Alice Corbet

Isabelle Desportes

Depoliticizing

Keywords:

come

impacted populations?

The separation of humanitarian action from politics is one of the founding stances of humanitarianism.

It aims to maximize humanitarian independence and neutrality, thus increases acceptance and access, including in conflict settings. Yet, humanitarian claims of non-involvement in politics have been dismissed by some as misleading, naive, or counter-productive. Some political scientists have argued that depoliticization is just another form of politics.

Possible themes to be addressed include self-censorship dilemmas, the standardization, technicalization and bureaucratization of aid, and what it means to be 'neutral' when intervening in situations of protest and/or authoritarianism. We particularly welcome contributions that discuss the depoliticisation of aid in light of current humanitarian trends, such as localization and the expectations of less powerful civil society actors and populations in need, the increasingly blurred lines between humanitarianism, development and peace-building, or the neoliberalisation of aid and role of private sector entities.

The first panel slot (9-10.30 a.m.) will focus on case studies following an introduction to the topic, with paper presentations by Marina Sharpe, Jasmine Burnley on Myanmar and the Rohingya, and Andrew Cunningham on Belarus.

The second panel slot (11-12.30)

Abstracts :

The (Core) Humanitarian Principles: Content, Character and Contemporary Controversies Submitted by Marina Sharpe That all humanitarian action should be undertaken in accordance with the principles of humanity, impartiality, neutrality and independence has become a truism, which exempts the principles from fulsome critical scrutiny. What the humanitarian principles really mean and particularly where they come from is rarely addressed. This lacuna transcends academic disciplines and even affects practitioners. (For example, in the depiction of the 'core humanitarian standard' in the latest Sphere Handbook, 'humanity, impartiality, neutrality and independence' encircle beneficiaries, however the four principles are not otherwise discussed in the Handbook's 400 plus pages.) In scholarship, the lack of analysis is particularly pronounced in the (Anglophone) international law literature. There is little international law scholarship on the humanitarian principles, and seemingly none that addresses all four principles together in one work; the legal analyses that do exist focus on impartiality and neutrality, perhaps in part because the important judgement in Nicaragua addressed the former principle and both principles have autonomous international humanitarian law meanings. This paper responds to this gap by critically analyzing the historical origins, legal character and normative content of the humanitarian principles. It then invokes this analysis to demonstrate how stressing the principles' legal character is one way to depoliticise them and to frame discussion of whether such depoliticization even desirable.

Norms and ideas in the Rohingya Internally Displaced Persons Crisis

Submitted by Jasmine Burnley

The humanitarian sector has evolved in ways that have changed its size, significance, agents and subjects: its deepening complexity and the emerging commitment to bridge the divide between humanitarianism and development have created a heavily populated ecosystem of diverse actors, and blurred the boundaries of who participates in a response. But despite these transformations, humanitarianism continues to be governed ostensibly by the humanitarian principles, which promise some form of insulation from politics. The residue of this discrepancy and depoliticization, is a disconnect between how humanitarian responses are expected to operate, and their day to day political realities, cut through with multiple assumptions around what works and distinct ideas about what a crisis is. This research examines the case of the Rohingya Internally Displaced Persons (IDP) crisis and finds that normative interpretations of the principles as well as powerful ideas around the construction of the crisis played a significant role in determining how the response was shaped, contributing to the depoliticization of human rights violations; and preventing the international community from recognising the overtly political project of discrimination orchestrated by authorities, with implications for its ability to strategically coordinate or advocate to Government. This case raises questions for how interpretations of the principles intersect with political realities on the ground, and of the power of ideas in formulating and typecasting crises in ways that depoliticise and crowd out non-conforming evidence for why an emergency has occurred and what may be sustaining it.

How do INGOs approach depoliticizing aid in politic ized environments? Case study of

Belarus and TB

Submitted by Andrew Cunningham

A recurrent theme for the few international humanitarian NGOs working in the Former Soviet Union involves how to 'depoliticize' themselves within highly political contexts. One of the most difficult cases is Belarus, where the issue of complicity is a major question as the government leans towards increased authoritarianism.

There is a sense by many that working on a technical-medical issue such as TB treatment is somehow 'safe' from political manipulation. But in a context like Belarus, is this possible? Isn't the mere presence of an INGO, and one which works very closely with the regime, in itself a political statement or act? If it were possible to depoliticize the organization's actions in such a context, how would this work?

This paper explores how INGOs go about discussing and debating the issue of complicity, and the role the concept of depoliticization plays in these debates. A triangle is imagined --between the organization's mandate to provide assistance, the agenda of the government, and the medical requirements of the disease. An interesting question is --what role does the disease itself play in the depoliticization process? The focus is only the role of medical programming in the depoliticization process.

When neoliberal rationality transforms humanitarianism into an ideology that depoliticizes

Submitted by Bertrand Bréqueville

The research question is: how does the impregnation of neoliberal rationality within humanitarianism make it a vector of depoliticization?

The intellectual environment of the 1970s -based on anti-totalitarianism -allowed humanitarianism to take off and neoliberalism to become hegemonic concomitantly. There is a historical relationship between the two. Humanitarian NGOs are increasingly permeable to the influences and interests of neoliberalism.

We analyse different situations: those bringing social rights into a humanitarian logic, those inducing an uninhibited relationship with the profit sector, and those imposing on humanitarian NGOs concepts that neoliberalism has appropriated and shaped (social protection, resilience, empowerment…).

Resilience is a good example of a concept, seemingly neutral, but which tends to depoliticize.

Resilience refuses to see the systemic nature of crises. It places responsibility for their own situation on the people concerned.

Humanitarianism is even more easily influenced as its doctrine, based on neutrality, is a factor of inertia. In contact with neoliberalism, the humanitarian doctrine turns into ideology in the pejorative sense of the term.

Humanitarian actors may well present their interventions as "not political". The main problem is the depoliticization that humanitarianism itself generates in different ways.

It is time for the humanitarian sector to question the underlying nature of its relationship to neoliberalism.

Bibliography: Bertrand Bréqueville, L'humanitaire sous l'emprise du néolibéralisme, Editions Charles

Leopold Mayer

How expert new media depoliticize disasters involving Indigenous peoples

Submitted by Anuszka Mosurska

Attempts to shift the ways disasters have traditionally been managed away from authoritarian, topdown approaches toward more bottom-up and inclusive processes involve viewpoints from marginalized groups. Therefore, there have been calls for greater inclusion of Indigenous peoples in disaster management. In theory, this suggests a shift in power structures, where the role of experts is contested. However, in popular imagination Indigenous peoples are often misrepresented, for instance through tropes of Indigenous peoples as custodians of the environment or especially vulnerable to environmental change. These matter because they can result in policies and practices in disaster risk reduction that do not capture the complex realities of Indigenous peoples' lives. To understand these. we present a critical discourse analysis of the news media from 2015-2020 focusing on how Indigenous peoples in disasters are represented. We find a principal framing of disasters as natural phenomena that are addressed through humanitarianism and technocratic interventions, which are justified by framing governments and NGOs as caring for Indigenous peoples. This renders Indigenous peoples helpless and depoliticizes disasters. However, we also identify weaker, competing discourses that focus on systems of oppression and self-determination. These frame disasters as political by discussing the role of colonialism in disaster creation. political support for the protesters and adherence to humanitarian and medical ethics, giving a new interpretation to the merging of politics and care provision in humanitarianism. This merger, which I have coined midani humanitarianism, after the field workers on the squares (Midan/Maidan), which are often the centre of protests, is recognisable across a wide array of settings in recent antigovernment protests. In doing so, it elaborates the coming about of a humanitarian ethic that is developed from the ground up, with a more flexible approach towards the overt incorporation of politics within humanitarian practice.

  approaches the topic of depoliticisation through thematic angles: neoliberalism with a presentation by Bertrand Bréqueville, inclusion of indigenous people in disaster management with a presentation by Anuszka Mosurska, to finally open up with possible alternatives to depoliticisation, with a presentation by Sophie Roborgh on ´humanitarians of the revolution´. The (Core) Humanitarian Principles: Content, Character and Contemporary Controversies: Marina Sharpe ; Norms and ideas in the Rohingya Internally Displaced Persons Crisis: Jasmine Burnley ; How do INGOs approach depoliticizing aid in politicized environments? Case study of Belarus and TB Author(s): Andrew Cunningham ; When neoliberal rationality transforms humanitarianism into an ideology that depoliticizes: Bertrand Bréqueville ; Humanitarians of the revolution -Politics and principles in grassroots medical humanitarianism in protest settings: Sophie Roborgh ; How expert new media depoliticize disasters involving Indigenous peoples: Anuszka Mosurska.

  Humanitarians of the revolution -Politics and principles in grassroots medical humanitarianism in protest settings Submitted by Sophie Roborgh This paper engages with the question of neutrality among grassroots medical humanitarians in protest environments. It explores medical volunteers' conceptualisation of their engagement and the nature of their activities in numerous settings subjected to recent protests, including in Chile to Myanmar, Belarus, Sudan, and Hong Kong. The paper explains how volunteers reconcile their (often explicit)


Hao Zhou

Ricardo A Lebensohn

Péter Reischig

Wolfgang Ludwig

Kaushik Bhattacharya

Imposing equilibrium on measured 3-D stress fields using Hodge decomposition and FFT-based optimization

Keywords: X-ray diffraction, experimental mechanics, stress equilibrium, Hodge decomposition, fast Fourier transform, micromechanics

We present a methodology to impose micromechanical constraints, i.e. stress equilibrium at grain and sub-grain scale, to an arbitrary (non-equilibrated) voxelized stress field obtained, for example, by means of synchrotron X-ray diffraction techniques. The method consists in finding the equilibrated stress field closest (in L 2 -norm sense) to the measured non-equilibrated stress field, via the solution of an optimization problem. The extraction of the divergence-free (equilibrated) part of a general (non-equilibrated) field is performed using the Hodge decomposition of a symmetric matrix field, which is the generalization of the Helmholtz decomposition of a vector field into the sum of an irrotational field and a solenoidal field. The combination of: a) the Euler-Lagrange equations that solve the optimization problem, and b) the Hodge decomposition, gives a differential expression that contains the bi-harmonic operator and two times the curl operator acting on the measured stress field. These high-order derivatives can be efficiently performed in Fourier space. The method is applied to filter the non-equilibrated parts of a synthetic piecewise constant stress fields with a known ground truth, and stress fields in Gum Metal, a beta-Ti-based alloy measured in-situ using Diffraction Contrast Tomography (DCT). In both cases, the largest corrections were obtained near grain boundaries.

Introduction

Emerging non-destructive X-ray diffraction (XRD) techniques for 3-D material characterization performed at light sources, collectively known as 3DXRD (e.g. [START_REF] Henning Friis Poulsen | Three-dimensional X-ray diffraction microscopy: mapping polycrystals and their dynamics[END_REF][START_REF] Ludwig | Three-dimensional grain mapping by x-ray diffraction contrast tomography and the use of friedel pairs in diffraction data analysis[END_REF][START_REF] Lienert | High-energy diffraction microscopy at the advanced photon source[END_REF]), provide previously inaccessible in-situ microstructural and micromechanical information on polycrystalline materials at grain and sub-grain scales. In-situ 3-D measurements are enabled by the collection of multiple diffraction patterns as the sample is rotated and deformed. These patterns are then used in combination with forward-modelling of the diffraction experiment to optimize the agreement between those measured detector images and simulated patterns corresponding to every possible orientation and micromechanical field configuration. Near-field (nf) techniques [START_REF] Suter | Forward modeling method for microstructure reconstruction using x-ray diffraction microscopy: Single-crystal verification[END_REF][START_REF] Li | Adaptive reconstruction method for three-dimensional orientation imaging[END_REF][START_REF] Viganò | Three-dimensional full-field x-ray orientation microscopy[END_REF] are used to obtain crystal orientation fields, resulting in spaceresolved voxelized microstructural images with intragranular resolution. Further, far-field (ff) techniques [START_REF] Henning Friis Poulsen | Three-dimensional maps of grain boundaries and the stress state of individual grains in polycrystals and powders[END_REF][START_REF] Vincent Bernier | Far-field high-energy diffraction microscopy: a tool for intergranular orientation and strain analysis[END_REF][START_REF] Oddershede | Determining grain resolved stresses in polycrystalline materials using three-dimensional x-ray diffraction[END_REF] provide information on local micromechanical fields. Due a trade-off between direct and reciprocal space resolutions, the original ff data inversion methods were able to deliver average stresses/elastic strains in single crystal grains and the volume and location of the centers of mass of those grains, but not voxelized intragranular fields.

Recently, 3DXRD data inversion methods were improved to provide not only voxelized crystal orientation but also and stress fields with intragranular resolution using a variety of experimental procedures, including a microbeam (1D), line beam (2D) or box beam (3D) illumination, each demanding a tailored optimization-based data processing approach [START_REF] Reischig | Three-dimensional sub-grain mapping of elastic strain state and orientations in the bulk of polycrystals[END_REF][START_REF] Hayashi | Intragranular three-dimensional stress tensor fields in plastically deformed polycrystals[END_REF][START_REF] Reischig | Three-dimensional reconstruction of intragranular strain and orientation in polycrystals by near-field x-ray diffraction[END_REF][START_REF] Shen | Voxel-based strain tensors from nearfield high energy diffraction microscopy[END_REF][START_REF] Henningsson | Reconstructing intragranular strain fields in polycrystalline materials from scanning 3DXRD data[END_REF]. However, these types of adopted optimization methods have been exclusively based on imposing diffraction constraints, by minimizing the difference between experimental and simulated detector images obtained by forward-modelling of the diffraction experiment, and retaining as measured fields the orientation and stress fields that produced that minimum. In either case, using the original data inversion based on grain averages, or the more advanced methods that deliver voxelized information, the resulting stress fields are in general unbalanced (not divergence-free, violating the stress equilibrium condition).

Since the micro-mechanical field that directly affects diffraction patterns-by changing the local lattice spacing-is elastic strain, accounting for stress equilibrium at the sub-grain level requires a fairly accurate knowledge of the single crystal properties, such as zero-stress lattice parameters, and single crystal elastic constants. Thus, failing to fulfill mechanical equilibrium in the solution may be unavoidable. In other instances, ignoring the latter allows for a simplified and faster solution process, where large-scale optimization is already challenging and computationally intensive. However, when the material properties are known, taking into account mechanical equilibrium clearly provides a more constrained solution, and may significantly improve the stress/strain fields determination, and consequently, the spatial resolution, reliability, and application range of 3DXRD methods. The elastic deformation solvers are typically iterative, thus mechanical constraints would be employed either to correct the final stress field solution, or in every iteration step, by strictly enforcing equilibrium in the latest solution, or as an additional component in an optimization target function.

Given the aforementioned problem, an approach to solve the lack of equilibrium of the measured stress field is to combine micromechanical modelling and simulations with experimental observations. In different contexts, this combination has been recently done by Pokharel and Lebensohn [START_REF] Pokharel | Instantiation of crystal plasticity simulations for micromechanical modelling with direct input from microstructural data collected at light sources[END_REF] using a fast Fourier transform (FFT)-based elasto-viscoplastic (EVPFFT) model, and by Chatterjee et al. [START_REF] Chatterjee | Study of residual stresses in ti-7al using theory and experiments[END_REF] adopting a field dislocation mechanics finite element model, to complement X-ray diffraction based experimental information and characterize residual and internal stresses. Also, Pagan and Beaudoin [START_REF] Darren | Utilizing a novel lattice orientation based stress characterization method to study stress fields of shear bands[END_REF] used lattice orientation and crystal plasticity kinematics to recover the geometry, and further calculated the stress through finite element simulation.

While the above authors solve physics-based field equations, McNelis, Dawson and Miller [START_REF] Mcnelis | A two-scale methodology for determining the residual stresses in polycrystalline solids using high energy x-ray diffraction data[END_REF] proposed a new approach where they seek to impose equilibrium to match the lattice stresses inferred from XRD and elasticity. In particular, they were interested in large mechanical parts (at the order of meters) where the detection spots are limited. Therefore, they proposed a two-scale method, where the continuum scale stress field imposes equilibrium to match the lattice scale stresses. The approach was extended to three dimensions by Demir et al. [START_REF] Demir | A computational framework for evaluating residual stress distributions from diffraction-based lattice strain data[END_REF], and has found successful applications [START_REF] Park | Quantifying threedimensional residual stress distributions using spatially-resolved diffraction measurements and finite element based data reduction[END_REF][START_REF] Park | Non-destructive characterization of subsurface residual stress fields and correlation with microstructural conditions in a shot-peened inconel component[END_REF]. However, this approach does not impose equilibrium at the grain and sub-grain scale, which is the focus of our work.

This paper presents a novel methodology to impose micromechanical constraints, i.e. stress equilibrium at the subgrain and grain scale, to an non-equilibrated voxelized stress field obtained, for example, by means of synchrotron X-ray diffraction techniques. The main idea is to find the equilibrated stress field closest (in L 2 -norm sense) to a measured (and possibly) non-equilibrated stress field, via the solution of an optimization problem based on the Hodge decomposition of a symmetric 2nd rank tensorial field. The Hodge decomposition is a generalization of the classical Helmholtz decomposition that states that any sufficiently smooth vector field can be decomposed into the sum of an irrotational vector field and a solenoidal vector field. We use the version of the Hodge decomposition for symmetric tensorial fields developed by Geymonat and Krasucki [START_REF] Geymonat | Hodge decomposition for symmetric matrix fields and the elasticity complex in lipschitz domains[END_REF]. We present an efficient numerical implementation of the Hodge decomposition of symmetric 2nd rank tensorial fields using FFTs, and apply it to the analysis/determination of equilibrated stress fields in polycrystalline materials, including 3-D space-resolved stress measured by X-ray diffraction. As mentioned before, these stress fields are in general not divergence-free, due to piecewise approximation based on grain averages and/or absent the consideration of any micromechanical constraint in the data inversion procedure. Using the proposed decomposition and formulating an optimization problem, unbalanced stress fields can be filtered to extract their divergence-free part. The proposed methodology has been further modified/extended to deal with interfaces and discontinuities occurring at grain boundaries. We begin by demonstrating that the method is capable of good recovery of synthetic data obtained in the elastic regime with an FFT-based micromechanical model that provides an objective ground-truth (a validation methodology originally used in [START_REF] Shen | Voxel-based strain tensors from nearfield high energy diffraction microscopy[END_REF]), and then apply it to a stress field actually obtained from diffraction data.

The outline of the paper is as follows. In Section 2 we present a summary of the 3D orientation and strain mapping approach developed by Reischig and Ludwig [START_REF] Reischig | Three-dimensional sub-grain mapping of elastic strain state and orientations in the bulk of polycrystals[END_REF], and its application to Gum Metal, a beta-Ti-based alloy, deformed and measured in-situ within the elastic regime. In Section 3 we recall the fundamentals of the Hodge decomposition and formulate the optimization problem that allow us to extract the divergence-free part of a general non-equilibrated stress field. In Section 4, we present the FFT-based method that enables an efficient numerical resolution of the previously formulated optimization problem when applied to a voxelized field, the type of data structure that naturally results from data reduction of 3DXRD experiments. Section 5 shows applications of the proposed method to synthetic piecewise constant stress fields with a known ground truth, and stress fields in Gum Metal measured by 3DXRD. In both cases, we show that the largest corrections are obtained near grain boundaries. In Section 6, we draw conclusions and give perspectives of the adoption of the proposed method in 3DXRD data inversion packages.

Experimental characterization of elastic strain tensors

Data acquisition

The experimental dataset used in this work to demostrate the proposed stress filtering method was recorded in a regular Diffraction Contrast Tomography (DCT) scan [START_REF] Ludwig | Three-dimensional grain mapping by x-ray diffraction contrast tomography and the use of friedel pairs in diffraction data analysis[END_REF][START_REF] Reischig | Advances in Xray diffraction contrast tomography: Flexibility in the setup geometry and application to multiphase materials[END_REF].

The method uses a parallel monochromatic synchrotron X-ray beam and records diffraction spots from individual grains of the polycrystalline sample on a near-field area detector while the sample is rotated continuously over 360 • . The image stack is subject to a number of pre-processing steps, at the end of which the diffraction spots are segmented and indexed according to their grain of origin and their (hkl) Miller indices. The indexing is based on crystallographic principles and the knowledge of the crystal structure and approximate undeformed lattice cell parameters of the one (or possible several) phases present in the polycrystal. Each diffraction spot is stored as a (u, v, w) volume array representing the diffracted intensity distribution as a 3D scalar field: u and v being the horizontal and vertical image coordinates, and w is the rotation angle. The use of monochromatic X-rays and a high-resolution detector provides sensitivity to the local orientation and local unit cell parameters within the crystallites. Local variations in lattice parameters, i.e. a change of the shape and dimensions of the unit cell, in this case is interpreted as mechanical elastic strain, and any effects from a possible local change of chemical composition is neglected.

The results presented in this study were obtained from a polycrystalline Gum Metal sample with a composition of Ti-36Nb-2Ta-3Zr-0.3O wt% and mean grain diameter of 61µm, from the "low load" scan presented in [START_REF] Reischig | Three-dimensional reconstruction of intragranular strain and orientation in polycrystals by near-field x-ray diffraction[END_REF]. Gum Metal is able to sustain elastic strains up to 2% elongation and above, thus its nickname. A 500µm tall gauge volume of a tensile specimen with an approximately 600µm wide rectangular cross-section was illuminated in its entirety with a box beam during the 360 • rotation, using 40 keV beam energy, 1.5 sec exposure time per image, 0.05 • angular step size, resulting in 7200 images in total. The detector pixel size was 1.4µm, and the rotation axis to detector distance was 7 mm. The sample was mounted in a miniature tensile rig [START_REF] Gueninchault | Nanox: a miniature mechanical stress rig designed for near-field X-ray diffraction imaging techniques[END_REF] and a DCT scan at a uniaxial external tensile stress of 34 MPa was acquired. On average 29 diffraction spots from the first three families of reflections could be recorded for each of the ∼ 1430 grains in the illuminated sample volume, which served as the input data for the reconstruction.

Reconstruction of the sub-grain elastic deformation field

The first stage of the data analysis follows the now standard DCT processing route, with a calibration procedure based on the dataset itself [START_REF] Reischig | Three-dimensional sub-grain mapping of elastic strain state and orientations in the bulk of polycrystals[END_REF][START_REF] Reischig | Three-dimensional reconstruction of intragranular strain and orientation in polycrystals by near-field x-ray diffraction[END_REF]. The centroid positions and mean orientations of the grains were determined based on their diffraction spot centroid metadata. The initial 3D shapes of the grains were reconstructed iteratively from the diffraction spots using the SIRT algorithm, and assuming a constant orientation and elastic strain distribution throughout the grain volume.

The deformation analysis applied here corresponds to an earlier version of the novel Iterative Tensor Field (ITF) Reconstruction method described in detail in [START_REF] Reischig | Three-dimensional reconstruction of intragranular strain and orientation in polycrystals by near-field x-ray diffraction[END_REF], the main difference being that here no Tikhonov regularization term was applied in the elastic deformation optimization target function (Equation (39) in [START_REF] Reischig | Three-dimensional reconstruction of intragranular strain and orientation in polycrystals by near-field x-ray diffraction[END_REF]). The ITF method was used to retrieve the complete local elastic strain tensor (6 parameters), the local lattice misorientations from a reference grain orientation (3 parameters) and refine the grain boundaries over the entire 3D voxelated gauge volume in the specimen. ITF is based on a vector representation of the 3D distribution of X-ray diffracting powers of the crystallites p (one scalar per voxel) and a 3D intragranular deformation field d (9 deformation components per voxel) of their crystal lattice, and aims to reconstruct this F (x, y, z) : R 3 → R 1+9 tensor field on a grain-by-grain basis in a static deformed state, measured in a single scan. Mosaicity, i.e. a variation of crystal lattice orientations without associated stress, is allowed in the solution, hence the compatibility equations are not enforced. A diffraction spot is effectively treated as a projection of the 3D grain shape into the 3D detector domain (u, v, w), where the sub-grain deformations (local misorientations and elastic strains) determine the projection geometry that is unknown. ITF builds on the principles of kinematical diffraction and ray tracing, and the solution for the elastic deformation field is found by minimizing the differences between the measured pixel intensities of the detector q mes (ordered as a column vector) and the simulated intensities q in the diffraction spots of a grain in an iterative large scale optimization. This is a strongly non-linear and usually under-determined problem but in each iteration step a local linear problem at the latest solution can be formulated in matrix form:

∆q = q mes -q = A d ∆d (1)
where ∆d (a column vector) contains the corrections sought to be applied to each elastic deformation component of each active grain voxel in the current iteration step, either positive or negative. The A d matrix contains the intensity contributions to each pixel (one row per pixel) from each elastic deformation component correction of the active grain voxels. The A d matrix is non-sparse, and it needs to be recomputed frequently due to the strong nonlinearity of the model. Translated into a least-squares problem, the solver minimizes the objective function Γ which in this case is the square of the L2 norm (denoted by . 2 ) of the residual vector:

Γ = ∆q -A d ∆d 2 2 (2) ∆d * = arg min ∆d (Γ) (3)
The elastic deformation corrections ∆d * found in the latest iteration step are applied to the current elastic deformation field of the grain. A 3D smoothing operation is then performed on the elastic deformation field in each step, which effectively acts as regularization in the optimization and results in a more realistic field. This smoothness suppresses but does not eliminate the deviations from local mechanical equilibrium across a grain volume.

The corrections to the diffracting powers, which determine the grain shapes, are recomputed in a separate step in a similar way but less frequently, using the latest elastic deformation field (i.e. the latest projection geometry).

Errors in the elastic deformation fields result from a number of sources in the acquisition and data processing. The strain sensitivity here is expected to be in the high single digits of 10 -4 . The overall deformation sensitivity of the setup was limited by the angular step size, pixel size and detector distance. The relatively small number of diffraction spots meant the problem was more ill-posed. Some systematic deviations in intensity were caused by the simplifications in the model of the image formation process. Random noise from the photon counting statistics, fluorescence and scattering processes was often significant. The grainby-grain reconstruction approach is inherently prone to higher errors near grain boundaries. These aspects and potential future improvements of the acquisition and data processing, including the handling of all grains simultaneously and ways to include the local mechanical equilibrium, are described in more detail in [START_REF] Reischig | Three-dimensional reconstruction of intragranular strain and orientation in polycrystals by near-field x-ray diffraction[END_REF]. The fit quality in the ITF deformation solver can be monitored through the residual. In this study, the fitting is considered to have failed in several percent of the grains, mainly at the top and bottom of the gauge volume and at the free surface. Although the dataset does not constitute the state-of-art measurement capabilities, it is perfectly suitable to demonstrate the proposed stress filtering method.

Proposed Method

Hodge decomposition

We use the following notation. Let Ω ⊂ R 3 denote a rectangular domain and H 1 per (Ω, R) be the set of all functions u ∈ H 1 loc (R 3 , R) that are periodic with period Ω. The Helmholtz decomposition is a fundamental theorem in vector calculus that has applications in many fields including elasticity, incompressible flows and electromagnetism (see for example, [START_REF] Cantarella | Vector calculus and the topology of domains in 3-space[END_REF]).

Theorem 1. Given any vector field v ∈ L 2 (Ω, R 3 ), there exists a scalar field ϕ ∈ H 1 per (Ω, R) and a vector field w ∈ H 1 per (Ω, R 3 ), such that,

v = ∇ × w + ∇ϕ, ∇ • w = 0.
Further ϕ, w are unique up to a constant.

The Hodge decomposition is a generalization to tensor fields, and we use a version due to Geymonat and Krasucki [START_REF] Geymonat | Hodge decomposition for symmetric matrix fields and the elasticity complex in lipschitz domains[END_REF] for symmetric fields. Let M 3 denotes the linear space of all second-order matrices and M 3 sym the linear space of all second-order symmetric matrices.

Theorem 2. Given a symmetric matrix field A ∈ L 2 (Ω, M 3 sym ), there exist a vector field y ∈ H 1 per (Ω, R 3 ), a symmetric matrix field H ∈ H 2 (Ω, M 3 sym ) and a constant matrix field c 0 such that,

A = ∇ × H × ∇ + 1 2 (∇y + ∇y T ) + c 0 , ∇ • H = 0
where (∇ × H ×∇) kn = ijk lmn ∇ i ∇ l H jm in indicial notation. Further y, H up to a constant, and c 0 are unique.

Problem formulation

Given a symmetric matrix field S exp ∈ L 2 (Ω, M 3 sym ), we want to find the closest field

S equil ∈ H 1 (Ω, M 3 sym ) in L 2 norm, such that ∇ • S equil = 0.
In other words,

S equil = arg min ||S -S exp || 2 : S ∈ H 1 per (Ω, M 3 sym ), ∇ • S = 0 .
It is easy to verify using the divergence theorem that (∇y + ∇y T ), S H 1 = 0 for any y ∈ H 1 per (Ω, R 3 ) and S ∈ H 1 per (Ω, M) that satisfies ∇ • S = 0. Since ∇ • S equil = 0, it follows that the Hodge decomposition of S equil has y = 0, or

S equil = ∇ × H × ∇ + c 0 , ∇ • H = 0 (4)
for some H ∈ H 2 per (Ω, M 3 sym ) and c 0 ∈ M 3 sym . It follows that

S exp = S equil + 1 2 (∇y + ∇y T ) = ∇ × H × ∇ + c 0 + 1 2 (∇y + ∇y T ). (5)
Now, taking the average over Ω and again using the divergence theorem, c 0 = S exp where we use the overhead bar to denote spatial average A = 1 |Ω| Ω AdX. Putting all this together, the original minimization problem can then be written as

H = arg min L (A) : A ∈ H 2 per (Ω, M 3 sym ), ∇ • A = 0 where L (A) := ||∇ × A × ∇ -S exp || 2 . ( 6
)
where tilde denotes the demeaned part S exp = S exp -S exp . We use this variational problem to find S equil .

The first variation

δL = Ω (∇ × H × ∇ -S exp ) • (∇ × δH × ∇) dX = 0 ∀ δH ∈ A where A = {A ∈ H 2 per (Ω, M 3 sym ) : ∇ • A = 0}. Notice that for any A, B ∈ H 1 per (Ω, M 3 sym ), Ω (∇ × A) • B -A • (∇ × B) dX = Ω ∇ • (A × B) dX = 0.
where the second equality follows from the divergence theorem. Thus,

Ω (∇ × A) • B dX = Ω A • (∇ × B) dX.
Applying this equality twice to the functional, we obtain

δL = Ω (∇ × ∇ × H × ∇ × ∇ -∇ × S exp × ∇) • δH dX = 0 ∀ δH ∈ A .
We obtain the Euler-Lagrange equation,

∇ × ∇ × H × ∇ × ∇ = ∇ × S exp × ∇. Since ∇ × (∇ × H) = ∇(∇ • H) -∇ 2 H and ∇ • H = 0, the Euler-Lagrange equation simplifies to ∇ 4 H = ∇ × S exp × ∇. (7)
We solve this equation to find H and then S equil . Recall that H is unique up to the constant, but the constant does not affect S equil : so we choose the constant to make H = 0.

Solution strategy

Given S exp , we average over the volume to find c 0 and S exp . We then solve Eq. ( 7) to find H. Finally, we use Eq. ( 4) to obtain S equil . The solution of Eq. ( 7) poses a challenge since it involves the fourth order biharmonic equation for H ∈ H 2 , which requires continuous functions and derivatives. This in turn requires either higher order elements using finiteelement discretization, or higher order differences in finite-difference methods. In either situation, it gives rise to stiff numerical problems [START_REF] John | The finite element method applied to fluid mechanics[END_REF][START_REF] Murli | Direct solution of the biharmonic equation using noncoupled approach[END_REF]. Another approach is to break it up into two harmonic problems and to solve them iteratively [START_REF] Cheng | Some mixed finite element methods for biharmonic equation[END_REF]. Here, we solve Eq. ( 7) using Fourier transforms. This poses two challenges. The first is that the experimental stressfield 4 is not periodic. To overcome this, a buffer is filled around the voxelized representation of the material to make the domain a periodic box. The properties of the buffer near the lateral surfaces are chosen to give zero stress. We refer to a detailed discussion in section 2. Second, the stress may suffer discontinuities across grain boundaries, interfaces and across the boundaries of the periodic unit cell. A Fourier series does not converge to discontinuous functions in the L ∞ norm, and this in turn leads to ringing artifacts in practice. We address this in the next section.

Different operators in Fourier space

Let Ω = (0, L) 3 to be a cube and let us discretize it with a N × N × N uniform cubic grid for N even. The corresponding domain in the Fourier space is (-π/L, π/L) 3 and again discretized uniformly with a N × N × N grid. For any f : Ω → R 3 , the discrete Fourier transform (DFT) f satisfies

f (i, j, k) = N l=1 N m=1 N n=1 f (l, m, n)exp( 2πi L ((i -1)(l -1) + (j -1)(m -1) + (l -1)(k -1)), (8) f (l, m, n) = N i=1 N j=1 N k=1 f (i, j, k)exp( -2πi L ((i-1)(l -1)+(j -1)(m-1)+(l -1)(k -1)), (9)
where i = √ -1 (distinct from index i).

Continuous differential operator

The continuous differential operator (CDO) of f in Fourier domain is obtained point-wise multiplication. For example, the partial derivative of f with respect to x is

∂f (i, j, k) ∂x = N l=1 N m=1 N n=1 iξ x f (l, m, n)exp( 2πi L ((i-1)(l-1)+(j-1)(m-1)+(l-1)(k-1)) (10)
where

ξ x =      2π L l, if 1 ≤ l < N 2 0, if l = N 2 2π L (l -N ), if N 2 < l ≤ N (11)
Higher order derivatives are a direct composition of first derivatives, with modification on the coefficients of the highest frequency term (see for example [START_REF] Steven | Notes on fft-based differentiation[END_REF]).

Discrete differential operator

We are interested in studying situations with grain boundaries and interfaces across which the stress may suffer discontinuities. A Fourier series does not converge to such a function in the L inf norm even as N → ∞. In practice, we encounter ringing artifacts or spurious oscillations at finite N . A solution to this problem is to construct the differential operator by taking finite differences and then using DFT [START_REF] Berbenni | A numerical spectral approach for solving elasto-static field dislocation and g-disclination mechanics[END_REF][START_REF] Vidyasagar | Predicting the effective response of bulk polycrystalline ferroelectric ceramics via improved spectral phase field methods[END_REF]. The first order central difference approximation reads,

∂f (i, j, k) ∂x = f (i + 1, j, k) -f (i -1, j, k) 2∆h (12)
etc., and therefore the discrete differential operator is given by Eq. ( 10) with the substitution

iξ x ← i ∆h sin( 2π(m -1) N ),
and analogously for the other partial derivatives. For example,

(∇ × H × ∇) kn = ijk lmn ∇ i ∇ l H jm ,
we substitute ∇ x with i ∆h sin( 2π(m-1)

N

) for the discrete approximation. Note that this first order operator approximates the continuous differential operator for small ξ x , but then decays thereby acting as a low-pass filter. This resolves the overshooting and ringing at the interfaces. However, the convergence with respect to mesh size is reduced to second order from the exponential convergence of the CDO.

The biharmonic operator adopts the following 25 point stencil that is consistent,

∇ 4 f (i, j, k) = 42f (i, j, k) -12(f (i -2, j, k) + f (i + 2, j, k) + f (i, j -2, k) + f (i, j + 2, k) + f (i, j, k -2) + f (i, j, k + 2)) + f (i -4, j, k) + f (i + 4, j, k) + f (i, j -4, k) + f (i, j + 4, k) + f (i, j, k -4) + f (i, j, k + 4) + 2(f (i -2, j -2, k) + f (i -2, j + 2, k) + f (i + 2, j -2, k) + f (i + 2, j + 2, k) + f (i, j -2, k -2) + f (i, j -2, k + 2) + f (i, j + 2, k -2) + f (i, j + 2, k + 2) + f (i -2, j, k -2) + f (i -2, j, k + 2) + f (i + 2, j, k -2) + f (i + 2, j, k + 2)).
Notice that this is based on the second neighbor to make it consistent with the first order difference , Eq. ( 12). We obtain the corresponding operator in Fourier space from the usual Fourier space biharmonic operator with the substitution

ξ 4 x + ξ 4 y + ξ 4 z ← (42 -24(cos(4π(m -1)/N ) + cos(4π(n -1)/N )) + cos(4π(l -1)/N )) + 2(cos(8π(m -1)/N ) + cos(8π(n -1)/N ) + cos(8π(l -1)/N )) + 8(cos(4π(m -1)/N )cos(4π(n -1)/N ) + cos(4π(n -1)/N )cos(4π(l -1)/N ) + cos(4π(m -1)/N )cos(4π(l -1)/N ))) 1 16∆h 4 .

Numerical Examples

We study two examples, one using synthetic data with a known ground truth and another with experimental data.

Synthetic data with known ground truth

We begin with synthetic data from the results of a simulation using the FFT-based method [START_REF] Ricardo A Lebensohn | An elasto-viscoplastic formulation based on fast fourier transforms for the prediction of micromechanical fields in polycrystalline materials[END_REF] to obtain an equilibrated stress field on a periodic unit cell representing a Cu polycrystal deformed in tension in the elastic regime. This polycrystalline unit cell is discretized by a 64 3 grid on a [0, 64]µm 3 box, and contains 100 Cu single crystal grains generated by periodic Voronoi tesselation. The Cu single crystal elastic constants reflect their cubic anisotropy, with C 11 = 168.4 GPa, C 12 = 121.4 GPa and C 44 = 75.4 GPa, which determines heterogeneous, piece-wise constant elastic properties associated with the different crystal orientations, and, therefore, a non-uniform stress field, when the polycrystal is subjected to load in the elastic regime. For this analysis, the unit cell was loaded to a strain of 10 -5 in uniaxial tension along axis x 3 , corresponding to a longitudinal stress Σ 33 = 6.32 MPa, and zero lateral stresses: Σ 11 = Σ 22 = 0, applied to the unit cell. Figure 1(a) shows the σ 11 component of the local stress field (in MPa), which fluctuates with respect to the macroscopic value Σ 11 = 0. Since this synthetic data was generated using a physically meaningful model that uses CDO to solve the governing equations of micro-mechanics (stress equilibrium and strain compatibility), the resulting stress field is indeed equilibrated (and thus divergence-free with respect to the CDO).

We start with this equilibrated stress field and obtain a grain-averaged stress field by averaging the stress over the grains -Figure 1(b) shows the σ 11 component of the grainaveraged stress. This piece-wise constant field is not divergence-free, and the divergence is concentrated at the grain boundaries. We compute the divergence of the grain-averaged stress using both CDO and DDO -the results are shown in Figure 1(c) and Figure 1(d) respectively. We observe that the divergence is generally concentrated at the grain bound- aries, but CDO smears it into the grain interior. This is a result of the spurious oscillation and overshooting that is characteristic of CDO on discontinuous functions.

We now study if it is possible to obtain the original stress field from the grain-averaged data using our method. We apply the proposed algorithm to the grain-averaged stress field using both CDO and DDO. The results are shown in Figure 2. We see that the recovery is imperfect, and this is confirmed in Table 1 which lists the L 2 and L ∞ norms of the error (|f (x)| L ∞ = sup{f (x)∀ x in Ω}). However, both CDO and DDO yield similar errors. In fact, Figure 2(g) shows the σ 11 component of the stress along the line 0.5 × 0.5 × [0, 1]). Both operators recover similar fields though the CDO has more oscillations within the grains and an overshoot at grain boundaries (see for example the grain boundary around z = 0.9).

To understand the error in recovery, recall that our algorithm only filters out the symmetrized gradient (curl-free) part and keeps all the divergence-free part of the given stressfield. So, if the difference between the given data and the ground truth deviate by both curl-free and divergence-free fields, our algorithm will filter out the symmetrized gradient portion of the deviation but retain the divergence-free part of that deviation. In our example, the difference between grain-averaged stress field and ground truth contains both curl-free and divergence-free components. Our algorithm filtered out the former but not the latter and this is the error in both Figure 2 and Table 1. We have verified this by taking the difference between grain-averaged stress field and ground truth, and applying our algorithm to it. The resulting residual equals (close to machine precision) to the recovery error (difference between the ground truth and recovered) in both CDO and DDO.

We further verify this in Figure 3. We perturb the ground truth by adding the symmetrized gradient ∇a + (∇a) T of some vector field a -see Figure 3(a). For a comparison of the scale, |σ 11 | L ∞ of the original field is 6.02 MPa. |σ 11 | L ∞ of the perturbation is 1.00 MPa. Our algorithm is able to recover the ground truth from the perturbation -see Figure 3(b) with minimal error: 9.016 × 10 -6 in L 2 norm and 0.014 in L inf norm. On the other hand, when we perturb the ground truth by adding a divergence free field (Figure 3(c)), our algorithm returns the perturbed field and not the ground truth (Figure 3(d)). For a comparison of the scale, |σ 11 | L ∞ of the original field is 6.02 MPa. |σ 11 | L ∞ of the perturbation is 4.81 MPa. We note that whether the perturbation remains does not depend on the scale of perturbation.

In conclusion, our approach finds the best projection to divergence-free fields. Therefore, it is able to filter out artifacts due to fields that are symmetrized gradients/curl-free, but unable to filter out artifacts that happen to be divergence-free. Indeed, it is not possible to filter out divergence-free artifacts without additional knowledge in terms of material behavior. For this reason, describing the average stress in each grain only provides limited information about the actual state of stress in the material.

Experimental Sample

In this section, we apply the proposed filtering methodology to the DCT experimental data on Gum Metal processed with the ITF method to obtain voxelized stress fields as described in Section 2. We focus on the measured stress field from a sample subjected to an average normal stress of 34 MPa along the z-axis while the other two lateral surfaces are stress free. In order to use our algorithm that assumes periodicity, we introduce a buffer region around the specimen. The buffer region on the side (along x-axis and y-axis) has zero stress to accommodate the free surface. We also note that we implicitly make the top and bottom to be periodically connected. This spurious connection between top and bottom in the longitudinal direction evidently deteriorates the stress balance near those regions. However, we should note that the experimental measurement is also less reliable in those locations and in grains on the free surface of the sample that have high mosaicity as a result of the spark cutting process. This treatment still preserves the information in the bulk of the sample. The resulting unit cell of [0, 640µm] 3 , resolved at 256 3 voxels is shown in Figure 4(a,b) along with the σ 11 and σ 33 components of the measured stress. We note that the state of stress is quite heterogeneous. Figure 4(c) shows the magnitude of the divergence of the measured stress-field calculated using DDO, and we observe high values of divergence at the grain boundaries and triple junctions. This shows the the difficulty of accurate measurement of lattice strains and thus stresses where lattice structure is not preserved.

We present the filtering results with DDO (Fig. 5). After filtering, the peak value is reduced. Most of the profile patterns are preserved. Comparing the difference, we notice that most of the region undergoes minor modification. Significant differences mostly occur around the grain boundaries. We also observe a significant correction of stress in the upper left grain, likely because the deformation solver failed for this grain and some other grains close to the surface. The main trends in Figure 5(g) are well preserved though and most of the smaller features within the grains are retained, even if the stress magnitudes changed. This is an encouraging result, and suggests that the elastic deformation solver based on diffraction constraints returns realistic trends in the stress fields. The difference in the actual magnitudes of this measurement are somewhat less concerning, as these were less reliable due to the deformation resolution of the experimental setup not being fully adequate. The deformation sensitivity of the setup will be improved in the future.

There are a number of outlier grains where the strain/stress state is extreme and significantly deviates from their neighbours. The ITF deformation fitting has potentially failed in these grains. A notable result of the filtering is that most of those grains are retained as outliers. Such high local deviations do not seem realistic, although their validity cannot be ruled out. Thus, it is not clear whether the filtered field is indeed more accurate in those regions. The measurement errors might be so high in those grains that the filtering cannot be expected to correct for them.

A detailed comparison of the two operators is conducted by looking closely at the stress distribution along line 320µm×320µm×[0, 640]µm) (Fig. 5)(g). Both methods yield similar results. In some grains, the stress is not corrected much. But in the grain near z ≈ 300µm, there is a significant correction. The mean difference between these two operators lie in grain boundaries. CDO and DDO both have some oscillations around the boundaries (e.g. z ≈ 250µm). But CDO provides a higher estimation at the boundaries. In contrast, DDO has a smoother transition but it is underestimating the values at the interface.

Conclusions

We developed a FFT-based Hodge decomposition method for symmetric matrix fields. To solve the problem of oscillations at interfaces and preserve the orthogonality of operators, we further developed a modified discrete differential operator. We applied the Hodge decomposition to recover a equilibrated stress field from experimental stress measurements. Since the method is based on a projection, it is able to filter out artifacts due to fields that are symmetrized gradients (curl-free), but unable to filter out artifacts that happen to be divergence-free. Indeed, it is not possible to filter out divergence-free artifacts without additional knowledge in terms of material behavior. We show that the algorithm can be effectively applied to experimentally obtained stress fields. We have also investigated the difference between two discrete differential operators, CDO and DDO. They provide quantitatively similar results overall. However, DDO performs better at interfaces damping out spurious oscillations.

The proposed methodology requires knowledge of the single crystal elastic constants. If they are not precisely known, the method could be extended to refine their values as part of the optimization procedure. 3DXRD reconstruction where grain centroids and average elastic strains/stresses are determined is now a fairly standard post-experimental analysis, and the proposed stress filtering can be directly combined with those reconstructions techniques as post-optimization correction. On the other hand, sub-grain methods are novel and still under development, and the proposed methodology could be combined with them in different ways. The sequential methodology demonstrated here, based on the application of the FFTbased micromechanical filter to stress fields resulting from diffraction-based optimization, could be extended to fulfill the diffraction and micromechanical constraints concurrently during data inversion and refinement. In both cases, but specially in the latter (concurrent optimization), the extension and application of the method has a great potential to increase both deformation and spatial accuracy, as they are interlinked, including the quantitative estimation of errors of the diffraction measurements.

In terms of numerical performance of the proposed method, since it is based on the efficient FFT computations, a single application of the filter to a 256 3 voxel image takes 473 seconds in an Intel Core i7-8700 CPU machine. This high efficiency of a single application of the filter bodes well for standing up the more numerically involved (but still realizable) concurrent optimization, mentioned above.

As a final comment, while the proposed method is based on imposing one type of micromechanical constraint on the measured stress field, i.e. stress equilibrium, the dual constraint is strain compatibility. Such condition can also be expressed in terms of differential operators acting, in this case, on the total strain field. Total strains are not directly accessible by the existing diffraction-based data reduction methods, since the non-elastic part of the deformation does not directly affect lattice spacing and, consequently, diffraction patterns. However, if new methods for determining total strain fields using 3DXRD experiments are further developed, e.g. using evolving 3D microstructural images to perform DCT-based Digital Volume Correlation (DVC), in what would be an extension of similar techniques based on Computed Tomography (CT), strain compatibility imposed via FFT-based methods should be certainly considered as part of the data inversion procedures.

Figure 1 :

 1 Figure 1: Synthetic data: (a) lateral stress field σ 11 obtained with FFT-based model; (b) grain averaged σ 11 ; (c) divergence based on CDO; (d) divergence based on DDO.

Figure 2 :

 2 Figure 2: Recovery from grain averaged data, σ 11 . (a-c) Results of recovery with CDO: (a) recovered stress field, (b) difference between ground truth an recovered stress field and (c) difference between grain averaged and recovered stress field. (d-f) Results of recovery with DDO: (a) recovered stress field, (b) difference between ground truth an recovered stress field and (c) difference between grain averaged and recovered stress field. (g) σ 11 along the line 0.5 × 0.5 × [0, 1]

Figure 3 :

 3 Figure 3: (a,b) Recovery from a perturbation that is a symmetrized gradient (curl-free) (a) perturbed and (b) recovered fields. Compared to the ground truth in Figure 1(a), the recovery is very good -9.016 × 10 -6 in L 2 norm and 0.014 in L inf norm. (c,d) Recovery from a perturbation that is divergence-free (c) perturbed and (d) recovered fields. The recovered field agrees with the perturbed field.

Figure 4 :

 4 Figure 4: Stress field as measured by DCT-ITF: (a) σ 11 (transparent box is the computational domain); (b) σ 33 . (c) Magnitude of the divergence based on DDO.

Figure 5 :

 5 Figure 5: Stress distribution: (a-c) σ 11 ; (d-f) σ 33 ; (g-i) σ 11 , σ 22 , σ 33 along the line 320µm×320µm×[0, 640]µm

Table 1 :

 1 Error in the recovery from grain averaged data

		Diff with ground truth Diff with average
	CDO (L 2 /L ∞ )	0.0024/4.6041	0.0018/2.6945
	DDO (L 2 /L ∞ )	0.0025/4.6007	0.0018/2.6222


Cong Wang

Erwan Paineau

Hynd Remita

Mohamed Nawfal Ghazzal

Cellulose Nanocrystals in Spherical Titania-Sol Microdroplet: From Dynamic Self-Assembly to Nanostructured TiO x /C Microsphere Synthesis

come

Introduction

Cellulose nanocrystals (CNCs) are spindle-like rods derived from a variety of natural resources via strong acid hydrolysis, [START_REF] Vanderfleet | Production routes to tailor the performance of cellulose nanocrystals[END_REF] which can spontaneously organize into a cholesteric structure, called chiral nematic (CN) structure. [START_REF] Kose | Unwinding a spiral of cellulose nanocrystals for stimuli-responsive stretchable optics[END_REF][START_REF] Giese | Functional materials from cellulose-derived liquid-crystal templates[END_REF][START_REF] Tran | Understanding the Self-Assembly of Cellulose Nanocrystals-Toward Chiral Photonic Materials[END_REF] The self-organization of CNCs into well-ordered CN liquid crystals are ubiquitous in nature, [START_REF] Wilts | Natural Helicoidal Structures: Morphology, Self-assembly and Optical Properties[END_REF] inspiring widespread interest in the development of green functional materials with versatile applications, such as biomimetic nanomaterials, soft nanotechnology, optoelectronics and photocatalysis technology. [START_REF] Wang | Nature-Inspired Emerging Chiral Liquid Crystal Nanostructures: From Molecular Self-Assembly to DNA Mesophase and Nanocolloids[END_REF][START_REF] Wang | A sol-gel biotemplating route with cellulose nanocrystals to design a photocatalyst for improving hydrogen generation[END_REF][START_REF] Nguyen | Self-Assembly Route to TiO2 and TiC with a Liquid Crystalline Order[END_REF][START_REF] Xiong | Biopolymeric photonic structures: design, fabrication, and emerging applications[END_REF] The transfer of such natural CN nanostructures into artificial spherical functional materials on the nano-or micro-scale, however, is challenging. [START_REF] Liu | Assembly of cellulose nanocrystals in a levitating drop probed by time-resolved small angle X-ray scattering[END_REF] Typically, studies have focused on obtaining long-range ordered CN structure in planar functional films by using evaporation-induced self-assembly (EISA) method. [START_REF] Hiratani | Transparent Depolarizing Organic and Inorganic Films for Optics and Sensors[END_REF][START_REF] Tardy | Optical Properties of Self-Assembled Cellulose Nanocrystals Films Suspended at Planar-Symmetrical Interfaces[END_REF][START_REF] Shopsowitz | Free-standing mesoporous silica films with tunable chiral nematic structures[END_REF][START_REF] Shopsowitz | Biopolymer Templated Glass with a Twist: Controlling the Chirality, Porosity, and Photonic Properties of Silica with Cellulose Nanocrystals[END_REF] To obtain this CN structure in a small film area, i.e. microfilms, EISA is not applicable and the use of a sophisticated printing method is required. [START_REF] Zhao | Printing of Responsive Photonic Cellulose Nanocrystal Microfilm Arrays[END_REF] Evaporation is essential to achieve equilibrium during isotropic-to-anisotropic transition. The phase transition starts with the nucleation of helical twist in localized spaces, which are called tactoids. As a key component for fundamentally understanding of the self-assembly process of CN structure, tactoids can dynamically bridge the disordered isotropic phase and the ordered CN phase. [START_REF] Wang | Liquid crystalline tactoids: ordered structure, defective coalescence and evolution in confined geometries[END_REF] The importance of tactoids in constructing functional material by controlling the liquid crystal properties has been highlighted, since Lagerwall and coworkers showed that the optical properties of the photonic films could be improved by applying shear forces to suspensions of CNCs during drying. [START_REF] Park | Macroscopic control of helix orientation in films dried from cholesteric liquid crystalline cellulose nanocrystal suspensions[END_REF] The shear forces improved the alignment of the tactoids, and consequently ameliorated the quality of the films. In planar geometries, water evaporation of the CNCs suspension will initiate the coalescence and sedimentation of tactoids, and eventually compress them into longrange lamellar liquid crystalline phase. [START_REF] Wang | Structure and transformation of tactoids in cellulose nanocrystal suspensions[END_REF] Significantly, such planar organization mechanism is not applicable to three-dimensional constrained geometry, especially to target materials with spherical morphology that exceed the restriction of film shape. Spherical confinement of CN liquid crystals are of particular interest in potential applications of sensing, actuators or lasers, optical devices and chiral molecular switches, due to their triggered novel topological structures and special geometries. 19- 24 However, only a few attempts have been dedicated to investigate the CN structure of CNCs aqueous suspension in confined microdroplets. [START_REF] Wang | Polymer and Mesoporous Silica Microspheres with Chiral Nematic Order from Cellulose Nanocrystals[END_REF][START_REF] Parker | Hierarchical Self-Assembly of Cellulose Nanocrystals in a Confined Geometry[END_REF][START_REF] Li | Colloidal cholesteric liquid crystal in spherical confinement[END_REF][START_REF] Li | Nanoparticle-laden droplets of liquid crystals: Interactive morphogenesis and dynamic assembly[END_REF][START_REF] Li | Periodic assembly of nanoparticle arrays in disclinations of cholesteric liquid crystals[END_REF] Although these studies have macroscopically described the evolution of CNCs under spherical constraint as a simple isotropic-to-anisotropic phase transition, the exhaustive organization process remains largely unexplored, in particular with regard to the involvement of inorganic precursors.

Moreover, the stability of precursors (metal chlorides or alkoxides) is highly sensitive to moisture, leading to a fast condensation which destabilizes the CN structure into the isotropic phase. [START_REF] Wang | A sol-gel biotemplating route with cellulose nanocrystals to design a photocatalyst for improving hydrogen generation[END_REF] Thus, further investigations aimed at providing additional insights in the organization process of CNCs in water or in the presence of precursors are meaningful and necessary.

Objectives

The study and control of CNCs self-assembly in spherical confinement remains a challenging and unexplored area. It is difficult to directly observe the self-organization of CNCs in such geometry. In this chapter, we reported a dynamic observation of the self-organization process of CNCs in confined titania-sol microdroplet. The isotropic to cholesteric transition undergoes nucleation, coalescence and migration of tactoids through two coalescence mechanisms, namely "head-to-tail" and "side-to-side" types.

The solubility and self-organization of CNCs in titania-sol enable a successful transfer of the chiral nematic structure into hybrid TiOx/C microspheres. TiOx/C microspheres show higher photocatalytic H2 generation compared to TiO2-P25. This chapter adds to the understanding of self-assembly process of CNCs in spherical geometry.

Preparation of microdroplets and microspheres

Materials

All solvents and reagents including Span 80 (viscosity 1000-2000 mPa.s, 20 °C), cyclohexane (99.7%), TAA (Titanium diisopropoxide bis(acetylacetonate), 75 wt.% in isopropanol), TiO2-P25 (80% anatase, 20% rutile) and CNCs (Cellulose Nanocrystals, obtained from University of MAINE) were used without further purification.

Preparation of TAA/CNC mixtures

Different volumes of TAA solutions (Table S5-1) were added in 1 mL ethanol stirred for 10 min, then mixed with 5 mL CNCs aqueous suspension (pH = 6.3).

Homogeneous transparent yellow mixtures can be obtained after 30 min stirring under room temperature.

Fabrication of TAA/CNC microdroplets and TiO x /C microspheres

Typically, Span 80 (0.47g) was mixed with cyclohexane (30 mL) and sonicated for 10 min to obtain a transparent solution, then transferred to a 100 mL round-bottom flask with magnetic stir bar. The prepared TAA/CNC solution was added dropwise into the flask and maintained a stirring speed at 500 rpm. After rapid-term emulsification at ambient temperature (20 °C), allowing the original state to be studied under POM. The effect of temperature on the CN structure was investigated by changing the emulsification temperature (varied from 20 to 100 °C). To synthesize the TiOx/C microspheres, TAA/CNC6% microdroplets were emulsified in a sealed flask at 60 °C for 90 min, then introduced into a Teflon-lined autoclave (50 mL) and heated at 180 °C for 18 h. The black solid microspheres were washed with ethanol and collected by centrifugation, then dried overnight in oven.

Optimization of CNCs concentration

To understand the role of spherical confinement, it is necessary to investigate first the influence of initial CNCs concentration before encapsulation. The main objective is the optimization of CNCs content in titania-sol, ensuring the spontaneous phase transition from isotropic to chiral nematic. This phase transition depends on the concentration of CNCs, but also on the pH, ionic salt and residual sulfur ratio. [START_REF] Araki | Effect of Trace Electrolyte on Liquid Crystal Type of Cellulose Microcrystals[END_REF][START_REF] Dong | Effects of Ionic Strength on the Isotropic-Chiral Nematic Phase Transition of Suspensions of Cellulose Crystallites[END_REF][START_REF] Li | pH dependence of the chirality of nematic cellulose nanocrystals[END_REF] Considering these facts, variable concentrations of CNCs suspension blended with different volumes of Ti(acac)2(OiPr)2 (TAA) precursor under constant pH (~ 6) were prepared (Figure 5-1c, Figure S5-1 andTable S5-1), where acetylacetonate is used as a chelating agent to inhibit fast condensation of the precursor. [START_REF] Wang | A sol-gel biotemplating route with cellulose nanocrystals to design a photocatalyst for improving hydrogen generation[END_REF] The CNCs suspensions exhibited birefringence under polarized light (Figure 5-1a). Typical fingerprint-like structures with alternated bright and dark lines can also be observed under POM, which indicated the formation of CN phases (Figure 5-1b and Figure S5-2). [START_REF] Giese | Functional materials from cellulose-derived liquid-crystal templates[END_REF] The addition of TAA precursor led to the dilution of CNCs suspensions, resulting in the reversion from CN phase to isotropic phases (shown black domain under POM) (Figure 5-1d). [START_REF] Thomas; Midhun | a Versatile Green Platform: From Biosources to Materials and Their Applications[END_REF][START_REF] Tran | Tactoid Annealing Improves Order in Self-Assembled Cellulose Nanocrystal Films with Chiral Nematic Structures[END_REF][START_REF] Parker | The Self-Assembly of Cellulose Nanocrystals: Hierarchical Design of Visual Appearance[END_REF] The birefringence and fingerprint patterns reappeared when the CNCs concentration further increased. The distances between bright and dark lines correspond to half of the helical pitch (P, defined as the distance for a 360° rotation of CNCs rod), which is the key characteristic of CN phase. [START_REF] Frka-Petesic | Retrieving the Coassembly Pathway of Composite Cellulose Nanocrystal Photonic Films from their Angular Optical Response[END_REF][START_REF] Schutz | Rod Packing in Chiral Nematic Cellulose Nanocrystal Dispersions Studied by Small-Angle X-ray Scattering and Laser Diffraction[END_REF] The value of P tightened from 6.2 to In this concentration range, the magnitude of the P should scale with the inverse of the aspect ratio, P ∝ D/L, which is usually observed for colloidal solution. [START_REF] Schutz | Rod Packing in Chiral Nematic Cellulose Nanocrystal Dispersions Studied by Small-Angle X-ray Scattering and Laser Diffraction[END_REF][START_REF] Grelet | What is the origin of chirality in the cholesteric phase of virus suspensions?[END_REF] In our case, the P is inversely proportional to concentration, P ∝C -1 , and the fraction of the anisotropic phase to the isotropic phase, as expected from Straley modelling of chiral nonflexible rods. [START_REF] Straley | Theory of piezoelectricity in nematic liquid crystals, and of the cholesteric ordering[END_REF] Moreover, two main mechanisms could be involved in rationalizing the decrease in P as the CNCs concentration increased. [START_REF] Lagerwall | Cellulose nanocrystal-based materials: from liquid crystal self-assembly and glass formation to multifunctional thin films[END_REF] The first one is related to the helix formation. Basically, the chirality transfer in liquid crystals is driven by the transfer of the chirality of the molecular cellulose to the macroscopic ordering of the liquid crystal phase. The larger the amount of CNCs in the solution, the stronger the total cohesion of the twist. [START_REF] Yarovoy | Effect of chiral polymers on lyotropic liquid crystals[END_REF] The second one is driven by the increase of the ionic strength (due to the negatively charged CNCs rods) upon increasing the CNCs concentration. This leads to an exponential decay of the electrostatic repulsion and thus allows CNCs rods closer to each other. Notably, the value of P tended to level off at high concentrations (> 6 wt%), accompanied by obvious birefringence (Figure 5-3). 44

Production of TAA/CNC microdroplets

With these results, inverse microemulsion method was employed to encapsulate CNCs and TAA/CNC microdroplets with diameter (Ф) of 110 ~ 170 μm (Figure 5-4a and Table S5-3). Compared with microfluidic, the size distribution of micodroplets through microemulsion method is less well controlled. However, this simple method enables the rapid emulsification of microdroplets and is easy to implement, offering the opportunity to capture and investigate the primitive states of tactoids by using POM.

The use of water-in-oil microemulsion avoids the evaporation of water from the microdroplets, allowing the self-organization in confined microspheres to be studied over time scales from seconds to hours. In standard procedure, 3 mL of the sonicated CNCs suspensions or TAA/CNC mixtures were added dropwise into cyclohexane (30 mL, containing 2 wt% Span 80) under constant stirring speed (500 rpm) to obtain relatively uniform microdroplets. Fresh microdroplets were collected into an unsealed capillary after 1 min emulsification at ambient temperature (20 °C), allowing the primitive state to be studied under POM. The concentration of CNCs suspensions varied from 4 to 8 wt%, in order to assess the threshold at which the isotropic-to-anisotropic phase transition occurred in microdroplets. The optical characteristics of CNCs in microdroplets are similar to that observed in homogeneous suspensions (Figure 5-4bg and Figure S5-3). However, for newly collected microdroplets, especially at low CNCs ratio, the isotropic phase was predominant. CNCs rods start to self-organize in birefringent tactoids that nucleate and grow over time (Figure 5-4 b, f). Onsager, in his pioneering work, [START_REF] Onsager | The effects of shape on the interaction of colloidal particles[END_REF] predicted that homogeneous dispersions of rod particles with high aspect ratios may undergo a phase separation spontaneously. This is due to a competition between orientational entropy and free volume entropy due to excluded-volume interactions. Intuitive observation of POM showed that the dark areas were gradually replaced by bright bands, indicating the onset of isotropic to CN phase transition (nucleation of tactoids). At this primitive nonequilibrium state, the value of P in tactoids is generally higher than that in suspensions (Table S5-1 and S5-3). In addition, the P value for TAA/CNC microdroplets is further expanded compared to CNCs microdroplets, owing to the insertion of the TAA precursor. Tactoid formation progressed weakly (may last several hours) under a low concentration of CNCs. The low local density reduces the probability of interactions between the CNCs rods, which slows down their interactions and phase transition kinetics. Comparatively, nucleation and coalescence can proceed simultaneously at higher concentrations (Figure 5-4 f, h). Based on these findings, we focus on following the case of TAA/CNC6% microdroplet to ensure the acquisition of detailed information on the nucleation and coalescence mechanism of tactoids in confined titania-sol microdroplet for a short period.

Revealing the evolution process

To further investigate the evolution process from isotropic to Ch nematic phase and reveal the growth mechanism of tactoids in spherical confinement, the internal structure progress of one TAA/CNC6% microdroplet (Ф ~ 110 μm) was tracked under POM after ~30 seconds of rapid-term emulsification at 20 °C (Figure 5-4). The use of a lambda retardation plate allowed us to determine the preferred orientation of CNCs directors in the CN phase within the microdroplet (Figure 5-4 b1-b8). At the beginning, isotropic phases dominated the microdroplet, in which small tactoids started to nucleate with one or two periodic bands (Figure 5-4 a1-c1). After 60 seconds of evolution, newborn tactoids grew into larger ones with more bands through coalescence (Figure 5-4 a2-c2). Note that the orientations of the directors of each tactoid were well aligned, but they were randomly oriented with respect to each other (Figure 5-4

b2 and c2).

Two different types of coalescence mechanisms were evidenced, which will be further discussed later. These newborn tactoids had spherical or ellipsoidal shapes, attributing to the interfacial tension between isotropic and anisotropic phases. [START_REF] Chen | Interfacial tension between isotropic and anisotropic phases of a suspension of rodlike particles[END_REF] After several minutes of growth, CNCs tactoids progressively migrated to the periphery of the microdroplet (water/oil interface), driving by the phase separation induced by both the spherical confinement and Marangoni flows (Figure 5-4 a3-c3). [START_REF] Li | Colloidal cholesteric liquid crystal in spherical confinement[END_REF][START_REF] O'keeffe | Boundary Geometry Effects on the Coalescence of Liquid Crystalline Tactoids and Formation of Topological Defects[END_REF] The fusion and rearrangement of adjacent tactoids were driven by the water/oil interface tensions (Figure 5-4 a4-c4), generating large distinct CN domains. Interestingly, the periodic stripe patterns also bent to accommodate the spherical boundary of microdroplet (Figure 5-4 a5-c5). As the process progressed, the ends of flexural bands continue to extend along with the interface until they bordered contiguous domains, triggering the onset of domain integration. Due to larger volume than tactoids, the integration process between domains tended to require a longer time, which is usually accompanied by the onset of topological defects (Figure 5-4 a6-c6). The vanishing of defects indicated the completion of the integration process. Spherical packing of CNCs started at the water/oil interface with the assistance of the tangential surface anchoring, and maintained the propagation toward the microdroplet centre (Figure 5

Mechanism discussion

Distinct Maltese cross originated from the radial alignment between isoclines of the CN helix axis with respect to the polarizers. [START_REF] Parker | Hierarchical Self-Assembly of Cellulose Nanocrystals in a Confined Geometry[END_REF] Obviously, incomplete phase separation may leave an anisotropic core, exhibiting a variable pseudo-Maltese cross patterns. Since the anisotropic core equipped with a specific orientation, the central area of the pseudo-Maltese cross will alternately brighten and darken when the microdroplet remained mobile or change its orientation with respect to the polarizer. We noticed that the central region showed parallel stripes with variable widths during the brightening process (Figure S5-4). There were two defects at the ends of the core, which splitted one S = +1 topological charge into two S = +1/2 charges, likely contributing to minimizing the free energy of the highly frustrated liquid crystalline. [START_REF] Xu | Self-Assembly of Two-Dimensional Colloids in Spherical Space[END_REF][START_REF] Verhoeff | Tactoids of plate-like particles: size, shape, and director field[END_REF][START_REF] Aletta | Shape and Director Field Deformation of Tactoids of Plate-Like Colloids in a Magnetic Field[END_REF] This ubiquitous phenomenon existed during the phase separation, while smaller microdroplets (further confinement) often biased towards the appearance of bipolar stripe pattern (Figure S5-5). [START_REF] Li | Nanoparticle-laden droplets of liquid crystals: Interactive morphogenesis and dynamic assembly[END_REF] Maltese cross pattern with a birefringent feature covered the core of the microdroplet, which changed as the width of the lamellar core increases (Figure S5-4), evidencing the roles of defects. After adequate phase separation and a sufficient equilibrium period, the microdroplet will form a core-shell structure, that is, a CN shell with concentrically packed CNCs pseudo-layers and an isotropic core (Figure 5-5 a8 and Figure 56), where the size of core is dependent on the volume fraction of CNCs. [START_REF] Li | Nanoparticle-laden droplets of liquid crystals: Interactive morphogenesis and dynamic assembly[END_REF][START_REF] Li | Periodic assembly of nanoparticle arrays in disclinations of cholesteric liquid crystals[END_REF] Remarkably, the equilibrium state of the microdroplet has to be accompanied by a specific number of defects, such as the dislocation of spherical pseudo-layers at the microdroplet periphery (Figure 5-7a). [START_REF] Li | Colloidal cholesteric liquid crystal in spherical confinement[END_REF] The point of marked difference between planar and spherical geometries lies in the driving mechanism. Evaporation and sedimentation play the main roles in planar geometry, whereas in confined microdroplet, the coalescence process is driven by the phase separation induced by spherical confinement and the Marangoni flow. The conformation of tactoids stems from nucleation and rearrangement of mobile CNCs rods, while growth depends mainly on coalescence. In our observations, two different types of coalescence mechanism were depicted, namely "head-to-tail" and "side-toside" types. The "head-to-tail" concerned two ellipsoidal-like tactoids (~ 10 μm) with quite linear helical directors. The tactoids rotated until their helical directors perfectly matched via electrostatic attraction, resulting in a larger tactoid with multiplied periodic bands (Figure 5-8a, Figure S5-6). For "side-to-side" coalescence, the helical directors of two tactoids were parallel to each other, and then tilted to form a 25° angle by the attraction between the CNCs rods, allowing the insertion of the bands and eventually forming a larger tactoid with long periodic bands (Figure 5-8b, Figure S5-7). This coalescence type usually produced Y-shaped defects, and lasted longer than "head-totail" type due to the involvement of bands fusion and rearrangement. In addition, another form was sometimes observed, that is, aligned "side-to-side" coalescence (Figure S5-8), which tended to occur between tactoids with similar volume and the same number of bands. We do not rule out the possibility of another coalescence mechanism, such as the "head-to-side" type observed during the formation of planar films. [START_REF] Wang | Structure and transformation of tactoids in cellulose nanocrystal suspensions[END_REF] This case commonly requires high kinetics energy (e.g. during evaporation or heating) to fold the periodic bands of one tactoid and merge into another tactoid.

Coalescence processes are accompanied by the appearance of defects, while they are not easy to be captured due to the small number of tactoids in early growth state and relatively short coalescence time. As evolution progressed, larger tactoids will gradually migrate to the periphery of the microdroplet as a result of phase separation, and will be anchored on the water/oil interface, with their helical axes pointing to the centre as well as local directors consistent with the planar anchor on microdroplet surface (Figure 56789). The orientation of planar anchor was orthogonal to the helical axis and has the capability to impose spherical curvature to the periodic bands. These anchored tactoids or larger CN domains preferentially fused through the "side-to-side" pathway. Since the initial states of these domains were out of synchronization, dislocation defects were usually formed in the director field (Figure 5678910). Such defects, however, can be restored through the resynchronization and reorientation of the director axis due to their thermodynamic destabilization. The integration process between the two domains will last for a few minutes depending on the relaxation rate of CNCs, and will always be accompanied by obvious birefringence (Figure 5

Transfer of CN structures to TiO x /C hybrid microspheres

The concentration of CNCs plays a critical role in its self-organization process under a spherical constraint. Apart from this, an appropriate increase in temperature can promote the shortening of the evolution period. At the same time, higher temperature produced heat stress inside microdroplet, which assisted the phase separation, but also squeezed the P (Figure 5-11a and Figure S5-9). To demonstrate the feasibility of transferring such CN structures from microdroplets to solid microspheres, the TAA/CNC6% microdroplets were emulsified at 60 °C for 90 min to ensure the thoroughly phase separation, and then converted into nanostructured TiOx/C microspheres (Ф ~ 115 μm) by a hydrothermal reaction (180 °C for 18 h). Scanning electron microscopy (SEM) was used to reveal the morphology and the structure of solid TiOx/C microspheres. The buckling and deformation of the microspheres were caused by the gradient of the pressure between TAA condensation and water extrusion during the condensation process in the hydrothermal reaction. (Figure 5-11b). [START_REF] Parker | Hierarchical Self-Assembly of Cellulose Nanocrystals in a Confined Geometry[END_REF][START_REF] Cerda | Geometry and physics of wrinkling[END_REF] Besides, there was a noticeable hole in the centre of the microsphere, pointing out a preferential localization of the TAA during the CNCs self-organization in the spherical microdroplet. The TAA was assembled in the spherical shell instead of the core. More To verify our hypothesis, the microspheres were cracked in liquid nitrogen after being embedded in epoxy resins, enabling the direct observation of the cross section of hybrid microspheres. The SEM images showed CN structures with periodic lamellar bands, which were captured into the TiOx/C microsphere wall (Figure 5 Surface analysis by X-ray photoelectron spectroscopy (XPS) enabled to determine the surface compositions of hybrid microspheres. The survey showed the presence of Ti, C and O species (Figure S5-13). The Ti 2p spectrum exhibited two contributions, 2p3/2 and 2p1/2 arose from spin orbit-splitting, located at respectively 459 and 464.7 eV, which is consistent with Ti 4+ in TiO2 (Figure 5-12a). [START_REF] Ghazzal | Photocatalytic degradation of Rhodamine 6G on mesoporous titania films: Combined effect of texture and dye aggregation forms[END_REF] Likewise, the shoulder Ti 2p1/2 at 460.2 eV that corresponding to Ti 3+ can be confirmed, [START_REF] Bharti | Formation of oxygen vacancies and Ti(3+) state in TiO2 thin film and enhanced optical properties by air plasma treatment[END_REF]

Conclusions

In summary, we encapsulated CNCs in spherical titania-sol microdroplets using an inverse microemulsion method. We monitored the self-assembly process from seconds to hours with the aim to determine the key steps that drive the self-organization of CNCs from isotropic to chiral nematic phase in such aqueous sol. The optimization of concentration allowed to perform a real-time POM observation of the nucleation and growth of tactoids. Tactoids coalescence was the key process that guided the phase transition of mixtures from isotropic to chiral nematic phase. Two coalescence mechanisms of tactoids, "head-to-tail" and "side-to-side", were revealed during the growth process. The phase separation induced by the spherical constraint eventually leads to a microdroplet with chiral nematic shell and isotropic core. Moreover, the impact of temperature on the chiral nematic structure was investigated and a proof-ofconcept of the transfer of such structure into solid TiOx/C microspheres by hydrothermal method was demonstrated. The hybrid TiOx/C microspheres exhibited high absorption in the visible range and promising application in photocatalytic H2 generation. This work offers a deeper insight and substantial understanding of the assembly process of bioinspired materials in spherical geometry, and paves a promising way in the design of novel microspheres with unique chiral nematic structures, which could be widely applied in the fields of catalysis, photocatalysis and Mie scattering.

Figure 5 - 1 .

 51 Figure 5-1. (a) Photographs under polarized light and (b) typical POM images of CNCs suspensions (from left to right, CNC 4% , CNC 6% , and CNC 8% ). (c) Photographs under polarized light and (d) POM images of TAA/CNC mixtures (from left to right, TAA/CNC 4% , TAA/CNC 6%, and TAA/CNC 8% ). All scale bars correspond to 50 μm.

3. 2

 2 μm as the concentration increased from 4 to 6 wt %, which has been attributed to the close packing of the CNCs rods, thus resulting in stronger intermolecular interactions as well as more pronounced birefringence (Figure 5-1a, Figure 5-2 and TableS5-2).[START_REF] Miller; Donald | Imaging of Anisotropic Cellulose Suspensions Using Environmental Scanning Electron Microscopy[END_REF][START_REF] Abitbol | Surface Charge Influence on the Phase Separation and Viscosity of Cellulose Nanocrystals[END_REF]

Figure 5 - 2 .

 52 Figure 5-2. Variation of the helical pitch value with the CNCs mass fraction. The insets include a schematic representation of half pitch (P/2) and a representative POM image of CNC 5% (showing a P value of 5.15 μm) recorded with a retardation λ-plate (530 nm). Scale bar = 50 μm

Figure 5 - 3 .

 53 Figure 5-3. Schematics of chiral nematic structure (from isotropic phase to chiral nematic phase and obvious birefringence).

Figure 5 - 4 .

 54 Figure 5-4. (a) Cartoon illustration of inverse microemulsion system. POM images of microdroplets of (b) CNC 4% , (c) CNC 6% , (d) CNC 8% , (e) TAA/CNC 4% , (f) TAA/CNC 6% and (g) TAA/CNC 8% (scale bars = 50 μm). (h) Schematic of microdroplet.

  -4 a7-c7). Newly formed spherical structures were asymmetric owing to stochastic nature in the fusion process. After approximately two hours of equilibration time, CNCs in the microdroplet exhibited uniform radial alignment (Figure 5-4 b8). Meanwhile, alternating bright and dark concentric rings were superimposed with a Maltese cross pattern (hedgehog point defect) that corresponded to the characteristic features of the CN phase (Figure 5-4 a8-c8).

Figure 5 - 5 .

 55 Figure 5-5. Typical POM images of time-series snapshots of TAA/CNC 6% microdroplet without (a1-a8) and under lambda retardation plate (b1-b8) (scale bar = 50 μm). Note that (a8) and (b8) was captured by using high magnification camera once equilibrium was reached. (c1-c8) Corresponding cartoon depiction of the microdroplet transforms from newborn tactoids to symmetrical concentric circles with Maltese cross structure.

Figure 5 - 6 .

 56 Figure 5-6. 3D model exhibits core-shell structure of TAA/CNC 6% microdroplet.

  Such dislocation stemmed from the inconsistent relaxation progress of CNCs, leading to a local coincidence between adjacent bright spherical pseudo-layers (Figure5-7b).[START_REF] Khadem | Relaxation dynamics in bio-colloidal cholesteric liquid crystals confined to cylindrical geometry[END_REF]

Figure 5 - 7 .

 57 Figure 5-7. (a) POM image and (b) corresponding 3D model of a dislocation occurring in the chiral nematic layers (Figure 5-7a is derived from Figure 5-5 a8, scale bar = 10 μm).

Figure 5 - 8 .

 58 Figure 5-8. POM images and corresponding 3D models showing two different types of coalescence mechanism of tactoids: (a) head-to-tail and (b) side-to-side (scale bar = 10 μm).

Figure 5 - 9 .

 59 Figure 5-9. (a) POM image of tactoids anchored on the water/oil interface with their helical axes to the centre of microdroplet (scale bar = 10 μm). (b) Corresponding 3D models of anchored tactoids.

  -10 d, e). Spherical pseudo-layers could be observed after the periphery of the microdroplet was completely packed by the CNCs domains (Figure 5-5 a7).

Figure 5 - 10 .

 510 Figure 5-10. POM images revealing the dislocation and fusion occurring during the domain integration (scale bar = 20 μm) and corresponding 3D models of dislocation and fusion.

  importantly, we observed parallel stripe patterns on the surface of microspheres (Figure S5-10), suggesting a potential transfer of CN structure to TiOx/C microspheres.

Figure 5 - 11 .

 511 Figure 5-11. The effect of temperature on P. The inset is a 3D schematic diagram of P compressed at higher temperature. (b) SEM image of TiO x /C microsphere with wrinkles on the surface and a hole in the centre. (c-g) Cross-section SEM images of microspheres showing CN structures with periodic lamellar bands.

  Figure S5-11). This further demonstrated the successful replication of the CN structure from TAA/CNC microdroplets to hybrid TiOx/C microspheres. The thermogravimetric analysis (TGA) was employed for identifying the titania content in TiOx/C (Figure S5-12). The weight loss before 200 °C was mainly caused by the evaporation of moisture. The sharp weight decrease ranging from 200 to 420 °C corresponded to the combustion of carbon species, which implied ~ 23.5% of TiO2 remained in TiOx/C microspheres.

  photocatalytic hydrogen evolution activity (196.7 µmol g -1 h -1 ) compared to commercial TiO2-P25 (134.7 µmol g -1 h -1 ) (Figure5-12c and FigureS5-15). The increase in photocatalytic activity could be attributed to the broad absorption of TiOx/C and the light scattering induced by the CN structure of hybrid nanostructured microspheres.[START_REF] Gesesse | Enhanced Photogenerated Charge Carriers and Photocatalytic Activity of Biotemplated Mesoporous TiO2 Films with a Chiral Nematic Structure[END_REF] These results provide a new perspective to obtain sub-stoichiometric titania oxides with improved light harvesting ability, showing great potential applications in photocatalysis field.

Figure 5 -

 5 Figure 5-12. (a) XPS spectrum of Ti 2p. (b) XRS patterns of TiO x /C microspheres. (c) Hydrogen generation constant rate of TiO x /C microspheres and commercial TiO 2 -P25.


Aude Lavayssière

Lise Retailleau

Evolution of the Volcano-Tectonic seismicity associated with Mayotte's active magmatic system

INTRODUCTION

EVOLUTION OF THE PROXIMAL CLUSTER

v Cluster in place since 2018 v Seismicity is decreasing with time (Fig. 3 and4) v Activity started in the center and moved to the edges of the cluster (Fig. 3 and4) v Events highlight different structures (Fig. 3) v At depths, events are shallower in the center than at the edges (Fig. 3) v Events further from center generally deeper (Fig. 5)

v Aseismic circular region in the top center (Fig. 3) v Aseismic vertical zone along the center (Fig. 3) v On the cross-section at 45°, we can distinguish two linear aseismic zones v The linear aseismic zones all converge to the aseismic top center (Fig. 3)

v Recent (end of 2020) seismicity is deep (> 30km) (Fig. 3 and4) v Mainly to the West-South West and to the South East (Fig. 3 and4)

Zones of interest: v Sub-cluster 1 (black) = linear almost-vertical structure that slowly dissipates through time v Sub-cluster 2 (red) = linear structure that slowly dissipates through time v Sub-cluster 3 (green) = continuous activity -slight increase with time in the West v Sub-cluster 4 et 4' (brown & purple) = deep activity; slightly dissipates but still active v Sub-cluster 5 (cyan) = very small cluster but shallower than the rest; appears slowly over time

EVOLUTION OF THE DISTAL CLUSTER

v Seismicity relatively stable with time but has moved (Fig. 6 and7) v 2 main sub-clusters: sub-cluster 1 disappears -sub-cluster 2 increases in activity (Fig. 6 and7)

Sub-cluster 1 v Activity slowly decreases (Fig. 7)

Mayottev

  Figure 2: Earthquakes detected manually compared to events detected with PhaseNet neural network

Figure 8 :

 8 Figure 8: Events in the distal cluster represented by its position along the cross-section trace through time, and color-coded by its depth.

Figure 5 :

 5 Figure5: Events in the proximal cluster represented by its depth and its azimuth compared to the center of the cluster, and color-coded by its distance to the center.

Figure 4 :

 4 Figure 4: Evolution of the seismicity through time. Each line of graphs represents the evolution on each traces (see Fig.3). Information on the location and evolution of surface degassing is indicated.

Figure 7 :

 7 Figure 7: Evolution of the seismicity through time.

Figure 3 :

 3 Figure 3: Cross-sections of the seismicity color-coded by time. Symbol at the top of each subplot represents the cross-sections trace.

Figure 1 :

 1 Figure 1: Map of Mayotte's seismicity. Each colored node represents the mean depth of the events located in the node, with a minimum of 2 events per node. Green inverted triangles are the land stations. The red volcano symbol is the location of the New Volcanic Edifice (NVE) and the red contour is the extent of new material from the eruption.