Theresia Tabchi
email: theresia.tabchi@gmail.com

Hussein Sabra
email: hussein.sabra@univ-reims.fr

Investigating forms of relation between teaching and research in the case of graph theory

Keywords: Generic examples, Graph theory, Relation between research and teaching, Resources, University teachers' practices

This paper deals with university teachers' practices. It seeks to characterize the forms of relation that can take place between their teaching practices and their research activity in the domain of graph theory. Our exploratory study draws on an interview with a university teacher and observations of her classes that focus on her use of generic examples in the classroom. Generic examples are particularly important in proofs in graph theory, and we therefore estimate that their use by teachers can shed light on the potential influence of their research activity on their teaching.

Introduction, context, and research question

Our work deals with the practices of university mathematics teachers. It is a contribution to the growing body of research on university teachers' practices [START_REF] Biza | Research on Teaching and Learning Mathematics at the Tertiary Level: State-of-the-Art and Looking Ahead[END_REF]. This same study raised a main issue as a field that requires exploration: the relation between evolution in practices, evolution in resources, and evolution of teachers' professional knowledge. This paper is a contribution to the work around the mentioned issue. We are particularly interested in the impact of research on teaching in mathematics. We suggest seeking this issue through the lens of teachers' interaction with resources. Some studies tried to characterize the influence of the research activity of university teachers on their teaching practices [START_REF] Neumann | Perceptions of the teaching-research nexus: a framework for analysis[END_REF][START_REF] Madsen | Relations between teaching and research in physical geography and mathematics at research-intensive universities[END_REF]. These studies consider research to be potentially important in university education where teaching practices can be informed by research practices. [START_REF] Neumann | Perceptions of the teaching-research nexus: a framework for analysis[END_REF] suggested criteria to describe the connections between teaching and research. She underlined the need for further research on the topic that considers disciplinary specificities. [START_REF] Madsen | Relations between teaching and research in physical geography and mathematics at research-intensive universities[END_REF] proposed a theoretical model based on specific research tasks to describe the teaching-research nexus that takes disciplinary specificities into account.

We will be using, throughout the paper, "teaching practices" and "research activity" to refer respectively to the "teaching" and "research" aspects of the university teachers' work. We designate by "university teachers" the university teachers who occupy teaching/research positions.

In a previous research we presented in this Thematic Working Group [START_REF] Sabra | The connectivity in resources for student-engineers: the case of resources for teaching sequences[END_REF], we identified, from case studies, factors related to research activity that influence teaching practices. We mentioned the need to characterize the relation between the research activity and teaching practices of mathematics teachers to better understand some of the choices made in a given teaching context. In a related perspective, [START_REF] Speer | Collegiate mathematics teaching: An unexamined practice[END_REF] highlighted that the rationales behind some teacher choices are largely covert. They underlined that teachers' choice of examples seems to be an important dimension of their teaching practices; [START_REF] Speer | Collegiate mathematics teaching: An unexamined practice[END_REF] specified that the selection of examples by teachers can be influenced by their pedagogical knowledge and their knowledge of the mathematical content that can evolve due to their interaction with this content in their activity of research. Indeed, the forms of relation between teaching and research depend closely on the profile of the university teachers. Several characteristics of the profile enhance those forms. Among them, if the teacher is an active mathematician or a trained mathematician (simply having a PhD in mathematics), their research field, the taught courses (the mathematics taught to engineers would differ from the mathematics taught to future mathematicians), the software used in their research, etc. Therefore, we consider that examining teachers' use of examples can reveal elements about the knowledge that impacted their choices of examples, which can contribute to our understanding of their teaching practices and the possible influence of their research on these practices. We hypothesize that characterizing the influence of research on teaching practices provides a deeper understanding of these practices. We will further underpin our claim at a later stage in this section.

We consider in our study a specific type of examples: the generic examples. A generic example is an example presented in a way that focuses on key features and ignores others to bring out its role as carrier of the general [START_REF] Mason | Generic examples: Seeing the general in the particular[END_REF]. In a study that seeks to characterize university teaching, [START_REF] Mali | Characteristics of university mathematics teaching: Use of generic examples in tutoring[END_REF] studied the influence of research on teaching through focusing on a teacher's use of generic examples in tutorials. They found that the teacher relates her use of generic examples to the practice of "decoding and encoding" that mathematicians use in research, where decoding refers to understanding all parts of a problem and encoding means writing it down formally.

To study the teaching-research relation, we choose to consider generic examples in the case of graph theory. Many reasons motivate our choice. Graphs are useful tools for modeling real-life situations and play a significant role in the mathematical training of undergraduate students in engineering, economics, and mathematics [START_REF] González | Characterizing Levels of Reasoning in Graph Theory[END_REF]. [START_REF] Rosen | Discrete mathematics and its applications[END_REF] points out that graph theory trains students to construct proofs, by developing their abilities of abstraction and reasoning. In this context, generic examples allow chains of reasoning about particular objects that make explicit the reasons for the truth of a statement [START_REF] Rowland | Beliefs and actions in university mathematics teaching[END_REF] making them an important aspect of producing proofs. Indeed, in this paper, we will try to bring elements of answer to the following research question: how can we characterize, through the use of generic examples in proofs, the forms of relation between the teaching practices of graph theory and the research activities in the same field?

We expect that studying the relation between research and teaching at university level can help gain a deeper understanding of teachers' choices, in and out of classroom. It helps identifying reasons that can enhanceor reducethe dissemination of research contents and skills in the context of university teaching. Hence, it can offer university students better learning opportunities by fostering a learning environment, connected -as much as possible -to the reality of the development of mathematics and its applications. In what follows, we will present our theoretical framework followed by the methodology and the collection of data along with the results obtained and some perspectives.

Theoretical Framework

We consider the model developed by [START_REF] Neumann | Perceptions of the teaching-research nexus: a framework for analysis[END_REF] to describe the teaching-research nexus. In addition, we draw on the Documentational Approach of Didactics (DAD) (Gueudet et al., 2012) with a focus on operational invariants to study teachers' interactions with resources in their teaching practices. We also rely on [START_REF] Chevallard | Analyse des pratiques enseignantes et didactique des mathématiques : L'approche anthropologique[END_REF] to define the two institutions, teaching and research.

Teaching-research nexus: forms of relation between teaching and research

Some researchers in science education attempted to find evidence of "positive" or "negative" correlations between research and teaching without considering a specific discipline [START_REF] Neumann | Perceptions of the teaching-research nexus: a framework for analysis[END_REF]. They tried to characterize possible relations between teaching and research activities (symbiosis, conflict, tension, etc.). [START_REF] Neumann | Perceptions of the teaching-research nexus: a framework for analysis[END_REF] presents three aspects of what she calls "teaching-research nexus": tangible, linked to a transfer of knowledge from research into teaching; intangible, related to the actions of a researcher in the teaching activity and vice versa; and global, referring to the nexus between teaching and research at departmental level. She underlines that these three types of nexus interrelate and intermingle, they are not necessarily separate or distinguishable.

In a study based on interviews focusing on the specificities of physical geography and mathematics, [START_REF] Madsen | Relations between teaching and research in physical geography and mathematics at research-intensive universities[END_REF] explore the nature and conditions of a positive "link" between teaching and research. They distinguish between two institutions, teaching and research. An institution is defined by Chevallard (1998) as being a "total" social entity, which allows and imposes on its subjects -people who occupy different positions offered within it -ways of doing and thinking. Chevallard sheds light on the cognitive dimension of institutions and on the relations of a subject (teacher or student) and of objects (such as mathematical knowledge) to institutions. He underlines that the objects of knowledge and the tools to produce them are considered differently when moving from an institution to another, from the research institution to the teaching institution in our case.

In what follows, we consider the tangible, intangible and global links between research and teaching as defined by [START_REF] Neumann | Perceptions of the teaching-research nexus: a framework for analysis[END_REF]. We investigate the potential of this model as a lens to study the teaching-research nexus in the specific domain of graph theory. We rely on this model that was developed in the context of educational sciences; we investigate, in the specific domain of graph theory, its potential as a lens to study the teaching-research nexus.

Documentational Approach of Didactics: a framework to study interactions with resources

The DAD considers the activity of a teacher as a continuous process. It distinguishes between resources and documents [START_REF] Gueudet | University Teachers' Resources Systems and Documents[END_REF], where resources are defined as all the things that could re-source a teacher's activity and documents are the result of the association of resources and schemes of use of these resources.

A scheme is defined by [START_REF] Vergnaud | Towards a Cognitive Theory of Practice[END_REF] as the invariant organization of conduct for a class of situations which is a set of situations having the same aim. It is a dynamic structure with four interacting components: An aim that indicates intentionality in the organization of the activity; Rules of actions which are the ways of acting generated by the scheme to achieve a specific aim; Operational invariants that are the reasons behind the rules of action and the underlying justifications; and possibilities of inferences which are the adaptations brought by a subject to his activity to respond to the specificities of a situation corresponding to an aim. In a study based on interviews with university teachers, [START_REF] Gueudet | University Teachers' Resources Systems and Documents[END_REF] constructed "documents tables" where each row comprises a class of situations, and the corresponding, resources, rules of action and operational invariants.

In our work, we seek to characterize teachers' interactions with resources in the teaching institution. We draw on the DAD to study teachers' interactions with resources in teaching graph theory. Hence, we specifically consider teaching aims and the corresponding situations classes. We hypothesize that characterizing teachers' interactions with resources and the operational invariants that drive and justify them interactions can provide us with elements about the research-teaching relation. This relation could take the form of a migration of the resources between the teaching and research institutions and the adaptations that follow, or of a dissemination by a teacher of the professional knowledge and mode of teaching (the operational invariants). We seek to identify operational invariants that guide teachers' use of generic examples which seem to have an important role in proofs in graph theory. Their use by teachers can shed light on their underlying mathematical and pedagogical knowledge, hence provide elements to characterize the nature of the teaching-research nexus (tangible, intangible, global) as defined by [START_REF] Neumann | Perceptions of the teaching-research nexus: a framework for analysis[END_REF].

Methodology

To identify forms of relation between teaching and research, we consider the case of a teacher, who we are going to call Chiraz. She is a trained mathematician; whose research was on applications of graph theory in telecommunication. We consider her interactions with a resource she uses to teach graph theory for second year students majoring in mathematics at the Lebanese University (Click here to view the resource). The resource, designed collectively by Chiraz and two colleagues, focuses on the theoretical aspect of graph theory. We consider her interactions with the resource in her teaching (planning and implementation), with a focus on the use of generic examples. We expect that her choice of examples can inform us about her relation to graph theory in the teaching and research institutions, hence about the possible influence of her research on her teaching. Our methodology is based on an interview (Click here to view the grid) with Chiraz and observation of her classes.

In the first stage, we transcribed the interview and constructed a documents-table. For that, we identified classes of situations in Chiraz's teaching practices from the analysis of the interview. We particularly considered classes of situations related to Chiraz's use of generic examples in her teaching. For each class of situations, we identified the corresponding rules of action and operational invariants. We highlight that a contrast can exist between Chiraz's relation to graph theory in the teaching and research institutions, where her teaching focuses on the theoretical aspect while her research targets the applied aspects of the domain. Identifying Chiraz's rules of action and operational invariants in her teaching can help us gain a deeper understanding of the choices she made in the design of the resource and its use in the classroom. Therefore, it can help us detect the influence, if it exists, of her research activity on her teaching, and the institutional conditions that impact her choices.

The second stage consists in classroom observations which can allow us to characterize forms of use of the resource in class, hence complete and specify the results obtained in the first stage. We chose to attend three sessions, consisting of lectures and tutorials, that deal with the concepts of "paths" and "cycles". Several theorems involving these concepts are presented in the resource with their proofs. We selected episodes where Chiraz makes use of generic examples. We were particularly interested in her approach when writing an argument to prove a statement or when solving an exercise such as reasoning on small objects that have a characteristic representative of a broader class of objects before considering larger cases. These lines of reasoning are closely related to the epistemology of graph theory and can therefore indicate an influence of Chiraz's research activity on her teaching in the classroom. Our role was limited to observing the progress of the sessions (with video recording) and the implementation of the resource, taking into account the interactions with students.

Data Analysis

Analysis of the interview

Chiraz claims that her research activity does not influence her teaching because her research was on applied mathematics while she teaches in a mathematics major. In her PhD work, she modeled telecommunications systems using graph theory, which assumes the optimization of costs and the number of operators to be used. This work requires to juggle between particular and general cases in connection with the phenomena to be modeled. She says: "in teaching graph theory, we start with the simple concepts, and then bit by bit, we increase the level". Her perception of graph theory in the teaching institution as content where knowledge and skills are built in a cumulative way allows us to hypothesize the absence of a tangible teaching-research nexus in her teaching practices.

The analysis of the interview with Chiraz shows that most of the identified operational invariants in teaching situations classes are related to didactical knowledge and institutional conditions. Presenting contents in the resource in a theoretical manner with no links to other domains and disciplines seems influenced by institutional constraints related to the training course. Chiraz underlines that in graph theory, students have difficulty with proofs. Hence, she divides the proofs of the theorems into several parts to facilitate their understanding, which seems due to her personal experience with graph theory in the teaching institution. She points out that examples help students get to the proof of a theorem, she says "graph theory is more about sketching to be able to analyze the ideas, specially that it is the first introduction of students to graph theory, they cannot see it directly, they have to sketch". Eventhough we did not identify a tangible teaching-research nexus in the analysis of the interview with Chiraz, we estimate that an intangible nexus may exist between her research activity and her inclass teaching practices. This nexus can be identified in her use of the resource during class, her use of generic examples when reasoning on proofs, the attitude she tries to develop in students.

Analysis of classroom observations

In the first session we observed, Chiraz gave propositions and worked on their proofs with the students. We take as an example one proposition as presented in the resource (cf. figure 1). The proof of the proposition consists in showing that if two longest paths of a graph have no intersection, then there will be a path longer than these two paths. Chiraz co-constructed the proof with the students, by asking questions about the steps to follow such as "what do we need to prove? What if it makes plenty of intersections?" (cf. figure 2). She started by reasoning on particular cases. She first considered a case where the path between two vertices taken respectively on the two longest paths does not intersect with any of these two paths (except in the two vertices) (cf. figure 2). Then she considered a more general case where the path between the two vertices has several intersections with each of the two longest paths. Focusing on the key features of these two cases allowed Chiraz to establish the truth of the proposition. Taking small cases as a starting point before working on general cases can be justified by Chiraz's comment during the interview: "Particular cases, simpler cases, are closer to students' logical thinking, and once they understand the particular cases, they can generalize". Reasoning on smaller/particular cases and moving to larger/general cases is a 'classic' research approach in graph theory [START_REF] Goldin | Problem Solving Heuristics, Affect, and Discrete Mathematics[END_REF] and can therefore indicate the presence of an intangible nexus between Chiraz's activity of research and her teaching practices in class.

Figure 1 :

 1 Figure 1: Our translation of proposition 21 and its proof, as shown in the collectively designed resource

Figure 2 :

 2 Figure 2: Extract of the transcription of Chiraz's explanation of proposition 21 in class

Results and discussion

In our attempt to characterize the forms of relations between research and teaching in graph theory, we conducted an interview with a graph theory teacher followed by classroom observations. The analysis of the interview did not reveal the presence of a teaching-research nexus. However, the inclass teaching, particularly the use of generic examples when reasoning on proofs by the teacher, showed an intangible nexus between the teacher's research activity and her teaching practices. It seems to us that the nature of the teaching-research nexus detected using Neumann's model (1992) depended on whether the teacher was describing her activity globally, or whether she was involved in a particular activity. It was also affected by the specificity of the discipline involved.

The choice of graph theory as a mathematical domain seems relevant to identify the possible forms of relation between the teacher's research activity and her teaching in the classroom. We were able to identify moments when the teacher in the classroom tried to develop in students' ways of reasoning that are used by mathematicians when working on research problems in graph theory which indicates the presence of an intangible teaching-research nexus. This nexus is specially identified in the teacher's use of generic examples. We highlight the fact that this study focuses on the case of a teacher with a profile characterized by a contrast between her PhD work (applied mathematics) and the mathematics she teaches (rather theoretical). Observing a teacher with a different profile in the same institutional context can shed more light on the observed phenomena.

The interview, as methodological tool, has limits in elucidating the intangible teaching-research nexus. There might be sometimes a disparity between the declared practices and the effective practices in the classroom. Despite the teacher's perception of no existing relation between her research and her teaching, and despite the choices made in the collective design to present contents (high level of formalism, and no mention of applications), an intangible teaching-research nexus was identified in her implementation of the resource in the classroom.

Considering the resources of a university teacher opens up new perspectives to improve our understanding of the relation between research activity and teachers' practices. Considering the interactions with students can be crucial to understand the relation between teaching and research. This consideration, in the particular case of intangible teaching-research nexus, may offer the possibility for identifying correlations between research activity and the expected activity of students.


Schleich Jean-Marc

Houyel Lucile

Almange Claude

Anderson Robert

A new dynamic 3D virtual methodology for teaching the mechanics of atrial septation as seen in the human heart

Keywords: embryology, atrial septation, computer graphics, medical education

Background

Learning embryology remains difficult, since it requires understanding of many complex phenomena. The temporal evolution of developmental events has classically been illustrated using cartoons, which create difficulty in linking spatial and temporal aspects, such correlation being the keystone of descriptive embryology.

Methods

We synthesized the bibliographic data from recent studies of atrial septal development. On the basis of this synthesis, consensus on the stages of atrial septation as seen in the human heart has been reached by a group of experts in cardiac embryology and paediatric cardiology. This has permitted the preparation of three-dimensional (3-D) computer graphic objects for the anatomical components involved in the different stages of normal human atrial septation.

Results

We have provided a virtual guide to the process of normal atrial septation, the animation providing an appreciation of the temporal and morphologic events necessary to separate the systemic and pulmonary venous returns.

Conclusion

We have shown that our animations of normal human atrial septation increase significantly the teaching of the complex developmental processes involved, and provide a new dynamic for the process of learning.

INTRODUCTION

It is axiomatic that accurate knowledge of normal cardiac development is essential for properly understanding the morphogenesis of congenital cardiac malformations. Appreciation of the events occurring during cardiac development, however, remains difficult. It requires a clear perception of the appearance, spatial location, and function of a number of complex anatomical structures. This information is usually provided by sketches or drawings, which are designed to facilitate mental visualization, and hence understanding, of the spatio-temporal evolution of the developmental sequence ( ; ;

). Such [START_REF] Harvey | Heart development[END_REF][START_REF] Houyel | Embryologie cardiaque[END_REF]Schoenwolf et al., 2008 drawings and sketches, nonetheless, may not be the most effective means of communication, since cartoons prepared of necessity in two dimensions give only a partial impression of the spatial geometry of the objects illustrated. Moreover, the dynamics of structural evolution are rendered in less than complete fashion by a limited sequence of static images. This inadequacy in presentation of complex processes of spatial development is a major limitation to understanding, with significant efforts required graphically to illustrate the temporal sequence of development of most cardiac structures.

Use of advanced computer graphics has now been shown to provide substantial improvements for medical teaching in general (Dev, ; ), and cardiac morphogenesis in particular ( ; ). We have found that computer 1999 [START_REF] Higgins | Virtual bronchoscopy for three-dimensional pulmonary image assessment: state of the art and future needs[END_REF][START_REF] Laan | Deformation-corrected computer-aided three-dimensional reconstruction of immunohistochemically stained organs: application to the rat heart during early organogenesis[END_REF][START_REF] Whiten | Computer-aided interactive three-dimensional reconstruction of the embryonic human heart[END_REF] graphics provide solutions for the design, modelling, visualization, and manipulation of three-dimensional (3-D) objects, as well as for generating animated film sequences ( ). With such techniques, it is now possible to show evolutions of events that were [START_REF] Berlage | Augmented-reality communication for diagnostic tasks in cardiology[END_REF] previously inaccessible (

). This dynamic aspect is the cornerstone of descriptive embryology. Our current aim, Schleich and Almange, 2000 therefore, was to create teaching material in the form of an animated sequence of virtual images, along with a commentary, so as to improve the understanding of the complex events occurring during septation of the atrial chambers in the human heart. We describe the construction of a sequence of virtual images, explaining the options taken during the design of our model, and the presentation of the results. We also describe the formal evaluation undertaken to validate our opinion that such use of virtual animations can improve substantially the understanding of the complex events occurring during normal atrial septation.

CONSTRUCTION OF THE MODEL

Our overall objective was to design and develop a 3-D model showing the dynamics of the various stages of normal development and septation of the atrial chambers in the human heart. Such embryological modelling is achieved conventionally through reconstruction of serially sectioned murine hearts ( ; ). The mouse heart, however, is unsuitable to use as a model of [START_REF] Brune | A three-dimensional model of the mouse at embryonic day 9[END_REF][START_REF] Dhenain | Three-dimensional digital mouse atlas using high-resolution MRI[END_REF] human development. This is because the pulmonary vein remains a solitary channel as it enters the left atrium in the mouse, while the left superior caval vein remains as a patent channel. The structure of the oval foramen is also markedly different in the mouse compared to man (

). Moreover, reconstructions of serially sectioned embryos provide no more than a view at fixed periods, losing the [START_REF] Webb | The structure of the mouse heart in late fetal stages[END_REF] continuity required over time. Although we have access to serially sectioned human hearts, as yet we have not developed the facilities to reconstruct these sections. Based on examination of this material, however, along with study of previous descriptions, we chose to model the developing heart, rather than reconstructing it. The work required close cooperation between paediatric cardiologists and the graphic modeller in charge of designing the three-dimensional animated sequence. We created, therefore, a pattern of development to integrate our anatomic knowledge into the final product. This pattern can be summarized as follows:

Creation of the scene

In a first step, the current state of understanding of atrial septal development was reviewed by a multidisciplinary group of physicians, including cardiologists, paediatric cardiologists, and morphologists. The approach took into account early descriptions, forgotten subsequent to their publication, such as the account given by His for formation of the vestibular spine (

), and the study by R se His, 1880 ö which described the infolding of the superior rim of the oval fossa (

). These important studies themselves point to the R se, 1899 ö deficiencies in accounts of atrial septation as they appear in currently popular textbooks of embryology, such as the one edited by Larsen ( ) and renewed by Schoenwolf et al. (

). The group also took note of more recent studies correlating [START_REF] Larsen | Human embryology[END_REF]Schoenwolf et al., 2008 studies of development with knowledge of the morphology of the atrial septum in the human ( ; ; [START_REF] Anderson | Development and structure of the atrial septum[END_REF][START_REF] Anderson | Development of the heart: (2) Septation of the atriums and ventricles[END_REF] ). [START_REF] Kim | Development of the myocardium of the atrioventricular canal and the vestibular spine in the human heart[END_REF][START_REF] Wessels | Atrial development in the human heart: an immunohistochemical study with emphasis on the role of mesenchymal tissues[END_REF]

Development of the atrial septum in the human heart

Septation of the atrial component of the primary heart tube begins at the end of the 4 week of gestation, and continues well into the 3 th month. During that time, the embryo grows from 4 to 6 mm to around 50 mm in crown-rump length. Concomitant with the septation of rd the atrial chambers, the apical parts of the ventricles balloon in sequence from the ventricular loop, the muscular ventricular septum appearing between them in consequence of the ballooning.

The first evidence of septation of the developing atrial component of the heart tube comes with the appearance of the primary atrial septum. This develops as a crescentic muscular partition, growing from the upper part of the atrial component of the primary heart tube, and taking its origin to the left of the right pulmonary ridge, this being the rightward margin of the persisting dorsal mesocardium. By the time of appearance of the primary septum from the atrial roof, the systemic venous tributaries have already remoulded from being bilaterally symmetrical to opening exclusively to the right side of the atrial component. Only subsequent to this rightward shift do the boundaries of the systemic venous component become evident as the venous valves, and only subsequent to the appearance of these valves is it possible to recognise a discrete systemic venous sinus (

). By this stage, the pulmonary vein has also canalised as [START_REF] Achtemeier | Considerations for developing evaluations of online courses[END_REF] a solitary channel, which opens into the left side of the atrial component of the heart tube adjacent to the left side of the atrioventricular junction (

). As the primary atrial septum increases in size, it grows towards the atrioventricular endocardial cushions, themselves Figure 1 approaching each other so as to divide the atrioventricular canal (

). As the primary atrial septum grows towards the cushions, Figure 1 passing between the systemic venous sinus and the opening of the solitary pulmonary vein, it carries on its leading edge a prominent cap of mesenchymal tissue (

). This mesenchymal cap fuses initially with the superior atrioventricular cushion. At the same time, a Figure 2 protrusion of mesenchymal tissue from the dorsal mesocardium expands through the right pulmonary ridge. This structure, initially described by His as the spina vestibuli (

), and now termed the vestibular spine, grows towards the inferior atrioventricular

" " Figure 4 cushion (
), reinforcing the area of attachment of the primary atrial septum to the superior cushion. During this growth of the Figure 3 primary atrial septum, and the appearance of the vestibular spine, the superior and inferior atrioventricular cushions have themselves fused, dividing the atrioventricular canal into the presumptive mitral and tricuspid valvar orifices.

Prior to fusion of the various mesenchymal components, there is a communication distal to the leading edge of the primary atrial septum. This is the primary interatrial foramen, or ostium primum ( ). Before this hole is closed, multiple fenestrations appear in " " Figure 2 the cranial part of the primary septum. Coalescence of these fenestrations produces the secondary interatrial foramen, or ostium secundum " . At this stage, the cranial margin of this second foramen is no more than the roof of the atrial chamber (

). Subsequent to

" Figure 3
formation of the secondary interatrial foramen, the mesenchymal components all fuse together to close the primary foramen. Evidence 3 8

from studies in the developing mouse heart show that dorsal mesocardial protrusion then muscularises to form the plump antero-inferior rim of the oval foramen ( ; ;

). It is only with subsequent migration of the orifice of [START_REF] Snarr | Origin and fate of cardiac mesenchyme[END_REF]Snarr et al., 2007aSnarr et al., 2007b the pulmonary vein from a position close to the atrioventricular junction to its final position in the roof of the left atrium, with concomitant appearance of four pulmonary venous orifices, that the deep infolding in the roof of the atrium typically described as the septum " secundum becomes evident (

). In reality, this partition between the right and left atrial chambers is a deep groove between the " Figure 5

orifices of the caval veins and the right pulmonary veins, as described long since by R se ( ). Subsequent to birth, it is this fold ö R se, 1899

ö that provides the buttress against which the flap valve, representing the primary atrial septum, abuts to close the oval foramen ( ). Figure 6 Proposal for Sequence of Shapes From the developmental sequence described above, which correlates well with the structure of the definitive atrial septum in the human heart (

), the experts defined several development steps involving the main structures necessary for atrial septation, namely Figure 6 the primary atrial septum, its mesenchymal cap, the atrioventricular endocardial cushions, the vestibular spine, and the superior interatrial fold. The experts also discussed the changes in shape occurring with the presumed sequence of atrial development.

The modeller then produced a storyboard to define the origins, forms, and dimensions between the different elements involved in atrial septation (

). The proposed storyboard was submitted to the medical experts for validation, and was modified as deemed necessary. Figure 7 When the proposed three-dimensional shapes were considered too complex to be represented in two dimensions, plaster models were constructed to improve the understanding of the spatial relationships of the different structures involved in the process of septation. Once the modifications were agreed and validated, further details were added to elucidate the shapes, spatial extension, and the relationship in space of the main structures during the temporal evolution of septation. A new storyboard was then generated and reviewed again by the experts. It was these final static shapes that formed the final storyboard.

3-D Modelling

The three-dimensional sequence was designed by the graphic modeller, and reviewed by the group of experts. On the basis of the storyboard, the overall sequence was organized around a limited number of static three-dimensional, representing the agreed crucial stages of atrial development. The modelling process then took place as follows (

): [START_REF] Schleich | Understanding normal cardiac development using animated models[END_REF] Modelling of the agreed shapes. Having agreed the basic patterns, the modeller used control points to produce a surface model, assigning basic colours to the various structures, choosing relatively unsophisticated surface attributes. The shapes generated were submitted to the medical experts for validation.

Animations to link the shapes. Animation was produced by point-to-point linear morphing from one main shape to another (

). Figure 8 The shapes at the beginning and end of the animation were modelled in parallel to achieve a point-to-point correspondence. In instances of significant deformation, intermediate shapes were produced to smooth the morphing.

The group of medical experts evaluated the animations on a regular basis, feeding back potential modifications, such as the speed of development of one element of the septal complex compared to the others. It also proved necessary to incorporate new developments, such as the appearance of the secondary foramen.

Once the three-dimensional sequence was complete, it was refined with surface attributes, such as colour, texture, and shadows, so as to produce a realistic end product. Colours were selected to appear as natural as possible, or to follow conventional rules. The realistic aspect was also enhanced by used of textures extracted from medical images, such as photographs of operative procedures ( ). Figure 9 We then created a film showing the dynamics of atrial septation, using Newtek s LightWave product (Newtek, San Antonio, TX, ' ™ USA). The final animation was edited using Adobe Premiere software. Oral commentary was added, explaining the main steps of development. Arrows were added when necessary to pinpoint certain specific anatomical changes.

The final animation and commentary are available in AVI and mpeg video format, which can be displayed on any standard Personal

Computer using an interactive CD or DVD. This animation can also be viewed on a videotape (in PAL or NTSC format) or DVD-Video.

These latter options may be better suited for demonstrating normal atrial septation in the lecture room.

EVALUATION

We took several considerations into account when seeking to evaluate the performance of our developed multimedia project ( ; ). We acknowledged that it was necessary to evaluate first the effectiveness of its usability, and [START_REF] Achtemeier | Considerations for developing evaluations of online courses[END_REF][START_REF] Huang | Designing high-quality interactive multimedia learning modules[END_REF] second the outcomes in terms of learning. Since the answers to both points are subjective, evaluation has to be performed by questioning attitudinal and open responses. Our questionnaire designed to take note of both aspects is shown in the . We measured attitudinal appendix responses using a 5 point Lickert rating scale from strongly disagree to strongly agree.

We have distributed the animations to several institutions worldwide, sending 34 evaluations to those studying the animations, and obtaining 25 responses. We were able to draw several inferences from the answers received:

Use of tool

Most of the users made moderate use of the tool, usually from 2 to 5 times each year, albeit 4 responders used the product more than 5 times each year. The animation was used for teaching small groups on 15 occasions, for main courses in classrooms in 9 instances, and for self-learning in 5. In 8 instances, the tool was also used by the instructor for personal education.

Effective usability

Except for one neutral answer, all agreed that navigation through the modules was easy, a response confirmed by the open answers. Some responders, nonetheless, suggested that a learning phase was needed, while others suggested improvements, especially for navigating within the animations. Only one of our responders encountered technical problems produced by the operating system.

Learning outcomes

All agreed that the animation was useful for understanding the mechanics of atrial septation, with more than half agreeing strongly. All also agreed, or strongly that the module helped conceptual understanding. We received open comments such as very didactic ,

"

" "

good balance between simplification and realism , really useful for the spatial and temporal understanding , appropriate for summarizing " " " " "

, and stimulated the motivation of the students for learning . Feedback from the instructors was also very encouraging, especially

" "
regarding the use of the sequences during main courses.

DISCUSSION

Most publications on three-dimensional reconstruction have essentially focused on constituting databases ( ). [START_REF] Ringwald | A database for mouse development[END_REF] Such reconstructions have originated either from serial sections obtained using optical microscopy ( ; ; [START_REF] Brune | A three-dimensional model of the mouse at embryonic day 9[END_REF][START_REF] Laan | Deformation-corrected computer-aided three-dimensional reconstruction of immunohistochemically stained organs: application to the rat heart during early organogenesis[END_REF] ), from electron microscopic images ( ), or more recently from data acquired using magnetic Whiten et al., 1998 Wenink and[START_REF] Wenink | The value of graphic reconstructions-comparison with scanning electron microscopy[END_REF] resonance imaging ( ), optical projection tomography ( ), or episcopic fluorescence imaging [START_REF] Dhenain | Three-dimensional digital mouse atlas using high-resolution MRI[END_REF][START_REF] Sharpe | Optical projection tomography as a tool for 3D microscopy and gene expression studies[END_REF]. Such studies have improved the understanding of cardiac structure at a given time of development, but [START_REF] Weninger | Phenotyping transgenic embryos: a rapid 3-D screening method based on episcopic fluorescence image capturing[END_REF] representing static points in time, have of necessity not provided the dynamics required to appreciate the temporal sequence of cardiac development. The use of virtual images permits three-dimensional modelling of all steps of cardiac embryology, and the creation of dynamic sequences. Moreover, when dealing with the early stages of the cardiac development, the techniques discussed above have thus far been performed only on experimental animals, producing results which are less than ideal when considering atrial septation in the human heart. To our knowledge, our study is the first attempt to use techniques of virtual imaging to elucidate the development of the atrial septum in the human heart. Morphing techniques, which display events in four dimensions, are perfectly adapted to this specific type of teaching. Using commercial graphics software, other teams can involve themselves in embryology, as well as anatomy, physiology, and so on. Moreover, as we have the 3D shapes, they could be manipulated by a 3D viewer. This animation has been used in the University in Rennes for several courses on atrial development, and this experience has demonstrated that virtual imaging significantly improves the understanding of complex phenomena. To validate this opinion, we designed a questionnaire to compare the classical and dynamic approaches. The assessment of the teachers themselves, combined with the remarks and comments of the students, showed that this type of animation aided significantly in the understanding of normal atrial septation, both spatially and temporally. Another positive aspect is that feedback suggested that memorizing the overall mechanisms of atrial septation became much easier compared to classical lessons based on cartoons. Animation, nonetheless, has a major drawback if used in isolation.

The attention of the students is focused mainly on the animation itself, in other words on the spatio-temporal aspects of the overall events involved in septation. Precise anatomical details, such as the names and functions of the various developmental components, are usually not simultaneously integrated during visualization of the sequence, and must be learnt afterwards. This type of animation, therefore, is an aid to understanding, but still requires learning the mechanics of atrial septation.

A few errors or inaccuracies may have entered the creation of the 3-D objects and/or 3-D morphing scenes, but the ultimate purpose of the animation was to show that it is possible, with virtual images, to improve training for the dissemination of the steps involved in cardiac septation. The virtual nature of the animation makes it amenable to change should new data appear, or if the experts deem that modifications are needed. Moreover, rigorously exact modelling would have been counterproductive when judged against our aim to facilitate the understanding of elementary embryological mechanisms based on correlations with the definitive atrial structures. We elected to simplify the situation so as to produce a model that would be readily accessible to students. It is now our intention to integrate the animation and three-dimensional objects into a computed-aided teaching tool, which will include exhaustive explanatory texts, interactive commentaries, and a set of bibliographic references. It is our hope that this tool can be expanded to help self-learning of overall normal cardiac development. With this in mind, we are now exploring ways of applying new technologies to higher education.

Figure 5

This section comes from a human embryo after the completion of cardiac septation. It shows that formation of the superior interatrial fold is not possible until the pulmonary veins have migrated to the atrial roof. Note the muscularisation of the mesenchymal mass to form the antero-inferior buttress of the atrial septum.

Figure 6

This section is from an adult human heart, and shows that the so-called septum secundum is, in reality, a deep interatrial fold. It also shows

" "

the muscularised antero-inferior buttress.

Figure 7

Drawing of the storyboard in colour: The origins, forms, dimensions, spatial extension and the relationship in space and temporal changes between the different elements involved in the atrial septation are defined. In red: the primary atrial septum; in yellow: the vestibular spine; in blue: the atrioventricular endocardial cushions and the mesenchymal cap of the primary atrial septum and the mesenchymal tissue of the vestibular spine. A: anterior, P: posterior, S: superior, I: inferior.

Figure 8

Successive pictures showing the deformations of the shapes over time. A: anterior, P: posterior, S: superior, I: inferior.

Figure 9

The hypothesis used to explain atrial septation is designed as follows: the primary septum (in yellow) grows from the cranial part of the primary atrium, leaving a foramen distal to its leading edge called the primary foramen. The vestibular spine (in purple) enters the heart and grows towards the inferior atrioventricular cushion on the right side of the primary septum. A second partition, in brown, is produced by folding of the atrial roof. The primary foramen is closed by an expansion of tissue coming from the endocardial cushions, in white. The successive pictures are particularly relevant, permitting students to visualize the chronological and spatial changes that occur within the atrial chambers. A: anterior, P: posterior, S: superior, I: inferior.

Figure 1

 1 Figure 1This sagittal section is from a human embryo of Carnegie stage 14, taken in a plane paralleling the so-called parasternal long axis section. It shows the initial location of the solitary pulmonary vein adjacent to the atrioventricular junction. Note the unfused cushions in the atrioventricular canal.

Figure 2

 2 Figure 2This frontal section, again from a human embryo, but at the early part of Carnegie stage 16, shows the primary atrial septum, with its mesenchymal cap, growing towards the superior endocardial cushion.

Figure 3 Figure 4

 34 Figure 3 This section, again in the frontal plane, is from a human embryo in the later part of the 16 Carnegie stage. The mesenchymal structures have th fused, but the superior roof of the atrial component is flat (black line), with no evidence of formation of a septum secundum ." "

Ackowledgements:

We thank St phane Andru, who designed the three-dimensional objects and the animated sequence, and Pierre Galluc for the development of é the Open GL 3-D viewer.


Joachim Trosseille
email: joachim.trosseille@espci.fr

Anne Mongruel

Laurent Royon

Daniel Beysens

Effective substrate emissivity during dew water condensation

Keywords: Dew, Condensation, Radiative cooling, Emissivity 2010 MSC: 00-01, 99-00

Dew formation is due to the radiative cooling of a surface. Surface emissivity is classically considered without the contribution of water condensation. However the latter, with an emissivity of w =0.98, can occupy more than 80% of the surface and strongly affect is radiative properties. Here we present experimental and theoretical studies of the effective surface emissivity, taking into account the presence of the condensing water. The main results are concerned with the effect of the dry surface emissivity on condensation. On the one hand it appears that there is a significant difference between high and low emissivity substrates in the duration of a transient regime at the begining of condensation, making the overall condensation yield smaller for low emissivity substrates. On the other hand, in permanent regime of condensation, the effective surface emissivity is dominated by the presence of drops and the condensation only weakly depends on the dry surface emissivity. These results are of practical importance and can be extended to all application involving radiative heat transfer with condensation.

Introduction

Dew is ubiquitous in nature. Its formation involves the cooling of a substrate (plant, soil or artificial substrate) below the dew point temperature of the surrounding humid air. Such a process is ensured by the radiation deficit between the substrate and the atmosphere and is, as a consequence, thought to be all the more efficient that the substrate emissivity is close to unity. As a matter of fact, most of dew condensation studies assume for radiative exchange a substrate emissivity without condensed water [START_REF] Clus | Comparison of various radiation-cooled dew condensers using computational fluid dynamics[END_REF][START_REF] Nikolayev | Water recovery from dew[END_REF][START_REF] Tomaszkiewicz | Dew as a sustainable non-conventional water resource: a critical review[END_REF][START_REF] Nilsson | Condensation of water by radiative cooling[END_REF]]. It appears, however, that condensation alters this picture as the presence of water droplets should modify the emissivity of the substrate according to the surface coverage of the droplet pattern. One interesting measurement was reported by Maestro-Valero et al. [START_REF] Maestro-Valero | Comparative analysis of two polyethylene foil materials for dew harvesting in a semi-arid climate[END_REF] concerning the spectral emissivity of dry and wet plastic substrate.

They showed by spectral measurements that substrate emissivity in the 7-14 µm band was increased when wet. Unfortunately, they did not characterize further this effect nor what were the wet conditions.

Figure 1 shows a thermal map of drops (darker spots) condensed on a reflective substrate (brighter spots). As explained in section 3.2, each color correspond to a different value of emissivity. The darker is the color, the higher is the emissivity. The different zones (A,B,C,D), explicited in section 5.1, highlight the non-homogeneity and the diversity of emissivity pattern on a condensing surface. At a local scale, emissivity depends strongly on the presence of water on a specific point and only affects the local energy balance. At a larger scale, the global energy balance of a condensing surface depends on what we call effective emissivity, which takes into account the coverage of water on the surface.

For practical reasons (heat transfer calculations, condensation rates, etc...), the global energy balance is more often preferred, which requires the knowledge of the interplay between water surface coverage and surface effective emissivity.

It is therefore the object of this study to investigate quantitatively the effect of condensation on surface effective emissivity during condensation. For this purpose we use two substrates with small and large emissivities while show- ing the same wetting properties. A specially designed experimental setup is used to create a radiative deficit between the substrate and a cold source, under controlled air temperature and humidity [START_REF] Trosseille | Radiative cooling for dew condensation[END_REF]. An infra-red camera is used to measure emissivity of the samples in the [7.5-14] µm range, the so-called atmospheric window where atmosphere is nearly transparent to IR radiations [START_REF] Berdahl | The thermal radiance of clear skies[END_REF].

Drop surface coverage

The surface coverage of dew or breath figures pattern during dropwise condensation is defined as e d = πR 2 i /S where R i is the individual drop radius and S the area of the considered surface. It evolves during condensation following different stages of growth, this evolution depending only on the drops and substrate dimensionalities [START_REF] Beysens | Dew Water[END_REF]. For diffusion limited condensation and for the general case of 3D droplets condensing on a 2D substrate, the different stages are the following:

(i) Nucleation of droplets on substrate inhomogeneities.

(ii) Individual droplet growth with time t as R i ∼ t 1/2 when e d is low, followed by a growth in t 1/3 , (iii) Self-similar growth when drops start to coalesce with each other. Individual drops grow as R i ∼ t 1/3 between coalescences, while the mean pattern radius evolves as < R >∼ t. The surface coverage e d reaches there a limit e 0 d due to the balance between drop growth, which increases e d , and drop coalescence, which lowers e d . The value of e 0 d depends on the substrate wetting properties (drop contact angle θ) [START_REF] Zhao | From droplets growth to film growth on a heterogeneous surface, condensation associated with a wettability gradient[END_REF], as

e 0 d ≈ 1 - θ(deg.) 180 (1)
(iv) Late stages, characterized by the re-nucleation of new droplet patterns on the bare surface between large drops with large interdistances (see zone D in figure 1 for instance), leading to the increase of the global surface water coverage. When p different drop patterns that have nucleated at p different times coexist (see [START_REF] Beysens | How does dew form ?[END_REF] and Fig. 1, zones B, C, D)), one gets

e d ≈ 1 -(1 -e 0 d ) p (2)
(v) Gravity makes the drops above a critical radius to slide down [START_REF] Elsherbini | Retention forces and contact angles for critical liquid drops on non-horizontal surfaces[END_REF] and limits the drop sizes.

Note that one has to differenciate between e d , the drop surface coverage of the substrate corresponding to the footprint of the drop on the substrate (dotted line in figure 2), from e e , the projected drop surface coverage as seen from above (full line in figure 2). The relevant quantity to consider for radiative exchanges is e e as defined by

e e =    e d if θ < 90 • 0.5 if θ > 90 • (3)

Experiments and methods

In contrast with usual condensation experiments where cooling is ensured by contact, radiative condensation experiments requires some specific devices and materials. Samples are elaborated to provide materials with various emissivities but same wetting properties. An infrared camera is used to measure local and mean emissivities. Water condensation is performed in both radiative and conductive cooling devices.

Samples

Two samples of low and high emissivities but with same wetting properties are prepared in the following way. Thin sheets of different materials are prepared, with the same 6 µm thick transparent low density polyethylene (LDPE) sheet on top of each stack (figure 3). The monochromatic light transmission coefficient τ λ of a thick transparent media can be calculated from equations 6 and 7 below using the extinction coefficient k λ , i.e. the complex part of the refractive index of the material. Datas from [START_REF] Al-Mahdouri | Evaluation of optical properties and thermal performances of different greenhouse covering materials[END_REF] show that for LDPE in the wavelength range [START_REF] Clus | Comparison of various radiation-cooled dew condensers using computational fluid dynamics[END_REF][START_REF] Nikolayev | Water recovery from dew[END_REF][START_REF] Tomaszkiewicz | Dew as a sustainable non-conventional water resource: a critical review[END_REF][START_REF] Nilsson | Condensation of water by radiative cooling[END_REF][START_REF] Maestro-Valero | Comparative analysis of two polyethylene foil materials for dew harvesting in a semi-arid climate[END_REF][START_REF] Trosseille | Radiative cooling for dew condensation[END_REF][START_REF] Berdahl | The thermal radiance of clear skies[END_REF][START_REF] Beysens | Dew Water[END_REF][START_REF] Zhao | From droplets growth to film growth on a heterogeneous surface, condensation associated with a wettability gradient[END_REF][START_REF] Beysens | How does dew form ?[END_REF][START_REF] Elsherbini | Retention forces and contact angles for critical liquid drops on non-horizontal surfaces[END_REF][START_REF] Al-Mahdouri | Evaluation of optical properties and thermal performances of different greenhouse covering materials[END_REF][START_REF] Gustavsen | Spectral emissivity of anodized aluminium and the thermal transmittance of aluminium window frames[END_REF][START_REF] Försth | Absorptivity and its dependence on heat source temperature and degree of thermal breakdown[END_REF][START_REF] Minkina | Atmospheric transmission coefficient modelling in the infrared for thermovision measurements[END_REF][START_REF] Wiecek | Thermovision in infrared basics and applications[END_REF][START_REF] Howell | Thermal Radiation Heat Transfer[END_REF][START_REF] Downing | Optical constant of water in the infrared[END_REF][START_REF] Rees | Angular variation of the infrared emissivity of ice and water surfaces[END_REF][20] µm the extinction coefficient is constant at k ∼ 10 Emissivities of samples are measured with an IR camera by the same technique as described in Section 3.2. The first sample, S-, is a stack of aluminium (emissivity ε alu 0.04 [START_REF] Gustavsen | Spectral emissivity of anodized aluminium and the thermal transmittance of aluminium window frames[END_REF]) and polyethylene sheets. The latter being highly transparent, the overall emissivity of the sample will be close to the emissivity of aluminium. In fact the emissivity of S-, measured by the same technique as explained in section 3.2 is ε S-= 0.05 ± 0.05. The second sample, S+, is 250 µm thick PVC sheet of emissivity ε P V C = 0.95 [START_REF] Försth | Absorptivity and its dependence on heat source temperature and degree of thermal breakdown[END_REF] sandwiched between an aluminium sheet and the same polyethylene sheet above. As for S-the resulting emissivity of sample S+ will be close to the emissivity of the PVC sheet. As a result the overall emissivity of S+, measured as explained in section 3.2, is ε S+ = 0.88 ± 0.01.

As the contact angle of water varies between its advancing (θ a ) and receding (θ r ) values due to drop coalescences, we choose to use an overall contact angle θ m to characterise the wettability of the samples. The latter is calculated from eq. 1, which relates the contact angle to the experimental measure of the surface coverage during the self-similar regime of condensation [START_REF] Zhao | From droplets growth to film growth on a heterogeneous surface, condensation associated with a wettability gradient[END_REF]. From the analysis of IR images (see fig. 1 at t=4000 s concerning the permanent regime of condensation) we get e 0 e =e 0 d =0.63 ± 0.02 and θ m = 65.9 • ± 3.6 • for water on low density polyethylene.

Emissivity measurements

In order to measure the emissivity of the samples, we use a cooling device made out of a Peltier element homogenized by a thick electrolytic copper plate.

The sample is stuck on the free surface of the Peltier stage which acts as a cold source. A contact cooling is thus used, as opposed to dew condensation which uses radiative cooling of the substrate. However, the emissivities as obtained from this method will be the same as gained with radiative cooling because the spatial properties of the condensation pattern, which are directly linked to the effective emissivity (see section 1), do not depend on the cooling process but only on the dimensionality of the droplets and the dimensionality of the substrate (see Ref. [START_REF] Beysens | Dew Water[END_REF]). Experiments are performed at room temperature T a =26 • C and relative humidity (ratio of ambiant vapour pressure to the saturation pressure at ambiant temperature) RH = 48 %. The power of the Peltier element is adjusted in order to induce or avoid, depending on the experiment, condensation on the sample.

Emissivity measurements are made by means of a FLIR A655SC infra-red camera in the range [7.5-14] µm. The sensor of the camera is a 640x480px wide uncooled microbolometer. A 25 µm close-up lens is used to provide a spatial resolution of 42.8 px/mm at working distance. The IR camera does not perform direct emissivity measurements but measures the total radiance L m received by each pixel of the sensor. The latter comes from (i) the radiance emitted by the object εL 0 (T ) with ε the emissivity of the object and L 0 (T ) the radiance of a black body at same object temperature T in the range [7.5-14] µm, and (ii) the radiance reflected by the object from the ambient environment, (1 -ε)L 0 (T a ).

Here L 0 (T a ) is the radiance emitted by a black body at the temperature T a of the ambient environment in the range [7.5-14] µm. In this expression the atmosphere between the sensor and the object is considered transparent to infrared radiations considering the small distance between object and camera (< 0.5 m) and the spectral range of the camera [START_REF] Minkina | Atmospheric transmission coefficient modelling in the infrared for thermovision measurements[END_REF][START_REF] Wiecek | Thermovision in infrared basics and applications[END_REF] . It thus follows

L m = εL 0 (T ) + (1 -ε)L 0 (T a ) (4)
Equation 4 then gives access to the emissivity :

ε = L m -L 0 (T a ) L 0 (T ) -L 0 (T a ) (5)
The measurement being performed on each pixel of the sensor, the radiance can then be averaged on a specific area. If one assumes that temperature is uniform on the surface, equation 5 gives a mean value of the emissivity on the area. As radiance measurements are performed between 7.5 and 14 µm, all values of measured emissivity in the manuscript will be total emissivity between 7.5 and 14 µm.

Two kinds of measurements are performed. The first measurement is the determination of the dry sample emissivity while the second measurement is a determination of the effective emissivity of the sample under condensation.

Condensation by radiative cooling

The substrate has to be cooled below the dew point temperature, that is the saturation temperature of a humid air at a given water vapor pressure. For that purpose, we use the same radiative cooling setup as described in [START_REF] Trosseille | Radiative cooling for dew condensation[END_REF] and hereafter denoted as "radiative chamber". Figure 4 reports its main features.

The device is based on radiative interactions between a cold source (1) and the object (2) under study. An infrared heat transfer takes place between the cold source and the object. The object ( 2) is settled on a sample holder [START_REF] Tomaszkiewicz | Dew as a sustainable non-conventional water resource: a critical review[END_REF] weighed by a precision balance to monitor the evolution of the condensed mass.

Results

For each sample described in section 3.1, two seprate experiments are carried out. In one experiment, the experimental setup described in section 3.2 is used. The sample is cooled by conduction and condensation is observed with an IR camera, in order to determine the evolution of the effective emissivity of the condensing surface during the first times of condensation. In a second experiment, the sample is introduced into the radiative chamber described in section 3.3. The condensed mass of water is recorded in order to measure the dynamics of condensation by radiative cooling.

Effective emissivity evolution

The condensation is obtained by conductive cooling using a Peltier element, as noted in Section 3.2. The evolution of the radiance coming from the condensing surface is observed by IR camera. Figure 6a shows the radiance map received by the thermal camera from surface S-(emissivity ε=0.05) during condensation. As well as described in the previous experiment, shown in figure 5, the radiance received by the camera decreases as drops condense on the surface, replacing part of the low emissivity dry surface by the emissivity of the drops. Figure 6b shows the radiance map received by the thermal camera from surface S-(emissivity ε=0.88). As opposed to the experiment on S-, the received radiance at t=0 s is low and mainly comes from the emission of the substrate. As condensation occurs on the surface, little modification of the received radiance is observed, despite condensation (same surface temperature and same material as for figure 6a.), attested by slight spots of lower radiance visible after t=5000 s.

Figure 7 shows the evolution of the mean effective emissivities calculated from equation 5 using the measured radiance shown in figure 6 averaged on the substrate area (≈ 500 mm 2 ). Curve S-corresponds to the effective condensation emissivity on substrate S-and curve S+ on substrate S+. Curve W represents the emissivity ε w = 0.98 of an infinitely thick layer of water (opaque, τ w =0) Mean substrate emissivity varies from 0.05 at t=0 s to 0.87 at t=10000 s for S-and from 0.88 at t=0 s to 0.95 at t=10000 s for S+ (see Fig. 7). Field of view : 15x12.5 mm 2 . After about 1000 s, S-and S+ emissivities become of comparable magnitude (see also Fig. 7). Color bar : mesured radiance (W.m -2 .sr -1 ) between 7.5 and 14 microns and curves S 0 -,+ represents the dry substrate emissivities. Both curves S+ and S-show similar evolution. At t=0 s curve Sexhibits a value ε S-≈0.03, close to the emissivity of the dry substrate (0.05).

Curve S+ shows the value ε S+ ≈ 0.87, close to the emissivity of dry substrate (0.88). The slight decrease at very short time is due to the sharp temperature drop at the beginning of the experiment. In fact as temperature T drops, black body radiation at T and measured radiation (resp. L 0 (T ) and L mes in eq.5)

decrease. The measured radiation decrease is however slowered by the dynamic of droplets condensation on the substrate which modifies the surface emissivity while growing. Measured radiation thus decreases slower than black body radiation and the result is a decrease of the calculated mean emissivty of the substrate (see eq.5). As condensation proceeds, temperature becomes stable and the mean emissivity of the substrate is increasing due to increasing water occupation of the substrate. Both curves S+ and S-increase rapidly until t ≈5000 s due to water droplet condensation. After this time they slowly continue to increase to reach a stable value close to the emissivity of an infinitely thick layer of water (0.98). Note that S+ reache after about 5000 s its asymptotic value while S-continues to slowly increase. The emissivity of S-anyway remains always smaller than S+, itself lower than the pure water emissivity (W). Finally one can note that the dynamics of S-curve shows a much larger amplitude than curve S+, meaning that the effective emissivity of substrate S-is much more modified by condensation than S+. One will see in the following that the above behavior can be understood in terms of both threshold in droplet size for IR absorption and drop surface coverage evolution.

Evolution of the condensed mass

When cooling is radiative, the modification of the effective emissivity by droplet condensation influences in turn the condensation dynamics. In order to evidence the effect, two 30 mm disks made of S-and S+ materials are introduced in the radiative chamber. Figure 8 reports the evolution of the condensed mass of water on both substrates. The different curves exhibit similar shape and present two growth regimes: a transient regime characterized by a non-linear increase of the condensed mass, followed by a permanent regime with constant condensation rate. Both condensation rates in permanent regime are very close with ṁ S+ =3.95×10 -5 ± 0.05 g.s -1 and ṁ S-=4.13×10 -5 ± 0.05 g.s -1 . However, condensation does not start at the same time for both experiments. In particular, condensation on substrate S-is delayed by ∆t (≈ 1000 s) as compared to substrate S+. During this delay, water droplets that have nucleated on substrate defects become large enough to increase the effective emissivity and to ensure an efficient cooling. Ta=25.1 • C, RH=95 %, ṁ S+ =3.95×10 -5 ± 0.05 g.s -1 , ṁ S-=4.13×10 -5 ± 0.05 g.s -1

Discussions

Water droplet effective emissivity

In order to understand the difference of behavior between S-and S+ substrates, one needs to determine the radiative properties of each materials. One of the materials is the substrate, whose emissivity and reflectivity are known and the other is water drops. Thus, we are concerned with the emissivity of spherical caps of water on a substrate.

Transmittivity τ λ of a water layer with thickness L can be expressed as a function of κ λ , the spectral absorption coefficient, as:

τ λ = exp(-κ λ L) (6)
With n λ the real part and k λ the imaginary part of the water refractive index (n * = n λ + ik λ ) [START_REF] Howell | Thermal Radiation Heat Transfer[END_REF],

κ λ = 4πk λ λ (7)
Note that Equation 6is only valid for an isothermal, isotropic and homogeneous material, which is considered to be the case for water in this study [REF ?].

The reflection coefficient of the interface air/water in the direction normal to the interface can be expressed as [START_REF] Howell | Thermal Radiation Heat Transfer[END_REF]

ρ λ = (n λ -1) 2 + k 2 λ (n λ + 1) 2 + k 2 λ ( 8
)
Using the refractive index spectral values n λ and k λ of water from [START_REF] Downing | Optical constant of water in the infrared[END_REF] one can thus deduce the variation with λ of reflectivity ρ λ (Fig. 9a) and transmittivity τ λ , the latter for different sample thickness values (Fig. 9b).

Let us now consider a water droplet on a substrate. The drop at the air-water interface partially reflects the radiation from the surrounding environment. Depending on drop thickness, radiation is partly transmitted from the substrate and from the surrounding environment. The radiation trajectory inside a drop is quite complex. Depending on the incidence of radiation, refraction and total reflexion are encountered. In order to treat the problem analytically, the drop is therefore schematized as a cylinder of radius r and height Z m (figure 10a). In addition, only radiation in the direction normal to the substrate is considered because it is the radiation captured by the camera. L e is the radiance emitted

Z m = r e (1 -cosθ) = r(1 -cosθ) sinθ (9)
This model obviously oversimplifies the problem because it excludes the angular variability of water emissivity. However it is supported by the relative homogeneity of drops radiations as shown in figure 1. Moreover, Rees and James (1992) [START_REF] Rees | Angular variation of the infrared emissivity of ice and water surfaces[END_REF] show that water angular emissivity only drops for angles higher than 60 • which means that only the peripherical region of the drop is concerned with angular emissivity dependance. This region is also concerned with water transparency due to drop thinning approching the contact line and would require a detailed study to quantify its radiative features. The effective radiative properties of emission and reflection of a substrate covered with water drops are calculated as follows. Figure 10c shows the trajectory of radiation through the layer of water (radiation is perpendicular to the substrate but an angle is made for the sake of clarity). Radiation L i λ corresponds to light incoming at the air/water interface. The fraction ρ λw is reflected while the other fraction 1 -ρ λw penetrates into water. The fraction τ λw of the later radiation is transmitted to the substrate, which in turn reflects a fraction ρ λs . It follows a sequence of reflections in the substrate and at the water/air interface.

Reflected radiance is thus the radiation leaving water after an infinity of successive reflections. From the definition of the reflection coefficient R λ = L r λ /L i λ it comes, with k an integer number:

R λ = ρ λw + ρ λs (1 -ρ λw ) 2 τ 2 λw ∞ k=0 (ρ λs ρ λw τ 2 λw ) k (10)
This is a geometric serie whose limiting value is

R λ = ρ λw + ρ λs (1 -ρ λw ) 2 τ 2 λw 1 -ρ λs ρ λw τ 2 λw ( 11
)
Using Kirchoffs law and the conservation of energy, the effective emissivity of a layer of water on a reflective substrate can be written as

E λ = 1 -R λ . Note
that R λ being a function of τ λw and then of the water thickness Z m (Eq. 6), E λ and R λ also depend on Z m . The integration of the spectral emissivity E λ between two wavelength x1 and x2 is made by equation 12 with L 0 λ (T ) being the spectral radiance of a black body at temperature T .

E = x2 x1 E λ L 0 λ (T )dλ x2 x1 L 0 λ (T )dλ (12)
In figure 11 is plotted the integration of the emissivity E of a slab of water between two different wavelength ranges as a function of the thickness of the drop for a substrate of low emissivity ε s = 0.05. The first wavelength range, (I), is [7.5-14] µm corresponding to the atmospheric window and the working range of the IR camera used for the measurements of emissivity. The second range, (II), is [2 -50] µm and corresponds to the whole range where data were available [START_REF] Downing | Optical constant of water in the infrared[END_REF]. As most of the energy of a body at temperature around 10 • C is emitted in this wavelength range, the integrated emissivity can be considered as the emissivity at all wavelengths.

It can be clearly seen from figure 11 that there is a thickness threshold Z 0 m above which the effective emissivity E is constant, with a value equal to the emissivity of a non-transparent material of reflectivity ρ w . The value is Z 0 m ≈ 20 µm for range (I) and effective emissivity stabilizes at E = 0.98, while threshold is Z 0 m ≈ 12 µm and effective emissivity stabilizes at E = 0.96 for range (II). Above Z 0 m the water transmittivity τ λw tends to 0, which implies from (11) that the reflection coefficient of the layer of water, R λ , tends to the reflectivity of the interface air/water ρ λw . As

E λ = 1 -R λ we get E λ (Z m → ∞) = 1 -ρ λw .
Then, the discrepancy between the effective emissivities for range (I) and (II) can be explained by the spectral distribution of ρ w in ranges (I) and (II) (figure 9a). Below Z 0 m , the effective emissivity decreases with the drop thickness until reaching a value close to the substrate emissivity (see inset in figure 11.), yet being slighlty higher because of the reflections on the air/water interface.

The above results help to understand the differences of local emissivity from the IR image of drops condensing on a substrate. Figure 1 shows a close up view of condensation on substrate S-(note that experimental conditions are somewhat different from figure 6). Drops with height larger than the threshold Z 0 m are nearly opaque with large effective emissivity E = 0.98 and appear dark (A). As most of the radiance received by the camera come from the emission of the drops, mean radiance is low. Clearer areas on the IR image (B), with high measured radiance, correspond to the dry substrate revealed because of drop displacement induced by coalescences. The mean radiance of these areas is thus composed of a small contribution from dry substrate emission (low substrate emissivity 0.05) and a large contribution corresponding to the emission of the surrounding environment reflected on the substrate (high substrate reflectivity 0.95). Regions of intermediate radiance (C) and (D) correspond to smaller drops covering the substrate. These drops, not thick enough to be completely opaque, only transmit a fraction of the reflected radiance.

Mean surface effective emissivity

The surface averaged effective emissivity can be calculated by considering only the completely opaque drops (drops thicker than Z 0 m ), the semi-transparent small drops contribution (Z m < Z 0 m ) being neglected. What matters is the surface fraction of the projected surface of drops (drop view surface), e e , calculated from equation 3.

In order to obtain the mean effective emissivity < E > one has to consider the effective emissivity E corresponding to the surface fraction e e of opaque drops, to which is added the contribution of the dry fraction of the substrate, 1 -e e , with emissivity ε s :

< E >= e e E + (1 -e e )ε s (13)
Figure 12 shows the variations of < E > as a function of the contact angle and for different substrate emissivities, as calculated from equations 13, 3 and the mean effective emissivity, except for substrate emissivities higher than the emissivity of bulk water. As condensation occurs, e e approaches unity and the part of the mean effective emissivity due to the substrate emissivity is reduced.

This effect explains the evolution of the mean effectivity during condensation (figure 7). At the beginning of the experiment only the dry substrate emissivity is measured. During the first moments of condensation, drops nucleate on substrate defects and start to occupy the surface. But as they are not thick enough to be opaque, the emissivity of the drop on the substrate is still close to the dry substrate emissivity. Then, as drops grow larger, they become less transparent and the substrate effective emissivity increases. Eventually, as the surface coverage stabilizes itself, a pattern appears with opaque large drops, semi-transparent smaller drops and zones clear of condensation, leading to a stabilization of the mean effective emissivity close for all substrates to the high water emissivity. As can be seen on fig.

12. The mean effective emissivity for substrate S+ and S-(ε S+ = 0.88, ε S-= 0.05, θ LDP E = 65.9 • ) are respectively < E S+ > = 0.975 and < E S-> = 0.934, close to the experimental values in the permanent regime (see figure 7), the discrepancy being due to the transparency of smaller droplets.

Condensation rate

Figure 8 shows that the dynamics of condensation are similar in both S+ and S-substrates but the delay of condensation is larger on the less emissive substrate S-. In order to understand this dynamics one can consider the following energy balance on the substrate + droplets, supposed to be at the same temperature T [REF?], and thus temperature being uniform on the surface [START_REF] Beysens | Dew Water[END_REF].

(M C o + mC w ) dT dt = φ r + φ cv + φ c + φ l (14)
Here M is the mass of the substrate, with specific heat C o , and m is water The mass balance on water in diffusion limited condensation reads

dm dt = a w S [p v (T a ) -p v (T )] ≈ a w S [p v (T a ) -p s (T )] (16)
The above equation gives the condensation rate dm/dt as a function of the difference in water vapor pressure at substrate temperature T (supposed to be the temperature of water drops), p v (T ) and air temperature at T a , p v (T a). S is the area of condensation and a w is the water vapor transfer coefficient [START_REF] Beysens | Dew Water[END_REF].

At time t = 0, humid air enters into the chamber. Since T S-> T S+ and thus p v (T S-) < p v (T S+ ), according to Eq. 16, the condensation rate dm/dt on S-is lower than on S+ (Fig. 8).

For t > t 0 S+ and substrate S+ water starts to condense (Fig. 8). After a small diminution due to the temperature drop at the beginning of the experiment coming from the release of latent heat (see sec. 4.1), effective emissivity increases slightly (Fig. 7) due to the growth of water droplets that become opaque to IR.

The corresponding increase of emissivity also induces an increase of radiative cooling flux, resulting after about 1000 s to a stationary state. For t < t 0 S-and substrate S-the initial condensation rate is small and the growth of the first condensed droplets is limited. The effective emissivity of the surface as well as the radiative heat flux is not substantially modified by the small drops which are nearly transparent to IR. Thus the temperature of the surface is constant and the condensation rate keeps small, which explains the very low increase of the condensed mass on curve S-in fig. 8. After ∆t = 1000 s drops reach the threshold size Z 0 m and become opaque to IR radiation. The same dynamic as for substrate S+ occurs with increase of emissivity and cooling radiative flux, leading to the decrease of the surface temperature and the increase of the condensation rate.

In the permanent regime the condensation rates of both experiments are stabilized. As explained in section 4.1 the effective emissivities of both experiments are close to pure water emissivity because of the high surface coverage of opaque drops whose radius is above threshold Z 0 m . It follows that in the steady state regime condensation rates becomes similar and independent on the particular value of the substrate emissivity. The only significant difference between low and high emissivity substrates is the time delay ∆t to reach the steady state, larger for low substrate emissivities. Once condensation starts the importance of the substrate emissivity decreases because of the presence of drops with high emissivity. At equilibrium the effective emissivity thus weakly depends on the substrate emissivity and condensation rate becomes independent of the radiative properties of the substrate.

For low substrate emissivity the transient regime however limits condensation yield, because, for same time of condensation, the condensed mass on low emissivity substrates will be smaller than on high emissivity substrates. Moreover, by limiting initially the cooling radiative heat transfer, a low emissive substrate can even prohibit condensation if the threshold in drop thickness Z 0 m is never reached.

Conclusion

In this study we have addressed the contribution of water condensation to the radiative cooling of substrates. Through the lens of what we called the effective emissivity of a condensing surface, we have explored the importance of the emissivity of the base substrate, partially hindered by the presence of condensed droplets which are caracterized by their contact angle on the substrate and their surface coverage ratio. From this study two important results emerge. i) The base substrate emissivity has a major impact at the begining of condensation as it determines the length of a transient regime of condensation when only small drops are present. This transient regime induces a smaller condensation yield for low emissivity substrates. ii) In permanent regime of condensation, the effective emissivity of a condensing surface is dominated by the presence of drops and the contribution of the base substrate radiative properties in the radiative cooling of the surface is small. The condensation rate in permanent regime is thus nearly independent of the base substrate radiative properties. These results of practical importance can be generalized to any fluid condensation on substate of different emissivity.

Figure 1 :

 1 Figure 1: Close up view of droplets condensed on a reflective substrate( ε S-= 0.05) . HR=62%, Ta=27 • C, Ts=7.6 • C. t=4000 s. A, B, C, D : zones of interest (see text). Color bar : mesured radiance (W.m -2 .sr -1 )

Figure 2 :

 2 Figure 2: Variation with respect to the drop contact angle θ of surface and surface view coverages in the initial self-similar stage. Wet surface coverage e d : Dotted line, cases a, b, c. Drop view surface ee: Full lines a, b, d. (Adapted from [8] and [9])

- 3 .

 3 Equations 6 and 7 can thus be used for integrated values and we get τ 0.99 for a 6 µm thick film. The top LPDE thin sheet is thus highly transparent to IR and each composite material will present the same contact angle with water drops while having emissivity values depending on the material layers stacked under the first sheet.

Figure 3 :

 3 Figure 3: Stacks of different emissivities but same wetting properties (LDPE : low density polyethylene). Left: S-substrate (ε S-= 0.05), Right: S+ substrate (ε S+ = 0.88). Other notations : see text.

Figure 4 : 4 :

 44 Figure 4: Schematics of the experimental setup. 1: Cold source ; 2: Cooled object ; 3: Stand ; 4: Cylindrical humid chamber; 5: IR -transparent window; 6: Mirrors ; 7-8: Visualization windows; arrows: Flow direction . T 1 ,T 2 ,T 3 : Thermocouples. (figure not to scale)

Figure 5

 5 Figure5shows a close-up view of the evolution of condensation on substrate S-(ε S-=0.05) with RH = 62 %, an air temperature of T a =27 • C. After a transient regime between t=0 s (T c (t = 0) =18.5 • C) and t=1000 s in which temperature rapidly decreases under the action of the Peltier element, convection and heat release due to condensation compensate the cooling power. A thermal equilibrium is then reached after t=1000 s when the substrate temperature isT c =7.6 • C. At t=0 s,no drops are present on the substrate. Radiance received by the camera is high as light mainly comes from the reflection of the ambient radiation (factor 1 -ε = 0.95). Then radiance progressively diminishes (t=1000 s), corresponding to the nucleation of water drops on the substrate, which modify the local emissivity. At t=2000 s circular spots of lower radiance appear on the substrate, corresponding to condensed drops following the classical evolution discussed in Section 1, with a pattern showing regions of various radiances.These regions thus correspond to different local emissivities. This pattern then lasts until the end of the experiment, characterized by a self-similar growth of the droplets.In order to examine the evolution of radiance and emissivity on substrates Sand S+ we use a larger view to take into account more drops and have a better statistics. Experiments are performed on substrates S-and S+ in the same thermal and humid air conditions, that is a relative humidity of RH = 48 %, an air temperature of T a =26 • C (yet slightly different from those of fig5). As well as for the previous experiment, temperature decreases from T c (t = 0) =16 • C at

Figure 5 :

 5 Figure 5: Repartition of radiance received by IR camera sensor in the [7.5-14] µm range from substrate S-during condensation . RH=62 %, Ta=27 • C, Tc=7.6 • C. Color bar : mesured radiance (W.m -2 .sr -1 )

Figure 6 :

 6 Figure 6: Repartition of radiance received by IR camera sensor in the [7.5-14] µm range from substrate S-(top) and S+ (bottom) during condensation . RH=48 %, Ta=26 • C, Tc=7.6 • C.

Figure 7 :

 7 Figure 7: Mean emissivities < E > during condensation corresponding to S+, S-dry substrate.Range :[7.5-14] µm. The vertical lines correspond to the pattern at 1000 s, 5000 s and 10000 s in figure6and the horizontal lines So+ and So-to emissivity measured on the dry substrates. W is the effective emissivity of an infinite layer of water corresponding to filmwise condensation. RH=48 %, Ta=26 • C, Tc=7.6 • C

Figure 8 :

 8 Figure 8: Evolution during radiative cooling of the condensed mass on S+ and S-substrates.

Figure 9 :

 9 Figure 9: (a) Variation as a function of wavelength λ of ρ λ , the normal reflection coefficient of the interface air/water. (b): Variation of water transmittivity τ λ with respect to wavelengthλ for different thickness. τ λ and ρ λ are calculated from equations 6, 7 and 8 using water optical index values from[START_REF] Downing | Optical constant of water in the infrared[END_REF]

cFigure 10 :

 10 Figure 10: (a) θ < 90 • Drop is approximated by a cylinder of radius r and height Zm. (b) θ > 90 • Drop is approximated by a cylinder of radius re and height Zm. (c) Radiation propagation through a plane layer of liquid water (non-zero incidence angle has been considered for the sake of clarity) The subscript λ has been omitted in order to make the picture more readable.

Figure 11 :

 11 Figure 11: Effective emissivity E for different wavelength range of the assemblage (substrate + drop) as a function of the thickness of the drop (Zm) ( semi-log plot) for substrate S with emissivity εs = 0.05. Wavelength ranges (I): [7.5 -14] µm, (II) : [2 -50] µm. Vertical lines determine the characteristic length Z 0 m above which the drop exhibits a constant effective emissivity. Inset: effective emissivities of range (I) and (II) for smaller drop thickness.

1Figure 12 :

 12 Figure12: Mean effective emissivity < E > calculated from eq. 13 in the range[START_REF] Berdahl | The thermal radiance of clear skies[END_REF][START_REF] Beysens | Dew Water[END_REF][START_REF] Zhao | From droplets growth to film growth on a heterogeneous surface, condensation associated with a wettability gradient[END_REF][START_REF] Beysens | How does dew form ?[END_REF][START_REF] Elsherbini | Retention forces and contact angles for critical liquid drops on non-horizontal surfaces[END_REF][START_REF] Al-Mahdouri | Evaluation of optical properties and thermal performances of different greenhouse covering materials[END_REF][START_REF] Gustavsen | Spectral emissivity of anodized aluminium and the thermal transmittance of aluminium window frames[END_REF][START_REF] Försth | Absorptivity and its dependence on heat source temperature and degree of thermal breakdown[END_REF] µm, as a function of the contact angle θ and for substrate emissivities ranging from εs=1 to εs=0. Dotted lines correspond to substrate emissivities of S+, ε S+ and S-, ε S-(see figure7) while the vertical interrupted line shows values corresponding to the contact angle θ LDP E .

  mass, with specific heat C w . The term (M C o + mC w )dT /dt represents the variation of sensible heat of the substrate plus water condensate. φ r represents the net radiative heat flux ( difference between absorbed and emitted radiative heat flux), it is negative and corresponds to the cooling power of the system. φ cv is the heat flux exchanged by convection with the ambiant air and φ c is heat conduction with the support. Since the substrate is cooler than air during condensation, both are positive. φ l represents the flux of latent heat liberated by condensation to the substrate. At time t < 0 the substrate is dry, there is no condensation and the temperature is stable for both experiment. Thus dT /dt = 0 and φ l = 0. Temperature of the substrate is determined by the balance between the radiative and convective plus conductive heat fluxes, φ r + φ cv + φ c = 0. The net radiative flux is expressed byφ r = εS(φ i -σT 4 )(15)ε is the emissivity of the surface and φ i is the incoming radiative heat flux density. σ is the Stefan-Boltzmann constant. As φ i is fixed, the driving param-eter of φ r and thus temperature T , is emissivity ε. Low emissivity induces weak radiative cooling flux and large substrate temperature; large emissivity induces high radiative flux and a low subs trate temperature of the substrate. Since ε S-< ε S+ it thus comes T S-> T S+ .

Acknowledgements

This work was support by a PhD grant from the French Ministère de l'Enseignement Supérieur et de la Recherche.