressed in the same way is likely to elicit some negative and noticeable reactions (e.g., surprise, distrust, outright hostility…). Similarly, positive sanctions in the course of an interaction can be considered as "bonuses" for adherence to social norms: they relentlessly support the course of the exchanges underpinning a fluid and predictable "expressive order" [START_REF] Goffman | Interaction ritual: Essays on face to face behavior[END_REF], but go unnoticed. In the example above, compliance with a gendered dress code may sometimes elicit polite assent, but most of the time, this assent is completely invisible. Conversely, the reactions of disapproval that follow the transgression of a norm are salient, even voluntarily amplified to appear as conspicuous.

Bystanders may blame the counter-normative behavior, express skepticism through mimicry, disappointment or disgust, to ostensibly disassociate themselves from the author, etc. These visible marks of disapproval are intended to deter the transgressor from reoffending and any witnesses from committing an infraction in turn. In this sense, the negative sanctions that follow deviant behavior contribute to informal social control [START_REF] Chekroun | Social control behavior: The effects of social situations and personal implication on informal social sanctions[END_REF].

The integration of norms and peer pressure in cases of deviance has been observed in very young children. The case of gender norms is particularly illustrative since their transgression is judged negatively even by preschoolers. For example, negative reactions to atypical behavior (e.g., refusing to be friends with a boy who wears nail polish) appear as early as age 3 and increase until age 5 [START_REF] Stoddart | Children's concepts of cross-gender activities[END_REF][START_REF] Ruble | Gender development[END_REF][START_REF] Ruble | The role of gender constancy in early gender development[END_REF]. Moreover, social control by peers is itself gendered, since it is greater when: 1) it comes from male witnesses than from female witnesses (boys are often more negative about violations of gender norm than girls who have more egalitarian attitudes), and 2) the transgressor is a boy (e.g., [START_REF] Blakemore | Children's beliefs about violating gender norms: Boys shouldn't look like girls, and girls shouldn't act like boys[END_REF][START_REF] Ruble | Gender development[END_REF]. As in adults, it is likely that negative sanctions such as rejection or teasing are more visible than positive sanctions that reinforce gender-conforming behavior.

Embarrassment and shame as observable traces of emotional reactions to a transgression

Reflexive social emotions as integrated "thermostat" supporting the expressive order of interactions?

Disapproving witnesses often have no particular legitimacy to sanction [START_REF] Osgood | Routine activities and individual deviant behavior[END_REF], but take on the temporary role of "Robin Hood" for the situation, becoming temporary vectors of informal social control. While effective in "bringing deviants into heel", these types of negative sanctions require the active intervention of on-the-ground witnesses, making the process costly and relatively uncertain. Indeed, witnesses may be inhibited or reluctant to act due to fear of retaliation, lack of resources, motivation, cognitive resources, or courage, etc. Moreover, the social control exercised by the witnesses necessarily acts "downstream", once the infraction has been committed. To these external sanctions, internal sanctions must be added in order to optimize the maintenance of the expressive order. The interactants not only incorporate norms (among others in the form of habitus, [START_REF] Bourdieu | Habitus, code et codification[END_REF], they also internalize the penalizing potential that could result from their violation. Thus, people tend not to deviate from the norm to avoid sanctions such as the fear of ridicule, which is quite mild, to social opprobrium, which can be much more harmful.

In addition, psychological and social motivations related to self-presentation, reputational concerns and impression management also encourage people to shape their own public image according to social norms. Indeed, human beings early 1) assume implicitly that their behavior or appearance could be evaluated by others positively or negatively, and 2) have a default preference to elicit positive instead of negative evaluations from others [START_REF] Botto | Sensitivity to the evaluation of others emerges by 24 months[END_REF], 2019). Consequently, they use reputational tactics like conforming to majority opinion (Haun & Tomassello, 2011), or appearing to be cooperative: for example, adults tend to be more generous in public compared to in private [START_REF] Dana | Exploiting moral wiggle room: Experiments demonstrating an illusory preference for fairness[END_REF] and children become more altruistic when a third-party observer is watching them (i.e., sharing more resources, being more helpful, stealing less or cheating less; [START_REF] Engelmann | Five-year olds, but not chimpanzees, attempt to manage their reputations[END_REF][START_REF] Leimgruber | Young children are more generous when others are aware of their actions[END_REF][START_REF] Piazza | Princess Alice is watching you": Children's belief in an invisible person inhibits cheating[END_REF][START_REF] Shinohara | Strategic reputation management: Children adjust their reward distribution in accordance with an observer's mental state[END_REF]Shinohara et al., , 2021)). Both avoiding deviant conducts and building one's positive public image are "upstream" self-regulation that contributes to informal social control, in addition to that exercised by others. Taken together, these mechanisms (i.e. reactive or "downstream" vs. preventive or "upstream", originating from others vs. oneself) ensure an anticipatory filtering of counter-normative behaviors.

Nevertheless, these filtering mechanisms are not always sufficient to prevent certain deviations from occurring. Among internal mechanisms, in addition to preventive sanctions, some reactive sanctions, acting a posteriori (i.e., once the norm has been transgressed), are also necessary. To be effective, they must act quickly, almost automatically and in a way that is relevant to the situation. Certain emotions may play this "thermostat" role in any potential transgressor, allowing upstream self-regulation. For example, when pairs of 3-, 4-and 5-yearold children were confronted with a norm violation such as inequity in resource distribution, namely they receive either more or less stickers than their social partner, they expressed negative emotions, such as unhappiness (LoBue et al., 2010). Specifically, reflexive social emotions such as shame, guilt, pride or embarrassment, participate in early embodied social control [START_REF] Semin | The acquisition of reflexive social emotions: The transmission and reproduction of social control through joint action[END_REF]. Indeed, it is well established that these reflexive social emotions motivate strategies to avoid rejection and facilitate relational appeasement and social cohesion (e.g., [START_REF] Chobhthaigh | Children's understanding of embarrassment: Integrating mental time travel and mental state information[END_REF][START_REF] Miller | Embarrassment: Poise and peril in everyday life[END_REF][START_REF] Muris | Small or big in the eyes of the other: On the developmental psychopathology of self-conscious emotions as shame, guilt, and pride[END_REF]. Children acquire these emotions later than basic emotions (happiness, fear, sadness, anger, surprise, disgust) which exist from the first year of life (e.g., [START_REF] Colonnesi | Development in children's attribution of embarrassment and the relationship with theory of mind and shyness[END_REF][START_REF] Lewis | Differences in shame and pride as a function of children's gender and task difficulty[END_REF][START_REF] Tangney | Shame and guilt across the lifespan. The development of moral emotions[END_REF]. Their development goes hand in hand with the acquisition of social norms, in a double movement. It is because children assimilate social norms that they are gradually able to evaluate their own behavior as deviant or not, and thus become capable of feeling shame or embarrassment when they transgress (or pride when their conduct is noticeably exemplar or conforms to the norm perceived as important). Moreover, it is also because shame and embarrassment are unpleasant that children internalize norms effectively. In line with the image proposed by [START_REF] Harris | Embarrassment: A form of social pain[END_REF], just as physical pain warns the organism of a threat to its physical integrity, shame and embarrassment are a kind of "social warning" that allows the maintenance of the individual's "social integrity". In addition, their display after violating a norm communicates the transgressor's knowledge of the violated norm and a submissive apology to appease others [START_REF] Keltner | The signs of appeasement: Evidence for the distinct displays of embarrassment, amusement, and shame[END_REF][START_REF] Tangney | Are shame, guilt, and embarrassment distinct emotions[END_REF]. In short, these painful emotions prevent deviance (upstream, with the avoidance of transgressions) or minimize it (downstream, with reparative behaviors and the message of "repentance" for one's conduct expressed via the simple manifestation of one's emotion).

Indicators of shame and embarrassment, emotions typical of public norm transgressions

When violating a norm, transgressors may react emotionally either to the violation itself or the sanctions from witnesses. Their emotional responses encompass a variety of observable states, ranging from simple clumsiness to an inability to express oneself. Among reflexive social emotions, embarrassment and shame are of particular interest for four reasons.

First, they always occur in front of a real or imaginary audience (e.g., [START_REF] Cova | Honte[END_REF][START_REF] Miller | Embarrassment: Poise and peril in everyday life[END_REF][START_REF] Smith | The role of public exposure in moral and nonmoral shame and guilt[END_REF]. Second, they result from a norm transgression. In the case of embarrassment, the infringed norm is a social convention: there is a discrepancy between the real image that one gives by one's conduct and a status or a role that one is supposed to assume (for example, arriving in disguise at a party where no one else is, using familiar terms when talking to one's superior). In the case of shame, a shortcoming contravenes a personal standard often associated with a real or projected status, that is damaged or lost (e.g., failing through incompetence in an oral presentation that is important to us) or a norm underpinned by moral values (e.g., allowing someone to be punished instead of oneself, laughing with others at someone who is absent) [START_REF] Lewis | Self-conscious emotions: Embarrassment, pride, shame, and guilt[END_REF][START_REF] Maire | L'embarras, émotion ambivalente et complexe: une présentation théorique[END_REF]. Third, they are associated with an evaluation of the Self, either in its central aspects for shame, or in more peripheral ones for embarrassment. Since norms are incorporated, their violation could elicit emotional states related to self-evaluation and to the (in)adequacy to the ongoing interaction. Finally, these two emotions are not only felt, but they are also expressed. Their expression serves a communicative purpose, as they signal to others and adherence to group social and/or moral norms.

A key issue for interactants is to identify expressions of shame and embarrassment.

Verbally, they are often exemplified by silence and show few typical manifestations, only some negative self-evaluations (e.g., "I'm no good at this" for shame), in contrast to positive selfevaluations for pride (e.g., "aah!", "I did it!") [START_REF] Alessandri | Parental evaluation and its relation to shame and pride in young children[END_REF][START_REF] Miller | Embarrassment: Poise and peril in everyday life[END_REF]Edelman & Hampson, 1981;[START_REF] Lewis | Differences in shame and pride as a function of children's gender and task difficulty[END_REF]. In speech, embarrassment can be identified by paraverbal cues such as stuttering, an unusually low-or high-pitched voice, quavering or slurred speech, nervous laughter or giggling. Finally, shame and embarrassment share certain non-verbal displays: blushing, blanching, blinking, sweating, lowering of the eyes, gaze aversion, attempts to hide face bowing of the head, tremor of the hand and hesitating or vacillating movement [START_REF] Botto | Sensitivity to the evaluation of others emerges by 24 months[END_REF][START_REF] Goffman | On facework: An analysis of ritual elements in social interaction[END_REF][START_REF] Goffman | Embarrassment and social organization[END_REF][START_REF] Keltner | The signs of appeasement: Evidence for the distinct displays of embarrassment, amusement, and shame[END_REF][START_REF] Ogien | Embarras[END_REF][START_REF] Rochat | Shame and self-knowledge[END_REF][START_REF] Stipek | Self-evaluation in young children[END_REF][START_REF] Tracy | Development of a FACS-verified set of basic and selfconscious emotion expressions[END_REF]. However, embarrassment is characterized in adults by a dynamic and orderly 5-second sequence of downward gaze, head movement and a non-Duchenne, nervous, coy or silly smile, as well as increased body motion (Edelman & Hampson, 1981;[START_REF] Keltner | The signs of appeasement: Evidence for the distinct displays of embarrassment, amusement, and shame[END_REF]. Nervous laughter and hands covering the mouth have been observed in young children [START_REF] Buss | The development of embarrassment[END_REF]. Shame, on the other hand, has a static and non-smiling display, with, in children, body collapsed, corners of the mouth downward, lower lip tucked between teeth, eyes lowered with gaze downward or askance, withdrawal from task situation (in contrast to pride, which is defined as erect posture with shoulders back and head up, smile, eyes directed at others, points at outcome or applauds; [START_REF] Alessandri | Parental evaluation and its relation to shame and pride in young children[END_REF][START_REF] Miller | Embarrassment: Poise and peril in everyday life[END_REF][START_REF] Ferguson | Children's understanding of guilt and shame[END_REF][START_REF] Lewis | Differences in shame and pride as a function of children's gender and task difficulty[END_REF]. The main similarities and differences between shame and embarrassment are summarized in Table 1.

Shame Embarrassment Triggers

Public transgression (real or imagined) of a norm... ...underpinned by (global) moral values or personal standards ...linked to a (local) social convention [START_REF] Smith | The role of public exposure in moral and nonmoral shame and guilt[END_REF] Link with the status in interaction Associated with a loss of status Associated with a change in status or with a counter-normative status Evaluation… …of the core self …of the presentation self [START_REF] Miller | Embarrassment: Poise and peril in everyday life[END_REF][START_REF] Tangney | Shame and guilt across the lifespan. The development of moral emotions[END_REF] Age From 3 years Primitive embarrassment (sensitive to an audience reaction): from 5 years Mature embarrassment (less sensitive to an audience reaction, resulting from a discrepancy between the desired self-image and the effective self-image): from 6 to 10 years? [START_REF] Bennett | Children's self-attribution of embarrassment[END_REF][START_REF] Chobhthaigh | Children's understanding of embarrassment: Integrating mental time travel and mental state information[END_REF][START_REF] Colonnesi | Development in children's attribution of embarrassment and the relationship with theory of mind and shyness[END_REF][START_REF] Lewis | Differences in shame and pride as a function of children's gender and task difficulty[END_REF][START_REF] Seidner | A developmental analysis of elementary school-aged children's concepts of pride and embarrassment[END_REF] Experience Self-evaluation … … always negative: depression, regret, feeling of being worthless, incompetent, inferior or powerless … often negative: feeling of being clumsy, awkward, unsuited to the situation, uncomfortable, unmasked [START_REF] Maire | L'embarras, émotion ambivalente et complexe: une présentation théorique[END_REF] Verbal cues Silence, little/no typical verbal expression [START_REF] Alessandri | Parental evaluation and its relation to shame and pride in young children[END_REF][START_REF] Miller | Embarrassment: Poise and peril in everyday life[END_REF][START_REF] Buss | The development of embarrassment[END_REF]Edelman & Hampson, 1981;[START_REF] Ferguson | Children's understanding of guilt and shame[END_REF][START_REF] Lewis | Differences in shame and pride as a function of children's gender and task difficulty[END_REF] Negative self-evaluation de soi négative (e.g., "I'm no good at this") ?

Paraverbal cues

Stuttering, quavering/disturbed speech or breaking of the voice, [START_REF] Ferguson | Children's understanding of guilt and shame[END_REF][START_REF] Goffman | On facework: An analysis of ritual elements in social interaction[END_REF][START_REF] Keltner | The signs of appeasement: Evidence for the distinct displays of embarrassment, amusement, and shame[END_REF][START_REF] Ogien | Embarras[END_REF][START_REF] Tracy | Development of a FACS-verified set of basic and selfconscious emotion expressions[END_REF] ?

Unusually low-or high-pitched voice Nervous laughter and giggling

Non-verbal cues

Blushing, blanching, blinking, sweating, lowering of the eyes, gaze aversion, attempts to hide face bowing of the head, tremor of the hand, and hesitating or vacillating movement (Alessandri & Lexis, 1993[START_REF] Miller | Embarrassment: Poise and peril in everyday life[END_REF][START_REF] Botto | Sensitivity to the evaluation of others emerges by 24 months[END_REF][START_REF] Buss | The development of embarrassment[END_REF]Edelman & Hampson, 1981;[START_REF] Ferguson | Children's understanding of guilt and shame[END_REF][START_REF] Goffman | On facework: An analysis of ritual elements in social interaction[END_REF][START_REF] Keltner | The signs of appeasement: Evidence for the distinct displays of embarrassment, amusement, and shame[END_REF][START_REF] Lewis | Differences in shame and pride as a function of children's gender and task difficulty[END_REF][START_REF] Seidner | A developmental analysis of elementary school-aged children's concepts of pride and embarrassment[END_REF]

Static display No smile

Body collapsed, corners of the mouth downward/lower lip tucked between teeth, eyes lowered with gaze downward or askance, withdrawal from task situation

Dynamic display in an ordered and brief sequence (coy, silly, non-Duchenne smile, head movement)

Increased body motion Nervous laugh and hands covering the mouth

Interpersonal functions

Inhibition of morally undesirable/inappropriate behavior

Communication of adherence to social norms and a desire for appeasement [START_REF] Keltner | The signs of appeasement: Evidence for the distinct displays of embarrassment, amusement, and shame[END_REF][START_REF] Tangney | Are shame, guilt, and embarrassment distinct emotions[END_REF] Table 1: Summary of similarities and differences regarding triggers, self-evaluation, age, associated experiences, interactional cues (verbal, paraverbal, nonverbal), and interpersonal functions of shame and embarrassment. Items in italics refer to results obtained in children.

Thus, embarrassment and shame, insofar as they produce observable traces in the interaction which serve to signal to others that one has internalized the social norms, constitute indirect but effective means to study social norms. Since even young children know social norms and are able to express social reflexive emotions, below we propose to examine how they emotionally respond to a violation of a gender norm.

Empirical proposal

3.1. Asymmetrical situations as highly normed situations useful to study the emotional reactions of a transgressor?

Although norms permeate social life from its inception, their influence on behavior varies according to context. Asymmetrical situations, in which an individual has power over another, may be good candidates to study norms transgressions, for at least three reasons. First, this type of interactions is often associated with highly codified communication rules, which make deviations from the norm very salient. Second, power relations often carry high stakes because they are often related to symbolic, economic or identity-related issues. As a result, actors may be particularly motivated to ensure that exchanges run smoothly, and therefore the sanctions for failure may be more marked than in egalitarian interactions. Third, power relations involve a verticality that engages social self-evaluation. A person's social image is gauged 1) as a dominant or subordinate individual, and 2) as a representative of one's own social group whose status varies within the social space (cf. "ingroup self", [START_REF] Pinto | L'Effet Brebis Galeuses : réactions à la déviance en contextes entre groupes[END_REF]. Such a public negative self-evaluation is at the heart of the reflexive social emotions mentioned above.

In sum, since power asymmetrical interactions provide an example of easily detectable, heavily sanctionable and potentially emotionally charged transgressions, they appear as an opportune context to identity some traces of embarrassment and shame, as reflections of internalized social control and incorporation of norms.

Ongoing study

Here we present the experimental design of an ongoing study with 8-9 year old children, which is part of a larger project investigating how the conceptualization of gender hierarchies emerge in childhood. People typically view the gender distinction as a status distinction [START_REF] Carli | Gender, hierarchy, and leadership: An introduction[END_REF], and by preschool age, children are already aware of male power [START_REF] Charafeddine | How preschoolers associate power with gender in male-female interactions: A cross-cultural investigation[END_REF]. In a cultural context imbued with a gender norm associating power with masculinity, one can ask about children's readiness to identify with the dominant or subordinate character, not only in situations where the interaction conforms to the norm (M>F) but also when it contravenes the norm (F>M). Children's reactions may be an indicator of the degree to which the male power norm permeates the construction of their social self. We therefore seek to investigate the social reflexive emotions that arise in these situations.

In this work in progress, the experiment consists of presenting children with an image of two characters interacting, one in a posture of dominance (head up, pointing the finger like someone giving an order) and the other in a posture of subordination (head down, contrite posture, see Figure 1). Children are first asked to identify with one of the two characters, by associating their own picture with the chosen character (i.e., dominant or subordinate). Then, they are shown the face of the other character, revealing their gender. The resulting power interaction can be either mixed (F-M) or unmixed (F-F, M-M), and either normative (M>F) or counter-normative (F>M). A hidden camera films the children as they discover the gender of the other character in the image, in order to capture any possible facial, gestural and vocal emotional expressions. Based on the respective indicators identified above (Table 1), these different expressions will be rated using the Facereader software. Children are also asked to indicate, on illustrated scales, their level of embarrassment, shame, surprise, pride and fear immediately after the discovery of the other character's gender. In addition, self-perception (Harter, 1982;[START_REF] Maintier | Validation d'un questionnaire d'auto-évaluation de Soi destiné aux enfants[END_REF] and general emotional state are assessed before and after this sequence through direct questionnaires. Shyness and competitive behavior are also rated through parental questionnaires [START_REF] Crozier | Shyness and self-esteem in middle childhood[END_REF][START_REF] Tassi | Task-oriented versus other-referenced competition: Differential implications for children's peer relations[END_REF].

Our main hypothesis is that the counter-normative mixed-gender condition (F>M) will elicit more shame in boys than the same-gender condition (M>M), and more embarassment in girls than the same-gender condition (F>F). Indeed, boys will consider the F>M interaction as a loss of status, that may elicit shame. In contrast, girls will see it as a discrepancy between their gender and the high status that they endorsed in the interaction, that may create embarrassment (no pride). A secondary hypothesis is that variations in emotional state and self-esteem (i.e., pre-vs. post-evaluations) will be greater in the mixed-gender condition than in the same-gender condition. In addition, shyness is expected to be associated with more shame and embarrassment, particularly in the counter-normative situation, and competitive behavior is expected to be associated with more pride.

Avenues of discussion and questions in progress

The present paper proposed that everyday transgressions, especially those involving power relations, may be useful for understanding how social norms govern interactions because transgressions often produce observable emotional reactions. However, many questions regarding gender identity and cognitive development remain unanswered.

In the ongoing study as well as in most publications, gender identity is considered as stable, objective and treated as a binary measure (i.e., participants are either boys or girls).

However, recent literature indicates that some children might not identify with a binary gender.

It suggests that gender identity may also be treated as a subjective and continuous measure, for example by using a continuum ranging from 'feeling totally like a boy' to 'feeling totally like a girl' [START_REF] Gülgöz | Between a boy and a girl: Measuring gender identity on a continuum[END_REF]. Adding a measure of felt gender identity to our current protocol may be a mean to capture some gender differences in responses to counter-normative situations in a more fine-grained way.

Moreover, will 8-9-year-old children be old enough to have integrated gender norms and hence be sensitive to the proposed counter-normative situation? In addition, comparisons between different age groups would also be useful to see if with age, emotional reactions change. On the one hand, it can be hypothesized that the experienced emotions might increase because internalized social control becomes more powerful as social norms are acquired. On the other hand, it can also be hypothesized that the expressed emotions might decrease because the control of emotional expressions develops, thus making it possible to mask them if necessary. Moreover, the asymmetrical situation presented here induces social power via postures related to physical dominance, that are salient aspects of the interaction. But the cues that children use to infer power relations are numerous and go far beyond the physical manifestation of power. Individuals considered as powerful are those who 1) access to or control over resources, 2) achieve their intended goals at the expense of others, 3) control others' outcomes, for example by granting and denying permission, 4) giving others orders, for example to divide labor, and/or 5) setting norms, that may be explicitly stated (e.g., powerful children deciding the rules to a game) or implicitly manifested (e.g., peers imitating powerful children's behaviors or appearance) [START_REF] Charafeddine | How preschoolers associate power with gender in male-female interactions: A cross-cultural investigation[END_REF][START_REF] Gülgöz | Who's the boss? Concepts of social power across development[END_REF]. Thus, could we sketch the same hypotheses by inducing power asymmetry through other power cues, that might be less immediately readable than physical dominance? Finally, we can also ask whether children are aware of their emotional reaction. If so, how do they explain it? Can they invoke i) personality traits vs. gender-specific characteristics, ii) stable vs. temporary explanations, iii) global vs. local, limited to the situation presented?

Figure 1 :

 1 Figure 1: Time course of the study (NB: in the example, the participant is a girl, the interaction is mixed and the situation is counter-normative since the participant has chosen the dominant character).

project supported by the French National Research Agency.


A Chierici

L Chirco

V Le Chenadec

R Scardovelli

Ph Yecko

S Zaleski

An optimized Vofi library to initialize the volume fraction field

Keywords: implicit function, numerical integration, volume fraction function, VOF method

The Vofi (Volume Of Fluid Initializer) library has been developed to initialize the volume fraction field determined by implicitly-defined interfaces. The major conceptual changes in the numerical algorithms of the library are discussed and a few new features, including the computation of the reference phase centroid and of the interface length/area, are presented and applied to grid cells that are cuboids. Several numerical tests are considered to demonstrate both the accuracy of the new features, as the grid resolution and the number of integration points are varied, and the considerably improved efficiency of the library with respect to its previous version. A few of these tests are also included in the software distribution written in C, examples of C++ and Fortran interfaces are also provided.

Introduction

Direct numerical simulation (DNS) is a popular method for detailed studies of multiphase flows which are encountered in many scientific disciplines, such as thermo-fluid dynamics, oceanography, chemistry and engineering. The Volume-of-Fluid (VOF) method is one of the numerical techniques that is used to predict the evolution of interfaces in multiphase flows within the framework of DNS [START_REF] Scardovelli | Direct numerical simulation of free-surface and interfacial flow[END_REF][START_REF] Tryggvason | Direct Numerical Simulations of Gas-Liquid Multiphase Flows[END_REF]. The method is based on the characteristic function χ(x, t), a multidimensional Heaviside step function, with value 1 in the reference phase and 0 elsewhere, at time t. The integral of the function χ over a computational domain D provides the volume of the portion of D that is occupied by the reference phase. The volume fraction C is associated to the function χ and it represents the fraction of a grid cell that is occupied by the reference phase, as defined by Eq. ( 2).

An accurate representation of the volume fraction field is necessary for an exact initialization of the total mass at the beginning of a numerical simulation. It is also required to compute convergence rates with grid spacing of geometric properties of the interface, such as its normal vector and cur-vature, and of instability growth rates [START_REF] Bagué | Instability growth rate of two-phase mixing layers from a linear eigenvalue problem and an initial-value problem[END_REF]. In a previous paper [START_REF] Bnà | Vofi -a library to initialize the volume fraction scalar field[END_REF], we have presented the first release of the Vofi library that can be used to initialize the volume fraction C in a computational grid with cubic cells of edge l 0 , given a user-defined implicit equation of the interface, f (x) = 0. The library is based on two main assumptions: 1) in a cell cut by the interface the implicit equation can be locally written explicitly as x 1 = h(x 2 , x 3 ), where each x i is one of the three coordinate directions; 2) the interface cannot intersect a cell edge more than twice. Both assumptions are verified if the local radius of curvature R of the interface is greater than the spatial step l 0 .

The Vofi algorithm is divided into three parts, as shown in the flowchart of Fig. 1. In the first part each grid cell is investigated to determine if it is empty, full or cut by the interface. In a cut cell, the function gradient is estimated in the cell with finite differences to change from the reference system with coordinates (x, y, z) to a local coordinate system with (x 1 , x 2 , x 3 ). In the second part the limits of integration are computed. In the third part the local height function h(x 2 , x 3 ) is computed numerically along the first direction x 1 at each node (x 2 , x 3 ) and the cut volume is computed with a double Gauss-Legendre quadrature rule. The function h satisfies the relation 0 ≤ h ≤ l 0 . The internal integration is performed along direction x 2 with n 2 nodes, while the external integration along direction x 3 with n 3 nodes.

In this paper we discuss the major changes in the numerical algorithms, demonstrate the improved efficiency of the library with respect to its previous version, and present a few new features. Most of the routines have been completely rewritten in order to considerably reduce the number of calls of the implicit function, where the CPU-time is mostly spent. The grid cell analysis and the numerical integration scheme are based on different algorithms, while the new features include the computation of the reference phase centroid and of the interface length in 2D and area in 3D. Moreover, grid cells now can be cuboids.

The library is written in C, as was the first release of Vofi, and a few two-dimensional and three-dimensional tests are provided with the software distribution, including visualization of the height data and interface triangulation. A version of the numerical tests, written both in Fortran and in C++, is also included to show how to call the library routines from these two languages.

Revised numerical algorithms and new features

Let D ⊂ R n be a computational domain and Ω the portion occupied by the reference phase

Ω = {x ∈ D : f (x) ≤ 0} , (1)
where points on the interface satisfy f (x) = 0, and f (x) < 0 for points in the interior of Ω. The interface Γ is a planar curve when n = 2, and a surface when n = 3. The characteristic function χ(x, t 0 ) at the initial time t 0 is equal to 1 inside Ω and 0 elsewhere. We consider a Cartesian subdivision of the domain D with cuboids with edges of length l x , l y and l z . The edges length may change from cell to cell. The volume fraction C(t 0 ) is defined by the integral

C(t 0 ) = 1 V 0 V χ(x, t 0 ) dV , ( 2
)
where V is the grid cell and V 0 its volume, V 0 = l x l y l z . We use the implicit function f (x) to determine the cell type. If the cell is empty or full with respect to the reference phase, the function f (x) does not change its sign inside the cell, and the integration in (2) is straightforward. On the other hand if the interface cuts the cell, we compute the limits of integration and perform the integration to evaluate the measure of the volume delimited by the cell boundary and the interface itself. A two-dimensional configuration is shown in Fig. 2 to illustrate the relation among the different variables. The reference phase, where the characteristic function χ is equal to 1, is inside the droplet and below the wavy interface. The implicit function f is negative inside the reference phase and its sign determines whether a cell with no interface is empty or full. In the cells cut by the interface local heights h are computed to approximate the two-dimensional integral of Eq. ( 2) with a one-dimensional numerical integration.

Cell analysis

In the first release of the library the user was required to call a routine to estimate a threshold value f th , a characteristic function value for the whole interface. This routine has been removed from the library and the value f th is now computed in each cell. In the two-dimensional example of Fig. 3a the function is evaluated at the N ve vertices of the cell (N ve = 2 2 in 2D, N ve = 2 3 in 3D) to approximate the components of the local gradient ∇f with centered finite differences

∂f ∂x ≈ (f 2,2 + f 2,0 ) -(f 0,2 + f 0,0 ) 2l x , ∂f ∂y ≈ (f 2,2 + f 0,2 ) -(f 2,0 + f 0,0 ) 2l y . (3)
The threshold value is then given by the expression f th = l m |∇f |, where l m = max(l x , l y )/2 is the maximum distance of a point on the cell boundary from a vertex. Afterwards, we compute f min , the minimum in absolute value of the N ve function values. If the function has the same sign at the vertices and f min > f th then the cell is either full, when its sign is negative, or empty. In 3D the approximation of the local gradient follows from (3) in a straightforward manner, for example

∂f ∂z ≈ (f 2,2,2 + f 0,2,2 + f 2,0,2 + f 0,0,2 ) -(f 2,2,0 + f 0,2,0 + f 2,0,0 + f 0,0,0 ) 4l z . (4)
and l m = max(l x , l y , l z )/ √ 2. For empty or full cells with f min > f th the analysis ends with 2 n function evaluations, whereas in the previous release of the library the function was evaluated at N no = 3 n nodes of a local submesh, as shown in Fig. 3b for n = 2.

More computations are required when the cell is closer to the interface and f min < f th . The two points in Fig. 3a with positive values f 0,0 and f 0,2 satisfy that inequality and we have to check if the configuration is compatible with a sign change in the function value. We consider the lower-left vertex x 0 , the unit horizontal vector i x , and compute f α = f (x 0 + i x ε), with ε l x . The following condition is verified

|f α | > |f 0,0 | , (5)
therefore we assume that the interface does not cut the horizontal edge. As a matter of fact, condition [START_REF] Bnà | Numerical integration of implicit functions for the initialization of the VOF function[END_REF] implies that points along the horizontal edge are initially moving away from the interface, then a sign change in the function value along that edge would be associated with a perturbation in the interface line with a characteristic lengthscale smaller than l x . This feature is not consistent with the requirement that the relevant lengthscales of the implicit function should be a few times greater than the spatial step. Afterwards, along the vertical edge we compute f β = f (x 0 + i y ε), but now |f β | < |f 0,0 |. In these conditions the analysis proceeds with a minimum search along that edge. For the interface Γ 1 a sign change is detected at point P , therefore the interface intersects the edge and the search is stopped; for the interface line Γ 2 a positive minimum is found and the edge is not cut by the interface. In 3D we check again for a sign change along the cell edges, but we have to consider the cell faces as well. The minimum search routines along a cell edge and on a cell face have not been changed and we refer to [START_REF] Bnà | Numerical integration of implicit functions for the initialization of the VOF function[END_REF] for their detailed description.

Next we consider a cut cell and we assume that the implicit function f (x) = 0 can be written in the explicit form x 1 = h(x 2 , x 3 ), where each x i is a different coordinate direction. In the previous release of the library we computed the gradient with centered finite differences in the N pt points of the local submesh of Fig. 3b where the function value f satisfied |f | < f th , with 1 ≤ N pt ≤ N no . The numerical computation of a gradient required four more function evaluations in 2D and six in 3D. The cell gradient was then defined as the average of the point gradients, and the main coordinate direction x 1 was associated to the cell gradient component with the greatest absolute value, while the second and third directions, x 2 and x 3 , to the other two components in decreasing absolute value.

In the new release, we consider again the function values at the N no nodes of the submesh, but the cell gradient is now the average of the gradients computed in the 4 subcells of Fig. 3b (8 subcells in 3D) with centered finite differences and no further function evaluation. To decrease the influence of nodes far from the interface, we assign a weight w to the gradient components in each subcell. For example for the lower-left subcell of Fig. 3b we have

∂f ∂x ≈ w (f 1,1 + f 1,0 ) -(f 0,1 + f 0,0 ) l x , ∂f ∂y ≈ w (f 1,1 + f 0,1 ) -(f 1,0 + f 0,0 ) l y .
(6) The value of the weight w is related to the number of sign changes of the function at the subcell vertices and is given in a lookup table. For the cell of Fig. 3a and interface Γ 1 the first direction x 1 is along the x-axis, the second direction x 2 along the y-axis.

The final step in the analysis of a cut cell is the determination of a tentative number of integration nodes n * , which is now based on an estimate of the local curvature κ. For an implicit equation of the interface line the curvature κ in absolute value is given by the expression

|κ| = -f 2 y f xx + 2f x f y f xy -f 2 x f yy (f 2 x + f 2 y ) 3/2 = 1 R , (7)
involving the first and second partial derivatives of the function f , with R being the local radius of curvature. We approximate the partial derivatives with centered finite differences in a nondimensional fashion by defining a characteristic length l c . For example the two terms f x and f xy are approximated by the expressions

f x ≈ (f 2,1 -f 0,1 ) l x l c , f xy ≈ (f 2,2 -f 0,2 ) -(f 2,0 -f 0,0 ) l x l y l 2 c . (8)
We have considered three different expressions for l c that provide the same value in the case of square cells

l c = l x l y , l c = (l x + l y )/2 , l c = max(l x , l y ) .
and several elongation ratios l x /l y . The geometric mean provides a number of tentative nodes very close to the value obtained with square cells, while the third expression considerably overestimates it. We have decided to select the intermediate expression, i.e. the aritmetic mean, because of the elongation of the cell and the mild overestimate. In 3D we compute the curvature on the two cell faces perpendicular to the third direction x 3 and in the midplane between them and then average the results. We have considered circles and spheres of different radius, center position and resolution of the unit domain for a total of 1200 test cases in both 2D and 3D. For each test we have computed the error E in absolute value between the analytical value of the area/volume and the numerical integration with a fixed number of points in each cell. We have fixed a target error E ≈ 10 -14 in 2D and E ≈ 10 -12 in 3D and computed a cubic relation with a least-squares method between the nondimensional curvature |κ * | and the number of integration nodes n * . These two relations have been implemented in the library and are shown in Fig. 4.

Internal and external limits of integration

In 2D the limits of integration are defined by the intersections of the interface line with the cell edges along the second direction x 2 . A single intersection is computed by the root-finding routine, a double intersection, as shown on the right of Fig. 5, requires the detection of a sign change at point P and then the computation of the two zeros x 21 and x 22 . The measure of the colored area in the central rectangle is computed with a numerical integration.

In 3D the internal limits of integration are computed as in the 2D problem, while the external limits of integration include the intersections of the interface with the four cell edges along the third direction x 3 . However, there is also the possibility of a very small intersection of the interface with a cell that is represented by the cap on the left of Fig. 5, with the two external limits x 31 and x 32 , that are computed starting from point S.

To illustrate the changes in the new release we consider the ellipsoid f (x , y , z ) = (x /4) 2 + (y /5) 2 + (z /6) 2 -1 = 0. The axis z is parallel to the coordinate axis z, while the other two axis x and y are rotated 60 degrees counterclockwise with respect to x and y. The ellipsoid center in Fig. 6a is at (0.35, 0.35, -5.97) and the ellipse represents the intersection of its surface with the plane z = 0, while in Fig. 6b the center is at (0.26, 0.26, -5.97). In both cases the cap height is ∆h = 0.03 and the grid cells are cubes of edge length l 0 = 1.

In the top-right cell of Fig. 6a the function is positive at the four vertices of the face at z = 0 and the minimum search routine detects a sign change at point S. Another routine, which is based on the root-finding algorithm, is then called to get a sequence of approximations converging to the external limit x 33 . This routine has been discussed in [START_REF] Bnà | Numerical integration of implicit functions for the initialization of the VOF function[END_REF] and the first two steps of the iterative procedure are shown graphically in Fig. 6a. The other two integration limits x 31 and x 32 are found as intersections of the interface with a cell edge.

In the top-right cell of Fig. 6b the function value is negative at the bottom-left vertex and positive at the other three. In the first release of the library the routine to compute the external limit x 32 was not called in this case, and the tiny volume between the two limits x 31 and x 32 was not computed. A possibile solution to this issue is to call the routine starting from any internal point S 1 and converging to the external limit x 32 . If we do not discriminate among cells, we should also call the routine from a point S 2 in the bottom-right cell that converges to x 31 , an integration limit which is also found as an intersection of the interface with a cell edge. Since an external limit in the interior of a face is a rather rare situation, we would waste a considerable amount of CPU time. To avoid this, from the external limits on the edges, such as x 31 in Fig. 6b, we now compute the function value at two points along the second direction x 2 , the first one very close to x 31 and the other on the opposite edge. In the top-right cell the function has a different sign at these two points, hence the routine to compute the external limits will be called, while in the bottom-right cell the sign is the same and there will be no further action.

Numerical integration

The root-finding algorithm implemented in the previous release was a hybrid secant-bisection method. We now consider three consecutive iterations (y 0 , f 0 ), (y 1 , f 1 ) (y 2 , f 2 ) and compute the next one in the following way

y 3 = y 2 - f (y 2 ) n k (y 2 ) (9)
where n k (y) is Newton's divided-differences polynomial of order k and n k (y) its derivative. For k = 1 we recover the secant method with n 1 (y

2 ) = f 21 = (f 2 -f 1 )/(y 2 -y 1 ). For k = 2 we define f 10 = (f 1 -f 0 )/(y 1 -y 0 ) and f 210 = (f 21 -f 10 )/(y 2 -y 0 ), then n 2 (y 2 ) = f 21 + f 210 (y 2 -y 1 )
. The method requires only one function evaluation at each iteration, and as k → ∞ the order of convergence p approaches 2 from below [START_REF] Sidi | Generalization of the secant method for nonlinear equations[END_REF]. With k = 2 the order of convergence increases from the value p = 1.6180 of the secant method to p = 1.8393, both values rounded to four significant digits. The bisection portion of the root-finding algorithm has not been changed. At the beginning of the algorithm or after a bisection step a secant iteration is performed.

In Fig. 7 we illustrate an integration with 8 nodes in 2D. In the previous version each height h i was computed independently from the neighboring ones. The function value was evaluated at two points on opposite edges (blue squares of Fig. 7a) to initialize the root-finding routine and compute the local height h i (red circles). In the new release we consider three consecutive couples (x 2,i , h i ), i = 0, 1, 2, and again Newton's divided-differences polynomial of order 2

h * (x) = h 0 + h 10 (x -x 2,0 ) + h 210 (x -x 2,0 )(x -x 2,1 ) (10)
where h 10 and h 210 are defined as f 10 and f 210 , respectively. The starting point for the root-finding routine is the extrapolation h * 3 of the polynomial (10) at x = x 2,3 . In Fig. 7b the blue squares represent the starting points h * i : for the first height at x 2,0 we use the old method, at point x 2,1 h * 1 = h 0 , at x 2,2 we use a linear extrapolation and from the fourth point x 2,3 the extrapolation is quadratic, thereafter the three points for the extrapolation (10) are shifted by one position. To start the iteration (9) we also need to approximate the derivative, that we extrapolate from three consecutive couples x 2,i , n 2 (x 2,i ) and the associated Newton's polynomial. With the polynomial extrapolation and the generalized secant method we usually save at least a couple of iterations per height calculation.

New features: cut volume centroid and interface area

In the first release of the library the grid cells were cubes of edge l 0 , now they can be cuboids of edges that may vary from cell to cell. Once the external limits of integration are computed, the cell is further subdivided into subcells and a double Gauss-Legendre integration is performed when the subcell is cut by the interface to compute the cut volume and its centroid. Let us consider a cell of edges l 1 , l 2 and l 3 along the three ordered coordinate directions x 1 , x 2 and x 3 . Furthermore, we assume that the cell has been divided in three subcells: the first is full and fills up the region 0 ≤ x 3 ≤ x 31 , the second is a cut subcell with x 31 ≤ x 3 ≤ x 32 , the third is empty with x 32 ≤ x 3 ≤ l 3 . The volume of the first subcell is V 01 = l 1 l 2 x 31 and its centroid is positioned at x c1 = l 1 /2, l 2 /2, x 31 /2 . The empty subcell gives no contribution. The cut volume V 02 and the centroid x c2 of the central subcell are given by the following expressions

V 02 = V 2 χ(x, t 0 ) dV ; x c2 = 1 V 02 V 2 x χ(x, t 0 ) dV (11)
where V 2 is the domain of integration of the subcell. To illustrate how the numerical integrations are done we consider the central subcell of Fig. 5 and let ∆x 3 = (x 32x 31 ). The internal limits x 22 and x 21 and the length of the interval ∆x 2 = (x 22x 21 ) vary with the third coordinate x 3 , even when not explicitly stated in the following expressions. The cut volume V 02 is then approximated by

V 02 = V 2 χ(x, t 0 ) dV = x 32 x 31 A(x 3 ) dx 3 ≈ ∆x 3 2 n 3 k=1 ω k A k , (12)
where the coefficients ω k are the integration weights, and the areas A k are the integrals of the local height function h(x 2 , x 3,k ) along the second direction x 2

A k = x 22 x 21 dx 2 h 0 dx 1 = x 22 x 21 h(x 2 , x 3,k ) dx 2 ≈ ∆x 2,k 2
n 2 j=1 ω j h j,k . ( 13
)
The number of nodes n 2 of the internal integration is the minimum value between the previously-defined parameter n * and that of a linear function of the nondimensional length δ 2 = ∆x 2,k /l n , with l n = max(l 1 , l 2 , l 3 ), while n 3 of the external integration is given by a linear function of the length δ 3 = ∆x 3 /l n . Furthermore, the user can also set a minimum and a maximum value for both n 2 and n 3 , which should be in the range between 3 and 20. The user-defined values are compared with and can overseed those determined by the library routines.

The coordinates of the centroid x c2 are then given by the following expressions

x c2;1 = 1 V 02
x 32

x 31 dx 3

x 22

x 21

dx 2 h 0 x 1 dx 1 = ∆x 3 2 V 02 ne k=1 ω k ∆x 2,k 2
n i j=1 ω j 2 h 2 j,k x c2;2 = 1 V 02
x 32

x 31 dx 3

x 22

x 21

x 2 dx 2 h 0 dx 1 = ∆x 3 2 V 02 ne k=1 ω k ∆x 2,k 2
n i j=1 ω j x 2,j,k h j,k x c2;3 = 1 V 02
x 32

x 31

x 3 dx 3 x 22 x 21 dx 2 h 0 dx 1 = ∆x 3 2 V 02 ne k=1 ω k x 3,k A k
Finally, the centroid x c of the cut cell is

x c = V 01 x c1 + V 02 x c2 V 01 + V 02 (14)
In 2D an n-point Gauss-Legendre quadrature rule uses n nodes in the interior of the interval of integration [x 2a , x 2b ]. In order to calculate the length of the interface line which is inside a two-dimensional cell, we first compute the height at the two endpoints x 2a and x 2b . In this way we divide the interval [x 2a , x 2b ] in n + 1 subintervals. We have implemented a nodebased method that approximates the interface arc in each subinterval with a polynomial and then computes its length by quadrature [START_REF] Floater | Point-base methods for estimating the length of a parametric curve[END_REF]. The method is fourth-order accurate and it requires the computation of the height h j+1/2 at the midpoint x 2,j+1/2 between two consecutive nodes x 2,j and x 2,j+1 . The Newton's polynomial approach of Section 2.3 is used to initialize the rootfinding routine to compute h j+1/2 . The method consider three points in each subinterval, x 2,j = x 2,j , h j , x 2,j+1/2 = x 2,j+1/2 , h j+1/2 and x 2,j+1 = x 2,j+1 , h j+1 , and the arclength L of the subinterval is approximated by the expression

L x 2,j , x 2,j+1 ≈ |x 2,j+1 -r| + |r -x 2,j | , (15)
where the shifted intermediate point r is

r = 1 2 x 2,j + x 2,j+1 + 1 3 √ 3 -x 2,j + 2 x 2,j+1/2 -x 2,j+1 . (16)
To illustrate the surface triangulation in three-dimensions we consider again the ellipsoidal cap of Section 2.2 but with center at (0.5, 0.45, -5.97), then the cap is inside a single grid cell and the coordinates ordering is x 1 = z, x 2 = y and x 3 = x. We compute the cap volume with n 2 = n 3 = 6 nodes for both the internal and external integrations and show the position of all the nodes in Fig. 8a (blue squares). From this nodes distribution and considering the internal integration at x 3,k , we compute the height at the two endpoints (x 21 , x 3,k ) and (x 22 , x 3,k ), and substitute them into the first and last elements of the array containing nodes position and height at x 3,k . We do the same for the external integration, we remove the first and last arrays of internal nodes and substitute them with the nodes computed at x 31 and x 32 . The new nodes are shown as red circles in Fig. 8b, in particular in the external limits at x 31 and x 32 only a single node is present. We have decided upon this choice for a more homogeneous distribution of the triangles position and size. Indeed, by simply adding the new nodes, the triangles tend to accumulate near the boundaries and become much smaller.

We then consider two consecutive arrays of nodes, say at x 3,k and x 3,k+1 . If the number of internal nodes n 2 is the same we take couples of consecutive nodes on both arrays, and define intermediate nodes by computing their average position, shown as green squares in Fig. 8b, and their average height to start the iteration (9) to calculate the local height. The two couples of consecutive nodes on both arrays and their intermediate node are then connected to form four triangles, that share a vertex at the intermediate node, as shown in Fig. 8b. If the number of internal nodes is not the same, a few triangles will be defined with two consecutive nodes on the array with more nodes and one on the other array, until the number of remaining nodes is the same. This is what is done with the single node in the external limits at x 31 and x 32 . The number of external nodes n 3 can change from cell to cell and nodes may not match across the cell boundary, therefore the triangulation is continuous within a cell but it may have holes at the boundary between cells where the edges of the triangles can connect different nodes. The measure of the interfacial area is approximated by the sum of the triangles area.

User manual

In the previous release the implicit function f (x) was defined by the user, possibly with a set of fixed parameters, for example the center coordinates and the radius of a sphere. These parameters now can be passed also dynami-cally as an additional argument of the function call. Furthermore, we provide the possibility to print heights and triangulation data in TECPLOT ASCII file format that can be easily visualized with the help of graphics software, as shown in Fig. 10 with the ParaView visualization application.

C/C++ calls

The Vofi library has been developed in C and the user can call two functions. The first function vofi get cc computes the volume fraction in a given cell and returns a real number (in double precision) cc = vofi get cc(impl func, par, x0, h0, xext, next, npts, nvis, ndim);

The interface position is often used in adaptive mesh refinement as a criterion to increase the local resolution. In this case it is required only to know if the cell is full, empty or cut, and not the actual value of the volume fraction. The second function vofi get cell type returns an integer with one of the three values 1, 0 and -1 if the cell is respectively full, empty or cut by the interface icc = vofi get cell type(impl func, par, x0, h0, ndim);

The input arguments are defined as follows

• impl func: the external function that computes f (x) and is expected to be declared as double impl func(const double x0[], void * const par) {...}

• par: parameters to be passed to impl func (pointer to a data structure defined by the user)

• x0: coordinates of cell vertex with smallest values (1D array of 3 real numbers)

• h0: cell edges (1D array of 3 real numbers)

• xext: center of mass coordinates and interface length/area (1D array of 4 real numbers, the length/area is always the fourth element)

• next: switches either to compute or not to compute the center of mass and/or the interface length/area (1D array of 2 integer numbers, with each value respectively 1 or 0)

• npts: minimum and maximum number of nodes allowed by the user for both the internal and external integrations (1D array of 4 integer numbers, to be effective each integer n should be in the range 3 ≤ n ≤ 20)

• nvis: switches either to print or not to print heights and triangulation data (1D array of 2 integer numbers, with each value respectively 1 or 0). This feature should be used sparingly, as the size of the output files grows very rapidly with grid resolution and number of integration nodes.

• ndim: space dimension (integer number, either 2 or 3)

The The prototypes are contained in the header file vofi.h that should be included when using the Vofi library. To link the library it is necessary to add -lvofi to the compiler command line. More details are given in the README file of the software distribution.

Fortran calls

The Vofi library has been developed in C and for its usage in Fortran codes we have written the interface module fvofi.f90 that calls the corresponding C functions. The examples in Fortran use the ISO C BINDING module that defines intrinsic procedures for C interoperability. The user should read the README file and look at the examples for different ways to pass the parameters to the external function that computes f (x).

Testing and validation

To illustrate the new features of the library we consider the circle with radius r = 0.25 and center of coordinates (x C , y C ) = (0.623, 0.377), inside a unit square domain subdivided into N 2 square cells. As we double the edge resolution starting from N = 10 and up to N = 80, the number of cut cells m I doubles as well. The number of integration nodes n 2 is the same in all the cells and we compare the numerical area, centroid coordinates and circumference length with their analytical values. The results are presented in Tables 1 and2 where each error is defined as the difference in absolute value between the numerical and analytical values. The interface arclength changes from cell to cell and Gauss nodes are not evenly spaced inside each cut cell, as seen in Fig. 9a. Therefore, we define the convergence rate p of the length of the circumference in the following way

p ≈ log E L,1 E L,2 log k 2 k 1 (17)
In Table 1 the error E L,1 is computed with a fixed number n 2 = 4 of integration nodes, corresponding to n s = 5 sectors of the interface arc in each cell, and edge resolution N , and E L,2 with the same number of sectors n s , but edge resolution 2N . The integer k in the denominator of ( 17) is then the number of cut cells, k = m I . In Table 2 the edge resolution N is kept fixed as we change the number of sectors, n s = 5, 10, 20, in each cell, hence k = n s (in the usual definition of the convergence rate a step size appears in the denominator, which is inversely proportional to the resolution that is used in (17)).

The results of Table 1 suggest that the accuracy of the computation of the centroid of a geometric figure is limited from above by the accuracy of the area computation, but the error for one of the centroid components can be one or two orders of magnitude smaller in particular symmetrical conditions. This is what we observe in both 2D and 3D numerical tests. The theoretical convergence order of 4 for the length of the interface line is recovered both by doubling the edge resolution N at constant number of sectors n s in each cell and by doubling the number of sectors n s at constant N .

In Fig. 9b we show the heights with n 2 = 4 of the same circle in a rectangular grid with N = 10. The area error is E A = 5.71 10 -7 which is similar to the error E A = 4.17 10 -7 of Table 1 with N = 5, but with the same number of cut cells, m I = 12.

In 3D we consider the sphere with radius r = 0.34 and center of coordinates (x C , y C , z C ) = (0.503, 0.451, 0.463), inside a unit cube with edge resolution N = 10 and cubic cells. In Table 3 we present the results with a different number of integration nodes, but keeping n 2 = n 3 . At this resolution the convergence rate for the measure of the interfacial surface is linear. However, as we increase the grid resolution N we observe that the measure of the surface tends to saturate as the convergence rate decreases. The triangulation of the interface with N = 10 and n 2 = n 3 = 4 is shown in Fig. 10, together with the heights and triangles of a selected cell.

Finally we compare the number of function calls that are required to initialize circular and spherical droplets. In 2D the edge resolution of the unit square is N and the total number of grid cells N 2 , while the number of cut cells is proportional to N and the total number of heights to the product N n 2 . The ratio of cut cells to the total number of cells scales as 1/N , therefore the number of empty or full cells becomes more and more predominant with N and the ratio of implicit function calls between the old and new versions should tend to the value 9/4 = 2.25, where 9 and 4 are the minimum numbers of function calls in the two library versions to determine the cell type. This asymptotic behavior is clearly observed in the results on the left of Table 4. The value of the ratio remains always greater than 2, while the average number of function calls to initialize a cut cell in the new version increases from an average value of 33 when n 2 = 4 to 70 when n 2 = 16. It is also interesting to note that the ratio increases slightly with the value of n 2 as seen on the right of Table 4. The reason for this behavior is that when n 2 = 4 the quadratic extrapolation (10) is applied only once, while with n 2 = 16 it is applied to 13 nodes. Therefore, the distance between two consecutive nodes diminishes, the initial guess (10) is more precise, and the average number of iterations to compute the local height is slightly reduced.

In 3D the edge resolution of the unit cube is still N and the total number of grid cells is now N 3 , while the number of cut cells is proportional to N 2 and the total number of heights to the product N 2 n 2 n 3 . The asymptotic value at large N is now 27/8 = 3.375, and the results on the left of Table 5 show that the numerical ratio is slowly decreasing towards this value. The library optimization is more efficient in 3D than in 2D, as the ratio between the function calls is a bit less than 5 for values of N up to 20 and n 2 = n 3 = 4, which implies the computation of 16 local heights. However at fixed N , the numerical ratio decreases with n 2 and n 3 . First, we observe that the number of computed heights divided by the number of grid cells is now n 2 n 3 /N (it was only n 2 /N in 2D) so that the CPU time spent in their calculation becomes predominant in 3D, as there are 256 local heights when n 2 = n 3 = 16. More precisely, with the optimized algorithm to compute the local height, that is described in Section 2.3, we gain at most a factor of order 2, and the results on the right of Table 5 clearly demonstrate this behavior. To summarize our findings in 3D, at moderate edge resolution N of the unit cube and number of local heights in each cut cell, we observe that the implicit function calls in the new version of the Vofi library are reduced by a factor close to 5 with respect to its first release. If we increase only the resolution N , the initialization of empty or full cells becomes predominant and the factor tends to the asymptotic value 3.375. On the other hand, if we increase only the number of local heights, that factor tends to the value 2.

Conclusions

Several numerical algorithms of the Vofi library have been rewritten in order to reduce the number of calls of the user-defined function that defines implicitly the interface. In particular, the grid cell analysis and the numerical integration scheme have been considerably optimized. New features of the software distribution include the subdivision of the computational domain in cuboids and the calculation of the reference phase centroid and the interface length in 2D and area in 3D in each cut cell. The library is written in C, and its functions can be called from Fortran routines using a supplied interface module and the ISO C BINDING module.

χ = 1 f < 0 χ = 1 f < 0 χ = 0 f > 0 local heights h

  prototypes for the external implicit function and the two library functions are typedef double (*integrand) (const double [],void * const); double vofi get cc(integrand,void * const,const double [],const double [],double [],const int[],const int[],const int [],const int); int vofi get cell type(integrand,void * const,const double [], const double [],const int);

Figure 2 :Figure 3 :

 23 Figure 2: A droplet impacting on a wavy interface in two dimensions. Inside the reference phase χ = 1 and f < 0, in the secondary phase χ = 0 and f > In the cells crossed by the interface the local heights h are used to compute numerically the cut area.

Figure 4 :

 4 Figure 4: (a) Error E as a function of the nondimensional curvature |κ * | for a fixed number of integration nodes, n 2 = n 3 = 4, 8, 16. Each point corresponds to an average over 10 different spheres with the same radius R. For a few points near the corresponding value of the cubic fit (black squares) the minimum and maximum errors are shown as vertical bars. (b) Cubic relations between the tentative number of integration nodes n * and the nondimensional numerical curvature |κ * |.

Figure 5 :

 5 Figure5: The grid cell is subdivided into three cuboids after the computation of the two external limits of integration x 31 and x 32 , starting from point S (left); each internal integration requires the computation of two internal limits of integration x 21 and x 22 , starting from point P (right).

Figure 6 :

 6 Figure 6: Intersection of the ellipsoid surface with the plane z = 0. (a) Ellipsoid center at (0.35, 0.35, -5.97): in the top-right cell the external limit x 33 is computed iteratively from point S. (b) Ellipsoid center at (0.26, 0.26, -5.97): in the top-right cell the function sign is evaluated at points u and v to start the iterative computation of x 32 from S 1 , while in the bottom-right cell the function sign at u and v is the same with no further action.

Figure 7 :Figure 8 :

 78 Figure 7: Integration with 8 nodes: a) in the previous hybrid secant-bisection method the algorithm is initialized with the function value at two points (blue squares), on opposite edges, that bracket the zero (red circles), b) in the new generalized secant method the starting value h * i (blue square) is extrapolated from previous height values.

Figure 9 :

 9 Figure 9: Integration of the circle with center at (0.623, 0.377) and radius r = 0.25 with n 2 = 4 nodes and n s = 5 sectors in each cell and edge resolution N = 10: (a) square cells, a cut cell at the highest resolution N = 80 is also shown; (b) rectangular cells.

Figure 10 :

 10 Figure 10: Integration of the sphere with center at (0.503, 0.451, 0.463) and radius r = 0.34 in the unit cube with edge resolution N = 10, and n 2 = n 3 = 4 in each cut cuboid: (a,b) heights and triangles in a selected cell subdivided in 3 cuboids; (c,d) heights and triangulation of the sphere (top face of the selected cell outlined in red).

Table 1 :

 1 Number of subdivisions N of the unit edge and number of cut cells m I for the circle of Fig.9; errors of the numerical computation of the circle area, E A , centroid coordinates, E x and E y , and circumference length, E L , with a fixed number of integration nodes, n 2 = 4, and of arc sectors, n s = 5, per cell; convergence rate p of the circumference length with edge resolution N .

	N m I	E A	E x	E y	E L (5)	p
	5 12 4.17e-07 1.12e-07 1.10e-07 2.08e-06	--
	10 20 4.68e-08 2.47e-09 5.52e-08 3.58e-07 3.446
	20 40 1.16e-10 1.02e-11 1.02e-11 2.26e-08 3.983
	40 80 5.04e-13 9.36e-14 9.36e-14 1.36e-09 4.059
	80 160 1.03e-15 1.55e-15 2.22e-16 6.59e-11 4.364

Table 2 :

 2 Number of subdivisions N of the unit edge and number of cut cells m I for the circle of Fig. 9; errors of the numerical computation of the circumference length, E L , with a fixed edge resolution N and different arc sectors, n s = 5, 10, 20, per cell; convergence rate p of the circumference length with number of sectors n s .

	N m I 5 12 2.08e-06 1.16e-07 6.74e-09 E L (5) E L (10) E L (20) p (5 → 10) p (10 → 20) 4.159 4.110
	10 20 3.58e-07 2.00e-08 1.16e-09	4.160	4.112
	20 40 2.26e-18 1.26e-09 7.28e-11	4.163	4.116
	40 80 1.36e-09 7.56e-11 4.36e-12	4.165	4.117
	80 160 6.59e-11 3.67e-12 2.12e-13	4.165	4.110

Table 3 :

 3 Number of internal and external nodes, n 2 and n 3 , for the sphere of Fig.10with edge resolution N = 10; errors of the numerical computation of the sphere volume, E V , centroid coordinates, E x , E y and E z , and interface surface, E S .

	n 2 , n 3	E V	E x	E y	E z	E S
	4	4.25e-09 6.49e-10 2.16e-10 1.81e-09 1.65e-03
	8	2.23e-14 5.33e-15 8.33e-16 6.36e-04
	16	0.00e-00 6.66e-16 2.22e-16 5.55e-17 2.96e-04

Table 4 :

 4 Ratio of implicit function calls between the old and new versions of the Vofi library in 2D: at constant number of internal nodes n 2 = 4 (left), and at constant number of subdivisions N = 5 of the unit edge (right).

	N n 2 ratio N n 2 ratio
	5 4 2.037 5 4 2.037
	10 4 2.133 5 8 2.058
	20 4 2.134 5 16 2.118
	40 4 2.213
	80 4 2.222

Table 5 :

 5 Ratio of implicit function calls between the old and new versions of the Vofi library in 3D: at constant number of integration nodes n 2 = n 3 = 4 (left), and at constant number of subdivisions N = 5 of the unit edge (right).

	3 ratio

N n 2 = n 3 ratio N n 2 = n

f 0,0 f 1,0 f 2,0 f 0,2 f 1,2 f 2,2 f 0,1 f 2,1 f 1,1 h y P

Acknowledgements

We thank Dr. S. Manservisi for useful discussions on the topics of this paper and in particular on its revision. The simulation data in 3D are visualized by the open-source software ParaView.


Tangi Aubert

Marco Cirillo

Stijn Flamée

Rik Van Deun

Holger Lange

Christian Thomsen

Zeger Hens

Homogeneously Alloyed CdSe 1-x Sx Quantum Dots (0 ≤ x ≤ 1): an Efficient Synthesis for Full Optical Tunability

Keywords: nanocrystals, solid solution, alloy, Raman

Over the last 20 years, semiconductor quantum dots (QDs) have attracted wide interest as versatile luminescent materials, finding numerous applications in fields ranging from photonics to biotechnology. [1][2][3][4] Tuning the emission color of QDs is usually done by varying their size. However, to meet the constantly increasing demands of their applications, a decoupling of QD size and emission color is needed. This is for example the case for LEDs that benefit from close packed QD layers, or for multicolor bioimaging. 2,5 To provide same-size-different-color QDs, syntheses of alloyed QDs (allQDs) have been recently developed. 6 Alloying provides an additional degree of freedom as it allows for a fine tuning of the emission wavelength by varying the QD composition rather than the size. In particular, II-VI alloys such as Cd(Se,Te), Cd(Se,S), Cd(Te,S) or (Zn,Cd)Se have been investigated, [6][7][8][9][10] while (Zn,Cd)(Se,S) proved to offer emission wavelength tunability from 440 to 650 nm. 11 Next to wavelength tunability, allQDs were also reported to have high photoluminescence quantum yield (PLQY), 12 and low blinking rates. 13,14 A major challenge in the synthesis of colloidal allQDs is to obtain a random distribution of the elements over the nanocrystal. Indeed, having true solid solution QDs constitutes a prerequisite to investigate the unique optical properties of these novel nanomaterials. According to the literature, this requires a balancing of the reactivity of the different precursors. 15 To demonstrate homogeneous alloying, electron-microscopy techniques are often combined with elemental analyses. 11,16 However, these methods are only reliable for larger QDs and can only address a small number of nanocrystals. Alternative, ensemble level methods that apply to small QDs as well are therefore still needed. In the case of II-VI allQDs, most reported syntheses make use of tri-octylphosphine (TOP) based chalcogen precursors, which are oxidation sensitive and toxic. An alternative approach is based on precursors with a low reactivity, resulting in a low yield reaction where the size of the QDs can only be controlled by the reaction time. 11 Here, we report on a new, fast and phosphine-free synthesis of homogeneously CdSe1-xSx allQDs (0 ≤ x ≤ 1) and we introduce Raman spectroscopy as a tool to analyze homogeneous alloying in colloidal QDs. The synthesis makes use of the hot injection of a heterogeneous dispersion of Se in ODE, which we recently reported and patented for making several kinds of metal selenide QDs. [17][18][19] We find that the reactivity of this heterogeneous ODE-Se toward the Cd precursor matches that of S homogeneously dissolved in ODE. Tracking the frequency and the width of the longitudinal optical phonon resonance as a function of composition using Raman spectroscopy, we show that combining both precursors yields homogeneously CdSe1-xSx allQDs. Importantly, the synthesis reaches close to full yield in no more than 5 minutes, even at high solid loading, and the size of the QDs at the end of the reaction can be reproducibly controlled by using carboxylic acids of different length. Growth of a CdS shell around CdSe1-xSx allQDs can raise the PLQY to 40% or more, yet the higher the sulfur content, the lower the PLQY increase, an effect we attribute to an enhanced hole delocalization.

In a typical synthesis, CdSe1-xSx allQDs are synthesized by injecting a mixture of heterogeneous ODE-Se and homogeneous ODE-S solutions under ambient conditions in a reaction mixture made by dissolving CdO in ODE using myristic acid and kept at 270 °C. After 5 minutes of reaction at 260°C, the reaction was quenched, the QDs were purified by precipitation and centrifugation and the ligands were exchanged for oleic acid (see Supporting Information (SI), paragraph 1.1). Figure 1a shows the (normalized) yield development -as obtained by UV-Vis analysis of weighted aliquots -for syntheses where only ODE-Se or ODE-S are used as precursors, yielding pure CdSe and pure CdS, respectively. It follows that even if the ODE-Se precursor has a higher initial rate, both reactions have an overall comparable development, reaching more than 92% of the respective final amount of moles after 5 minutes. One can thus expect that simply mixing the Se and S precursor in the reaction may lead to a homogeneous alloying of these two elements. Thus, CdSe1-xSx QDs were synthesized over the entire composition range (0 ≤ x ≤ 1) and systematically analyzed by X-ray fluorescence (XRF) and energy dispersive X-ray spectroscopy (EDX) to determine the actual S/Se ratio of the samples. We find that the obtained S/Se ratio was always equivalent to the S/Se ratio of the solution injected in the reaction (see SI, Figure S1).XRF was also used to determine the amount of unreacted precursors, from which we obtain reaction yields higher than 90%. As shown in Figure 1b for the case of CdSe0.5S0.5 QDs, the synthesis results in quasi spherical and monodisperse QDs. Although it appears that the size of the QDs slightly decreases when increasing the sulfur content (see SI, Figure S2), all the sizes were centered around 3 nm with a size dispersion always below 10%. X-ray diffraction (XRD) analyses showed that the synthesized CdSe1-xSx QDs have a zinc blende structure as previously reported for CdSe QDs synthesized with the same procedure (see SI, Figure S3). 17 In addition, the lattice parameter of these QDs decreases linearly with the amount of sulfur according to Vegard's law, 20 a first indication of homogeneous alloying.

To further confirm the homogeneity of the CdSe1-xSx alloying, we used Raman spectroscopy since the dependence of the phonon frequencies on the reduced atomic mass of the atoms in the unit cell allows extracting the average lattice configuration. We fitted the fundamental Raman bands with Lorentzian functions to account for the longitudinal optical phonon (LO) and surface optical phonon, while using a second order polynomial to fit the background. 21 Figure 1c represents a set of background-corrected spectra, normalized to the intensity of the LO related band (see SI, Figure S4 for larger figure, and Figure S5 for the fits to the LO-related band). Clearly, changing the S/Se ratio shifts the Raman bands, where incorporating the lighter S leads to an approximately linear shift of the LO frequency to higher energy (Figure 1d). Since the formation of minority domains would result in independent peaks at the CdSe or CdS frequency, this strongly supports a homogeneous distribution of Se and S atoms over the QD. Additionally, new bands appear in the Raman spectra at frequencies between the CdSe and CdS LO frequencies. This reflects the breaking of the translational invariance of the lattice in a random alloy, which lifts the q=0 Raman selection rule and thereby making previously forbidden vibrations afar from the LO phonon frequency Raman active. [22][23][24] Additionally, non-longitudinal alloy vibrations incorporating both anions could be Raman active and result in additional peaks, as observed for CdSe0.5S0.5 alloys and confirmed by DFT calculations. 21 These effects agree with published studies on alloyed semiconductors and isotope mixtures. Raman measurements of the zone-center optical mode in silicon with various isotopic compositions shows a linear frequency-mass dependence, whereas disorder-induced features appear. 25 The same is valid for alloyed semiconductors, such as Ga1-xAlxAs/GaAs, 22 ZnxCd1-xSe 23 and InAlGaN. 14 Homogeneous alloying is further confirmed by the overall increase of the peak full width at half maximum (FWHM) with respect to the pure compounds, except for the CdSe0.5S0.5 LO peak. This indicates a very homogenous distribution of S and Se for this composition, whereas other S/Se ratios result in an increased asymmetry and a subsequent increase of the FWHM, in agreement with the expectations for homogeneous alloys. 24,26 Size variations generally also result in a broadened Raman peak due to a varying phonon confinement. We expect the contribute of this to be small for our samples due to weak confinement and rather narrow distributions. For the lowest S content (x=0.2), we find a LO frequency close to that of CdSe QDs. Here, a partial segregation of Se and S cannot be fully excluded, which could actually be due to the initially higher reactivity of the Se precursor as seen in Figure 1a.

In terms of optical properties, all the CdSe1-xSx QDs showed well defined and narrow absorption and emission peaks (Figures 2a and2b). Both show a non-linear blueshift with increasing sulfur content, reflecting the concomitant change of the band gap. The non-linear dependence of the band gap on composition is well-known for alloyed materials, with a quadratic relation reported for bulk CdSe1-xSx. 27 After synthesis, the CdSe1-xSx QDs typically showed a PLQY below 10%. Apart from the band gap shift, increasing the sulfur content also raises the trap emission at longer wavelength, which is commonly found for CdS QDs 28 Similar to pure CdSe QDs synthesized with the same protocol, 17 we found that the size of CdSe1-xSx QDs with given composition decreases when increasing the chain length of the carboxylic acid (see SI, Figure S7). As this results in a blueshift of the absorption and emission features (see SI, Figure S8), the synthesis method introduced here makes size and composition two independent variables to tune the optical properties of the allQDs synthesized, while keeping a high reaction yield.

To further improve the PLQY of these QDs, up to 7 layers of CdS were grown on CdSe1-xSx QDs through a successive ion layer adsorption reaction procedure (SILAR, see SI, paragraph 1.2). 29 All the synthesized core-shell CdSe1-xSx/CdS QDs were quasi-spherical and monodisperse, with average sizes in agreement with the theoretical number of CdS layers grown and showing well defined and narrow absorption and emission spectra (see SI, Figures S9 andS10). Both the absorption and emission peaks shift to the red with the growth of the CdS shell, reflecting a decrease in the confinement energy. As expected, the PLQY of the CdSe1-xSx/CdS QDs increases with the growth of the first 3 CdS layers (Figure 3a) and then decreases with further growth of thicker CdS shells, probably reflecting the introduction of strain and defects in the heterostructures. 21 However, the initial increase of the PLQY is less pronounced with S-rich alloyed cores. Since it concurs with an enhanced reduction of the confinement energy (Figure 3b), we attribute this to an enhanced delocalization of the hole that results from a reduced valence band offset between core and shell with increasing S content of the core (see SI).

In conclusion, we report on a new, fast and phosphinefree method for the synthesis of CdSe1-xSx allQDs with 0≤x≤1. The approach makes use of a heterogeneous ODE-Se precursor which shows similar reactivity towards the Cd precursor as a homogeneous ODE-S precursor. As confirmed by Raman spectroscopy, the synthesized CdSe1-xSx QDs consist of random solid solutions, whose optical properties are composition-dependent. In addition, the emission wavelength can also be adjusted by changing the size of the QDs, which is readily achieved using carboxylic acids with a different chain length in the synthesis. In this way, the synthesis proposed here offers two independent parameters -size and composition -to tune the optical properties of QDs synthesized at close to full yield. We also showed that for CdSe1-xSx/CdS core-shell QDs, the effi-ciency of the shell in confining the exciton in the core decreases when increasing the sulfur content, most likely due to a smaller interfacial valence band offset.

Figure 1 .

 1 Figure 1. (a) Normalized amount of CdSe and CdS in moles as a function of the reaction time. (b) TEM micrograph of CdSe0.5S0.5 QDs. (c) Raman spectra and (d) LO frequency and FWHM of CdSe1-xSx QDs depending on the sulfur content.

Figure 2 .

 2 Figure 2. Normalized (a) absorption and (b) emission spectra of CdSe1-xSx QDs with different sulfur content (see SI, Figure S6 for additional spectra).

Figure 3 .

 3 Figure 3. (a) PLQY and (b) confinement energy of CdSe1-xSx/CdS QDs depending on the number of CdS layers (see SI, paragraph 3.2 for determination of the confinement energy).

ACKNOWLEDGMENT

The authors acknowledge BelSPo (IAP 7.35, photonics@be), FWO-Vlaanderen (project G.0760.12), German Research Foundation (project LA 2901/1-1), Max Kade Foundation, Hercules Foundation (project AUGE/09/024 "Advanced Luminescence Setup") for financial support. Mathieu Pasturel (UR1) is acknowledged for his help on XRD analyses.

ASSOCIATED CONTENT

Supporting Information. Full synthesis and analytical details, additional XRF, EDX, XRD, TEM, PL, UV-vis absorption, Raman analyses. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Authors

* E-mails: zeger.hens@ugent.be, tangi.aubert@ugent.be Present Addresses § Department of Physics, Columbia University, United States.