Linda Nanan Vallée

Sao Mai Nguyen

Christophe Lohr

Ioannis Kanellos

Olivier Asseu

HUMAN SKELETON DETECTION, MODELING AND GESTURE IMITATION LEARNING FOR A SOCIAL PURPOSE

Keywords: Imitation learning, Artificial intelligence, Gesture recognition, Autism spectrum disorders (ASD), Gaussian Mixture Model (GMM)

Gesture recognition is topical in computer science and aims at interpreting human gestures via mathematical algorithms. Among the numerous applications are physical rehabilitation and imitation games. In this work, we suggest performing human gesture recognition within the context of a serious imitation game, which would aim at improving social interactions with teenagers with autism spectrum disorders. We use an artificial intelligence algorithm to detect the skeleton of the participant, then model the human pose space and describe an imitation learning method using a Gaussian Mixture Model in the Riemannian manifold.

I. INTRODUCTION

Autism spectrum disorders (ASD) are linked with brain development [START_REF]What Is Autism?[END_REF]. Main symptoms of ASD are difficulties with communication and social interactions, repetitive behaviors and obsessive interests.

Autistic people also have talent [START_REF] Baron-Cohen | Talent in Autism: Hyper-Systemizing, Hyper-Attention to Detail and Sensory Hypersensitivity[END_REF]. For instance, an exceptionally good memory, a great attention to detail, an excellent ability to respect schedules, an exceptional level of honesty. Some of them are savants [START_REF] Hughes | Savant Syndrome Has a Distinct Psychological Profile in Autism[END_REF].

According to the World Health Organization [START_REF]Autism Spectrum Disorders & Other Developmental Disorders. From Raising Awareness to Building Capacity[END_REF], the global autism prevalence is around 1 in 160 children and autism is generally more common in boys than girls. Some autism prevalence studies were made per geographical area [START_REF] Franz | Autism Spec-trum Disorder in Sub-Saharan Africa: A Comprehensive Scoping Review[END_REF].

Still today, the exact cause of ASD is not known [START_REF] Shaw | Etiology of Autism Spectrum Disorders: Genes, Environment, or Both?[END_REF].

Several scientific studies have targeted improving the way autistic children communicate or interact with others [START_REF] Nadel | L'autisme, chapter Imitation et austisme[END_REF]. This is because these two functions are crucial. Other symptoms of autism can be seen mostly as consequences of impairments in social interactions or communication.

Additionally, the imitation process is known to be a pillar in learning, communication and social interactions. Imitation games can therefore prove useful in helping autistic people interact with others.

In the present work, the structure of a gesture imitation game is proposed, which shall improve social interactions with autistic teenagers and preteens. Furthermore, skeleton detection and imitation learning methods are described.

The following section of this paper presents existing work related to imitation learning as well as improvement of social interactions with autistic children.

Section III then describes the methodology: the main phases of the imitation game as well as the skeleton detection and human motion learning methods.

Simulation results for skeleton detection are presented in section IV.

Section V finally concludes this paper and suggests future work.

II. RELATED WORK

II.1. On imitation learning and gesture recognition

In computer science, imitation learning, also called programming by demonstration, is a technique for teaching a computer or a robot to perform new tasks, through generalization from observing multiple demonstrations [START_REF] Bendiouis | Imitation et communication chez le jeune enfant avec autisme[END_REF].

Within the framework of gesture recognition, a gesture would be performed several times by a human being and then, a method used for the system to be able to later recognize the task.

Different spatial gesture models exist (figure 1). Some are 3D-model based and others are appearance-based. Image sequences and deformable 2D templates are part of the latter group.

The former comprises of skeletal and volumetric models.

Subtypes of the volumetric model category are NURBS, primitives and super-quadrics. Since human movements are nonlinear, the Euclidian space is not really suitable to represent those. Human postures and motion are therefore often represented in alternative spaces such as the Riemannian one [START_REF] Jost | Riemannian Geometry and Geometric Analysis[END_REF], which has proved useful as shown in [START_REF] Zeestraten | An Approach for Imitation Learning on Riemannian Manifolds[END_REF].

Once the human body is modeled, gestures must actually be learned by the system by observation of several demonstrations. Probabilistic methods serve this purpose and can for instance be based on Hidden Markov Models or Gaussian mixture models. The skeletal model, the representation of the human body in the Riemannian space as well as GMM have been used within the framework of physical rehabilitation exercises [START_REF] Lin | Online Segmentation of Human Motion for Automated Rehabilitation Exercise Analysis[END_REF] [13].

In the context of our imitation game for teenagers with ASD, skeleton detection and gesture recognition methods are redirected to serve social purposes, as the aim is to improve the participants' ability to interact with others.

II.2. On imitation and autism

In [START_REF] Nadel | L'autisme, chapter Imitation et austisme[END_REF] the author indicates that autistic children are able to imitate, whereas the general opinion previously differed.

The imitation process is complex and consists of subcomponents: induced imitation, spontaneous imitation, recognition of being imitated. This process is fundamental for learning, communicating and interacting socially.

In [START_REF] Bendiouis | Imitation et communication chez le jeune enfant avec autisme[END_REF] an experimental work on imitation practice is presented in order to improve imitation abilities and reduce the autism level of 21 autistic children aged 4 to 10.

Nadel's imitation scale is used to evaluate the level of the three subcomponents of the imitation process.

These two studies are interesting on the psychological level because they show how imitation can positively impact autistic children, but they do not use modern techniques for imitation learning. The experimentation consists of human caregivers performing simple imitation games with the children.

In [START_REF] Sara | ECHOES: An Intelligent Serious Game for Fostering Social Communication in Children with Autism[END_REF] Bernardini describes a multi-site intervention where 46 children with ASD aged 5 to 14 improved their social interactions through playing games, among which imitation games, with an intelligent agent called Andy. In most cases, the probability of the child answering Andy's requests increased. However social interactions initiated by the child were not really impacted.

In the work [START_REF] Mazon | Effectiveness and Usability of Technology-Based in-Terventions for Children and Adolescents with ASD: A Systematic Review of Reliability, Consistency, Generalization and Durability Related to the Effects of Intervention[END_REF], the authors perform a review of studies using technological tools with autistic children and show that very few:

• aim at therapeutic effectiveness as well as technology usability;

• focus on teenagers;

• have a robust methodology.

III. METHODOLOGY

At the time of using artificial intelligence algorithms within our gesture imitation game for autistic teenagers, it is important to develop a robust methodology. Furthermore, the ease of use of the technology must be guaranteed while trying to improve the participants' initial condition in terms of social abilities.

Our gesture imitation game consists of two preliminary, three core and one final stages.

The two preliminary ones are the greetings and pairing stages.

Then come the three imitation modules: one based on induced imitation, another on spontaneous imitation and the third one on the recognition of being imitated.

This proposed structure follows multiple discussions with autism professionals.

In this paper we focus on potential methods for three core processes that will be useful throughout the game: skeleton detection, body representation, and finally recognition of previously learned gestures.

At game initiation, the skeleton of the participant is detected through the computer camera. Tensorflow is used to train and execute neural networks for element classification like in gesture recognition.

As for body representation, a human pose y at time t is represented by the orientation and position of all of the considered joints. The number of joints here is N. Therefore:

y t = [O 1 , P 1 , O 2 , P 2 ,. .., O N , P N ] (1)
where O N are joint orientations.

Joint positions P N are not absolute but normalized relative positions. They are computed from their absolute positions p n relatively to the absolute position p ss of the spine shoulder. Their normalization is done using the spine bone length L spine :

P N = (p n -p ss )/L spine (2)
Unlike joint positions, joint orientations cannot be viewed in the Euclidian space but they can be represented in a 3D Riemannian manifold.

Therefore the human pose space is modeled as the Cartesian product of position and orientation of all of the human joints:

H = ℝ ! x 𝑆 ! x ℝ ! x 𝑆 ! x … x ℝ ! x 𝑆 ! (3)
In such space, among the various available methods, the one that was chosen was the Gaussian Mixture Models in Riemannian manifolds, as explained in [START_REF] Devanne | Multi-Level Motion Analysis for Physical Exercises Assessment in Kinaesthetic Rehabilitation[END_REF].

Since the Riemannian space is nonlinear, tangent spaces at reference points are considered in order to be able to compute standard statistics, like mean and covariance. Paper [START_REF] Karcher | Riemannian Center of Mass and Mollifier Smoothing[END_REF] allows for the calculation of the mean µ of N points 𝑝 ! on the human pose space:

𝜇 = arg min ! ! !!! 𝑑(𝜇, 𝑝 ! ) ! (4)
where d(µ, p) is the geodesic distance on the manifold which can be written using logarithmic map as d(µ, p) = ‖ 𝐿𝑜𝑔 ! 𝑝 ‖.

µ is also called the Riemannian center of mass.

The covariance matrix can then be computed, allowing for the learning of a Gaussian Mixture Model:

p(x) = 𝜙 ! ! !!! 𝑁(𝑥 𝜇 ! , Σ ! ) (5)
where x encodes both the human pose 𝑦 ! and the timestamps t, K is the number of Gaussians, 𝜙 ! the weight of the k-th Gaussian, 𝜇 ! the Riemannian center of mass of the k-th Gaussian computed on the manifold and Σ ! the covariance matrix of the k-th Gaussian. The parameters 𝜙 ! , 𝜇 ! and Σ ! are learned using Expectation-Maximization on the human pose space [START_REF] Simo-Serra | 3D Human Pose Tracking Priors Using Geodesic Mixture Models[END_REF].

IV. EXPERIMENTAL RESULTS

For human pose detection, Tensorflow Pose Estimator was implemented. Below are the key command lines executed:

$ git clone https://www.github.com/ildoonet/tf-pose-estimation

$ cd tf-pose-estimation $ pip3 install -r requirements.txt $ cd tf_pose/pafprocess $ swig -python -c++ pafprocess.i && python3 setup.py build_ext --inplace $ python3 run_webcam.py --model=mobilenet_thin --resize=432x368 --camera=0

The git clone command copied the Tensorflow Pose Estimator file hierarchy into our work environment.

We then moved to the main directory and installed all of the required modules: The swig command connects the C++ programs with the Python ones.

As mentioned earlier, Tensorflow API is Python-based but for the execution of the applications, high-performance C++ is used.

We finally executed run_webcam.py with the model and format of our choice, and obtained the following results captured through the webcam (figure 3).

The body joints of the participant are represented by dots and connected through lines.

Different colors are used to distinguish the different body parts.

Elements represented by the dots are the ankles, knees, hips, wrists, elbows, shoulders and ears.

On the first capture (fig. 3a), we can see our participant from side-on, with the left arm behind him and the right arm raised in front of him, at head level. In spite of body occlusion (some body parts are superimposed), the body parts are correctly detected.

On a real-life background and even with low light, skeleton detection is functional using the Openpose algorithm.

This will be performed at the beginning of our gesture imitation game and useful throughout the different stages.

Each pose is represented by the position and orientation of the (fourteen) previously listed joints, as shown in formula [START_REF]What Is Autism?[END_REF]. The joint positions are calculated relatively to the spine as seen in formula (2).

The human pose space is represented in the Riemannian manifold as expressed in formula [START_REF] Hughes | Savant Syndrome Has a Distinct Psychological Profile in Autism[END_REF] presented in the methodology section. Equations ( 4) and ( 5) are then used for gesture learning and recognition.

Figure 1 :

 1 Figure 1: Spatial gesture model types

A

  Gaussian mixture model (GMM) is a probabilistic model that assumes all the data points are generated from a mixture of a finite number of Gaussian distributions with unknown parameters. model then represents a probabilistic description of the target (ideal) movement against which imitation attempts are compared.

  This is done using the Openpose algorithm with the open source Tensorflow library, which is used to develop Machine learning and Deep learning algorithms. Tensorflow allows for solving of high complexity mathematical issues using experimental learning architectures. It is similar to a programming system in which computations are represented by graphs where nodes are mathematical operations and arrow borders are interconnected multidimensional data called tensors. Tensorflow application programming interface (API) is Python-based but high-performance C++ is used for the execution of the applications.

Figure 2 :

 2 Figure 2: Illustration of the human pose space with threeGaussians computed on tangent space at means 𝜇 ![START_REF] Devanne | Multi-Level Motion Analysis for Physical Exercises Assessment in Kinaesthetic Rehabilitation[END_REF]

Figure 3 :

 3 Figure 3: Experimental results of the implementation of the Openpose algorithm

  Then from the tf_pose/pafprocess directory, we executed the swig command with the appropriate arguments and launched the setup file.

	fire
	matplotlib
	numba
	psutil
	pycocotools
	requests
	scikit-image
	scipy
	slidingwindow
	tqdm
	git+https://github.com/ppwwyyxx/tensorpack.git
	argparse
	dill

V. CONCLUSION

In this paper, the structure of a gesture imitation game was suggested and technical methods for skeleton detection and representation, as well as gesture recognition, were described. Initial results for skeleton detection were then presented.

The full serious game will be developed in order to practice induced and spontaneous imitation with autistic teenagers and assess the improvement of their imitation and social abilities.

Later on, it would be interesting to transform the initially hard-coded game into a smart adaptive tutoring system and to use a social robot as a game partner. This could increase participants' interest and involvement.

ACKNOWLEDGEMENT

Here is the opportunity to express deep gratitude to Dr Lavenne-Collot Nathalie, child psychiatrist at the university hospital of Brest, France, as well as to Mr. Shaw Joël, caseworker with autistic children in Abidjan, Cote d'Ivoire, for their availability and interest in this work.


Julio Dávila

Jan Eeckhout

César Martinelli

BARGAINING OVER PUBLIC GOODS

In a simple public good economy, we propose a natural bargaining procedure whose equilibria converge to Lindahl allocations as the cost of bargaining vanishes. The procedure splits the decision over the allocation in a decision about personalized prices and a decision about output levels for the public good. Since this procedure does not assume price-taking behavior, it provides a strategic foundation for the personalized taxes inherent to the Lindahl solution to the public goods problem.

Introduction

The private provision of public goods in general leads to inefficient allocations in a competitive market environment. This inefficiency is often attributed to a missing market. If personalized markets could be created that individually price the public good for each agent, then a competitive equilibrium could implement an efficient allocation. For an economy with public goods, this outcome is known as a Lindahl equilibrium. Typically, however, a Lindahl equilibrium is deemed unrealistic because of a serious shortcoming: in the personalized markets upon which it rests the agents are assumed to have a price-taking behavior. But unfortunately, by the personalized nature of those markets, there is only one single agent on the demand side in each of them, which makes price-taking behavior of this single agent an utterly unrealistic assumption. On contrast, we propose in this paper a bargaining procedure that leads (without the need of assuming price-taking behavior) to an outcome arbitrarily close to a Lindahl allocation as the cost of bargaining vanishes. As a matter of fact, the two only agents of our model (for the sake of simplicity) have quite on the contrary a lot of market power.

In the case of a missing market (as it happens in the presence of a public good), one way to allocate the surplus left unappropriated is through Coasian bargaining. As pointed by Coase, as long as there remain gains from trade the parties involved have incentives to get together and strike a deal. The main feature of such bargaining is that it is decentralized (no benevolent government must intervene), and the extent to which the surplus can be allocated to the parties depends on the details of the bargaining protocol and on whether the bargaining is costly or not.

The study of this type of bargaining in legislatures has already been addressed in [START_REF] Baron | Bargaining in legislatures[END_REF], where a model is presented in which one of the different possible ways to divide a given pie is chosen by a vote according to the majority rule. It turns out that typically there exist many equilibria for this procedure. [START_REF] Banks | A bargaining model of collective choice[END_REF] present a far more general model in which the space of alternatives is a compact, convex subset of a multidimensional Euclidean space. They consider arbitrary voting rules and prove the existence of stationary equilibria, the upper hemicontinuity of equilibrium proposals in structural and preference parameters, and a core equivalence result. While several of these bargaining set-ups encompass economies with public goods, they differ from our own set-up in that ours implicitly imposes constraints on what the proposer is able to offer to other agents. We believe these constraints reflect in a natural way the sharing of power in a bargaining situation over both the output level and the financing of a public good. At any rate, as a consequence of imposing such constraints on the offers, the bargaining protocol considered here allows to attain (in the limiting case of vanishing bargaining costs) the outcome that would result from completing the markets that are missing because of the presence of a public good, but without resorting to the heroic assumption of price-taking behavior with respect to personalized prices.

Specifically, in order to formalize the Coasian procedure of extracting rents, we propose a sequential bargaining as a collective decision making process. Its sequential nature reflects a realistic feature of the power of setting the agenda of the negotiation. The proposal and acceptance decisions are endogenous. Agents can reject a proposal and have it modified in their turn. Because of the cost of any delay in reaching an agreement, the bargaining outcome will in general be inefficient thus reflecting the power of setting the agenda: when there is impatience for reaching an agreement, the agent who makes an offer that is accepted will extract more rents at the cost of inefficiency. As the impatience or cost of a delay vanishes, those rents disappear and the allocation is efficient and independent of who had the power of setting the agenda.

In this paper we show that, despite the fact that output decisions for public goods and their mode of financing are often the result of a political process rife with opportunities for strategic behavior, the Lindahl allocations can be implemented without assuming price-taking behavior with respect to personalized prices. This is obtained precisely through a strategic bargaining of the parties over the financing of the public good.1 In order to show this we take here a first step towards modelling the political process behind output and financing decisions with regard to public goods as a sequential bargaining game of complete information.

Specifically, we consider an economy with any finite number of public goods and private goods (not necessarily the same number of each). There are two agents in this economy who take turns alternatingly (as in [START_REF] Rubinstein | Perfect Equilibrium in a Bargaining Model[END_REF]) to propose a maximum level of provision of each of the public goods and a way to split between them the cost of financing any level of the public goods to be provided up to the proposed maxima (this amounts to proposing personalized prices or taxes). The other agent can then either accept or reject the proposal. In case of acceptance, this other agent chooses the amount of each public good to be provided (subject to the maximum amount offered in the proposal). Each agent pays for the public goods according to the personalized prices agreed upon. The levels of public goods and their financing are fixed thereafter so the game is effectively over. If instead the other agent rejects the proposal, then it is his turn to make a proposal himself of a new maximum amounts and personalized prices, and so on.

Within this set-up we show that, as the discount factors of each agent in the economy converge to one (no matter how), the allocation of any stationary subgame perfect equilibrium converges to a Lindahl equilibrium allocation. Specifically, we show first that, for infinitely patient agents (i.e. for discount factors equal to 1), the set of Lindahl allocations coincides with the set of stationary subgame perfect equilibrium allocations of the alternating-offers bargaining game described above. Then we establish the upper hemicontinuity of the correspondence mapping, to each pair of discount factors, the set of stationary subgame perfect equilibrium allocations of the bargaining game, and in particular its upper hemicontinuity for discount factors δ A = δ B = 1. The conclusion then follows from these two results.

In this simple set-up, unanimity plays an important role in our modelling. In effect, a tax proposal can be adopted only if it is acceptable to every agent. This has the virtue of conferring a Wicksellian character to the Lindahl solution to the public goods problem. In effect, in a classical contribution to the theory of public finance, Wicksell [9] proposed unanimity as the criterion for just taxation. The game we propose incorporates the need of achieving consensus to effect tax proposals that to some extent may characterize budgetary procedures in political regimes with multiple checks and balances or in parliamentary democracies without a majority party. Inefficiencies associated with bargaining are shown to disappear as agents become increasingly patient. Thus, a Wicksellian procedure turns out to be consistent with a Lindahl result. Of course, this result depends on the assumption of complete information.

The model

We consider first a public good economy consisting of only two agents A and B, but an arbitrary number n + m of goods: n private goods x 1 , . . . , x n and m public goods y 1 , . . . , y m . The agents are infinitely lived and time is discrete. For each agent i = A, B, let x i ∈ R n + be i's consumption of the private good, and let y ∈ R m + be their common consumption of the public good. The agents have preferences over the two types of goods represented by standard utility functions u A (x A , y) and u B (x B , y). 2 The agents are endowed with amounts e i (with total endowment e = e A + e B ) of the private goods, and as a normalization, we assume the initial amount of the public goods is zero. A linear technology M ∈ R n×m allows to produce each public good y j by means of the private goods, requiring m ij units of private good x i for each unit of y j , for all i = 1, . . . , n.

We consider an alternating-offers bargaining game. In any given period prior to an agreement, an agent i makes an offer consisting of a vector p i = (p xi , p yi ) of prices for the other agent (in terms of, say, the private good x 1 ) and an upper bound q i to the contribution of private goods to the provision of public goods or, equivalently, to the provision of public goods itself. After receiving an offer the other agent can either accept it or reject it. In case of acceptance, the accepting agent chooses his consumption of private goods and the quantities of public goods to be provided subject to the accepted prices p i and upper bound on trades q i . This procedure is repeated until a proposal is accepted. The utility of each agent i is discounted in each iteration by a positive discount factor δ i not bigger than 1. The utility of never reaching an agreement is 0.

At this point it is worth observing that the right to make a proposal can be thought of as a property right over the surplus from the public good. The details of those property rights will certainly matter when bargaining is costly -as we will show below. However, one of our findings is that the way in which property rights are assigned does not matter when the cost of a delay disappears.

The stationary subgame perfect (SSP) equilibrium allocations

We consider first the stationary subgame perfect equilibria without delay of this game (i.e. an equilibrium in which no agent has incentives to reject the offer received). Such equilibria are characterized by an offer (p A , q A ) of a vector of prices and a maximum amount for the contributions of private goods or the provisions of public goods that A offers to B and a similar offer made by (p B , q B ), such that, for instance, (p A , q A ) maximizes the utility that A obtains from B's immediate acceptance of his offer (p A , q A ), subject to the constraint that it is indeed in B's interest to accept (p A , q A ), i.e. (p A , q A ) solves max u A (e -xB (p A , q A ) -M ỹB (p A , q A ), ỹB (p A , q A ))

(1) subject to the constraint

u B (x B (p A , q A ),ỹ B (p A , q A )) ≥ δ B u B (e -xA (p B , q B ) -M ỹA (p B , q B ), ỹA (p B , q B )) (2)
given p B , q B -where xB (p A , q A ), ỹB (p A , q A ) is the solution to max u B (x, y)

p A • (x -e B , y) ≤ 0 (x -e B , y) ≤ q A , 5 (3)
-and symmetrically for (p B , q B ). That is to say, A's offer maximizes the utility that A will derive from B's immediate acceptance of the offer, provided that the offer is such that B is interested indeed in accepting immediately the offer. Note however that a SSP equilibrium without delay can equivalently be characterized by the allocations effectively offered by the agents. In effect, conditional to immediate acceptance, an offer by A of (p A , q A ) amounts to offering B the bundle (x B (p A , q A ), ỹB (p A , q A )), which is characterized by satisfying the condition

(u B x (x B , y), u B y (x B , y)) x B -e B y ≥ 0. ( 4
(u B x (x B , y), u B y (x B , y)) x B -e B y ≥ 0 u B (x B , y) ≥ δ B u B (x B B , y B ) x A + x B + M y = e A + e B (5) given (x A B , x B B , y B ), and (x A B , x B B , y B ) solves max u B (x B , y) (u A x (x A , y), u A y (x A , y)) x A -e A y ≥ 0 u A (x A , y) ≥ δ A u A (x A A , y A ) x A + x B + M y = e A + e B (6) given (x A A , x B A , y A ).

The Lindahl allocations

A Lindahl equilibrium allocation (x A , x B , y) is characterized by satisfying that (1) the allocation is feasible, i.e. x A , x B , y ≥ 0 and

x A + x B + M y = e A + e B (7)
(2) it allocates to each agent i his demand (x i , y) at his personalized relative prices (implicitly equal to the marginal rates of substitution determined by his marginal utilities at (x i , y)), i.e. for i = A, B,

(u i x (x i , y), u i y (x i , y)) x i -e i y = 0. ( 8
)
For the case of one private good and one public good, Lindahl Equilibrium allocations can be represented in a Kolm triangle, the public goods equivalent of the Edgeworth box of a private goods exchange economy (see Figure 1 below). The orthogonal distance from any point within the triangle to each of its sides represents each agent's allocation of the private good (x A and x B ), while the vertical distance to the base of the triangle represents their common consumption y of the public good. The triangular form accounts for the linear production technology: an increase in the quantity of public goods produced implies an proportional decrease in the quantity of private goods that remain for consumption. Thus, at the initial endowment e there is no provision of the public good. The personalized prices p A and p B represent the terms of exchange of one unit of public good for p A units of the private good from agent A and p B units of private good from agent B. A balanced budget where the total contributions equal the total cost of production of the public good implies that for all y, I i=1 p i y = y. For a given price schedule p, the offer curve OC i gives the optimal amount of public good and private good demanded by agent i at those terms of trade. An intersection of the offer curves represents then optimal quantities of the public good and the private good consumed by the agents given a vector of personalized prices that balances the budget. This corresponds to a Lindahl equilibrium allocation. There are of course other efficient allocation represented by the Pareto set P , but they are not attainable by means of price schedules starting from the initial endowment e.

SSP equilibrium allocations are Lindahl allocations when δ

A = δ B = 1
First we show that for infinitely patient agents, that is to say, when the discount factors δ A and δ B are 1, the Lindahl equilibrium allocations, and only these allocations, are offered at a stationary subgame perfect equilibrium of the bargaining game above.

Theorem 1. When δ A = δ B = 1, at every SSP equilibrium with immediate acceptance the agents' offers lead to the same allocation which is, moreover, the allocation of a Lindahl equilibrium. Conversely, every Lindahl equilibrium allocation is the allocation offered by the two agents at some SSP equilibrium without delay.

Proof. Let (x A

A , x B A , y A ) and (x A B , x B B , y B ) be the feasible allocations resulting from B's (resp. A's) acceptance of A's (resp. B's) offer of price and maximum provisions of public goods at a SSP equilibrium with immediate acceptance for infinitely many patients. That is to say, let (x A A , x B A , y A ) and (x A B , x B B , y B ) be such that

Figure 1 .

 1 Figure 1

  If x, y satisfies Du(x, y)(x -e, y) > 0, letλ > 0 µ = Du(x, y)(x -e, y) (x -e, y)(x -e, y) p = 1 λ Du(x, y) -Du(x, y)(x -e, y) (x -e, y)(x -e, y) (x -e, y) ≥ 0, 6q 2 = (x -e, y)(x -e, y). We provide the proof for the case n = 2. The general case is then established applying this case in the 2-dimensional space spanned by the vectors a and b.Sincea ∈ R 2 ++ and ab > 0, then b / ∈ -R 2 + . Since b / ∈ -R 2 + and b / ∈ R 2 + , then b 1 b 2 < 0. Assume,without loss of generality that b 1 < 0 and b 2 > 0. Note first that, for all a ∈ R 2 ++ , the inequality a -a•b b•b b ≥ 0 holds if, and only if,

)

  Conversely, any bundle (x B , y) satisfying the previous inequality is solution to B's problem above for some offer (p A , y A ) by A. This fact is established in Lemma 1 in the Appendix.

	As a consequence, A SSP equilibrium without delay can also be characterized by
	allocations (x A A , x B A , y A ) and (x A B , x B B , y B ), proposed by A and B respectively, such that (x A A , x B A , y A ) solves
	max u A (x A , y)

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	A	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OC A p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OC B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P

The mixed competitive mechanisms proposed in[START_REF] Groves | Optimal Allocation of Public Goods: A Solution to the Free Rider Problem[END_REF] obviously can efficiently allocate private and public goods to coincide with the Lindahl allocation (see alsoTian (1989)). However, those direct mechanisms both rely on a centralized mechanism designer and they may involve complicated mechanisms. The virtue here is that an efficient allocation of private and public goods is obtained through a decentralized bargaining procedure. It is simple: it relies explicitly on personalized contributions (taxes) and incorporates the notion that agenda setters have power to extract more rents when bargaining is costly.

That is to say C 2 , monotone, differentiably strictly quasi-concave, non-negative utility functions that are well-behaved at the boundary of R n+m + .

The choice of the norm is inessential.

This is a consequence of Lemma 2 below

(1)

(2) (x A B , x B B , y B ) ∈ arg max

x A ,x B ,y u B (x B , y)

given (x A A , x B A , y A ).

(10)

Then necessarily, from the first-order conditions, there exist multipliers λ A , µ A , λ B , µ B ≥ 0 and ν A , ν B such that

where m t i• is the i-th row of M transposed as a column, or equivalently, eliminating the multipliers

and similarly

are binding 3 (and hence both allocations are on the same indifference surface for both agents ), 4 then none of these two allocations can be efficient while being different. Therefore the efficiency condition that

are collinear cannot hold neither at (x A A , x B A , y A ) nor at (x A B , x B B , y B ). In particular, there exist h, k such that

(a similar argument applies if the inequality holds the other way), then (13) above cannot be satisfied for non-negative multipliers (in the case the inequality is reversed it is condition (12) above that cannot be satisfied for non-negative multipliers).

In effect, since (λ B , µ B ) must solve (13) above, then it must also solve

(18) Since µ ≥ 0 necessarily and the numerator is strictly positive according to ( 16), so must be the denominator. But this denominator is equivalent to the following expression which in fact happens to have the opposite sign,

y) ≥ 0 (agent A will never choose at equilibrium to let B ask for a provision of public goods bigger than the one necessary to attain A's demand at the implicit prices), then necessary

4 Note that for this step to hold true it is crucial to have the equality of the discount factors δ A and δ B to 1. 8 since the first scalar product is null and the Hessian of u A at (x A B , y B ) is semi-definite negative in the space orthogonal to the gradient of u A at (x A B , y B ) and hence also to any (0, . . . , 0, x A hB -e A h , 0, . . . , 0, y kB , 0, . . . , 0) orthogonal to (u A x (x A B , y B ), u A y (x A B , y B )), i.e. for all (x A hB -e A h , y kB ) such that

) necessarily, i.e. whenever δ A = 1 = δ B at a SSP equilibrium with immediate acceptance the two agents offer the same allocation. Let then x A , x B , y be the common allocation offered at a SSP equilibrium when δ A = 1 = δ B , i.e. such that (1)

(3) and (x B , y) ∈ arg max u A (e -xB -M ŷ, ŷ)

Assume that

Then, since both u A and u B are concave,

Therefore, there exists λ > 0 such that

But since (1)

(2)

x A -e A y > 0 (28) and (3) for some λ > 0,

and similarly

i.e, the allocations proposed by both agents at a SSP equilibrium is on both agents' offer curves so that it is a Lindahl equilibrium allocation. Conversely, let x A , x B , y be the allocation of a Lindahl equilibrium, i.e. such that (1)

(2)

x A -e A y = 0 (34)

(3) and

Let (p A , q A ) be ((u B x (x B , y), u B y (x B , y)), y) and, similarly, let (p B , q B ) be the vec- tor ((u A x (x A , y), u A y (x A , y)), y), so that x A = xA (p B , q B ), x B = xB (p A , q A ) and ỹA (p B , q B ) = y = ỹB (p A , q A ). Then it can be easily checked that the Lindahl allocation (x A , x B , y) is the outcome of the following SSP equilibrium profile of strategies:

(1) A offers (p A , q A ) whenever he has the opportunity to do so, and accepts only offers (p, q) such that

(2) B offers (p B , q B ) whenever he has the opportunity to do so, and accepts only offers (p, q) such that

QED 6. Convergence of SSP equilibrium allocations to Lindahl allocations as δ A , δ B → 1

In the relevant case in which bargaining entails some frictions, so that the factors by which the agents discount future utilities are strictly smaller than 1, any SSP equilibrium allocation is still arbitrarily close to a Lindahl allocation if agents are patient enough. This is a consequence of the upper hemicontinuity of the correspondence of SSP equilibrium allocations with respect to the agents' discount factors, as Theorem 2 next establishes.

Theorem 2. Every SSP equilibrium allocation converges to a Lindahl allocation as δ A , δ B → 1.

Proof. Consider the correspondence Φ defined as

where Du i (x i , y) stands for (u i x (x i , y), u i y (x i , y)), for all i = A, B. Note that, by the Theorem of the Maximum, arg max

is a compact-valued, upper hemicontinuous correspondence that depends explicitly on x B B , y B , and δ B but also trivially on x A B and x A A , x B A , y A , δ A . 5 And similarly for agent A's problem. Therefore, Φ is the cartesian product of compact-valued, upper hemicontinuous correspondences, and hence it is compact-valued and upper hemicontinuous itself. 6 5 Since u A depends continuously on x A , y and also trivially on

, and the correspondence defined by the constraints

is continuous and compact-valued.

6 See Lemma A1 in [START_REF] Dávila | Competitive Bargaining Equilibrium[END_REF].

11

Consider the correspondence Γ of fixed points of Φ as a function of δ A , δ B , i.e.

Γ(δ

Since Φ is compact-valued and upper hemicontinuous, then the correspondence mapping to each pair (δ A , δ B ) the fixed points of Φ(

Finally, note that Γ is the correspondence of SSP equilibrium allocations (without delay). Since this correspondence is upper hemicontinuous, in particular at (δ A , δ B ) = (1, 1) and, according to Theorem 1, Γ(1, 1) is the set of Lindahl allocations, then the claim follows. QED (A.5)

Appendix

7 See Lemma A3 in [START_REF] Dávila | Competitive Bargaining Equilibrium[END_REF].


Elodie Mattio

Fabrice Lamadie

Isaac Rodriguez-Ruiz

Beatrice Cames

Sophie Charton

Photonic Lab-on-a-Chip analytical systems for nuclear applications: optical performance and UV-Vis-lR material characterization after chemical exposure and gamma irradiation

Keywords: Opto/microfluidic, Photonic Lab-on-a-Chip, Materials resistance, Irradiation, Chemical exposure igj

The use of microfluidics technology and the miniaturization of analytical techniques is of high interest for the chemical and nuclear industries. In the latter, the reduction of effluents deriving from actinides concentration monitoring along R&D and pilot-scale purification processes is a permanent concern. lndeed, the extremely harsh operation conditions limit the imple mentation of standard analytical techniques and methodologies, and in this regard, the use of spectrophotometric techniques for effluents characterization becomes advantageous in terms of robustness, implementation and sensitivity at the microfluidic scale. In this work, we report a study of the effect of exposure to different chemicals used in hydrometallurgical processes, and to gamma radiation typical of the Plutonium and Uranium Refining by Extraction process, on the optical and structural properties of different polymeric materials commonly used for the fabrication of microfluidic and optofluidic systems. This study shows that low-cost castable and/or engravable materials (e.g. polydimethylsiloxane and poly-methyl methacrylate) are ideal for the study and development of Photonic Lab on a Chip systems that will be used in a nuclear environment.

Introduction

With the generalization of fully automated and intelligent processes, microfluidic devices are more than ever at the heart of R&D. In particular, they are a relevant solution for the growing analytical needs inherent in process-control, without multiplying the dedicated tirnes and volumes. How ever, while the design and manufacturing issues of chips (assembly, connectors) are often mentioned in literature (see e.g. recent review of Mariet et al. [START_REF] Mariet | Microfluidics devices applied to radionuclides separation in acidic media for the nuclear fuel cycle[END_REF]), materials issues, especially regarding chemical and radiation exposure, are less discussed [START_REF] Rodríguez-Ruiz | Analysis of the structural integrity of SU-8-based optofluidic systems for small-molecule crystallization studies[END_REF][START_REF] Piruska | The autofluorescence of plastic materials and chips measured under laser irradiation[END_REF] as glass is frequently considered in these studies. Yet, hydrometallurgical processes in particular, implement strong acids and organic solvents, which are likely to interact with the chips manufacturing materials. Additional radiation damage is also possible in the nuclear industry. While glass is maybe the more-chemically resistant transparent material, the development cost of the chips and their low mechanical strength are potentially detrimental to a serial application. As a result, possible replacement materials need to be considered and are here studied.

In a previous work, the relevance of polydimethylsiloxane (PDMS) chips for Uranium (VI) concentration measurements by UV-Visible absorption on microliter volumes has been evidenced [START_REF] Rodríguez-Ruiz | Uranium(VI) onchip microliter concentration measurements in a highly extended UV-visible absorbance linearity range[END_REF]. Following the promising results obtained, and in anticipation of a possible industrial implementation of this micro-analysis, two questions related to the chip durability must be examined: (1) the resistance to chemicals, that is shared by many chemical industrial sectors, and (2) the resistance to irradiation, that is more specific to nuclear applications.

PDMS is among the most used materials for microchip fabrication by the classical cast moulding technique [START_REF] Mcdonald | Fabrication of microfluidic systems in poly(dimethylsiloxane)[END_REF]. A low cost fabrication process for the microfluidic moulds was recently compared to the conventional soft lithography one, which requires clean room infrastructure and hazardous chemicals, for the fabrication of moulds for PDMS optofluidic Photonic Lab-on-a-Chip platforms (PhLoCs) [START_REF] Rodríguez-Ruiz | Broadcasting photonic lab on a chip concept through a low cost manufacturing approach[END_REF]. These platforms, monolithically integrating micro-optical elements, allow for the implementation of optical spectroscopy for analytical purposes, in the vicinity of a microfluidic structure [START_REF] Rodríguez-Ruiz | Photonic lab-on-a-chip: integration of optical spectroscopy in microfluidic systems[END_REF]. The good optical properties presented by these low-cost devices open up promising perspectives for oneshot chips that could be e.g. a good option for analytical purpose. However, resistance to chemicals is not granted with PDMS structures [START_REF] Rodríguez-Ruiz | Analysis of the structural integrity of SU-8-based optofluidic systems for small-molecule crystallization studies[END_REF], and radiation damages need to be further investigated. Thus in order to improve the components durability, and the chips versatility (especially for analytical purposes), a right choice on fabrication materials must be done.

In this regard, many other optically transparent polymers [START_REF] Ren | Materials for microfluidic chip fabrication[END_REF] are seen as interesting candidates. OSTEMER, a recent thiol-alkene-epoxy co-polymer allowing for cast molding chip production, has shown high resistance to chemicals, as e.g. those used in solvent extraction [START_REF] Carlborg | Rapid permanent hydrophilic and hydrophobic patterning of polymer surfaces via off-stoichiometry thiol-ene (OSTE) photografting[END_REF]. On the other hand, other common optically transparent materials like poly-methyl methacrylate (PMMA), Polycarbonate (PC) and polystyrene (PS) have also been widely used for chip fabrication by different manufacturing techniques [START_REF] Ogończyk | Bonding of microfluidic devices fabricated in polycarbonate[END_REF]. Among these techniques, it is worth to mention not only the standard thermo-molding [START_REF] Becker | Polymer microfabrication technologies for microfluidic systems[END_REF] or engraving techniques (such as laser engraving and micro-milling [START_REF] Guckenberger | Micromilling: a method for ultra-rapid prototyping of plastic microfluidic devices[END_REF]), but also the emerging additive fabrication processes applied in 3D printing technologies [START_REF] Au | 3D-Printed microfluidics[END_REF]: stereolithography, laminated object manufacturing and more recently fused deposition modelling [START_REF] Bressan | A simple procedure to produce FDM-based 3D-printed microfluidic devices with an integrated PMMA optical window[END_REF]. These processes allow the serial production and reproduction of complex structures [START_REF] Mattio | 3D-printed lab-on-valve for fluorescent determination of cadmium and lead in water[END_REF] at different scales, and more particularly at very small scales, as microfluidic systems require.

In a previous work [START_REF] Rodríguez-Ruiz | Uranium(VI) onchip microliter concentration measurements in a highly extended UV-visible absorbance linearity range[END_REF], preliminary tests have been made to observe the possible effects of gamma irradiation on PDMS and OSTEMER, two candidates for the realization of Uranium measurements chips. In this short communication, the chemical and irradiation resistance of these two materials, as well as that of other possible candidates for the manufacture of PhLoCs, is further examined. In addition to a simple visual observation, that is sufficient to qualitatively evaluate the creation of defects and materials swelling, the precise evolution of the relevant optical properties is quantified using post-exposure UV-Vis and IR measurements. In a second step, two chips dedicated to absorptiometric analyses [START_REF] Rodríguez-Ruiz | Uranium(VI) onchip microliter concentration measurements in a highly extended UV-visible absorbance linearity range[END_REF] have been made in the two most cheap and widely used materials, PDMS and PMMA, and their robustness has been compared to a glass (fused silica) made chip, the reference material.

Materials and methods

Materials preparation

Optically transparent PC, PMMA and PS sheets 1-1.1 mm thick, typically used for micro-systems fabrication by thermo-molding, laser engraving or micro-milling techniques, were purchased from commercial distributors. Two commercial polymers for microsystems prototyping by cast moulding were also considered: PDMS, used for decades in microfluidics and optofluidics fabrication, and the latter OSTEMER. PDMS (Sylgard 184 elastomer kit, supplied by Dow Corning, MI, USA) sheets 1 mm thick were fabricated by single-step casting on a mould fabricated by using a flat plastic container with optical quality walls as a substrate, and creating a well-controlled 1 mm gap by using 1 mm thick glass slides as spacers. Analogously, OSTEMER sheets 1 mm thick (OSTEMER 322 crystal clear, supplied by Mercene Labs, Sweden, in the form of a bi-component resin to be mixed before cross-linking) were fabricated using the same process, but this time using a flat PDMS sheet as a moulding substrate, to prevent OSTEMER adhesion [START_REF] Pham | Development of high throughput microfluidic platforms for the measurement of the protein solution thermodynamic properties 17[END_REF]. Samples to be chemically-exposed or irradiated were prepared by cutting the purchased or fabricated sheets in 2 cm × 2 cm pieces. 13 samples, presumed identical, were prepared for each investigated material. In each series, one sample was left non-irradiated and non-exposed to chemical products, in order to serve as reference, three were exposed to chemical environment, and the other nine were irradiated (see Table 1).

Chemical exposure

In order to examine their possible optical properties degradation and their structural integrity in operation, samples were exposed to 4 aggressive media commonly used in hydrometallurgical processes, and particularly in the nuclear waste recycling process PUREX:

• Concentrated nitric acid (HNO 3 4M), used for lixiviation and dissolution steps; • Hydrogenated tetra propylene, TPH, which is a typical diluent used in solvent extraction analogous to dodecane; • Isane, which is also a diluent in solvent extraction, as a substitute to TPH; • Tributylphosphate, TBP, which is the reference solvent in the PUREX process (Note: the actual process uses a dilution of 30% TBP and 70% TPH).

Samples of each material were fully immersed for 24 h in each media, subsequently rinsed with abundant water, and finally dried out at room temperature before their optical characterization.

Gamma irradiation

Samples were irradiated using a Cs-137 gamma irradiation equipment, with a total activity of 171.7 TBq. They were placed in a 20 locations basket, so that their individual location, and dose, are perfectly known. Moreover to prevent any dose rate effect, locations have been chosen in order to keep dose rate as constant as possible during irradiation (from 0.628 to 0.851 kGy/h, depending on the distance from the gamma source). They were individually removed from the irradiator at different times, leading to different and controlled exposure doses (see Table 1).

UV-Vis and IR characterization

The UV-Vis transmission properties of all samples (reference, chemically-exposed and irradiated) were studied by means of the setup described in Fig. 1a. The coupling of light to the samples was achieved using solarisationresistant 400 µm fiber optics (Thorlabs, Newton, NJ, USA, NA = 0.22) directly connected to an ad-hoc fabricated sample holder by means of collimation lenses, and allowing for a correct and reproducible sample positioning. This sample holder is equipped with a slot (thickness gap 2 mm) that allows to set precisely the optical path (see Fig. 1a). A Ava-Light-D(H)-S-BAL balanced deuterium-halogen light source and a Avaspec ULS2048CL-EVO spectrometer (Avantes, Apeldoorn, Netherlands) were used for light coupling and subsequent spectrum analysis. Spectra were recorded from 200 to 1000 nm with a resolution of ± 0.56 nm fixed by the spectrometer in terms of pixel dispersion. Transmittance was classically calculated using the Beer-Lambert law. FTIR spectroscopy was performed on all irradiated and reference samples to characterize any possible sample chemical alteration caused by gamma irradiation degradation. To this end, a Thermo Nicolet FTIR 6700 was used, and spectra were collected in attenuated total reflection (ATR) mode.

Analytical functional benchmark

The objective of this final test was to evaluate a potential influence of the material and the manufacturing process (laser engraving + chemical edging for high-resolution glass structures, micro-milling for PMMA and cast melding for PDMS) on the performance of UV-visible spectroscopic measurements at the microfluidic scale. Thus, three geometrically identical chips have been manufactured, from the previously mentioned 3 different materials, based on the PhLoC design used in a previous work [START_REF] Rodríguez-Ruiz | Uranium(VI) onchip microliter concentration measurements in a highly extended UV-visible absorbance linearity range[END_REF].

The first device was made of glass (cf. Fig. 2a), more precisely of fused silica with an OH content of 1200 ppm. It was manufactured by a special laser engraving + chemical edging technique developed by FEMTOprint (Switzerland). As fused silica is well-known for its excellent optical properties, this chip was therefore chosen as the reference for comparative purposes. A second chip was engraved in PMMA by micro-milling (cf. Fig. 2b), and third one was fabricated by casting PDMS on a dry film mould manufactured by soft lithography techniques [START_REF] Rodríguez-Ruiz | Broadcasting photonic lab on a chip concept through a low cost manufacturing approach[END_REF] (cf. Fig. 2c). The common design for the 3 optofluidic chips consists of a fluidic continuous channel with three optical interrogation areas of increasing path length (highlighted in red in Fig. 2a). Each interrogation area comprised a set of collimating micro-lenses and auto-alignment channels, allowing for a correct positioning of fiber optics (solarisation resistant fiber optics, 220 µm, Thorlabs, Newton, NJ, USA, NA = 0.22) on the microlenses focal point, in order to efficiently couple/decouple light to the system. Fiber optics were coupled to the previously described light source and spectrometer, and the chips performances have been benchmarked in terms of light coupling efficiency and by absorptiometric measure-ments considering neodymium dosage. To this end, standard solutions of neodymium (III) nitrate hexahydrate (99.9%, Sigma-Aldrich, USA) in HNO 3 (GPR RECTAPUR ® , VWR Chemicals, USA) at 1 mol•L -1 have been prepared, in the range 0.1 ≤ C g•L -1 ≤ 50.

Results and discussion

Chemical compatibility

Figure 3 compares the UV-Vis optical transmittance of the reference sample with samples of the same material after solvent exposure. For each material, almost identi-cal transmittance spectra were measured for the reference and the exposed samples. The variations observed for PDMS and OSTEMER samples are a consequence of the manufacturing process leading to small thickness variations, microbubbles occurrence, etc., locally modifying the optical properties of the material regardless the chemical exposure.

All samples displayed good structural resistance and preserved their optical properties when exposed to HNO 3 4M. Regarding TPH exposure, only PDMS showed swelling, while no effect on transmittance spectra was observed. The other polymers were not affected by this solvent. For TBP exposure, PC, PDMS and OSTEMER remained stable after 24 h of total immersion, while the PMMA and PS samples were respectively partially (becoming opaque) or totally dissolved (see Fig. 1b,c). However, further tests performed on mixed TBP at 30% concentration in volume on TPH (common solvent in the PUREX process) revealed that both PMMA and PS remained unaltered after 24 h of immersion. Samples exposed to isane, displayed similar behaviour to those exposed to TPH (to be expected due to their close chemical composition), with a more pronounced swelling effect on PDMS, leading to a boost in size of about 30% (see Fig. 4).

This can be particularly detrimental for the structural stability of a microfluidic structure, especially considering the substantial changes in the optical path lengths in the case of absorptiometric PhLoC platforms. For the 1 mm thick samples considered, swelling was observed to result in a 10-25% decrease of transmittance on the UV-Vis spectra (depending on the wavelength). Moreover, due to the absorption of small molecules on the PDMS pores with coloured reagent, the use of isane or TPH solutions of coloured reagents leads to PDMS dyeing, which can directly influence the UV-Vis analysis. This last point may be an issue for the measurement of most of the elements involved in the PUREX process, particularly uranium and plutonium.

Effect of gamma irradiation

For each sample, 4 wavelengths, related to the maximum absorption peaks of Pu(III)-565 nm, Pu(IV)-475 nm, U(IV)-647 nm, and U(VI)-413 nm (of interest in the PUREX process for irradiated nuclear fuel treatment) have been selected, in order to assess the effects of gamma irradiation in the polymers optical properties. The effect of increasing irradiation doses on the transmittance is illustrated in Fig. 5 for each of the abovementioned wavelengths.

Additionally, the full UV-Vis transmittance spectra measured on each samples are presented in Figures s1 to s5 of the supporting information. It can be observed that all materials presented barely noticeable to slight optical degradation above 500 nm at the wavelengths of interest, regardless of the irradiation dose (Fig. 5a,c). However, as we move towards the UV wavelengths, a significant effect of the dose level is observed: the higher the dose level, the higher the transmittance drop. This effect is less pronounced for PC, which is a plastic material widely used in nuclear environments, for manufacturing sealed containers such glove boxes or prototyping chemical reactors suitable to a radioactive surroundings. However it is especially noticeable for PMMA and OSTEMER (see trend lines depicted in Fig. 5b,d), which in addition exhibit a natural transmission cut-off in the near UV spectrum. In the case of OSTEMER, direct examination reveals inhomogeneous color formation, leading to yellowing zones, mainly at the higher irradiation doses (Fig. 1d,e). The observation of a red shift of the UV cut-off and the formation of an absorption tail, leading to polymer color formation is in agreement with previously reported data [17,[START_REF] Clough | Color formation in irradiated polymers[END_REF]. Regarding OSTEMER, which is a heterogeneous material, the observed color inhomogeneities could be explained by an uneven distribution (due to insufficient reagent mixing) of the different monomers and additives within the samples, which are likely to promote radiation-induced color formation [START_REF] Klemchuk | Protecting polymers against damage from gamma radiation[END_REF].

In order to evidence possible radiation effects on the polymers structural and chemical composition, FTIR absorbance spectra were measured and are reported in Fig. 6 (the baselines correspond to non-irradiated materials, i.e. the reference value). All irradiated samples displayed (within the resolution of the spectrometer, < 1 cm -1 ) equivalent absorbance bands and peaks, without any deviation from the reference samples. Thus, interestingly no irradiation effects could be deduced from these analyses, even if some materials are obviously altered (see Fig. 1d).

To summarize, Table 2 condenses the results of irradiation and chemical exposure tests. In addition, the most common manufacturing methods for each material are recalled to allow a more in-depth comparison. Corresponding data for fused silica are also reported for the sake of comparison, as it is today one of the preferred material for microfluidic devices, especially in hostile environment. While its resistance to chemicals is excellent, γ-irradiated fused silica displays intense coloration (darkening), which is well reported in the literature [START_REF] Marshall | Induced optical absorption in gamma, neutron and ultraviolet irradiated fused quartz and silica[END_REF][START_REF] Zhong | Effect of γ-ray irradiation on the optical property and laser damage performance of silica[END_REF].

Functional benchmarking

We have observed that the exposure to γ-irradiation and chemicals of interest becomes limiting factors for the use of some of the studied polymers as construction materials for microfluidic/optofluidic chips to be used in a nuclear environment. But in order to assess microsystems optical quality (i.e. microsystems analytical performance), it is also important to account for the different chip fabrication techniques, which are strongly material-dependent. In this work we are benchmarking two widespread low-cost microfabrication techniques (micro milling and cast molding) against the highest optical standard for microfluidics microfabrication at the present date (i.e. fused silica, providing standard optical material, and laser engraving + chemical edging, providing the best possible optical grade wall finishing to the microfluidic and micro-optical structures).

Figure 7a depicts the coupling efficiency of the three optofluidic systems, as a function of the optical path. Intensity values have been obtained by measuring the transmitted light through each system and optical path at 577.6 nm (absorption peak for neodymium dosage analysis) for a reference solution of HNO 3 1M. Data has been normalized as a function of optical path length for comparison purposes. As expected, the light coupling efficiency is, in general terms, higher on the glass device compared to the PMMA and PDMS systems. This effect is more evidenced at the 10 mm interrogation channels (which is actually 15 mm long on the glass chip), as the largest the optical path, the more critical becomes the optical quality of the micro optical elements to avoid undesired scattering and thus transmittance losses. Nevertheless, the coupling efficiency for the three systems is comparable. The resulting calibration curves for neodymium analytical dosage in acidic media, performed in the 3 different optofluidic chips, are compared in Fig. 7b. For each considered concentration, 5 measurements have been performed in order to assess the dispersion around the average values.

In each case, a linear evolution can be achieved using only 2 of the 3 available optical paths (1 and 10 mm), as highlighted in Fig. 7b. Moreover, the three chips reveal analogous sensitivity as the slopes of the adjusted linear regressions are comparable (see Table 3). In all three cases, the R 2 ensures a good fit with the experimental data. Note that for the highest concentrations, it is theoretically required to account for the refraction effects induced by the solution and to correct the transmittance accordingly. However, for the sake of the comparison here considered, this effect can be neglected since it depends only on the sample optical path lengths (similar on the three chips) but not on the nature of the systems materials.

PMMA and glass exhibit a very close limit of detection (0.45 and 0.40 g•L -1 respectively), calculated here using the IUPAC definition [START_REF] Long | Limit of detection: a closer look at the IUPAC definition[END_REF]. According to this definition, LOD = k ⋅ s b ∕m where k is a coverage factor ensuring a confidence level, m is the sensitivity defined as the slope of the linear fitting and s b is the standard deviation of the blank. It can be noted that the PDMS has a slightly higher detection limit of 0.77 g•L -1 , due to a larger standard deviation of the blank and lower light coupling efficiency.

These results evidence that both PMMA and PDMS chips, although displaying a lower light coupling efficiency, exhibit analytical performances comparable to the reference fused silica chips for absorptiometric measurements. Furthermore, these two polymer materials are perfectly adapted for a Taylor made production of PhLocs with a lower cost.

Conclusions

Several polymeric materials used for the manufacturing of microfluidic chips have been investigated in view of their possible use in nuclear fuel reprocessing process: PDMS, PC, PS, PMMA and OSTEMER. The resistance to γ radiations, as well as to chemical products used in the PUREX process has been examined. For OSTEMER and PMMA, the direct visual observations reveal color changes after high dose of γ-irradiation. This effect, also reported for glass after following exposure to a few tens of kGy, is shared by most of transparent materials. As far as chemical compatibility is concerned, PDMS, PMMA, and PS respectively displayed swelling and partial/total dissolution in presence of the solvents, and thus they are not suitable for microfluidic applications involving organic solvents. They can however be used in an acidic environment and/or with time limited exposition to organic components. Besides these considerations, the manufacturing process itself must be considered, in terms cost and ease of fabrication of the chips: OSTEMER and PDMS chips, manufactured by cast molding, better fulfil these technical-economic criteria, as it does not require any specific and expensive equipment. The comparison between absorptiometric chips made of casted PDMS and engraved PMMA showed that the material and manufacturing processes has minor influence on the spectrophotometric measurements performances. The tests conducted here expose the materials to high doses of radiation or high TBP concentration. They are hence quite penalizing tests and not necessarily representative of the process conditions. Additional studies are required in order to test the effects in more representative conditions, especially those induced by flowing radionuclides and/or diluted solvent.

Fig. 1 a

 1 Fig. 1 a Operation scheme and image of the experimental setup for UV-Vis sample characterization, b PMMA and c PS samples after 24 h immersion in TBP solvent, displaying surface etching and partial dissolution, d, e OSTEMER samples respectively irradiated at a dose of 58.81 and 60.48 kGy, displaying visible alteration of optical properties (inhomogeneous yellowing). (Color figure online)

Fig. 2

 2 Fig. 2 Upper part-Photographs of the three microfluidic chips (a glass, b PMMA, and c PDMS) used for comparisons (the 3 optical paths are highlighted in red on the glass chip). Bottom part-Micro-

Fig. 3 Fig. 4

 34 Fig. 3 UV-Vis transmittance spectra of reference and solventexposed samples of the different polymers considered in this work. (Color figure online)

Fig. 5 Fig. 6

 56 Fig. 5 Optical transmittance properties for the irradiated samples, as a function of the dose, for the maximum absorption peaks of a Pu(III)-565 nm, b Pu(IV)-475 nm, c U(IV)-647 nm; and d U(VI)-413 nm, wavelengths of interest in the PUREX process for irradiated nuclear fuel treatment. (Color figure online)

Fig. 7 a

 7 Fig. 7 a Comparative of the light coupling efficiency through the 3 different interrogation channels for the glass, PMMA and PDMS optofluidic devices; b Calibration curves for neodymium analytical dosage in acidic media performed at λ = 577.6 nm on the 3 devices-(I) Optical path 10 mm (II) Optical path 1 mm. (Color figure online)

Table 1

 1

	Exposure doses versus materials	Material	Exposure doses (kGy)
		PC	6.3	20.0	26.0	39.7	45.4	59.1	65.7	78.6	87.5
		PMMA	6.1	19.0	25.1	38.0	43.4	57.1	62.4	75.3	82.0
		PS	5.8	18.1	23.9	35.4	40.5	54.4	59.3	72.0	81.1
		PDMS	5.2	16.1	21.1	32.2	36.8	47.9	53.0	64.9	73.2
		OSTEMER	5.1	15.5	20.7	31.4	35.9	47.1	51.0	62.8	64.5
		Mean	5.7	17.7	23.3	35.3	40.4	53.1	58.3	70.7	77.7

Table 2

 2 Summary of chemical and nuclear compatibilities of glass, OSTEMER, PC, PDMS, PMMA and PS (+ compatible, -deformation or coloration of material, --high degradation)

	Compatibility	Fused silica	OSTEMER	PC	PDMS	PMMA	PS
	Chemical exposure
	Nitric acid 4M	+	+	+	+	+	+
	TPH	+	+	+	-	+	+
	TBP	+	+	+	+	--	--
	Isane	+	+	+	-	+	+
	Irradiation	-[20, 21]	-	+	+	-	+
	Manufacturing	Laser engraving	Soft lithography	Engraving, 3D printing,	Casting	Engraving, 3D printing,	Engrav-
				Hot embossing		Hot embossing	ing, Hot
							embossing

Table 3

 3 Analytical features for the three chips (0-50 g•L -1 , measurements repeated 5 times)

	PhLoC material	Sensitivity (L g -1 )	R 2	LOD (g L -1 )
	PDMS	0.00593	0.9985	0.77
	PMMA	0.00577	0.9986	0.45
	Glass	0.00528	0.9974	0.40

Acknowledgements This work was supported by the Nuclear Energy Division of CEA (DISN/PRATA-PRONG). IR-R acknowledges the CEA-Enhanced Eurotalents Program for his Incoming CEA postdoctoral fellowship.


Diana M Hackenburg

Alison Adams

Katherine Brownson

Israel T Borokini

Tatiana M Gladkikh

Shannon C Herd-Hoare

Helina Jolly

Andrew N Kadykalo

Erika B Kraus

Kelsey R Mcdonough

Diana M Hackenburg Ab

Alison Adams Ab

Tatiana M Gladkikh Ab

Joshua W Morse Ab

Saroop S Sandhu

Navchaa Tugjamba

Améline Vallet

Meaningfully engaging the next generation of ecosystem services specialists

Keywords: Early-career specialists, Ecosystem services, Professional development, Young ecosystem services specialists

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Commentary

As the field of ecosystem services (ES) continues to grow, novel opportunities abound to engage with diverse disciplines [START_REF] Bennett | Research Frontiers in Ecosystem Service Science[END_REF][START_REF] Van Riper | Incorporating Sociocultural Phenomena into Ecosystem-Service Valuation: The Importance of Critical Pluralism[END_REF], social justice issues [START_REF] Dawson | Justice and equity: emerging research and policy approaches to address ecosystem service trade-offs[END_REF][START_REF] Schröter | Refocusing ecosystem services towards sustainability[END_REF], and emerging paradigms, such as nature's contributions to people [START_REF] Peterson | Welcoming different perspectives in IPBES: "Nature's contributions to people" and "Ecosystem services[END_REF], relational values [START_REF] Himes | Relational values: the key to pluralistic valuation of ecosystem services[END_REF], and shared values [START_REF] Irvine | Ecosystem services and the idea of shared values[END_REF]. Particularly, the ES field must usher in a new generation of scholars, practitioners, and policymakers who are beginning their careers. Harnessing the next generation of specialists' new perspectives, strengths, and ambitions presents challenges and opportunities for the ES community [START_REF] Lim | Early-career experts essential for planetary sustainability[END_REF]. Our team of 14 Early-Career Specialists (ECS) explores these risks and rewards, followed by recommendations for enhancing engagement and advancement of future ES leaders and experts. Despite ES's wide adoption as a sustainability framework, the field still struggles to embrace diverse perspectives [START_REF] Peterson | Welcoming different perspectives in IPBES: "Nature's contributions to people" and "Ecosystem services[END_REF]; we argue that this shortcoming extends to engaging ECS. In this community where founding members are still actively involved, finding a niche as a newcomer, where one can elevate his/her voice and contribute to theory and practice, can be difficult. Additionally, while calls for greater interdisciplinarity and community-engaged research may be critical to the uptake of ES for decision-making [START_REF] Bennett | Research Frontiers in Ecosystem Service Science[END_REF][START_REF] Costanza | Twenty years of ecosystem services: How far have we come and how far do we still need to go?[END_REF], this route can be risky because of inadequate institutional (political and social) support, especially for those early in their careers [START_REF] Dooling | Doctoral Student and Early Career Academic Perspectives on Interdisciplinarity[END_REF][START_REF] Hein | Overcoming early career barriers to interdisciplinary climate change research[END_REF]. These barriers, in conjunction with the common financial and mental stress associated with postsecondary education in science and academia [START_REF] Belluz | The 7 biggest problems facing science, according to 270 scientists[END_REF], may outweigh the benefits for future experts, limiting the influx of new talent and capacity [START_REF] Rhoten | Risks and Rewards of an Interdisciplinary Research Path[END_REF].

The world must "mobilize all of its talents" to solve global sustainability challenges [START_REF] Brück | Empowering Young Scientists[END_REF], and ECS have the potential to lead ES in new, exciting directions. Many ECS exhibit a strong interest in social justice and real-world problem solving, and seek to broaden the impact of ES on societal and ecological decisions, policies, and practices [START_REF] Rhoten | Risks and Rewards of an Interdisciplinary Research Path[END_REF]. Our exposure to diverse cultures, values, experiences, and epistemologies, alongside our familiarity with emerging communication and engagement technologies, provides us with the skills to bridge diverse disciplines, stakeholders, and sectors [START_REF] Costanza | Twenty years of ecosystem services: How far have we come and how far do we still need to go?[END_REF][START_REF] Lambini | Public engagement: Young scientists welcome at IPBES[END_REF]. Moreover, ECS already produce high-quality, innovative science [START_REF] Ebadi | How to boost scientific production? A statistical analysis of research funding and other influencing factors[END_REF]. One initiative already recognizing and building on this potential is the Young Ecosystem Services Specialists (YESS) network [START_REF] Böhnke-Henrichs | YESS -The network for Young Ecosystem Services Specialists[END_REF]. Since 2011, YESS has expanded to over 420 members worldwide and has hosted engagement-driven sessions at multiple conferences, along with facilitating a multitude of ongoing capacity-building activities (e.g., [START_REF] Bull | Strengths, Weaknesses, Opportunities and Threats: A SWOT analysis of the ecosystem services framework[END_REF].

Given these benefits, we argue that further investment in ECS' professional development and inclusion will strengthen the field's continuity and longevity, as well as enhance the use of ES in global conservation and sustainable development efforts. To this end, we present several actions that we consider critical stepping stones for meaningfully engaging the next generation:

 Remove barriers for engagement: Several opportunities exist within the ES community to increase equitable participation of ECS. We encourage all ES events and conferences to offer reduced rates for attendees five years or fewer post-graduate degree. We also propose establishing a research fund for ECS that could serve as a platform for showcasing innovative ideas (e.g., NSF CAREER and ESIP FUNding Friday). ES journals can reduce fees and expand opportunities for ECS, such as across-career special issues or present-to-publish programs (The [START_REF] Editors | The future of scholarly publishing: Paywalls and profits or a new plan?[END_REF]. The ES community also should continue to promote interdisciplinary work by increasing coverage in related journals and by advocating for academic institutions to recognize and reward community and policy-level contributions (e.g., the San Francisco Declaration on Research Assessment, 2013). Further, some early career-specific funding could be prioritized towards interdisciplinary work.

 Create direct, long-term mentoring and networking programs: Professional societies like ACES and ESP play a key role in providing networking opportunities, which pave the way for future job and collaboration opportunities [START_REF] Ansmann | Career development for early career academics: Benefits of networking and the role of professional societies[END_REF][START_REF] Daily | Developing a Scientific Basis for Managing Earth's Life Support Systems[END_REF]. We encourage growing mentor and network programs, particularly for community members in developing countries [START_REF] Dike | Obstacles facing Africa's young climate scientists[END_REF]. Conference fellowships, like ACES program for graduate students, are extremely valuable, and we encourage expanding such opportunities further to cover multiple experiences (e.g., the IPBES Fellowship Programme) and more directly facilitate networking between early career attendees and established professionals, for example, through reverse mentoring [START_REF] Morris | Reverse Mentoring: Untapped Resource in the Academy?[END_REF].

 Extend leadership positions: We recommend that ES advisory and journal boards, sciencepolicy initiatives, and event committees create permanent positions for ECS. For conferences, we recommend recruiting ECS to serve on the planning committee and to help develop activities specifically geared toward early career attendees. Further, we advocate future conferences mandate all activities feature diverse representation, to include ECS and local ES experts or practitioners.

 Increase learning opportunities: Recognizing many educational and training opportunities already exist, the ES community should evaluate potential areas of need and draw from other fields to guide future development of interdisciplinary courses, programs, and standards [START_REF] Gustafsson | Producing expertise: the Intergovernmental Science-Policy Platform on Biodiversity & Ecosystem Services' socialisation of young scholars[END_REF][START_REF] Welch-Devine | A pedagogical model for integrative training in conservation and sustainability[END_REF]. For instance, developing introductory-level modules for all ages-elementary through university-could introduce a wider range of students to the science and practice of ES. We highly encourage the development and expansion of formal trainings offered by ES organizations and experts, as well as more informal resources like those listed by ESP. In particular, we envision creating an immersive workshop that regularly brings together ECS and established leaders to allow mutual learning through discussion of the field's key ideas, techniques, and needs.

Together, these actions lay out a progressive strategy for meaningfully engaging and developing ECS in ES. Being early in our careers, we want to ensure our chosen field thrives, and believe a high investment in emerging experts now will yield high returns for the future planet and human well-being.

We hope these ideas will advance ongoing efforts to promote an inclusive ES community that welcomes and embraces diverse perspectives.

Acknowledgements

We would like to thank A Community on Ecosystem Services and the Young Ecosystem Services Specialists groups for bringing this group together at the 2018 ACES meeting.

Research data for this article

Data not available / No data was used for the research described in the article