Fabien Gandon
email: fabien.gandon@inria.fr

Artificial I ntelligence a nd I ntelligence Augmentation

Intelligent approaches to follow and support Web evolution

Initially, the Web was essentially perceived as a huge distributed library of linked pages, a worldwide documentary space for humans. In the mid-90s, with wikis and forums, the Web was re-opened in read-write mode and this paved the way to numerous new social media applications. The Web is now a space where three billion users interact with billions of pages and numerous software. In parallel, extensions of the Web were developed and deployed to make it more and more machine friendly supporting the publication and consumption by software agents of worldwide linked data published on a semantic Web. As a result of all its evolutions, the Web became a collaborative space for natural and artificial intelligence. This raises the problem of supporting these worldwide interactions and forming these hybrid communities. In my talk I presented some of the opportunities and challenges for Web Science in building this evolution of a Web toward a universal space linking all kinds of intelligence.

AI in classical tasks and problems of the Web

A first set of challenges can be directly identified from the classical tasks and problems we encounter on the Web e.g. help us search, browse, contribute, etc. The Web already is populated by Web bots but they usually are restricted to certain realms while they could be generalized. For instance we could generalize the bots as the ones of Wikipedia to bots on the open Web designed to monitor and preserve certain characteristics of the Web. We could imagine Web farms for Web AIs hosting autonomous agent that would study, monitor and report on the Web. Problems that could be targeted by these Web bots include: the detection of metrics manipulation, cross-language plagiarisms, centralization or digital divide; the prevention of vandalism or spamming; the generation of links, back links, navigational content beyond search results; etc. These agents would be based on policies and values important to the philosophy of the Web (e.g. seek decentralization, equality of access) to improve its resilience and quality.

The special relation of AI and data(sets) on the Web

The open and linked data facet of the Web is a special case of particular importance when considering the links between AI and Web (science) data (science). Artificial Intelligence can be used to assist Web Scientists and vice-versa. Intelligent agent can help produce, curate, share and maintain corpora and datasets. For instance AI techniques could be designed to check the quality of a dataset and look for bias in it. Inversely, Web Science could produce multidisciplinary methods and tools to certify the quality and characterize training sets to improve the quality of the learning and conclusion made by AIs using them.

Benevolent AIs for a resilient Web

The two previous ideas could be generalized to the goal of designing benevolent AIs for the Web. Web agents working to improve users' experience, understanding, awareness and control of their participation and contributions to the Web. For instance, educational AI could help educate Web users in many domains including Web literacy or ethical thinking. Agents could also provide customized descriptions of the context in which a user is, including security, neutrality and privacy notices or his human-computing participation when it occurs. AI could also help users burst our filter b ubbles a nd f oster s erendipity. O n t he l onger term, benevolent AIs could actively help enforce (human) rights on the Web and be scrutiny agents for important values of the Web.

AI to help us humans scale and face humanity on the Web

With the advent of the Web, human individuals also face humanity in all its scale and diversity. Web scientists could design AIs to help humans face humanity on the Web and help us scale to the world-wide web scale. These goal-driven agents could actively participate to the online activity and, for instance, foster linkage, interactions and convergence, bridge, translate, check, or augment our posts and maintain for us an overview of our social context and activity. They could also prevent or report problems such as bullying, harassment and polarization.

A variety of AIs to absorb the varieties of the Web

The force of these AIs could also be in their multiplicity and interactions. The law of requisite variety of W.R. Ashby says that "variety absorbs variety" and in our case a diversity of AIs could be a good way to address the many types of diversity we find of the Web (content, users, contexts, tasks, usages, resources, etc.). In fact more than AI, it is maybe distributed AI that has a rendezvous with the Web and its sciences [START_REF] Gandon | Distributed Artificial Intelligence And Knowledge Management: Ontologies And Multi-Agent Systems For A Corporate Semantic Web[END_REF]. Multi-agent systems and distributed AI blackboards are examples of distributed AI architectures which, if merged with the Web architecture would allow for many different kinds of AIs to collaborate worldwide to the benefit o f t he W eb. T he A Is a nd t he multi-agent s ystems would a lso i n r eturn b enefit from the Web, its resources and its methods. Following the wiki-way, AIs could be created, edited, crossed, and bred on the Web, socially maintained, copied and versioned: the Web way applied to AI with, for instance, "copy-paste-customize" based contribution to the population of agents. For this to happen, and just as it was the case for the Web, we would need a public domain Web-based AI architecture.

Explore and expand all the forms of intelligence on the Web

The multidisciplinary nature of Web Science also puts it in an ideal position to explore and expand the forms of intelligence on the Web. First, both Web Science and AI are highly multidisciplinary [START_REF] Gandon | The three 'W' of the World Wide Web call for the three 'M' of a Massively Multidisciplinary Methodology[END_REF] and multiple disciplines that are common to both fields a re a s many bridges to make them interact. AI could also be used to operationalize the expertise from each domain into agents that help us providing assistance, reporting or training from the domains they represent. These agents could help us find a nd s upport a m assively multidisciplinary method and allow us to scale to the multi-disciplinary interactions required by the design and study of the Web. One possibility, for instance, would be for these AIs to produce and maintain boundary artifacts at the frontiers of disciplines. The multidisciplinary domains of Web Science could also be leveraged to identify other ways of simulating, reproducing or engaging intelligence including emotional intelligence, communication skills, imagination, etc.

Studying and building the hybrid societies of the Web

Such an evolution as the one described in the previous sections would finally lead Web Science to consider the challenge of studying and designing hybrid societies of natural intelligence and artificial i ntelligence o n t he W eb. T his s tudy would h ave t o i nclude d ifferent f orms of natural intelligence (e.g. people, connected animals, connected plants) and different forms of artificial i ntelligence ( reasoning, l earning, i nducing, e tc.). T he challenge w ill a lso b e t o study their interactions with the resources of the Web (linked pages, linked data, connected objects, etc.) forming the environment of these forms of intelligences. Web Science will have to face the problem of this massive interaction design with the Web and everything it links [START_REF] Gandon | Paving the WAI: Defining Web-Augmented Interactions[END_REF] and AI will have to face the problem of engaging in very different types of interactions with different forms of intelligence including different kinds of AIs [START_REF] Gandon | Challenges in Bridging Social Semantics and Formal Semantics on the Web[END_REF]. Studying and designing these hybrid societies, from swarms to complex societies with their normative rules, their social constructs, their governance, etc. will be a highly challenging and multidisciplinary task.

Towards a Web linking all forms of intelligence

In Web Science, we should build our research program as a joint effort between Web Science and two research fields b orn i n t he 5 0s: " AI" f or A rtificial In telligence [START_REF] Mccarthy | A proposal for the dartmouth summer research project on artificial intelligence[END_REF] an d "IA"for Intelligence Amplification [START_REF] Ashby | Design for an intelligence-amplifier[END_REF] a nd I ntelligence Augmentation [START_REF] Douglas | Augmenting human intellect: A conceptual framework[END_REF].

To conclude this abstract in one sentence, I would say that a Web Science research agenda must account for the fact that the long term potential of the Web is to augment and link all forms of intelligence.


Guilherme Machado
email: guilherme.machado@hmg.inpg.fr

Denis Favier
email: denis.favier@hmg.inpg.fr

Grégory Chagnon
email: gregory.chagnon@hmg.inpg.fr

Membrane curvatures and stress-strain full fields of axisymmetric bulge tests from 3D-DIC measurements. Theory and validation on virtual and experimental results

come

multiaxial stress and strain fields [START_REF] Seshaiyer | Multiaxial mechanical behavior of human saccular aneurysms[END_REF], to study the mechanical properties of human tissues [START_REF] Miller | Determination of elastic parameters for human fetal membranes[END_REF][START_REF] Kriewall | Mechanical-behavior of fetal dura mater under large axisymmetric inflation[END_REF] and to measure the finite load-deformation behavior of a living cells sheet cultured on a mesoscopic freestanding elastomer membrane [START_REF] Selby | Apparatus for measuring the finite load-deformation behavior of a sheet of epithelial cells cultured on a mesoscopic freestanding elastomer membrane[END_REF]. More recently Grolleau et al. [START_REF] Grolleau | Biaxial testing of sheet materials at high strain rates using viscoelastic bars[END_REF] combined a Hopkinson bar system with a conventional bulge test to study the biaxial testing of sheet materials at high strain rates.

On the theoretical point of view, the bulge test was as well studied in detail. A seminal work was proposed by Adkins and Rivlin [START_REF] Adkins | Large elastic deformation of isotropic materials. IX. The deformation of thin shells[END_REF] that details the boundary value problem, in the field of nonlinear elasticity, associated with the Mooney isotropic membrane inflation. Other important work has come from Hill [START_REF] Hill | A theory of the plastic bulging of a metal diaphragm by lateral pressure[END_REF] introducing a theoretical analysis of plastic flow at the pole of a metal diaphragm, based on the Lévy-Mises stress-strain equations. In an analogous manner, Ross and Prager [START_REF] Ross | On the theory of the bulge test[END_REF] showed that the use of Tresca's yield condition and associated flow rule leads to a simple theory, where the basic equations can be integrated in closed form even for finite deflections.

Bulge test was mostly conducted as an easy way to determine mechanical behavior of isotropic materials subjected to equibiaxial tensile stress state. Indeed, due to the axial symmetry of the problem and respecting the primary assertions of the membrane theory, the equibiaxiality of the stress and strain is obtained at the top of the inflated sample. Found over the pole neighborhood, where a spherical cap is supposed to exist, the equibiaxial stress-strain state can be easily calculated.

To achieve this goal the knowledge of the curvature radius, tangential elongation and pressure recorded during the test are needed. Such approach was recently used to study rubber-like materials [START_REF] Meunier | Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber[END_REF][START_REF] Sasso | Characterization of hyperelastic rubber-like materials by biaxial and uniaxial stretching tests based on optical methods[END_REF], and NiTi superelastic sheets [START_REF] Grolleau | Assessment of tension-compression asymmetry of NiTi using circular bulge testing of thin plates[END_REF]. However, the current shape of the total bulge is rarely spherical, as shown by Dudderar et al. [START_REF] Dudderar | Measurement of the shapes of foil bulge-test samples[END_REF] using the two-source holographic technique. Therefore, if the pressure and principal curvatures fields are measured, much more information is available from axisymmetrical bulge test than only stress and strain states at the pole. The membrane stress tensor can be determined from these data using membrane theory and equilibrium equations without defining any explicit constitutive equation. However, the accurate experimental measurement of curvatures is not a simple task. To avoid this difficulty, a relation describing the material behavior was often used in the past. In this case, the principal curvatures were often written in terms of the principal stretch ratios and replaced in the Lagrangian form of the equilibrium equations. In a general manner, this procedure intends to reduce the problem to a first-order system by introducing new variables [START_REF] Wineman | Large axisymmetric inflation of a nonlinear viscoelastic membrane by lateral pressure[END_REF][START_REF] Yang | On axisymmetrical deformations of nonlinear membranes[END_REF]. As a consequence, there is also loss in generality in assuming an explicit form for the constitutive relation which is chosen to represent the material. See for example the works [START_REF] Klingbeil | Some numerical investigations on empirical strain energy functions in the large axisymmetric extensions of rubber membranes[END_REF] for hyperelastic potentials; [START_REF] Wineman | On axisymmetric deformations of nonlinear viscoelastic membranes[END_REF] and [START_REF] Feng | Viscoelastic behavior of elastomeric membranes[END_REF] for viscoelastic behavior.

In a classical manner, the bulge specimen is supposed to be deformed axisymmetrically and profiles are measured. In that case, meridional curvature is directly measured and circumferential curvature is deduced under axisymmetric assumption [START_REF] Hsu | A triplane video-based experimental system for studying axisymmetrically inflated biomembranes[END_REF]. The accuracy of this process heavily depends on the finite number of experimental measurements points, and information data is restricted to a meridional path only. This manner to compute the principal curvatures is restricted to axisymmetric problems and also a numerical procedure must be employed to solve the coupled nonlinear differential problem.

In this context, the present work intends to provide useful method to obtain the membrane stress and strain fields during the bulge test using the three-dimensional digital image correlation (3D-DIC) technique. The 3D-DIC is able to capture three-dimensional surface geometry and displacements of the deformed surface. This paper explains how these kinematical fields allow to evaluate the strain and curvatures fields in each point of the membrane. The stress field can also be deduced using the axisymmetrical hypothesis. Section 2 starts with a brief description of 3D-DIC technique, where the bases of contour and strain measurement principles are outlined. In the same section, the theoretical framework to evaluate the first and second fundamental forms of surface and relations between principal curvatures and membrane stress are emphasized: a method to determine the curvature fields from 3D-DIC data is thus presented. Then we show that curvature fields combined with the measurements of the inflation pressure allow calculating the stress fields for an axisymmetric bulge. In Section 3, a finite element calculation is realized on a Mooney hyperelastic membrane to emulate experimental data, i.e., generate a virtual 3D-DIC information. These data allow to validate the curvature calculation method and to estimate errors associated with stress determination. In Section 4 the method is applied to real experimental data. The experimental results obtained in the bulge test of a silicone rubber-like material are presented. All precautions concerning the experimental setup are emphasized. The tridimensional correlation process is not perfect. In view of missing and outlying values that can occasionally occur, a strategy is invoked to lead to these occurrences. Finally, the experimental data fields are compared to a finite element model carried out using a neo-Hookean hyperelastic law fitted on uniaxial and equibiaxial tensile tests. This allows to highlight the advantages of the proposed method compared to usual methods used to analyze bulge tests.

A brief outline of 3D-DIC

In the 3D-DIC technique, random gray-value dot patterns on specimen surfaces are observed by two cameras in a stereoscopic setup. Also called stereo digital image correlation (SDIC), the 3D-DIC is based on both stereo-vision technique and digital image correlation (DIC). A comprehensive description of metrological aspects of 3D-DIC technique can be found in the literature, e.g. [START_REF] Luo | Accurate measurement of threedimensional deformations in deformable and rigid bodies using computer vision[END_REF], [START_REF] Sutton | Digital image correlation for shape and deformation measurements[END_REF] and [START_REF] Sutton | Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts,Theory and Applications[END_REF]. See [START_REF] Orteu | 3-D computer vision in experimental mechanics[END_REF] for references on applications of stereo-vision and 3D-DIC to experimental mechanics.

The stereo-vision technique allows to recover the three-dimensional information of a scene from two different view points. Considering a physical point M of coordinates X i in a Cartesian reference frame to be measured (Fig. 1). Forward, Latin indices represent the values 1, 2, 3 and, the Greek ones the values 1, 2. The stereo-projections of the point M (X i ) on the images planes of two cameras

C1 and C2 are given by M C1 (X α S1 ), M C2 (X α S2 )
, where X α S are the camera sensor coordinate system.

Every deviation from the straightness of the projection can be related to the presence of radial and tangential distortions. Fig. 1 illustrates the epipolar geometry, i.e., the geometry of stereo vision defined by the pinhole positions Cα, focal lengths fα , and image plane centers Pα. It is convenient to define a set of coordinate systems to represent the imaging process. The first coordinate system is the global coordinate system (GCS), with axes (X 1 , X 2 , X 3 ). The second coordinate system typically defined is the camera coordinate system (CCS) located at the pinhole positions Cα. The third set of axes defines the sensor coordinate system (SCS), with X 1 S , X 2 S in terms of pixels.

The 3D-DIC uses a DIC algorithm to determine point correspondences between two images of a specimen acquired at the same time from two bounded cameras. The digitized images are compared to match subsets -so called facets -from one image to another. The facet is a square area f size ×f size pixels, centered on the evaluation grid point. This facet is used from the correlation algorithm in order to identify the corresponding area in the second image. The correlation scores are computed by measuring the similarity of a fixed facet in the first image to a shifting subset window in the second one. Knowing of the imaging parameters for each camera and the orientations of the cameras with respect to each other by a calibration procedure, the position of each object point in three dimensions is calculated. The minimal facet size (f size ) is limited by the size and roughness of the stochastic pattern on the object surface. Each facet must contain a characteristic part of the pattern with good contrast features.

As a result, the 3D-DIC provides a set of physical points M in terms of Cartesian coordinates These parameters are considered as intrinsic since the cameras remain in fixed positions relative to each other during the experiment. The evaluation algorithm computes the extrinsic parameters (translation vector and the rotation matrix between the GCS and the SCS) as well, which in turn are influenced by the intrinsic parameters and the positions of the cameras. The uncertainties of the calibration parameters, resulting from the deviations of the detected markers from the model positions are then calculated. A total number of eight images are typically sufficient to calculate all calibration parameters accurately [START_REF] Becker | Error estimations of 3D digital image correlation measurements[END_REF]. A comprehensive description of the 3D-DIC calibration method with the mathematical development can be found in [START_REF] Sutton | Digital image correlation for shape and deformation measurements[END_REF].

X i in
Note that the spatial resolution (Sres) of the discretized surface depends essentially on camera sensor resolution (Cres) and on the choice of the evaluation grid that defines the distance between the data points on the object. The grid spacing (Gs) is the distance between the grid points in pixel.

Thus, grid spacing limits the spatial resolution, as each grid point represents one single data point of the result. Finally, data output is arranged in a matrix representation in the form X i (u, v) where the integer values u, v ∈ [1, ⌊Cres/Gs⌉] are respectively lines and columns indices of the matrix representation for each coordinate (X i ). See the scheme in Fig. 1.

Once the 3D contour has been determined, the next step in digital 3D correlation is the determination of the three-dimensional deformation of the object surface. This process is carried out by correlation of the images, taken by both cameras in the deformed state, with their original reference images. To evaluate the 3D displacement field, 3D-DIC determines point correspondences is not appropriate to describe the complexity of the real transformation, e.g., in a curved specimen surface or for significant lens distortions [START_REF] Schreier | Systematic errors in digital image correlation due to undermatched subset shape functions[END_REF]. Systematical errors is one of the principal limitation for the resolution of the resulting data [START_REF] Becker | Error estimations of 3D digital image correlation measurements[END_REF]. Finally, uncertainties of the calibration parameters lead to errors when reconstructing the 3D coordinates. Calibration errors appear in a systematical manner as a function of the facet positions in the camera frames, causing local distortions of the reconstructed 3D space. However, results from multiple experiments performed in [START_REF] Sutton | Digital image correlation for shape and deformation measurements[END_REF] indicate a typical accuracy for the displacement field on curved or planar specimens on the order of ±0.01 pixels for the in-plane components and D/50000 in the out-of-plane component (where D is the distance from the object to the camera) for typical stereo-camera arrangements. A point-to-point accuracy of ±10 -4 for the in-plane surface strains is achievable even when the object is subjected to large rigid-body rotations and arbitrary amounts of rigid-body translations since these motions do not corrupt the strain measurements.

Curvature evaluation from coordinates fields measured by 3D-DIC

Using the 3D-DIC, the reference surface Ω 0 of the object in question is discretized by a regular grid that represents the surface curvilinear coordinates Θ α . These coordinates are obtained directly from the grid placed by 3D-DIC using the grid indexes Θ 1 = u and Θ 2 = v where u and v are respectively lines and columns indices of the matrix representation X i (u, v) . This grid maps any discrete point M 0 of the undeformed body, associating a set of curvilinear coordinates, so called Θ α -curves. Since the coordinates Θ α are convective, these curves are subjected to the same deformation as the body and are transformed at time t in a new set of curves that represents the deformed state Ω t , as presented in Fig. 2. In the same figure, an arbitrary surface in three-dimensional Euclidean space E 3 is considered. The mapping of Ω 0 into Ω t is such that the correspondence of the points M 0

and M is assumed one to one. The reference and deformed surfaces are defined by S = S(Θ α ) and s = s(Θ α , t), respectively, with S and s ∈ E 3 , where Θ α are the surface curvilinear coordinates. In the following, s is supposed to be single-valued and sufficiently differentiable with respect to both arguments Θ α as many times as required.

Fig. 2 The surface basic geometric variable definition: e i are the unit base vectors related with the orthogonal Cartesian coordinates X i and x i ; G i and g i are the covariant basis related with M 0 and M respectively.

In order to handle with surfaces, two important pieces of data associated with any surface are recalled: the first and second fundamental forms of surface. Let M be a point on the surface whose coordinates are Θ α and Q be a neighboring point of the surface with coordinates Θ α +dΘ α . Defining ds as the elementary distance between M and Q, the first fundamental form can be expressed by the quadratic form

(ds) 2 = g αβ dΘ α dΘ β . ( 1
)
where the elements g αβ = gα • g β are the covariant components of the surface metric tensor g = g αβ g α ⊗ g β . This symmetric and positive-definite tensor can be expressed on covariant, gα, or contravariant, g α , basis where g β • g α = δ α β and δ α β is the Kronecker symbol. The first fundamental form is an invariant of surface and encodes the intrinsic data about the surface without referring back to the ambient space where the surface lies.

The second fundamental form, on the other hand, is a surface invariant that encodes the information about how the surface is curved in the embedding space. From the analysis of the differential geometry, it follows that the covariant components of surface curvature tensor b = b αβ g α ⊗ g β , associated with the second fundamental, are given by

b αβ = ∂gα ∂Θ β • g 3 (2)
where

g 3 = g 1 × g 2 ∥g 1 × g 2 ∥ = g 3 . ( 3
)
The interest is focused in the two principal curvatures κ I,II at a given point M in the membrane surface. Considering now the eigenvalue problem in each point

M ( b -κ I,II g ) v I,II = 0 (4)
where the surface shape operator is given by bg -1 . Since the shape operator is self-adjoint on the tangent plane at the surface in M , the corresponding eigenvalues κ I , κ II are called the principal curvatures at M , and v I and v II are the principal curvature directions at M . Recalling the definition of the Gaussian-curvature, K, and mean-curvature, H, it follows that

K = det(bg -1 ) = κ I κ II (5)
and

H = - 1 2 trace(bg -1 ) = 1 2 (κ I + κ II ). ( 6
)
As consequence, the maximal and minimal curvatures resulting from the quadratic form κ 2 -2Hκ-K = 0 associated with the surface shape operator are given by

κ I , κ II = H± √ H 2 -K. ( 7
)
For comprehensive exposition of other definitions, concepts and applications on differential geometry, see for example [START_REF] Carmo | Differential geometry of curves and surfaces[END_REF], [START_REF] Ciarlet | An Introduction to Differential Geometry with Applications to Elasticity[END_REF] and [START_REF] Toponogov | Differential Geometry of Curves and Surfaces: A Concise Guide[END_REF].

Finally, a MatLab routine (see Appendix A) was developed to perform a two-dimensional numerical differentiation scheme over the Θ α coordinates for each time t. Thus, the terms of Eqs. [START_REF] Avril | Overview of identification methods of mechanical parameters based on full-field measurements[END_REF] to [START_REF] Miller | Determination of elastic parameters for human fetal membranes[END_REF] are computed and principal curvatures can be determined at each membrane point. Given that 3D-DIC is able to capture any 3D contour and the curvature tensor is calculated for each sur-face point, the presented scheme of curvature calculation is not restricted to shape characteristics like axisymmetry.

Meridional and circumferential stress determination

Considering now the axisymmetric bulge test, the equilibrium equations can also be derived directly by balancing the forces of deformed shape element. Employing the primary assertions of the membrane theory, every material point of the membrane in both reference and deformed configuration is in a plane stress state [START_REF] Green | Large Elastic Deformation[END_REF]. In Fig. 3(a) the initially flat thin planar circular sheet of radius a and with an uniform thickness h 0 is presented. For convenience, to describe the strain field, cylindrical coordinates (R, Φ, Z = 0) define the position of a material particle M 0 in the middle surface of the stress-free membrane in the reference configuration. Let the pressure p acts in the bottom surface of specimen, the membrane experiences a deformation. The point M 0 moves to the position M , and the deformed configuration is defined by the coordinates r(R, Φ), ϕ(R, Φ) and z(R, Φ). Note that in either configuration, the profile of the membrane has no circumferential dependence. Under the assumption of material isotropy, the principal directions of both stretch and stress at each material particle are known ab initio to be the meridional and circumferential directions to the membrane surface. From this point forward, these directions will be denoted by the subscripts m and c respectively, as shown in Fig. 3(b).

Assuming quasistatic motion, the equations of equilibrium, at point M , for a thin axisymmetric isotropic membrane, as adopted by Hill [START_REF] Hill | A theory of the plastic bulging of a metal diaphragm by lateral pressure[END_REF], can be expressed as

σm = p 2h κc ( 8
) σc = p 2h κc ( 2 - κm κc ) , ( 9
)
where p is the time-dependent normal pressure acting uniformly (dp/dR = 0) and h the current thickness. The stress components are expressed in Eqs. 8 and 9 directly in terms of pressure and the current bulge geometry. As mentioned in [START_REF] Wineman | Large axisymmetric inflation of a nonlinear viscoelastic membrane by lateral pressure[END_REF] and [START_REF] Humphrey | Computer methods in membrane biomechanics[END_REF] a remarkable consequence of membrane theory is that it admits equilibrium solutions without explicitly requiring a constitutive equation, since the equilibrium equations are derived directly by balancing forces of an element of the deformed shape.

As consequence they are valid for all classes of in-plane isotropic materials. As pointed out by Hsu et al. [START_REF] Hsu | A triplane video-based experimental system for studying axisymmetrically inflated biomembranes[END_REF], however, principal curvatures κm, κc can be experimentally computed, thus Eqs. ( 8)

Fig. 3 (a) The axisymmetric bulge test scheme. (b) A deformed differential element and principal directions of stresses.

and ( 9) are very useful for exploiting axisymmetric bulge test data. Given the magnitude of error associated with the measurement system, errors in the stress determination using these equations are overwhelmingly the result of membrane theory. This point will be clarified in Section 3.3 where membrane hypotheses are investigated.

3 Benchmark inflation test using a Mooney membrane

Experimental data emulation

In order to demonstrate some quantitative predictions of the presented method to obtain the membrane stress field presented in Sec. 2 the finite element method (FEM) was used to emulate experimental data similar to that obtained by 3D-DIC. The aim here is to use a regular finite element mesh that represents the grid used by 3D-DIC to describe the surface geometry. These virtual data are obviously free from experimental problems such as missing and outlying values usually encountered in real 3D-DIC.

Using the Mooney hyperelastic model [START_REF] Mooney | A theory of large elastic deformation[END_REF], the FEM outputs nodal coordinates and displacements are used as input data. An uniform mesh with an element edge length of 2 mm was used to discretize a square zone inscribed into the circular geometry to obtain a good spatial resolution for the curvatures calculation. The Mooney model was arbitrarily chosen; any other constitutive model could have been used as well. For details of Mooney model and basic equations on large deformations see [START_REF] Meunier | Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber[END_REF].

Finite element model description

Numerical simulations were performed using the finite element code ABAQUS. Assuming an incompressible isotropic hyperelastic material behavior, the [START_REF] Mooney | A theory of large elastic deformation[END_REF] strain-energy function W M can be written as

W M (I 1 , I 2 ) = C 10 (I 1 -3) + C 01 (I 2 -3) (10)
where C 10 and C 01 are material parameters. I 1 and I 2 are the first and second strain invariants of the right Cauchy-Green deformation tensor C (C = F T F, where F is the deformation gradient tensor).

Due to its mathematical simplicity as well as its prediction accuracy in the range of moderatelylarge strains, this hyperelastic model have been widely employed in the description of the behavior of rubber-like materials.

The model geometry consists in a thin circular plate, of initial radius R(a) = 90 mm with h 0 = 2 mm of thickness. The circular plate is simply supported and subjected to an uniform inflating pressure acting perpendicular to the current configuration. The Mooney hyperelastic model parameters are those given in [START_REF] Meunier | Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber[END_REF], where C 10 = 0.14 MPa and C 01 = 0.024 MPa for the RTV141 silicone rubber cured at 70 • C during 150 min. The finite deformation problem was modeled with the S4R shell elements (4-node doubly curved general-purpose shell, reduced integration with hourglass control, finite membrane strains) with eleven Gauss integration points through the thickness.

Fig. 4(a) depicts a sliced sequence of deformed configurations for two pressure loads of 7.5 kPa and 15 kPa, respectively.

Curvature Method: application to the virtual data

Using the FEM deformed nodal coordinates output, the principal curvatures κm, κc are computed over all surface, as described in Sec. are perfectly concentric. Stress components can thus be evaluated from Eqs. ( 8) and ( 9). Moreover, since boundary conditions are perfectly balanced over the circumferential perimeter at R(a), a perfectly spherical cap is obtained, i.e., surrounding the pole a cap of umbilical points κm=κc ̸ = 0 can be easily identified. At such points, both principal curvatures are equal and every tangent vector is a principal direction.

The output fields principal stretches (λm,λc), principal curvatures (κm,κc) and principal stresses (σm,σc) from the FEM are plotted for a single meridional path in Fig. 6(a), (b) and (c), respectively.

Afterward, the principal stresses (σm,σc) computed using Eqs. ( 8) and ( 9) were compared with the Mooney membrane FEM outputs. It can be seen that the results fit very well in Fig. 6(c).

The relative differences between both stress components using the presented method and FEM are plotted in Fig. 6(d) for 7.5 kPa and 15 kPa. Errors tend to be less than 2 % for both stress directions for maximum pressure level of 15 kPa. On the other hand, errors increase for lower pressure levels.

This result is natural knowing that principal curvatures vanish when the surface becomes flat.

However, the principal explanation in disagreement of the stress results comes from the fact that, no matter how thin, all structures are three-dimensional and possess some bending stiffness. It is important to bear in mind that the membrane theory is an approximation. Fig. 7 shows the variation of FEM principal stresses over the membrane thickness h in the pole (R = 0) and in the border (R = 0.95a), for an initial ratio radius/thickness of a/h = 45. Note that, bending effects are more pronounced nearest to the border (R → a) in both configurations. This fact justifies the form of curves presented in Fig. 6(d). The other observation from Fig. 7 is that the bending influence can increase for low pressure levels. A similar result was found by Kyriacou et al. [START_REF] Kyriacou | Inverse finite element characterization of nonlinear hyperelastic membranes[END_REF] for an inverse finite element characterization using a bulge test sample with a/h ≈ 30 stretched at the pole of about 1.3. The authors noticed that the error in the estimated parameter increased with decrease of the pressure at which the data were generated. Depending on the material and geometry of the sample, the amount of bending could be non-negligible at small strain levels.

A last important remark arises from the fact that curvatures estimated for real 3D-DIC data are very sensitive to experimental noise since estimates of partial derivatives of second order are necessary. Gaussian noise was then introduced into the X i coordinates of FEM data results to simulate potential experimental errors. Applying our method to these noisy data leads to very poor estimation of both circumferential and meridional stresses, as shown in Fig. 8. Smoothing can be used to reduce experimental additive noise and small-scale information while keeping the most important imprints of data set. For these cases, a discretized smoothing spline algorithm proposed by Garcia [START_REF] Garcia | Robust smoothing of gridded data in one and higher dimensions with missing values[END_REF] was implemented. The weighted robust version of the algorithm was used to deal with occurrences of missing and outlying values of gridded data. The perturbed data were used as input of post-treatment routines. Curvatures and stresses were recalculated and Fig. 8 presents the results. Smoothing allows to obtain very close estimated values of the stresses from the noisy data to those of the original FEM data.

h(mm) sm R(0)= sc R(0) sm R(0.95a) sc R(0.95a) sm R(0)= sc R(0) sm R(0.95a) sc R(0.95a)
Fig. 7 Bending stiffness effects in both deformed configurations (7.5 kPa and 15 kPa pressure levels) for the numerical benchmark. Principal stresses variation (i.e. σm/σ m(mean) analogous for σc) over the element thickness (h) at the sample pole (R(0) = 0mm) and border (R(a) = 90mm).

Fig. 8 Principal stress (σm,σc) results. Original element output (FEM), calculated using the artificially noised data (noised) and calculated using the smoothed data (smoothed).

4 Application of the method to experimental bulge test of a silicone elastomer

Experimental bulge test

The objective of this section is to experimentally validate the proposed method to determine curvatures and stresses during a bulge test of a silicone elastomer. Experimental difficulties are pointed out and results are confronted with a finite element model. The chosen silicone rubber is an unfilled formulation produced by Rhodia (RTV 141). Specimens were produced using the processing described in [START_REF] Meunier | Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber[END_REF], following the processing route: (i) mixing the two liquid components (uncured silicone and curing agent with a 10/1 ratio), (ii) putting the uncured mixture under vacuum for 30 min in order to eliminate undesirable entrapped bubbles, (iii) injecting the liquid mixture in sheet mold and (iv) putting mold inside an oven at 70 o C for 4 hours to cure the silicone.

Compared to [START_REF] Meunier | Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber[END_REF], the cure was augmented in order to increase the tear resistance of the material.

This allows to largely swell the membranes. Using the RTV141 silicone, experimental bulge tests were carried out to determine a non-uniform stress strain state over the membrane surface.

Bulge test setup

The specimen disk dimensions are the same as in Sec. 3 (initial radius R(a) = 90 mm; thickness h 0 = 2 mm) constrained between two clamping flanges. In the used test machine, a pressurized air line is connected to the bottom circumferential clamp. Given the small pressures used during the experiments, a regulator is required to reduce the line pressure to a maximum of 30 kPa. To decrease the pressure an unloading valve is opened to create a leak. The internal pressure is measured by a Foxboro 0 -500 mbar pressure sensor. A picture of the experimental setup is shown in Fig. 9(a).

3D-DIC setup and experimental difficulties

The Dantec Q-400 commercial 3D-DIC system was used to acquire and correlate the full-field bulge shape information. Since the silicone sample experiences a large out-plane displacement, an appropriate focal length setup is absolutely necessary. The CCD cameras, using 20 mm focal length lenses opened at f /16, are arranged so that the pole and border of the circular membrane are present in both camera image planes. It is important that the object surface shows enough structure to allow the algorithms to correlate identical points from both cameras. Note that, using only two cameras, information near to the border will be lost if the membrane experiences large displacements creating a self-shadow effect in this region. A parameter Rmax has been used to define the limit of the correlation zone, i.e., the maximum distance from the membrane center which is seen by the two cameras. A third camera could be added in this case to be able to correlate along all meridional direction, see [START_REF] Orteu | Multiple-camera instrumentation of a single point incremental forming process pilot for shape and 3D displacement measurements: Methodology and results[END_REF] for a multiple-camera instrumentation example.

High spatial resolution is needed to allow an accurate measurement of curvatures. The spatial resolution achieved in practice depends on a number of factors, including but not limited to camera resolution, lens optical quality, and marker size and quality. Facet size also limits the spatial resolution strains since the greyscale information within a facet is used for the correlation between

images. An increase of the facet size causes smoothing of the results, decreasing the spatial reso-

Test Results

Cyclic inflations were performed, applying p = 30 kPa several times, to pre-stretch the specimen in order to accommodate the material. This is due to the fact that silicone rubber can exhibit considerable stress softening phenomenon for different loading cases [START_REF] Machado | Analysis of the isotropic models of the mullins effect based on filled silicone rubber experimental results[END_REF]. After a recovery time of Fig. 10 Principal curvature κm plotted onto 3D-DIC reconstructed surface. Solid line represents the meridional path used in Fig. 11 1 h and starting from an initially flat surface, inflations were performed from 8 kPa to a maximum pressure of 27.1 kPa. These inflations yielded principal stretches at the pole of about 1.10 and 1.36 respectively. Fig. 10 shows the principal curvature κm plotted onto 3D-DIC reconstructed surface. This bulge test can be considered as axisymmetric. Plots of the deformed membrane, over the meridional path, for each of the two inflation states can be found in Fig. 11. The values of z direction displacements (in mm) and principal stretches (λm,λc) were obtained direct from 3D-DIC system. Principal curvatures (κm,κc) and principal stresses (σm,σc) were computed using Eqs. ( 7), ( 8) and ( 9) respectively.

For each of the two inflation states all material points, save the pole (R(0) = 0 mm) and the clamped boundary (R(a) = 90 mm), experience both axial and meridional displacements as a result of their respective finite deformations. The plots of the principal stretches (λm,λc) in Fig. 11(c) show that each inflation state involves a heterogeneous strain state, i.e., equibiaxial stretch at the pole, planar stretching at R(a). As expected, the circumferential stretch λc tends to one towards the clamped boundary, i.e., when R → R(a). With respect to principal curvature distributions (κm, κc), note that equibiaxial membrane deformations near the membrane pole are associated with an approximately spherical geometry, i.e., κm ≈ κc. The non perfect equality may be explained by the fact that the umbilical point may not lie exactly on the z direction axis. This subtile difference can increase for high deformed states. Stress state can be assumed to be approximately balanced-biaxial at the pole, following by the circumferential stress σc which steadily decreases when R → R(a).

Material characterization and numerical simulation

In the previous section, stress and strain fields have been obtained by the presented method which does not require any constitutive equation and parameter determination. The aim of this section is to compare these results with usual analysis of bulge tests. This analysis is done thanks to a numerical simulation implying a modeling of the mechanical behavior of the unfilled silicone rubber RTV141. It is known that different modes of deformation are required to obtain the constitutive equation and the parameters that define the stress-strain relationship accurately. The tests used in this investigation are cyclic uniaxial and equibiaxial tension, completely independent, i.e., they were performed in parallel to the bulge test described in the last section.

Uniaxial tensile test and equibiaxial test are performed at room temperature at λ = 1.25 × 10 -2 s -1 . The material presents a weak hysteresis and a very little strain rate dependency [START_REF] Meunier | Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber[END_REF].

Nevertheless, a stress softening phenomenon, called Mullins effect [START_REF] Mullins | Softening of rubber by deformation[END_REF] appears, contrary to [START_REF] Meunier | Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber[END_REF] using a smaller curing time. Mullins effect strongly depends on the maximum previous deformation.

To measure the influence of the previous maximum deformation, increasing cyclic uniaxial tests were performed at λ = 1.75 and λ = 1.95. Fig. 12(a) highlights that for moderate deformations λ < 1.40, the second loading curves are nearly the same whatever is the previous maximum strain.

This means that increasing the maximum strain has only a slight effect on the behavior at moderate deformations. Therefore, for the range of deformation, a hyperelastic constitutive equation is sufficiently accurate to model the mechanical behavior. The neo-Hookean constitutive equation [START_REF] Treloar | The elasticity of a network of long chain molecules (I and II)[END_REF] is chosen to describe mild deformations:

W N (I 1 ) = C 10 (I 1 -3) . ( 11
)
The material parameter is fitted on second loading curves for uniaxial and equibiaxial tests, C 10 = 0.335 MPa is obtained. The results are presented in Fig. 12(a) and (b). This shows that the model predicts satisfactorily the two loading stress for λ < 1.40.

Experimental and numerical comparison of bulge test results

Results of FEM using the neo-Hookean model are superposed with the experimental fields (displacement, principal curvatures, principal stretches and principal stresses) in Fig. 11. All numerical predictions clearly follow the trends calculated from the presented method to obtain the princi- pal curvatures applied to the experimental data. For the z displacement field (Fig. 11(a)) a pronounced difference, for the lower pressure value, is observed. Consequently the curvatures evaluation (Fig. 11(b)) for this same load state is obviously affected. Nevertheless, errors in curvatures determination decreased with an increase in the strain level as observed in the benchmark of Section 3.

Considering the principal stresses plots, a good agreement is observed in spite of the hyperelastic constitutive equation simplicity.

It is important to keep in mind the simplicity of the neo-Hookean model, the simple procedure used for material parameter identification and the precaution taken to accommodate the material before testing. An interesting characterization method for rubber is proposed in [START_REF] Guélon | A new characterisation method for rubber[END_REF], performing only one heterogeneous mechanical test, measuring the displacement and strain fields using DIC and applying an inverse method. In such experiment, a large set of heterogenous states was obtained and results are represented in the I 1 × I 2 invariants plane, between uniaxial tensile and planar shear tensile paths. Fig. 13 illustrates the strain states, in the I 1 × I 2 invariants plane, for the classical three homogeneous tests used in material parameter identification. In the same figure, the gray zone represents all the heterogenous states covered in the bulge test, with the advantage of knowledge about the stress field. Note that, for small values of I 1 and I 2 , bending effects could be non-negligible and membrane theory fails as explained in Section 3.3. However, there is no simple rule to estimate these bending effects independently of the tested material and of the geometry of the bulge sample. Given a material, an estimation of the magnitude of bending effects could be done experimentally by varying the geometry of bulge sample.

For the presented material test, the distribution of loading cases in this zone are limited to

Rmax achieved during the correlation step. This zone is represented by a circular hatch pattern in Fig. 13. As aforementioned, the Rmax depends on the membrane displacements levels and camera dispositions in order to acquire the information over the border, i.e., the zone dominated by planar shear tensile state.

Conclusion

The aim of the present work was to propose a method which enriches greatly the analysis of usual bulge tests by using the 3D-DIC technique. Firstly, this paper presented a general method to determine surface curvature tensors from 3D-DIC data. This requires no assumptions regarding axisymmetry or other shape characteristics of the membrane surface. Secondly, the determination of the curvature tensor fields allowed to check the axisymmetry of the bulge test, which had to be obtained for planar-isotropic material and perfect boundary conditions. Once this axisymmetry was proved, the proposed method allowed to determine the stress-strain fields associated with the membrane inflation problem without any constitutive equation. The method was applied in this paper both on virtual data generated by numerically simulating the bulge test and on real data of one experiment performed on a silicon elastomer. In this last case, experimental difficulties were pointed out, particularly the difficulty to obtain the information near to the border for the chosen material. These two examples showed that bulge test can be used as a very rich heterogeneous mechanical test which allows to obtain valuable information and can be used to different types of materials. A wide range of stress-strain states, ranging from equibiaxial to planar tensile stress state, can potentially be achieved improving the experimental setup to obtain the stress-strain fields at the border for very large displacements.

Fig. 1

 1 Fig. 1 Representation of the epipolar geometry and the coordinates output format.

2 . 3 ,Fig. 4

 234 Fig. 4 Numerical inflation of Mooney behavior membrane. (a) z direction displacements (in mm) for pressure loads of 7.5 kPa and 15 kPa. Mesh are represented only in reference configuration and the two pressure steps are superposed; (b) Principal in-plane stress (in MPa) are plotted onto deformed shape and only the regular square zone is represented. Solid circles represent the meridional path.

Fig. 5

 5 Fig. 5 Curvatures of Mooney behavior membrane at p = 15 kPa. (a) Meridional and (b) circumferential isovalues using the FEM deformed nodal coordinates output as data.

Fig. 6

 6 Fig. 6 Two deformed configurations 7.5 kPa and 15 kPa pressure levels for the incompressible Mooney (C 10 = 0.14 MPa and C 01 = 0.024 MPa) membrane. (a) Principal stretches (λm,λc); (b) Principal curvatures (κm,κc); (c) Cauchy stress (σm,σc) confronted with the finite element results (FEM); (d) Errors with respect to principal stresses in both deformed configurations.

  lution. In order to obtain independent data points, the facets of neighboring data points should not overlap. In this work, full camera resolution (Cres) is used (2048 pixels × 2048 pixels) with an evaluation grid size Gs = 12 pixels and a facet size f size = 21 pixels, resulting in a spatial resolution of Sr = 1.45 mm over the membrane surface.

Fig. 9 (

 9 Fig. 9(b) presents an example of reconstructed surface. Note that some data are missing in some small regions on surface. It occurs due to some correlation abnormalities, normally associated with locally irregular pattern.

Fig. 9

 9 Fig. 9 (a) The bulge test experimental setup using 3D-DIC technique; (b) Reconstructed RTV141 membrane surface.

Fig. 11

 11 Fig. 11 Distributions of experimental (Exp) and numerical (FEM) fields: (a) z direction displacements; (b) principal curvatures (κm,κc); (c) principal stretches (λm,λc); (d) principal stresses (σm,σc); corresponding to 8 kPa and 21.7 kPa inflation states.

Fig. 12

 12 Fig. 12 Experimental data and the hyperelastic fitting using the neo-Hookean hyperelastic model: (a) Uniaxial and (b) equibiaxial tensile tests at λ = 1.25 × 10 -2 s -1 .

Fig. 13

 13 Fig.[START_REF] Ross | On the theory of the bulge test[END_REF] Loading case in I 1 × I 2 The gray zone represents all heterogeneous deformations states covered by the bulge test. The Bulge (Rmax) curve refers to the heterogeneous path at Rmax point. The circular hatch pattern represents the distribution of loading cases for the presented material test.

Acknowledgements

We would like to thank the French ANR for supporting this work through the project RAAMO ("Robot Anguille Autonome pour Milieux Opaques")

Appendix A

MatLab routine used to evaluate the surface curvatures.

function [ K, H, Ki , K i i ] = s c u r v a t u r e (X, Y, Z , g s )

% s c u r v a t u r e -compute g a u s s i a n , mean and p r i n c i p a l c u r v a t u r e s o f a s u r f a c e % % [K, H, Ki , K i i ] = s c u r v a t u r e (X, Y, Z , g s ) , where : % -X, Y, Z a r e m a t r i x o f p o i n t s on t h e s u r f a c e . % -g s s p e c i f i e s t h e s p a c i n g b e t w e e n p o i n t s i n e v e r y d i r e c t i o n . ( D e f a u l t g s =1) % -K i s t h e g a u s s i a n c u r v a t u r e . % -H i s t h e mean c u r v a t u r e . % -Ki and K i i a r e minimum and maximum c u r v a t u r e s a t each p o i n t . % % F i r s t D e r i v a t i v e s

%% P r i n c i p a l C u r v a t u r e s .

Ki=H+sqrt (H. ˆ2 -K) ; K i i=H-sqrt (H. ˆ2 -K) ; %end s c u r v a t u r e


Expédite Yen - Pon

Pier Alexandre Champagne

Lucie Plougastel

Sandra Gabillet

Pierre Thuéry

Mizuki Johnson

Gilles Muller

Grégory Pieters

Frédéric Taran

K N Houk

Davide Audisio

Sydnone-based Approach to Hetero-helicenes Through 1,3-Dipolar-Cycloadditions

The first approach to pyrazole containing helicenes via sydnone-aryne [3+2]-cycloaddition is described. An unprecedented regioselectivity in the cycloaddition step towards the more sterically constrained product was observed in presence of extended aromatic scaffolds. DFT calculations enabled to understand the origin of this unexpected selectivity.

Ortho-fused aromatic rings belong to a class of helical-shaped molecules named helicenes. 1 Since their discovery, these variegated aromatics have fascinated chemist practitioners due to their elegant architecture, inherent chirality and structural complexity. Helical structures are particularly interesting as they are found in many biomacromolecules inspiring chemists involved in the field of asymmetric catalysis, 2 and their enhanced chiroptical properties have attracted much attention in material science. [START_REF]cycloaddition with Ntosylpyridinium Imides[END_REF] The presence of a heteroatom in the fused polycyclic system considerably alters the electronic structure and helps tuning various optoelectrical properties. 4 In particular, nitrogen-containing helicenes, including pyridine, 5 pyrrole, 6 pyrazine 7 and imidazolonium 4a,8 have attracted much attention. Despite the broad interest over this family of compounds, synthetic access still remains challenging and often requires cumbersome multistep approaches. 1 Ideally, the desired helical motif would be assembled in the last step of the sequence, thus enabling divergent opportunities for structure diversity. Sydnones are azomethine imines, well-known for their 1,3-dipolar-cycloadditions with linear and strained alkynes, generating pyrazoles, a pharmaceutically and agrochemically relevant heterocyclic scaffold. 9 We reasoned that properly designed prohelical sydnones 3, bearing ortho-extended aromatic substitutions, would be suitable partners with arynes bearing an extended aromatic core. [START_REF]Inner-outer ring 1,3-bis(trimethylsilyloxy)-1,3-dienes as useful intermediates in the synthesis of helicenes[END_REF] After cycloaddition, the subsequent loss of carbon dioxide would deliver the desired pyrazole-containing helicenes, a family of heterohelicenes so far unreported (Figure 1).

We now describe a novel disconnection allowing a direct access to a range of unreported helical pyrazoles, based on a key sydnonearyne cycloaddition. In the process, we discovered a unique example of selective cycloaddition involving these mesoionic betaines in favor of the sterically hindered helical product and determined the reasons behind such selectivity with analysis of DFT calculations. The implementation of this strategy relies on the prerequisite preparation of ortho-fused aromatic azomethine imine dipoles. N-methyl sydnone 1 was identified as key component to build up such ortho-aromatic structures. Readily available in two steps from commercial sarcosine, 1 provides a versatile handle for further derivatization. It was envisioned that metal catalyzed functionalization at the C4 position of the sydnone with 2-halobenzaldehyde would provide a key intermediate that might undergo intramolecular Knoevenagel condensation under basic conditions to give the desired product 3. Initial attempts showcased that the whole cascade could be performed in one single operation. Product 3a was first isolated in moderate yield together with the intermediate uncyclized aldehyde S3a. [START_REF] Barroso | For a nice overview on carbohelicenes racemization[END_REF] After some experimentation, it was found that in presence of catalytic amounts of Pd(OAc)2 and triphenylphosphine, with an excess of K2CO3 (4 equiv.), the desired mesoionic compound 3a could be obtained in 88% yield. Similarly, the tetracyclic ortho-sydnone 3b could be isolated in 66% yield starting from the corresponding 1-bromo-2-naphthaldehyde (Figure 2A).

With a reliable access to the prerequisite ortho-fused aromatic azomethine imines 3a and 3b secured, we turned our attention to the key 1,3-dipolar-cycloaddition between polycyclic sydnones and aryne precursors. The sydnone-aryne cycloaddition is a 50 year old transformation pioneered by Gotthardt, Huisgen, and Knorr and later by Kato and Tsuge in 1974, 12 but only recently the synthetic potential of this transformation has been studied in detail. 13 At first, we reacted sydnones 3a and 3b in presence of silyl triflate 4a (1.5 equiv.) and TBAF (1.5 equiv.) at room temperature. As expected, the desired products 5 and 6 were isolated in good yields, 77 and 70%, respectively (Figure 2C). It is worth mentioning that 5 and 6 Figure 2: A: One-pot synthesis of polycyclic sydnones 3a and 3b; B: aryne precursor; C: Synthesis of helical pyrazoles 5 and 6. D: Synthesis of helical pyrazoles 7a, 8a, 9a and 10a. * isomer ratio was measured by 1 H-NMR of the crude mixture.

have been synthesized in only two steps from readily available Nmethyl sydnone. [START_REF] Zhao | Synthesis of pyrido[1,2-b]indazoles via aryne[END_REF] The sequence could be extended to the orthofused 1,2-naphthyne precursor 4b (Figure 2D). In presence of tricyclic sydnone 3a the formation of the two possible cycloadducts, the desired hetero-[5]-helicene 7a and the S-shaped product 7b, was observed. As expected, no degree of selectivity was achieved and the cycloadducts were isolated in a 1:1 ratio and an overall 74% yield. The tetracylic sydnone 3b gave a low degree of selectivity in favor of the sterically hindered hetero- [6]-helicene 8a (8a/8b, ratio 63/37). The 1 H NMR analysis of the two regioisomers showed well-resolved signals, which could be clearly assigned with COSY and NOESY measurements. [START_REF] Barroso | For a nice overview on carbohelicenes racemization[END_REF] Intrigued by this unexpected result, we investigated the reactivity of ortho-sydnones 3a and 3b in presence of 3,4-phenanthryne precursor 4c (Figure 2D). This ortho-annulated aryne is particularly interesting because it should allow the formation of [6]-and [7]-helicenes. The reaction of 3a with 4c in presence of TBAF in THF delivered a crude mixture with a useful selectivity in favor of the helical product (crude 1 H-NMR ratio 90:10). After purification, the two components of the reaction could be isolated in 79% and 6% yield. The 2D-NMR measurements determined the identity of the major compound as the desired [6]-helical pyrazole 9a. 15 When 4c was reacted in presence of 3b, [7]-helicene 10a was isolated in 62% yield with high selectivity (10a:10b 96:4). Crystals of both isomers 10a and 10b were grown by slowly evaporating dichloromethane solutions and the structures were determined by single-crystal X-ray diffraction (Figure 3A).

The chiroptical properties of [7]-helicene 10a were subjected to a preliminary evaluation. Enantiomers of 10a were resolved from a racemic mixture using HPLC with a chiral-phase column. 16 In Figure 3B are shown the circular dichrograms of the enantiopure samples, which exhibit both positive and negative Cotton effects at 235 and 357 nm. The spectra of the enantiomers are mirror images of each other. The measurements conducted in degassed dichloromethane for the set of enantiomers led to the observation of a mirror-image CPL signal (Figure 3C). The glum values are -0.001 and +0.001 at about the maximum emission wavelength for 10a. 17 These values are of the same order of magnitude as those for other examples of organic CPL-active helicenes. 18 These results confirm that the solution of [7]-helicene 10a in degassed dichloromethane exhibits active CPL signals, and also that the emitted light is polarized in opposite directions for the two enantiomeric forms for this helicene-like structure.

The selectivities observed for these sydnone-aryne cycloadditions were unexpected, given precedents in the literature. First, sydnones are known to be poorly regioselective in their cycloadditions with asymmetrical alkyne dipolarophiles. 13,[START_REF]85/15[END_REF][20][START_REF] Mcmahon | Generation and regioselective trapping of a 3,4piperidyne for the synthesis of functionalized heterocycles[END_REF][START_REF] Shah | Expanding the strained alkyne toolbox: generation and utility of oxygen-containing strained alkynes[END_REF][START_REF] Noteworthy | Discovery of new mutually orthogonal bioorthogonal cycloaddition pairs through computational screening[END_REF] Moreover, the Houk-Garg distortion-based model to explain the preferred regioselectivity of attack of nucleophiles on strained alkynes, 24 which is based on the difference in internal angles of the alkyne carbons, predicts low selectivity with 1,2-naphthyne or 3,4-phenanthryne, albeit in the observed direction. DFT optimizations of the two structures (M06-2X/6-31+G(d,p)) [START_REF] Barroso | For a nice overview on carbohelicenes racemization[END_REF] reveal that the two aryne carbons have similar internal bond angles (Figure 4A). In order to highlight whether the peculiar structure of sydnones 3a and 3b is responsible for the unusual selectivity, sydnone 3c was synthesized. As shown in Figure 4B, when N-phenyl sydnone 3c was reacted with 4c the opposite selectivity was observed (ratio 33:66 in favor of 11b). [START_REF] Krenske | The nature of pyrazoles 11a and 11b was unambiguously established by NOE and X-ray crystallography (see SI for details). (26)[END_REF] This result suggests that the origin of the selectivity might be related to the structure of the polycyclic sydnone itself. To understand the origins of such a dichotomy, we calculated the free energy profiles for the reactions of 3a-3c with both 1,2-naphthyne and 3,4-phenanthryne, using the same DFT method described above. Profiles for the reaction of 3a with 3,4-phenanthryne are shown in Figure 4C. [START_REF] Barroso | For a nice overview on carbohelicenes racemization[END_REF] In all cases examined, the regiodetermining cycloaddition step is also rate-determining and fully irreversible, as the formation of the intermediate cycloadduct (int) is highly exergonic (ΔG = -33 to -58 kcal/mol). The CO2 extrusion step (TS 2) is then extremely facile and once again very exergonic. All cycloaddition transition states (TS 1) have very low activation barriers (ΔG ‡ between 9 and 12 kcal/mol). Our calculations also quantitatively reproduce the experimental selectivities, that is formation of the helical regioisomer is predicted to be favored for sydnones 3a and 3b, but unfavored for 3c.

Figure 4: A) DFT-optimized structures of polycyclic arynes derived from 4b and 4c. B) Reaction between non planar N-phenyl sydnone 3c and 3,4-phenanthryne precursor 4c. * isomer ratio was measured by 1 H-NMR of the crude mixture. C) Energy profile for the reaction of 3a with 3,4-phenanthryne to form 9a or 9b. Free energies (enthalpies) are in kcal/mol and were obtained at the M06-2X/6-311+G(2d,2p)/SMD (THF) // M06-2X/6-31+G(d,p) level of theory. D) Cycloaddition TSs leading to regioisomers 9a and 9b. M06-2X/6-311+G(2d,2p)/SMD (THF) // M06-2X/6-31+G (d,p). Free energies in kcal/mol. The TSs leading to the two regioisomers for the representative reaction of 3a with 3,4-phenanthryne are shown in Figure 4D. The TSs for the other pairs of reactants can be found in the SI and display similar arrangements. First, the TS leading to the major helical isomer is more asynchronous, with the shorter forming bond being between the more nucleophilic terminus of the azomethine imine of the sydnone and the more electrophilic (linear) carbon of the aryne. 26 Indeed, for the fused sydnones 3a and 3b, the nitrogen atom bears more HOMO character than the carbon, while for N-phenyl sydnone 3c, the opposite is found. Second, the forming bonds are fairly long, indicative of very early TSs, which are in accord with the high exergonicity of the cycloaddition steps and the Hammond postulate. Third, the major TS seems to benefit from stabilizing C-H•••π interactions (also called face-to-edge π-π interactions) between the polycyclic backbones of the reactants, while the minor TS does not. 27 Distortion/interaction analysis 28 confirms this behavior: as the TSs are so early, the reactants are barely distorted from their groundstate geometries, and total distortion energies are, at most, 4.0 kcal/mol. As such, even though the helical regioisomer of the cycloadducts and pyrazoles is more sterically-encumbered and always higher in energy than the S-shape isomer, this effect is not important in TS 1 since the reactants are still far from each other. Conversely, interaction energies range from -7.4 to -12.4 kcal/mol, and are greater for the helical vs the S-shape regioisomer. In fact, for the six systems studied, the activation energies correlate with interaction energies, but have no correlation with distortion energies. [START_REF] Barroso | For a nice overview on carbohelicenes racemization[END_REF] To confirm the stabilization offered by the dispersive aromatic-aromatic interactions, we computed the binding energies of aromatic dimers in the same geometries as the helical TSs. [START_REF] Barroso | For a nice overview on carbohelicenes racemization[END_REF] For the four combinations evaluated, the binding energies were between -0.4 to -1.9 kcal/mol, demonstrating the stabilization offered by the C-H•••π interactions for the helical TSs, in addition to the more favorable primary orbital interactions. Thus, the TSs that benefit from the most interactions are also earlier, further lowering the cost to distort the reactants. These results indicate that with other very reactive partners, regioselective cycloadditions might be possible when interactions with polycyclic backbones are present. Indeed, when substituted sydnones 3d-3l were reacted with 4c similar values of regioselectivity (> 85/15) were observed in favor of the and the corresponding [6] and [7]-helicenes 12a-20a were isolated in 49 to 76% yields (Figure 5). While the presence of electro-neutral and -withdrawing substituents on the sydnone does not affect the transformation, di-substituted electron-rich dipole 3h was poorly reactive and the desired [6]-helicene 16a was isolated in 18% yield. [START_REF]Unreacted sydnone 3h was preset in the reaction crude even after prolonged reaction time[END_REF] Derivative 20a with a the presence of a chloride substituent offered a useful handle for further functionalization. Under catalytic conditions, the products of Sukuzi cross-coupling reaction 21a and 22a were isolated in 96 and 56% yield. This preliminary proof-of-concept showed the possibility to further functionalize this helical scaffold by means of metal catalysis and could be of interest for potential applications.

In summary, we have developed a method to access [4],[5], [6] and [7]-helicenes containing pyrazoles through sydnone 1,3-dipolar cycloadditions. This process involves the design and synthesis of ortho-substituted polyaromatic sydnones, which are more nucleophilic than conventional ones, and highlights the first example of regioselective cycloaddition of such mesoionic dipoles with aryne derivatives. Calculations showed that primary orbital interactions and C-H•••π dispersive interactions control the regioselectivity of this transformation. This reaction will ultimately provide a modular access to substituted derivatives, and could be amenable to the synthesis of other helicenes families.

ASSOCIATED CONTENT Supporting Information

The supporting information is available free of charge via the Internet at http://pubs.acs.org. Experimental procedures and computational details NMR spectra for obtained compounds Crystallographic data for compounds 10a (CCDC 1544556), 10b (CCDC 1872990), and 11b (CCDC 1544557).

Figure 1 :

 1 Figure 1: Design of pyrazoles-based helicenes.

Figure 3 :

 3 Figure 3: A) Molecular structure of 10a: I) top and II) side views. Molecular structure of 10b: III) top and IV) side views. Ellipsoids are set at 40% probability; hydrogen atoms are omitted for clarity. B) UV (gray curve) and CD spectra of (+)-10a (black curve) and (-)-10a (red curve); C) CPL (upper curve) and total luminescence (lower curves) spectra of (+)-10a (black curve) and (-)-10a (red curve) in degassed dichloromethane at 295 K, upon excitation at 430 nm.

Figure 5 :

 5 Figure 5: Synthesis of helical pyrazoles 12a-22a. * isomer ratio was measured by 1 H-NMR of the crude mixture. n.i. not isolate. a See supporting information for detailed conditions.

ACKNOWLEDGMENT

This work was supported by CEA and ANR (ANR-17-CE07-0035-01) and the US National Science Foundation (CHE-1361104). P.A.C gratefully acknowledges the Fonds de recherche du Québec -Nature et Technologies for a postdoctoral fellowship. Computations were performed on the Hoffman2 cluster at UCLA. G.M. acknowledges the NIH Minority Biomedical Research Support (grant 1 SC3 GM089589-08) and the Henry Dreyfus Teacher-Scholar Award for financial supports. The authors thank C. Chollet, E. Marcon, A. Goudet, S. Lebrequier and D.-A. Buisson (DRF-JOLIOT-SCBM, CEA) for the excellent analytical support. We wish to acknowledge Dr. P. Dauban (ICSN, CNRS, France) for helpful discussion.

AUTHOR INFORMATIONS

Corresponding Author * davide.audisio@cea.fr

Notes

The authors declare no competing financial interests.


C E Munson

Q Gaimard

Kamel Merghem

S Sundaram

J Rogers

Jean-Paul Ramdane

A Salvestrini

Ougazzaden

D J Rogers

J De Sanoit

P L Voss

A Ramdane

J P Salvestrini

A Ougazzaden
email: abdallah.ougazzaden@georgiatech-metz.fr

Modeling, design, fabrication and experimentation of a GaN-based, 63 Ni betavoltaic battery

Keywords: betavoltaic, battery, GaN, 63 Ni

published or not. The documents may come

Modeling, design, fabrication and experimentation of a

GaN-based, 63Ni betavoltaic battery

Introduction

Numerous elds require long-life power sources on the microor milliwatt scale. Although lithium-ion batteries currently respond to many of the requirements, the technology only has a functional lifespan on the order of a few years and such batteries require frequent recharging. When replacement/ recharging is excessively expensive or impractical (such as in pacemakers, undersea sensing electronics and space applications), however, another option is required and betavoltaic batteries (which convert beta particle energy into electricity by means of radiation-resistant p-n junctions) have long been considered an attractive option in that they do not need recharging and they have a lifespan in excess of a few decades. Unfortunately, however, practical fabrication of betavoltaics with adequate power outputs has proven dif cult up till now.

Tritium ( 3 H), 63 Ni and 147 Pm are the most common radionuclei considered for use as sources in betavoltaic batteries. They offer (respectively) average beta particle energies of 5.6, 17.4 and 224.1 keV and half-lives of 12.3, 100 and 2.6 years [START_REF] Ulmen | Development of diode junction nuclear battery using Ni-63[END_REF]. The relatively low beta particle energies released from 3 H and 63 Ni, in particular, make them more suitable for use in consumer-grade batteries than 147 Pm since the beta particles can be selectively blocked by only a few micrometers of shielding and they can thus be safely and easily packaged. Amongst these two radionuclei, 63 Ni has a superior half-life, which makes it a better choice as a long-term energy source and it is, therefore, the source considered in this paper.

Previous studies on the conversion of 63 Ni beta particle energy into electricity by means of GaN-based p-n junctions reported output powers from 23 pW to 0.57 nW [START_REF] Kuruoglu | Betavoltaic study of a GaN p-i-n structure grown by metal-organic vapour phase epitaxy with a Ni-63 source[END_REF][START_REF] Cheng | High open-circuit voltage betavoltaic cell based on GaN pin homojunction[END_REF][START_REF] Cheng | A high open-circuit voltage gallium nitride betavoltaic microbattery[END_REF][START_REF] Cheng | Demonstration of a high open-circuit voltage GaN betavoltaic microbattery[END_REF]. Note that, in practice, the power output is typically lower than theor etical expectations, which are often above 2 nW [START_REF] San | Design and simulation of gan based schottky betavoltaic nuclear microbattery[END_REF]. Similar experiments with silicon and SiC based devices using 63 Ni achieved output powers of around 15.4 pW [START_REF] Chandrashekhar | Demonstration of a 4H SiC betavoltaic cell[END_REF] and 0.34 nW [START_REF] Gao | Demonstration, radiation tolerance and design on a betavoltaic micropower[END_REF], though some silicon designs approached 9 nW [START_REF] Wang | The effect of temperature changes on electrical performance of the betavoltaic cell[END_REF].

In this paper, a new state-of-the-art betavoltaic consisting of a p-GaN / i-GaN / n-GaN PIN structure on top of which a thin 63 Ni layer is deposited, is fabricated and tested. Preliminary modeling for such a structure was introduced in a previous paper published by our group, along with a series and shunt resistance (SSR) model [START_REF] Munson | Model of Ni-63 battery with realistic PIN structure[END_REF]. Similarly, there has also been MCNP-based simulations created to estimate the performance of 147 Pm sandwiched between p-n junction GaN layers [START_REF] Tang | Optimization design of gan betavoltaic microbattery[END_REF].

In this paper, more re ned versions of these models (for the 63 Ni betavoltaic absorption and electrical circuit) used to design the betavoltaic GaN PIN device along with its fabrication process, and the device performance, are presented. The establishment of a reusable betavoltaic cell test platform which was developed to conform with radiation safety norms is also reported.

Model

In the following, a 63 Ni absorption model is developed which takes into account both self-absorption and beta particle trajectory incidence angle. The SSR model for the PIN structure, (developed in previous work [START_REF] Munson | Model of Ni-63 battery with realistic PIN structure[END_REF]) is also outlined.

63 Ni absorption model

Monte Carlo (MC) simulations [START_REF] Ulmen | Development of diode junction nuclear battery using Ni-63[END_REF] were used to estimate the beta particle energy deposited into the GaN based on a xed surface area (1 cm 2 ) of 63 Ni. The increase in activity and emitted power with 63 Ni thickness, was investigated in order to take into account the interplay of self-absorption, incidence angle and beta particle energy. A thickness range of 0 to 4000 nm was used for 63 Ni so that saturation effects could be studied for both thick and thin layers. The simulation results yield the proportion of emitted beta particle energy that makes it into the GaN material (denoted as N(y, d, e, θ), in nm). These are then be combined to form an overall 63 Ni model. This model considers re ectance from the device surface and all angles of incidence (equation 1), for an energy emittance pro le mimicking that of 63 Ni (equation 2). It is also valid for various 63 Ni thicknesses as chosen by equation 3:

α θ = π θ=0 N(y, d, e, θ) × (1 -R θ ) × ∆θ (1
)

α e = 60 keV e=5 keV α θ (y, d, e) × Pr(E = e) × ∆e (2)
α T (y, t) = t d=0 α e (y, d) × ∆d (3)
where R θ is the percent re ectance at a given angle θ, Pr(E = e) is the probability of an electron being emitted from 63 Ni at the given energy e (eV), y is the y-coordinate of depth into the device (nm) and t is the total 63 Ni thickness (nm). Equation 4 provides an estimation for the total amount of energy P nm emitted from a 1 cm 2 × 1 nm volume of 63 Ni, not considering self-absorption (which is taken into account in the simulation data, N):

P nm ∼ ρ × A 10 7 × E avg × q ( W cm 2 × nm ) ( 4
)
where ρ is the density of 63 Ni (8.91 g cm 3 ) and A is the speci c activity of 63 Ni per gram (2.2 × 10 12 Bq/g) [START_REF] Cheng | A high open-circuit voltage gallium nitride betavoltaic microbattery[END_REF], E avg is the average energy released from 63 Ni (17 keV) and q is the electron charge. The results of these simulations, for a p-GaN / i-GaN / n-GaN PIN structure of 270 nm, 200 nm and 1 µm, respectively, can be seen in gure 1, where we show the absorption pro les of beta particles in GaN for various 63 Ni thicknesses. For very thin layers of 63 Ni the GaN absorption is more pronounced, while for thick 63 Ni layers the pro le smoothes out to a more impeded, exponential absorption curve. This is possibly due to self-absorption as the 63 Ni source becomes thicker.

Finally, gure 2 shows that the power absorbed in the GaN material saturates towards 2.85 µW cm -2 as the 63 Ni thickness increases. Assuming an unlimited device size, the optimal 63 Ni thickness is in nitely thin, since this would result in the least amount of self-absorption. It can be estimated that the optimal 63 Ni thickness (for a 1 cm 2 sized device) is around 1500 nm of 63 Ni, since the absorbed power quickly begins to saturate after this point. These results match well with results published in literature, which demonstrate a similar saturation towards 2.85 µW cm -2 with increasing 63 Ni thickness [START_REF] Honsberg | GaN betavoltaic energy converters Conf[END_REF].

Device model and series-shunt resistance model

The device modeling software used for this paper was Silvaco, although any device modeling software could be used. The SSR models were used in order to take into account the nonideal behavior of the PIN diode and contacts, which consider both recombination effects and defects introduced by the device, as well as imperfections due to contacts. Taking into account these series and shunt resistances, which can be determined experimentally or estimated based on similar devices, the current and voltage of the PIN diode can be rewritten as:

V = I d × R s + 1 + R s R sh × V d ( 5
)
I = I d + V d R sh (6)
where R s is the total series resistance, R sh the total shunt resistance, and I d and V d are, respectively, the current and voltage obtained using a Silvaco simulation of the ideal PIN structure.

Design, material growth, processing and packaging

Device design

The device design was governed by the total amount of 63 Ni activity available. A devices consisted of a p-GaN/i-GaN/n-GaN PIN structure on top of which a thin 63 Ni layer was placed. As it has been seen before, the optimal 63 Ni thickness, for a 1 cm 2 sized device, is around 1500 nm. This thickness corresponds to around 1.2 GBq of activity after electrodeposition (see subsection material growth). This means that six devices with 1 cm 2 in size could be realized. This number of devices is reasonable knowing that the nal mounting of the devices will be done in a sealed glovebox leading to a nonnegligible risk of having failed devices at the end (see subsection packaging). Considering a realistic p-type doping of 5 × 10 17 cm -3 and an unintentional intrinsic doping of 1 × 10 17 cm -3 , which are the doping levels we were able to achieve with our equipment during this experiment, this would result in an electric eld that spanned around 200-300 nm of the intrinsic region. Taking this into account, gure 3(a) shows the chosen device design which includes a 40 nm current spreading layer on top of the device (20 nm of gold and 20 nm of palladium) to form the p-contact. This is followed by a 270 nm thick p-GaN region and a 200 nm thick i-GaN region. The rest of the device consists of 1 µm thick n-type GaN. Figure 3(b) shows a sketch of the device after electrical contact processing. The current spreading layer allows for better carrier collection even if the lower electron energies (less than 5 keV) are almost entirely absorbed within it (see gure 4). The p-GaN layer absorbs electron energies around 8 keV, whereas the intrinsic GaN region absorbs electron energies around the 63 Ni emission peak (14 keV). Finally, most electrons having energies above 22 keV are absorbed in the n-GaN region and substrate.

The power conversion ef ciency of a perfect PIN device can be estimated as:

PCE(%) = η tr × η sp × η i × η be (7)
where η tr is the amount of energy that is not backscattered from the surface of the device (MC simulations indicate that 29% of the beta particle energy should be backscattered), η sp is the amount of energy that is transmitted through the spreading layer (9% of the beta particle energy is absorbed in the metallic spreading layer, again based on MC simulation), η i is the amount of electron energy falling within the i-GaN region (12% of the beta particle energy is absorbed in the i-GaN region, again based on MC simulation as seen in gure 1) and η be is the maximum betavoltaic current efciency for GaN [START_REF] Olsen | Review of betavoltaic energy conversion NASA Conf[END_REF] which is 27%. This leads to a highest possible PCE of around 2.09%. 3.2.2. 63 Ni deposition. 63 Ni was synthesized using neutron and gamma ray bombardment of a 62 Ni target followed by an acid dissolution and radiochemical separation in order to obtain 63 NiCl 2 radioactive salt. A modi ed Watts bath was then prepared using 63 NiCl 2 , H 2 O, Na 2 SO 4 and H 3 BO 3 . The 63 Ni material was then electrodeposited onto the two sides of a piece of copper foil approximately 1 cm 2 in surface area with 140 µm thick copper foil, which absorbed all energy that was emitted away from the devices, and using a Pt anode (see gure 5). The deposition yield was approximately 80% leading to 2.4 GBq of activity deposited on the two facets of the copper foil for each source. Assuming a 17 keV average beta particle energy, this equates to approximately 6.545 µW of energy emitted (i.e. 3.27 µW emitted from each side of the source).

Device processing

Three metallic patterns were de ned by lift-off using contact photolithography. A thin spreading bilayer of palladium and gold (20 nm / 20 nm) was rst deposited as a p-contact on the GaN p + type top layer, over an area of 8 × 8 mm 2 . A contact pad, made of palladium and gold bilayer (20 nm / 200 nm), was subsequently deposited on the side of this spreading layer to allow for wire bonding. The mesa was then dry etched, to a depth of 1 µm, so as to reach the n-type layer. This step was achieved using BCl 3 /Cl 2 /N 2 inductively coupled plasma reactive-ion etching (RIE), through a SiO 2 mask. The n-contact consisted of titanium / aluminum / gold (10 nm / 30 nm / 300 nm) deposition. To ensure good ohmic contacts, the diodes were subsequently rapid thermal annealed under N 2 atmosphere. A photograph of a fabricated device is shown in gure 6(a). The nal steps involved dicing the diodes, mounting on speci c holders and wire bonding (see gure 6(b)).

Packaging

In order to facilitate radioactive 63 Ni testing within a con ned glovebox, a 3D printed sample holder was developed ( gure 7). This holder offered the options of both reusability and ne tuneability of the device/source distance using a manual screw mechanism (as seen in gure 7(b)). Notice here that the 63 Ni holder can be slid in and out of the testing box via the tray ( gure 7(c), top right). It can be seen from gure 7 that each testing box was conceived to house two betavoltaic devices forming a sandwitch around the 63 Ni source. There is also a turnkey screw mechanism which allows us to vary the source/betavoltaic distance, and thus the power, delivered to the devices. A dagger insertion mechanism provides for easy loading/extraction of both the devices and the 63 Ni source, while preventing damage to the wire bonding upon loading. The testing box is equipped with 8 electrical connectors (4 per device) allowing the use of the 4-point probe method for I-V measurements.

Results

A total of nine 1 × 1 cm 2 , GaN-based PIN devices and three 63 Ni sources were fabricated. Hall measurements revealed p-type and n-type doping levels of around 1 × 10 17 cm -3 and 5 × 10 18 cm -3 , respectively. SIMS revealed the doping depth pro les to have relatively abrupt steps at the interfaces. The PIN devices exhibited a turn-on voltage of around 3.5 V and a leakage current on the order of 3-70 µA under a bias voltage of -5 V (this current was relatively high because of the comparatively large size of the device). A typical I-V plot of these devices is shown in gure 8.

Electron beam simulation

Before experimenting with actual 63 Ni, some preliminary experiments were performed in an SEM using an 80 µW beam power and a 17 keV accelerating voltage (so as to simulate the typical 63 Ni beta particle energy). Figure 9 shows both the dark and electron beam IV measurements for two of the devices, which are typical of the devices having high and medium current leakage, respectively. For both of them, the short-circuit electron beam induced currents (EBIC) are similar and approximately equal to 3.5-4 µA, whereas the open circuit voltages (V oc ) are quite different and equal to 150 mV and 1 V for the devices having high and medium current leakage, respectively. The inset depicts a typical EBIC image recorded during these experiments, showing a very uniform carrier collection ef ciency and thus betacurrent generation throughout the entire device. Since the EBIC signal is proportional to the generated current, the EBIC images did not vary signi cantly between the different devices.

Finally, gure 10 shows the collected current under 80 µW of e-beam irradiation, both through the metallic current spreading contact layer (green line) and directly on the p-GaN eld (without any metallization, blue line). The difference between the two curves can be attributed to the absorption of the impinging electrons in the metallic current spreading contact layer or backscattering at a higher amount, which leads to less generated electron-hole pairs in the PIN junction. Thus, the beam current losses in the metallic layer lead to a decrease by around 21% of both the short circuit current and open circuit voltage.

63 Ni experimental data

Among the nine fabricated devices, two were damaged during the mounting in the testing boxes. Among the seven devices that were left, two of them were placed closer to the 63 Ni source (around 0.5 mm gap) and the other ve were placed at around 5 mm from the 63 Ni source (wire bonding prevented closer positioning). Table 1 summarizes the results obtained for the different devices in terms of short circuit current (I sc ) and V oc . For all of the devices, the results weakly depended on the box in which the PIN device, and thus 63 Ni source, was mounted. It is clear that both the distance to the 63 Ni source and the current leakage value affected the performance of the device. As seen in the previous subsection, I sc is only slightly lower for the high leakage current devices (average I sc of 1.15 ± 0.2 nA) than for the medium leakage current devices (average I sc of 1.4 ± 0.2 nA), whereas, a one order of magnitude lower V oc was expected (see gure 11). Additionally, the I sc for devices placed closer to the 63 Ni source (3.05 ± 0.1 nA) is much higher than that obtained for devices positioned 4 mm (9 times) further away. The beta particles counted versus absorber thickness (air in this case), can be estimated by N(x) = N 0 × e - (µB×ρ×x) where N 0 is the initial beta particles counted (at zero absorber thickness), µ B is the beta absorption coef cient (which is 2071.16 cm 2 g -1 for air) and ρ is the density of the absorber (1.205 × 10 -3 g cm-3 for air). This predicts losses of 12 and 68%, respectively, for gaps of 0.5 and 4.5 mm. Accordingly, an I sc of 3.05 nA for a device placed 0.5 mm from the source should drop to 0.98 nA for a device 4.5 mm away from the source. This is in very good agreement with the measured value of 1.15 ± 0.2 nA.

Particularly noteworthy are the two devices with medium leakage current, which had an I sc of 1.4 ± 0.2 nA, a V oc of 0.79 ± 0.01 V and a ll factor (FF) of 47% (see gure 11), corresponding to a generated power ( P = V oc × I sc × FF ) of 0.52 ± 0.08 nW and leading to a PCE of 0.016%. As shown in table 2, the generated power can be easily doubled by connecting the two devices in series (i.e. placing them on either side of the 63 Ni source). This leads to a device with a generated power of 1.04 ± 0.1 nW with the same PCE. Furthermore, if the two devices in series are placed closer to the source this leads to an I sc of 3 nA, and the generated power and PCE can be increased to 2.23 nW and 0.034% , respectively.

Model comparison

The model prepared in section 2 can now be used to compare the predicted device performance with these experimental values. The series device described at the end of the previous subsection (in which devices are positioned in proximity to each side of the 63 Ni source) was adopted for modeling. Figure 12 shows both the experimental and simulated (both ideal and SSR models) dark current-voltage curves for this device. The SSR model, with 7 Ω and 910 MΩ as series and shunt resistance, respectively, and an ideality of 3.6, ts the experimental results well. Similarly, gure 13 shows the IV curves under 63 Ni irradiation in order to highlight the betacurrents. The green dashed curve is based on the experimental results under 63 Ni irradiation (6.545 µW total). The solid blue line shows the SSR model predictions, and the dashed gray line shows the results directly from the ideal. Again, the SSR is in very good agreement with the experimental results, with only a minor variation in the V oc expectations (though still within 0.1 V). Table 3 shows these results more quantitatively. The ideal device model predicts a generated power of 68.45 µW cm -3 with a 6.545 µW 63 Ni input. After adding the SSR model we can see that this drops to 12.25 µW cm -3 , due to a large decrease of both V oc and FF. This is rather close to the experimental results, (which shows 9.9 µW cm -3 ), the difference being mostly due to the smaller FF.

Conclusion

This paper described the design, fabrication and experimental testing of 1 cm 2 GaN/ 63 Ni based betavoltaic batteries and a versatile testing box which allowed for reusability and tuning of device/source distance. The optimal devices constructed in these experiments achieved an output power of 0.52 nW, which could likely be extended to 1.04 nW if the boxes are run as-designed in series con guration. We additionally feel a simple reduction of the sample/source gap should allow 2.23 nW output, however. It was proposed that the ef ciency of the experimental device was limited predominantly due to the relatively thick p-GaN region (270 nm) and current spreading layers (40 nm), combined with a relatively thin i-GaN region (200 nm) having excessive residual doping. These issues can be improved in the future by optimizing the active layer thicknesses, replacing the i-GaN with dilute BGaN alloy (which allows a several order of magnitude in resistivity [START_REF] Srour | Solar blind metal-semiconductor-metal ultraviolet photodetectors using quasi-alloy of BGaN/GaN superlattices[END_REF][START_REF] Ravindran | Dual-purpose BGaN layers on performance of nitride-based high electron mobility transistors[END_REF]), and decreasing the thickness of the current spreading layer down to 20 nm or using interdigitated electrodes.

The paper also presented the development of a 63 Ni betavoltaic simulation model that can be quickly and easily generated with standard Monte Carlo simulation software (such as CASINO) but can still provide high-quality and accurate results that match well with literature. This model was also combined with the device and SSR models, and gave a very good t with the experimental data.

Figure 1 .

 1 Figure 1. GaN beta particle absorption pro les for various 63 Ni thicknesses. 63 Ni layer not shown.

Figure 2 .

 2 Figure 2. Power absorption ef ciency in GaN versus 63 Ni thickness [12].

Figure 3 .

 3 Figure 3. (a) Designed 1 cm 2 area device structure with 200 nm thick intrinsic region and (b) PIN structure with electrical contacts. (not to scale).

Figure 4 .

 4 Figure 4. Simulation of the beta particle energy absorption by each layer of the PIN structure (top) and the same energy absorption graph, but now scaled by the probability of beta particle emission (according to the particle energy) from 63 Ni material (bottom).Higher energies are less likely to be emitted from 63 Ni (and 17 keV is most likely); the bottom graph is designed to visually re ect this.

3. 2 .

 2 Material growth 3.2.1. GaN epitaxy. All the PIN structures were grown at 1030 °C using a T-shaped MOVPE reactor under a pressure of 100 Torr and a V/III ratio of 8000. Trimethylgallium and ammonia were used as precursors for gallium and nitrogen, respectively. SiH 4 and CP 2 Mg were used as n-type and p-type dopants, respectively. The growths were performed on 2 sapphire wafers, starting with 1.8 µm thick buffer of intrinsic GaN. The PIN structures were structurally and morphologically characterized with high resolution x-ray diffraction, scanning electron microscope (SEM) and atomic force microscope. The dopant concentrations and their pro les were characterized by Hall effect measurements and secondary ion mass spectroscopy (SIMS) analysis, respectively.

Figure 5 .

 5 Figure 5. Photographs of a 63 Ni layer electrodeposited on a copper foil (left) and mounted in the holder (right).

Figure 6 .

 6 Figure 6. Photographs of (a) a processed device, and (b) a device mounted on a printed circuit board and wire bonded.

Figure 7 .

 7 Figure 7. (a) Schematic and photographs of (b) open and (c) closed testing box, with turnkey screw mechanism for varying the betavoltaic to 63 Ni source distance.

Figure 8 .

 8 Figure 8. Typical I-V plot of the fabricated PIN devices.

Figure 9 .

 9 Figure 9. Dark and electron beam exposed IV measurements for two of the devices recorded with a 17 keV beam energy and an 80 µW power. Inset: typical EBIC image, showing consistent current generation throughout the device.

Figure 10 .

 10 Figure 10. e-beam generated current under 80 µW of irradiation, both through the current spreading layer and directly on the p-GaN eld.

Figure 11 .

 11 Figure 11. Current-voltage measurements recorded in two medium leakage current PIN devices placed at 5 mm away from the 63 Ni source.

a

  Deduced from experiments led on the single device and extrapolated to two devices mounted in series b Deduced from experiments led on the single device and extrapolated to two devices mounted in series and placed as close as possible to the source.

Figure 12 .

 12 Figure 12. Simulated versus experimental, dark IV curves with two 1 cm 2 PIN devices mounted in series.

Figure 13 .

 13 Figure 13. Simulated versus experimental, IV curves for irradiation with a 6.545 µW 63 Ni beta source (3.27 µW each PIN) and two 1 cm 2 PIN devices mounted in series.

Table 2 .

 2 Mounting dependence of the device performances.

		I sc	V oc	P	PCE
	Device mounting	(nA)	(V)	(nW)	(%)
	1 device	1.4	0.79 0.52	0.016
	2 in series a	1.4	1.58 1.04	0.016
	2 in series closer
	to the source b

Table 1 .

 1 Betacurrent experimental measurements for all four devices and with three different 63 Ni sources.

				I sc (nA)/V oc (mV)
		Distance	Box	Box
	Device leakage	to source	#1	#2	Box #3
	Medium	5 mm	1.4/800	N/A	1.25/780
	High	5 mm	0.95/53	1.35/65	1.1/57
	High	0.5 mm	3.1/∼0	3/∼0	N/A

Table 3 .

 3 Betavoltaic device performance under a 6.545 µW 63 Ni source and two 1 cm 2 PIN devices mounted in series.

		63 Ni model
		Ideal device	SSR model	Exp. data
	I sc (nA)	3.03	3.03	3
	V oc (V)	5.6	1.7	1.6
	FF (%)	90.9	53.6	46.9
	P (nW)	15.42	2.76	2.23
	P D (µW cm -3 )	68.45	12.25	9.9
	PCE (%)	0.23	0.042	0.034

Acknowledgments

This work was supported by the Agence Nationale de la Recherche of France as part of the BATGAN project (ANR-11-BS09-0038).

ORCID iDs

J P Salvestrini https://orcid.org/0000-0002-0482-1178