domain: phys math

text:
V Monchiet

G Bonnet

FFT based iterative schemes for composite conductors with uniform boundary conditions

Keywords:

In the present paper, we extend the FFT method to deal with the homogenization problem of composite conductors with uniform boundary conditions. The principle of the approach consists of applying a transformation to build a periodic problem from the solution with uniform boundary conditions. It is shown that the related periodic problem must be applied to an extended domain obtained by mirror symmetry of the unit cell. The conductivity equation must then be solved on this extended domain under an applied periodic polarization field as a loading parameter. Illustrations are provided and the effective conductivity obtained with FFT is compared to finite element solutions to validate the approach. The proposed method can be applied to microstructure geometries of all kinds, including cells obtained through imaging devices.

Introduction

The determination of the effective properties of composite conductors involves the resolution of the heat equation with specific conditions at the boundary of the considered unit cell. Three kinds of boundary conditions are commonly used: periodic boundary conditions, uniform temperature gradient conditions (Dirichlet type), and uniform flux conditions (Neumann type). Uniform boundary conditions were introduced in the context of random composites and are due to the pioneering works by [START_REF] Hill | The elastic behaviour of a crystalline aggregate[END_REF][START_REF] Hill | Elastic properties of reinforced solids: some theoretical principles[END_REF] and [START_REF] Kröner | Das fundamental integral der anisotropen elastischen differential gleichungen[END_REF][START_REF] Kröner | Berechnung der elastischen konstanten des vielkristalls aus den konstanten des einkristalls[END_REF]. Alternatively, in the field of asymptotic homogenization methods (see [START_REF] Auriault | Study of the macroscopic behavior of a saturated porous elastic medium[END_REF], [START_REF] Sanchez-Palencia | Non-homogeneous media and vibration theory[END_REF] and [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF]), the effective properties of the composite are determined by solving the unit cell problem with periodic boundary conditions. While these boundary conditions have been introduced for specific microstructures, the periodic conditions can be applied at the boundary of a unit cell of a random microstructure and uniform boundary conditions to periodic composites. This raises fundamental questions about the comparison between the effective properties computed with these different kinds of boundary conditions. Also, the definition of the representative volume element over which the boundary conditions are applied has been addressed in many works [START_REF] Huet | Application of variational concepts to size effects in elastic heterogeneous bodies[END_REF][START_REF] Hazanov | Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume[END_REF][START_REF] Kanit | Determination of the size of the representative volume element for random composites: statistical and numerical approach[END_REF], but such problems are not central in the present work.

In this paper, we address the question of solving the unit cell problem with uniform boundary conditions. This question has been the subject of many recent studies in the case of composite elasticity.

For instance, [START_REF] Gélébart | A modified FFT-based solver for the mechanical simulation of heterogeneous materials with Dirichlet boundary conditions[END_REF] proposed to consider the problem with Dirichlet boundary conditions using the FFT method by embedding the unit cell in a larger domain to apply periodic boundary conditions. [START_REF] To | Fourier transform approach to non periodic homogenization problems in porous conductive media[END_REF] addressed the more general problem of Dirichlet and Neumann boundary conditions. Other approaches consist of applying a mirror transformation to the original unit cell [START_REF] Wiegmann | Fast Elliptic Solvers on Rectangular Parallelepipeds[END_REF][START_REF] Grimm-Strele | FFT-based homogenization with mixed uniform boundary conditions[END_REF]. [START_REF] Wiegmann | Fast Elliptic Solvers on Rectangular Parallelepipeds[END_REF] provides a general method for solving homogeneous cells with arbitrary Dirichlet and Neumann boundary conditions with Fourier series. However, it does not deal with the problem of composites and the FFT method of Moulinec andSuquet (1994, 1998). [START_REF] Grimm-Strele | FFT-based homogenization with mixed uniform boundary conditions[END_REF] extend the FFT method of Moulinec and Suquet to deal with mixed boundary conditions in the case of elasticity but fail to deal with the Dirichlet and Neumann boundary conditions. In the present paper we deal with the thermal conductivity problem whose equations are close to elasticity but for which transformations have been identified to deal with uniform (Dirichlet and Neumann) boundary conditions. In fact, the principle of the mirrors method is older and can be attributed, for instance, to [START_REF] Timoshenko | Theory of Plates and Shells[END_REF] who considered periodic alternated loadings to solve the problem of rectangular plates. It should also be noted that in the case of a symmetric microstructure, it can be shown that solving the periodic boundary conditions can be accomplished by applying known (mixed) boundary conditions [START_REF] Bornert | Homogenization in Mechanics of Materials[END_REF].

In this paper, we propose to handle the problem with Dirichlet and Neumann conditions for anisotropic composite conductors. The E-mail address: vincent.monchiet@univ-eiffel.fr (V. Monchiet). method is based on the mirror transformation already considered by [START_REF] Wiegmann | Fast Elliptic Solvers on Rectangular Parallelepipeds[END_REF] and Grimm-Strele and Kabel (2021) but introduce new transformation for the temperature field to derive a periodic problem on a larger domain. By doing so, it is shown that the classic FFT method can be applied considering new appropriate loadings fields. The advantage of our proposed method is that it can easily be implemented with existing home-made FFT codes without major modifications. More specifically, we extend the primal approach of Moulinec andSuquet (1994, 1998), the dual approach [START_REF] Bhattacharya | A model problem concerning recoverable strains of shape-memory polycrystals[END_REF][START_REF] Bonnet | Effective properties of elastic periodic composite media with fibers[END_REF], and the polarization method [START_REF] Monchiet | A polarization-based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast[END_REF] to deal with the Dirichlet and Neumann problems. Illustrations and comparisons with FE solutions are provided to validate the approach.

Statement of the problem

We consider a unit cell of a composite and we assume that this cell is parallelepipedic (rectangular for 2D problems) and we denote by 𝐿 1 , 𝐿 2 , 𝐿 3 its dimensions along each space directions of the cartesian frame (𝑂, 𝑥 1 , 𝑥 2 , 𝑥 3 ). The center of the cartesian frame, denoted ''O'', is located at the left bottom corner of the unit cell 𝛺 as shown in Fig. 1.

The homogenized conductivity properties of a composite material are classically determined by solving the set of partial differential equations: div(𝝈) = 0

(1)

𝝈 = 𝑲(𝒙).𝜺

(2)

𝜺 = ∇𝑢 (3)
with appropriate boundary conditions. Three kind of conditions are conventionally applied to the unit cell:

• Uniform temperature gradient at the boundary (Dirichlet boundary condition):

𝑢 = 𝑬.𝒙, ∀𝒙 ∈ 𝜕𝛺 (4)
• Uniform flux at the boundary (Neumann boundary condition):

𝝈.𝒏 = 𝜮.𝒏, ∀𝒙 ∈ 𝜕𝛺 (5)
• Periodic boundary conditions:

𝑢 -𝑬.𝒙 periodic, 𝝈.𝒏 antiperiodic (6)
The FFT method is based on Fourier series discretization, the periodic boundary conditions are then implicitly satisfied by the considered space of discretization functions. However the method cannot be applied to the two other kinds of boundary conditions. For instance, with the Dirichlet condition, the total temperature field can be decomposed into two parts: (i) a temperature associated with the applied temperature gradient 𝑬 that is 𝑬.𝒙, (ii) a fluctuation 𝑣 = 𝑢 -𝑬.𝒙 that is null at the boundary of the RVE. This field 𝑣 is then periodic and can be represented along Fourier series. However, the related normal flux 𝝈.𝒏 takes values on the boundary that are obtained from the solution of the Dirichlet problem and have no reason to be antiperiodic for this kind of boundary conditions. Similarly, in the case of the Neumann condition at the boundary, the value of temperature at the boundary has no reason to be periodic.

Dirichlet problem

Extension of the domain

We first deal in this section with the conductivity problem related to Dirichlet boundary condition. As already explained, the difficulty arises from the fact that the flux is arbitrary on the boundary of the cell while the FFT method is restricted to problems with the antiperiodicity for the normal flux. The principle of the approach is to solve a complementary problem over a larger unit cell obtained with mirror transformation. Considering the initial unit cell 𝛺 = [0, 𝐿 1 ] × [0, 𝐿 2 ] × [0, 𝐿 3 ], we build a new cell 𝛺 ′ by reflexion with respect to the three orthogonal planes 𝑂𝑥 1 𝑥 2 , 𝑂𝑥 1 𝑥 3 and 𝑂𝑥 2 𝑥 3 of the cartesian frame (see Fig. 2 for the representation of the extended cell 𝛺 ′ in the 2D case).

Let us denote by 𝐼 𝛼 (𝒙), for 𝛼 = 1, 2 and ∀𝒙 ∈ 𝛺, the characteristic functions of the phases : 𝛼 = 1 for the matrix and 𝛼 = 2 for the inclusions. The characteristic function 𝐼 𝛼 (𝒙) is equal to 1 in the phase 𝛼 and is null outside. The characteristic function over the extended domain 𝛺 ′ is given by:

𝐼 ′ 𝛼 (𝒙) = 𝐼 𝛼 (𝑺.𝒙) (7)
in which 𝐼 ′ 𝛼 (𝒙) is defined over the extended domain 𝛺 ′ while 𝐼 𝛼 (𝒙) only lies in 𝛺. The components of 𝑺 in the cartesian frame are:

𝑺 = ⎛ ⎜ ⎜ ⎝ 𝑠 1 0 0 0 𝑠 2 0 0 0 𝑠 3 ⎞ ⎟ ⎟ ⎠ (8)
in which 𝑠 𝑖 = sign(𝑥 𝑖 ) for 𝑖 = 1, 2, 3.

Compatibility equation

Denoting by 𝑣 = 𝑢 -𝑬.𝒙 the fluctuation in the Dirichlet problem, we introduce 𝑣 ′ (𝒙) defined by:

𝑣 ′ (𝒙) = 𝑠 1 𝑠 2 𝑠 3 𝑣(𝑺.𝒙), ∀𝒙 ∈ 𝛺 ′ (9)
Again, 𝑣 ′ (𝒙) is defined over 𝛺 ′ while 𝑣(𝒙) only lies in 𝛺. With the introduction of 𝑣 ′ , it is shown in the next that it is possible to introduce a flux that is 𝛺 ′ -antiperiodic while the flux related to 𝑣 is not 𝛺-antiperiodic.

Because 𝑣 is null at the boundary of 𝛺, 𝑣 ′ is null at the boundary of each individual cells (the original cell 𝛺 and its image by the reflexion). As a consequence, 𝑣 ′ is continuous from one individual cell to another. In addition, the temperature 𝑣 ′ is 𝛺 ′ -periodic because it is null at the boundary of 𝛺 ′ and has a null volume average over 𝛺 ′ (due to the coefficient 𝑠 1 𝑠 2 𝑠 3 behind 𝑣 in Eq. ( 9) whose value alternates between 1 and -1). Note that on the interface between the individual unit cells, the continuity of the flux at the interface between individual cells is also equivalent to the continuity of the normal gradient because the material has the same conductivity at both side of the interface. This is due to the mirror symmetry transformation applied to the original cell. Moreover, the temperature being continuous at the interface between individual cells, the tangential components of the gradient of temperature is also continuous. Finally, the normal and tangential components of the temperature gradient being continuous, it means that the total gradient is continuous across the interface. The continuity of the temperature gradient is then implicitly satisfied if the continuities of the temperature and normal flux are satisfied.

Let us defined by 𝒒(𝒙) and 𝒒 ′ (𝒙) the gradient of 𝑣(𝒙) and 𝑣 ′ (𝒙) respectively:

𝒒(𝒙) = ∇𝑣(𝒙), ∀𝒙 ∈ 𝛺, 𝒒 ′ (𝒙) = ∇𝑣 ′ (𝒙), ∀𝒙 ∈ 𝛺 ′ (10)
Applying the classic chain rule, one has ∇[𝑣(𝑺.𝒙)] = 𝑺.∇𝑣(𝑺.𝒙) and then 𝒒(𝒙) and 𝒒 ′ (𝒙) are related by:

𝒒 ′ (𝒙) = 𝑠 1 𝑠 2 𝑠 3 𝑺.𝒒(𝑺.𝒙), ∀𝒙 ∈ 𝛺 ′ (11)

Flux and thermal equilibrium

Denoting by 𝝈(𝒙) the flux in the initial cell 𝛺, we introduce 𝝈 ′ (𝒙) such that:

𝝈 ′ (𝒙) = 𝑻 .𝝈(𝑺.𝒙), ∀𝒙 ∈ 𝛺 ′ (12)
where the two order tensor 𝑻 is defined by:

𝑻 = 𝑠 1 𝑠 2 𝑠 3 𝑺 = ⎛ ⎜ ⎜ ⎝ 𝑠 2 𝑠 3 0 0 0 𝑠 1 𝑠 3 0 0 0 𝑠 1 𝑠 2 ⎞ ⎟ ⎟ ⎠ (13)
This field is divergence free in 𝛺 ′ - where  represents the interfaces between the individual cells. This distinction is needed because the components of 𝑺 are constant in each individual cell but are discontinuous at the interfaces between the individual cells (the equilibrium of the normal flux at the interface between the individual cells is examined below). The proof for the divergence free condition is easy, indeed, considering Eq. ( 12) and applying the chain rule, one has ∀𝒙 ∈ 𝛺 ′ -:

div(𝑻 .𝝈(𝑺.𝒙)) = 𝑻 ∶ [∇𝝈(𝑺.𝒙).𝑺] (14) Moreover, 𝑻 .𝑺 = 𝑠 1 𝑠 2 𝑠 3 𝑰, then ∀𝒙 ∈ 𝛺 ′ -: div(𝑻 .𝝈(𝑺.𝒙)) = 𝑠 1 𝑠 2 𝑠 3 𝑰 ∶ [∇𝝈(𝑺.𝒙)] = 𝑠 1 𝑠 2 𝑠 3 div(𝝈)(𝑺.𝒙) (15)
Then if 𝝈(𝒙) is divergence free, 𝝈 ′ (𝒙) is also divergence free in each individual cells. The flux 𝝈 ′ (𝒙) is in equilibrium in all the domain 𝛺 ′ if the normal flux is continuous across the interfaces between the individual cells. Otherwise, this would imply some line sources at the junction between individual cells, and would increase the difficulty when solving the problem with the FFT method. We consider, for example, the interface corresponding to 𝑥 1 = 0. We aim to verify that 𝜎 ′ 1 (0, 𝑥 2 , 𝑥 3 ) admits the same limit at the right and left on the considered interface. 𝑻 is diagonal, then, accordingly to Eq. ( 12), 𝜎 ′ 11 (0, 𝑥 2 , 𝑥 3 ) = 𝑇 11 𝜎 11 (0, 𝑠 2 𝑥 2 , 𝑠 3 𝑥 3 ). Moreover the component 𝑇 11 is independent of 𝑠 1 and then 𝜎 ′ 11 is then also independent of 𝑠 1 that ensures its continuity across the interface 𝑥 1 = 0. The proof is the same for the continuity of components 𝜎 ′ 22 and 𝜎 ′ 33 across the interfaces 𝑥 2 = 0 and 𝑥 3 = 0.

For completeness, one has to verify the antiperiodicity conditions. Consider for examples the opposite edges corresponding to 𝑥 1 = ±𝐿 1 that can be also written 𝑥 1 = 𝑠 1 𝐿 1 . The demonstration is very similar, one has to prove that 𝜎 ′ 11 takes the same value on these opposite edges. One has, accordingly to Eq. ( 12),

𝜎 ′ 1 (𝑠 1 𝐿 1 , 𝑥 2 , 𝑥 3 ) = 𝑇 11 𝜎 1 (𝐿 1 , 𝑠 2 𝑥 2 , 𝑠 3 𝑥 3 ) that is independent of 𝑠 1 ,
proving the antiperiodicity for the two opposite edges 𝑥 1 = ±𝐿 1 . The proof for the other opposite edges is similar. Finally, it has been shown that 𝝈 ′ is in equilibrium over 𝛺 ′ and is 𝛺 ′ -antiperiodic.

Thermal relationship

We now establish the relation between the gradient of temperature 𝒒 ′ (𝒙) and the flux 𝝈 ′ (𝒙) in all the domain 𝛺 ′ . In 𝛺 the flux 𝝈 and the temperature gradient 𝒒(𝒙) are related by the affine relation:

𝝈(𝒙) = 𝑲(𝒙).[𝒒(𝒙) + 𝑬]

(16)

It follows that, considering Eq. ( 12): Finally, because 𝑺.𝑺 = 𝑰, one can also read:

𝝈 ′ (𝒙) = 𝑠 1 𝑠 2 𝑠 3 𝑺. [
𝝈 ′ (𝒙) = 𝑲 ′ (𝒙).[𝒒 ′ (𝒙) + 𝑬 ′ (𝒙)] (19)
where:

• 𝑲 ′ (𝒙) is given by:

𝑲 ′ (𝒙) = 𝑺.

𝑲(𝑺.𝒙).𝑺 (20)

It must be noticed that 𝑲 ′ (𝒙) remains strictly positive definite by the transformation if 𝑲(𝒙) is also positive definite. Also, if both constituents are isotropic, 𝑲(𝒙) = 𝑘(𝒙)𝑰, then 𝑲 ′ (𝒙) is also isotropic and 𝑲 ′ (𝒙) = 𝑘(𝑺.𝒙)𝑰. When the constituents are anisotropic, the components of 𝑲 ′ (𝒙) in the cartesian frame are 𝐾 ′ 𝑖𝑗 (𝒙) = 𝑠 𝑖 𝑠 𝑗 𝐾 𝑖𝑗 (𝑺.𝒙) (here the Einstein convention on repeated indices must not be applied). It follows that the diagonal components of 𝑲 ′ (𝒙) remain unchanged by the transformation, 𝐾 ′ 𝑖𝑖 (𝒙) = 𝐾 𝑖𝑖 (𝑺.𝒙), while the off-diagonal components must be either positive or negative depending of the considered individual unit cell.

• The loading parameter 𝑬 ′ (𝒙) is given by:

𝑬 ′ (𝒙) = 𝑠 1 𝑠 2 𝑠 3 𝑺.𝑬 (21)
This loading parameter is consistent with:

𝑠 1 𝑠 2 𝑠 3 𝑺.𝑲(𝑺.𝒙).𝑬 = 𝑲 ′ (𝒙).𝑬 ′ (𝒙) (22)
To conclude, 𝑣 ′ satisfies the equations related to a conductivity problem over 𝛺 ′ with the conductivity tensor 𝑲 ′ (𝒙) under the applied periodic eigenfield 𝑬 ′ (𝒙). The resulting problem which has to be solved is remarkable by its simplicity, particularly in the isotropic case, since it only involves the resolution with the applied piecewise constant loading function 𝑬 ′ (𝒙) instead of the constant loading 𝑬 in the classic periodic problem. As a consequence, a classic FFT based iterative schemes can be used to solve that problem with very minor modification. The values of 𝑬 ′ (𝒙) taken in the period 𝛺 ′ is illustrated in Fig. 3 for 2D problems. At the left, we provide the components 𝐸 ′ 1 (𝒙) and, at the right, the component 𝐸 ′ 2 (𝒙). The period 𝛺 ′ is constituted of four quarters corresponding to the original unit cell 𝛺 and its mirrors images. The component 𝐸 ′ 1 (𝒙) is equal to 𝐸 1 for the upper quarters and equal to -𝐸 1 for the lower quarters. The component 𝐸 ′ 2 (𝒙) is equal to 𝐸 2 for the right quarters and equal to -𝐸 2 for the left quarters.

Effective conductivity

The identification of the macroscopic conductivity must be done considering the Hill-Mandel Lemma. The macroscopic potential is:

𝑊 = ⟨𝝈 ′ (𝒙).(𝒒 ′ (𝒙) + 𝑬 ′ (𝒙))⟩ 𝛺 ′ = ⟨𝑠 1 𝑠 2 𝑠 3 𝑺.𝝈 ′ (𝒙)⟩ 𝛺 ′ .𝑬 (23)
In the above equation, the average of the scalar product between 𝝈 ′ (𝒙) and 𝒒 ′ (𝒙) vanishes since 𝝈 ′ (𝒙).𝒏 is antiperiodic and 𝒒 ′ (𝒙) derivates from the periodic temperature field 𝑣 ′ (𝒙). Due to the symmetry, one has also: In virtue of the Hill-Mandell Lemma, we deduce from Eq. ( 24) together with Eq. ( 23) that the macroscopic flux is given by:

𝑊 = ⟨𝝈(𝒙).(𝒒(𝒙) + 𝑬)⟩ 𝛺 = 𝜮.𝑬 (24)
𝜮 = ⟨𝑠 1 𝑠 2 𝑠 3 𝑺.𝝈 ′ (𝒙)⟩ 𝛺 ′ (25)
Once the flux is computed in 𝛺 ′ , the macroscopic flux is derived from (25) allowing the computation of the components of the effective conductivity related to the Dirichlet problem.

The resolution of the Dirichlet problem over the extended domain 𝛺 ′ , can be solved considering classic FFT iterative schemes that are adapted in order to apply the loading field 𝑬 ′ (𝒙). The details are provided in Appendix.

Neumann problem

Compatibility

Consider the temperature field 𝑢(𝒙) solution of the Neumann problem and let us introduce 𝑢 ′ (𝒙) such that: Because 𝑺.𝑺 = 𝑰, it follows that:

𝑢 ′ (𝒙) = 𝑢(𝑺.𝒙), ∀𝒙 ∈ 𝛺 ′ (26)
div(𝒑 ′ (𝒙)) = div(𝒑)(𝑺.𝒙), ∀𝒙 ∈ 𝛺 ′ - (31)
Then, 𝒑 ′ (𝒙) is divergence-free if 𝒑 is also divergence-free. Again, the interface between individual cells, , has been excluded because 𝑺 is not differentiable. However, since 𝒑.𝒏 = 0 at the interfaces , 𝒑 ′ (𝒙).𝒏 is also null, and then the equilibrium is fulfilled at the interfaces  as well. Moreover, since 𝒑.𝒏 = 0 at the boundary of 𝛺, 𝒑 ′ (𝒙).𝒏 is also null on the boundary of 𝛺 ′ and is then anti-periodic.

Thermal relationships

The conductivity law in 𝛺 is:

𝜺(𝒙) = 𝑯(𝒙).[𝒑(𝒙) + 𝜮], ∀𝒙 ∈ 𝛺 ( 32
)
where 𝑯(𝒙) is resistivity tensor, the inverse of the conductivity tensor 𝑲(𝒙). Replacing 𝒙 by 𝑺.𝒙 and owing to Eq. ( 31) together with Eq. ( 28), one obtains:

𝜺 ′ (𝒙) = 𝑺.𝑯(𝑺.𝒙).[𝑺.𝒑 ′ (𝒙) + 𝜮], ∀𝒙 ∈ 𝛺 ′ (33)
Because 𝑺.𝑺 = 𝑰, the above relation can be also read:

𝜺 ′ (𝒙) = 𝑯 ′ (𝒙).[𝒑 ′ (𝒙) + 𝜮 ′ (𝒙)], ∀𝒙 ∈ 𝛺 ′ (34)
with the following definitions for 𝑯 ′ (𝒙) and 𝜮 ′ (𝒙):

𝑯 ′ (𝒙) = 𝑺.𝑯(𝑺.𝒙).𝑺, 𝜮 ′ (𝒙) = 𝑺.𝜮 (35)
As a conclusion, 𝒖 ′ (𝒙) is the solution of a conductivity problem over 𝛺 ′ with the resistivity tensor 𝑯 ′ (𝒙) and the prescribed eigenfield 𝜮 ′ (𝒙).

Note that 𝑯 ′ (𝒙) = (𝑲 ′ (𝒙)) -1 where 𝑲 ′ (𝒙) is the conductivity introduced in the Dirichlet problem (see Eq. ( 20)). The eigenfield 𝜮 ′ (𝒙) is a piecewise constant function. The value taken by 𝜮 ′ (𝒙) in 𝛺 ′ is illustrated in the 2D case in Fig. 4.

Homogenized conductivity

In virtue of the Hill-Mandell Lemma, the macroscopic temperature gradient is given by (the proof is very similar to that already given in Section 3.5):

𝑬 = ⟨𝑺.𝜺 ′ (𝒙)⟩ 𝛺 ′ (36)
The Neumann problem can be solved with the classic FFT iterative schemes adapted to account for the loading field 𝜮 ′ (𝒙), the details are provided in Appendix.

Applications

As an application purpose, we consider a composite conductor made up of circular inclusions. The conductivity of the matrix and the inclusions are denoted 𝑘 1 and 𝑘 2 respectively. Three unit cells are considered for the calculations and are illustrated in The unit cell (a) corresponds to a single centered circular inclusion of radius 𝑅 = 0.45 (the size of the inclusion is 1 along each space direction). The value 𝑅 = 0.45 of the inclusion radius is chosen to exhibit better the difference between the different kinds of boundary conditions. Indeed, with a smaller value of the radius, the effective conductivities obtained with the three boundary conditions become much closer to each other. The unit cell (b) is constituted of four quarters of a single circular inclusion of radius 𝑅 = 0.45. Note that unit cells (a) and (b) lead to the same value of the effective conductivity when the periodic boundary conditions are considered but are not equivalent for the Dirichlet and Neumann conditions. The unit cell (c) contains a population of 51 inclusions with the same radius 𝑅 = 0.05 (the size of the unit cell is still equal to 1).

Note that the exact expressions of Fourier coefficients of the loading parameter 𝑬 ′ (𝒙) and 𝜮 ′ (𝒙) are considered. For instance, the Fourier transform of 𝑬 ′ (𝒙) is:

Ê′ (𝝃) = 1 |𝛺 ′ | ∫ 𝛺 ′ 𝑬 ′ (𝒙) exp(-𝑖𝝃.𝒙)𝑑𝑥 1 𝑑𝑥 2 (37)
The field 𝑬 ′ (𝒙) being a piecewise constant function in 𝛺 ′ , the integration is trivial and leads to:

Ê′ (𝝃) = ( ŝ2 (𝝃)𝐸 1 ŝ1 (𝝃)𝐸 2 ) , ŝ1 (𝝃) = -𝑖 sinc(𝐿 2 𝜉 2 ) sinc ( 𝐿 1 𝜉 1 2 ) sin ( 𝐿 1 𝜉 1 ) , ( 38
)
ŝ2 (𝝃) = -𝑖 sinc(𝐿 1 𝜉 1 ) sinc ( 𝐿 2 𝜉 2 2 ) sin ( 𝐿 2 𝜉 2 )
The Fourier coefficients of the components of 𝜮 ′ (𝒙) are:

Σ′ (𝝃) = ( ŝ1 (𝝃)𝛴 1 ŝ2 (𝝃)𝛴 2 ) ( 39
)
The two fields 𝑬 ′ and 𝜮 ′ are piecewise constant over a set of rectangular domains. The characteristic functions of these domains can be defined either in real space or in Fourier space, leading to two different expressions of 𝑬 ′ and 𝜮 ′ . Obviously, the transition from one space to the other is easily achieved using the FFT. Both calculations are rigorously identical when considering an infinite number of wave vectors. When taking a finite number of wave vectors, this leads to differences due to the discretization and the truncature approximation.

We first provide the results corresponding to a single centered inclusion (cells (a) and (d) in Fig. 5). The convergence of the effective conductivity with the number of wave vectors is presented in Fig. 6. The effective conductivity is computed with 𝑁 = 32, 64, 128, 256, 512 wave vectors along each space direction considering the primal, dual and polarization based iterative schemes for Dirichlet condition (at the right), the Neumann condition (at the left), a phase contrast 𝑘 2 ∕𝑘 1 of 100 at the top and 0.01 at the bottom.

In the case of a high contrast, 𝑘 2 ∕𝑘 1 = 100, the dual and polarization iterative schemes have a better convergence with the grid refinement than the primal iterative scheme based on the temperature gradient. This is observed for both the Dirichlet and Neumann boundary conditions. Conversely, regarding the results in the case of a small contrast, 𝑘 2 ∕𝑘 1 = 0.01, a better convergence with the grid refinement is observed with the primal iterative scheme and the polarization. To conclude the results for the single centered inclusion, the polarization method has a better convergence with the grid refinement whatever the value of contrast.

Interestingly, we have extrapolated the value of the effective conductivity to estimate its value for any value of the number of wave vectors 𝑁 taken along each spatial direction, i.e., the resolution of the image. Assuming that the dependence of 𝐾 ℎ𝑜𝑚 on the resolution 𝑁 follows the form of a power series:

𝐾 ℎ𝑜𝑚 (𝑁) = 𝑘 1 𝐽 ∑ 𝑗=1 𝐴 𝑗
𝑁 𝑗-1 (40)

In order to determine the coefficient 𝐴 𝑗 for 𝑗 = 1, … , 𝐽 , it is necessary to assume that 𝐾 ℎ𝑜𝑚 (𝑁) is known for a finite number of 𝑁 values, denoted 𝑁 1 , 𝑁 2 , … , 𝑁 𝐽 such that 𝑁 1 < 𝑁 2 < ⋯ < 𝑁 𝐽 . The coefficients 𝐴 1 , … , 𝐴 𝐽 can then be identified by solving the following linear system:

𝑘 1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 1 𝑁 1 ... 1 𝑁 𝐽 -1 1 1 1 𝑁 1 ... 1 𝑁 𝐽 -1 1 ⋮ 1 1 𝑁 𝐽 ... 1 𝑁 𝐽 -1 𝐽 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐴 1 𝐴 2 ⋮ 𝐴 𝐽 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐾 ℎ𝑜𝑚 (𝑁 1 ) 𝐾 ℎ𝑜𝑚 (𝑁 2 ) ⋮ 𝐾 ℎ𝑜𝑚 (𝑁 𝐽 ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (41)
The effective conductivity 𝐾 ℎ𝑜𝑚 (𝑁 𝑗 ) is computed for resolutions 𝑁 1 , 𝑁 2 , … , 𝑁 𝐽 , where 𝑗 = 1, … , 𝐽 . In the present study, 𝐾 ℎ𝑜𝑚 is computed for five different resolutions, namely 32, 64, 128, 256, 512. The extrapolated function 𝐾 ℎ𝑜𝑚 (𝑁) is shown in Fig. 6 for the three iterative schemes. The solid line represents the extrapolated function, while the crosses correspond to the discrete values of 𝐾 ℎ𝑜𝑚 computed for the different resolutions. It is worth noting that the extrapolated function provides an estimate of the conductivity for an infinite resolution (𝑁 = +∞), which corresponds to 𝑘 1 𝐴 1 . Remarkably, the limits computed by the three iterative schemes converge to very close values as the resolution tends to infinity, allowing for a reliable estimation of the theoretical value of 𝐾 ℎ𝑜𝑚 for both kinds of boundary conditions.

In Fig. 7, we represent the effective conductivities corresponding to the Dirichlet and Neumann problem as functions of the phase contrast 𝑘 2 ∕𝑘 1 for the single centered inclusion at the left (unit cells (a) and (d)), and the population of inclusion at the right (unit cells (c) and (f)). As a validation purpose, the results obtained with the FFT are compared with those computed with the FEM for which a good accuracy is observed. As expected, it is noticed that the solution derived with the periodic boundary and the applied uniform temperature gradient 𝑬 (classic periodic solution) is bounded by the two other ones corresponding to Dirichlet and Neumann conditions.

The monotonicity of the estimated effective conductivity is observed only when the FFT method uses the shape function, i.e. the exact Fourier transform of the characteristic function of the phase. The method based on the shape function is explained in the work of [START_REF] Bonnet | Effective properties of elastic periodic composite media with fibers[END_REF]. The advantage is to describe exactly the microstructure geometry while in the original method of Moulinec andSuquet (1994, 1998), the unit cell is discretized with pixels that correspond to the inexactly integrated FFT methods pointed out by [START_REF] Monchiet | Combining FFT methods and standard variational principles to compute bounds and estimates for the properties of elastic composites[END_REF] and [START_REF] Vondřejc | Improved guaranteed computable bounds on homogenized properties of periodic media by the Fourier-Galerkin method with exact integration[END_REF]. However, the use of shape function is only restricted to some particular geometries for which it is possible to derive the exact Fourier transform of the characteristic function, the different problems for which the shape function can be used has been depicted in the recent work of [START_REF] Nguyen | Derivation of FFT numerical bounds of the effective properties of composites and polycristals[END_REF].

We now provide the solution with the unit cells containing four quarters of circular inclusion (unit cells (b) and (e) in Fig. 8). The effective conductivity is provided as a function of the phase contrast varying from 10 -3 to 10 3 for the three kinds of boundary conditions and considering the FFT and FE methods. It can be observed that the effective conductivity computed with the Dirichlet boundary conditions diverges for highly conductive inclusions. Such result has already been observed by [START_REF] Salmi | Apparent and effective mechanical properties of linear matrix-inclusion random composites: Improved bounds for the effective behavior[END_REF] for instance. When the conductivity 𝑘 2 is very large, we observe that the flux remains finite in the inclusions that do not intersect the edge, so that the temperature gradient is close to zero. At the limit, for infinitely conductive inclusions, the temperature gradient in these inclusions therefore tends towards zero. However, for the inclusions that intersect the edge of the cell, the temperature gradient cannot be zero as a consequence of the Dirichlet condition, which imposes 𝑢 = 𝑬.𝒙, on the boundary of the cell. Consequently, this implies a '' blow-up'' of the flux in these inclusions, as mentioned in the work of [START_REF] Salmi | Apparent and effective mechanical properties of linear matrix-inclusion random composites: Improved bounds for the effective behavior[END_REF], that leads to the divergence of the effective conductivity. Dually, i.e. for highly resistive inclusions and the Neumann condition at the edge, we observe a blowup of the temperature gradient in the inclusions that intersect the edge of the cell. As a consequence, the effective conductivity tends to zero (divergence of the effective resistivity) in the case of highly resistive inclusions when the Neumann boundary condition is used.

The distribution of the temperature gradient 𝜀 ′ 1 and the flux 𝜎 ′ 1 corresponding to the Dirichlet problem (with 𝐸 1 = 1 and 𝐸 2 = 0) and the Neumann problem (with 𝛴 1 = 1 and 𝛴 2 = 0) are provided in Fig. 9. The phase contrast is 𝑘 2 ∕𝑘 1 = 1000. It is observed, for the Dirichlet problem, that the flux in the inclusions is of the same magnitude as the contrast: this explains the divergence of the effective conductivity for highly conductive inclusions. In Fig. 10 another microstructure is tested to support our conclusion. At the left, we show a unit cell 𝛺 ′ obtained by the mirror symmetry of the original cell 𝛺 corresponding to a population of inclusions for which some of them cross the boundary. At the right, we provide the distribution of 𝜀 ′ 1 applying the Dirichlet condition with 𝐸 1 = 1, 𝐸 2 = 0 and we choose the contrast 𝑘 2 ∕𝑘 1 = 100. It is observed that the temperature gradient concentrates in the inclusions that cross the top and bottom edges of the individual cells and the amplitude is still of the Fig. 7. Variations of the effective conductivity as a function of the phase contrast 𝑘 2 ∕𝑘 1 . Comparisons between the FFT and FEM solutions for the three kinds of boundary conditions. At the left : one single centered inclusion (unit cell (a) in Fig. 5), at the right : a population of circular inclusions (unit cell (c) in Fig. 5). same magnitude as the contrast. Again, in that situation, the solution diverges for highly conductive inclusions. This problem is inherent with the use of the Dirichlet condition that is not suited when the unit cell boundary is crossed by the inclusion. When the Dirichlet condition is used, the unit cell must be chosen such that the inclusions do not cross the boundary or must be considered only if the contrast is inferior to 1. Also, the problem with Neumann condition must be applied to unit cell for which the inclusion do not cross the boundary or if the contrast is superior to 1.

Finally, we derive the homogenized conductivity for a composite with anisotropic phases. The problem under consideration corresponds to the unit cell (c) in Fig. 5. Random values for the conductivity in the matrix and the inclusions are used (but ensuring the positiveness of the conductivity tensor in each phase). The conductivity tensors (expressed in the cartesian frame) in 𝛺 1 and 𝛺 2 regions are: with the grid refinement are provided in Figs. 11,12,13 for the Dirichlet and Neumann problems. The conductivities computed with the primal, dual and polarization based iterative schemes for 𝑁 = 32, 64, 128, 256, 512 and the results are also extrapolated in order to evaluate the limit at 𝑁 → +∞. Again, the interpolation with the different iterative schemes leads to very close values of the limit.

𝑲 1 = ( 1.
The results are summarized in Table 1. We provide the limit at 𝑁 = +∞ obtained for the three iterative schemes and for the two kind of boundary conditions. Also we compare our solutions with FE ones, the relative error is given under parenthesis in Table 1 FE solutions. The FE solutions are computed considering 123 260 triangular elements. The proposed FFT methods leads to very accurate values of the conductivities in the case of an anisotropic material.

Conclusion

FFT methods have been generalized to handle the problem of composite conductors with uniform boundary conditions. The resulting iterative schemes involve only minor modifications and existing codes are easily adapted to solve these homogenization problems. The computation is performed on an extended cell obtain by mirror symmetry that is easily built from the initial image of the microstructure. Moreover, these unit cells are loaded with a piecewise constant eigenfield for which simple analytic expressions are derived in the paper. The method has been illustrated for 2D microstructures with circular inclusions and the results have been compared with FEM with a good accuracy. The method could be extended for other problems such as elastic composites that would constitute future developments.

Declaration of competing interest

All authors declare have no any financial and personal relationships with other people or organizations that could inappropriately influence their work.

Data availability

No data was used for the research described in the article

Appendix. Resolution with FFT algorithm

A.1. Dirichlet problem

Consider the Dirichlet problem that involves the temperature gradient 𝒒 ′ (the dependence of the fields with the coordinates 𝒙 is omitted in this section for the sake of simplicity) whose volume average over the domain 𝛺 ′ is null, ⟨𝒒 ′ ⟩ 𝛺 ′ = 0. The associated flux, 𝝈 ′ = 𝑲 ′ .(𝒒 ′ + 𝑬 ′ ) is divergence free. Note that 𝑬 ′ is a piecewise constant field whose volume average is also null, ⟨𝑬 ′ ⟩ 𝛺 ′ = 0. The flux being divergence free, it implies that it satisfies to:

𝜞 0 * 𝝈 ′ = 0 (A.1)
where 𝜞 0 is the classic periodic Green tensor for the thermal conductivity problem (the reader can refer to [START_REF] Eyre | A fast numerical scheme for computing the response of composites using grid refinement[END_REF], [START_REF] Milton | The Theory of Composites[END_REF] and [START_REF] Monchiet | A polarization-based fast numerical method for computing the effective conductivity of composites[END_REF] for the definition of the Green tensor) and '' * '' represent the convolution product. An iterative scheme associated with the Dirichlet problem can then be read:

𝒒 ′ 𝑖+1 = 𝒒 ′ 𝑖 -𝜞 0 * [ 𝑲 ′ .(𝒒 ′ 𝑖 + 𝑬 ′ ) ] (A.2)
This iterative scheme is initialized with 𝒒 ′ 1 = 0. Note that it is possible to introduce 𝜺 ′ 𝑖 = 𝒒 ′ 𝑖 + 𝑬 ′ and to reformulate the iterative scheme as follows:

𝜺 ′ 𝑖+1 = 𝜺 ′ 𝑖 -𝜞 0 * [ 𝑲 ′ .𝜺 ′ 𝑖 ] (A.3)
In this second form, the iterative process is initialized with 𝜺 ′ 1 = 𝑬 ′ . At convergence of the iterative scheme the macroscopic flux is computed from Eq. (25).

Many other alternative methods have been provided in the literature to improve the convergence of the method. For instance, a dual method based on the flux can be employed to solve the Dirichlet problem. The temperature gradient 𝒒 ′ being curl free, it satisfies to:

𝜟 0 * 𝒒 ′ = 0 (A.4)
where 𝜟 0 is the dual Green tensor whose expression can be found in the same aforementioned references. Eq. (A.4) can be also expressed with

Fig. 1 .

 1 Fig. 1. Unit cell of the composite conductor.

Fig. 2 .

 2 Fig. 2. Extended unit cell of the composite conductor.

Fig. 3 .

 3 Fig. 3. Values of components of 𝑬 ′ (𝒙) taken in the unit cell 𝛺 ′ for the 2𝐷 case.

  The temperature field 𝑢 ′ (𝒙) is continuous across the interfaces between individual cells. We denote the temperature gradient in 𝛺 by 𝜺 and the temperature gradient in 𝛺 ′ by 𝜺 ′ :𝜺(𝒙) = ∇𝑢(𝒙), ∀𝒙 ∈ 𝛺, 𝜺 ′ (𝒙) = ∇𝑢 ′ (𝒙), ∀𝒙 ∈ 𝛺 ′ (27)By applying the chain rule, one can obtain the following relation between 𝜺(𝒙) and 𝜺 ′ (𝒙):𝜺 ′ (𝒙) = 𝑺.𝜺(𝑺.𝒙), ∀𝒙 ∈ 𝛺 ′ (28)4.2. EquilibriumThe flux related to this problem is divergence-free and satisfies the Neumann condition 𝝈.𝒏 = 𝜮.𝒏 at the boundary of the unit cell 𝛺. Let us decompose 𝝈 as follows: 𝝈 = 𝜮 + 𝒑 where 𝒑 is divergence-free and satisfies to the Neumann condition 𝒑.𝒏 = 0 at the boundary of the unit cell 𝛺. Let us introduce 𝒑 ′ the vector field over the extended domain 𝛺 ′ such that: 𝒑 ′ (𝒙) = 𝑺.𝒑(𝑺.𝒙), ∀𝒙 ∈ 𝛺 ′ (29) 𝒑 ′ (𝒙) is also divergence-free. Indeed, applying the divergence to 𝒑 ′ (𝒙) taking into account Eq. (31) yields: div(𝒑 ′ (𝒙)) = div(𝑺.𝒑(𝑺.𝒙)) = 𝑺 ∶ [∇𝒑(𝑺.𝒙).𝑺] , ∀𝒙 ∈ 𝛺 ′ (30)

Fig. 4 .

 4 Fig. 4. Values of 𝜮 ′ (𝒙) taken in the unit cell 𝛺 ′ .

  Fig. 5. The first line (cells (a), (b) and (c)) correspond to the original cells on which the Dirichlet or Neumann boundary conditions are applied. The second line correspond to the extended cells obtain by the reflexion with respect to the axes 𝑂𝑥 1 and 𝑂𝑥 2 over which the periodic boundary conditions and the applied loading field 𝑬 ′ (𝒙) (for the Dirichlet condition) or 𝜮 ′ (𝒙) (for the Neumann condition). Also, for comparison, we also compute the classic periodic solution under the applied uniform temperature gradient 𝑬 to the cells (a), (b) and (c). Our results are also compared with FE solutions to validate the approach considering the periodic, Dirichlet and Neumann conditions to the unit cell (a), (b) and (c).

Fig. 5 .

 5 Fig. 5. Unit cells of the composite conductor.

Fig. 6 .

 6 Fig. 6. Variations of the effective conductivity as a function of the number of wave vectors for one single centered circular inclusion. Figures at the top: problem with Dirichlet condition (at the right) or Neumann boundary condition (at the left) for 𝑘 2 ∕𝑘 1 = 100. Figures at the bottom: the same boundary conditions but for a contrast 𝑘 2 ∕𝑘 1 = 0.01.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Variations of the effective conductivity as function of the phase contrast 𝑘 2 ∕𝑘 1 . Comparisons between the FFT and FEM solutions for the three kind of boundary conditions.

Fig. 10 .

 10 Fig. 10. Unit cell with a population of inclusions crossing the boundary (at the left). Distribution of 𝜀 ′ 1 (at the right) for the Dirichlet problem (with 𝐸 1 = 1 and 𝐸 2 = 0) and the contrast 𝑘 2 ∕𝑘 1 = 100.

Fig. 11 .

 11 Fig. 11. Variations of the conductivity 𝐾 ℎ𝑜𝑚 as functions of the grid refinement for the Dirichlet problem (at the left) and the Neumann problem (at the right). Case of a composite with a population of inclusions and local anisotropic phases.

Fig. 12 .

 12 Fig. 12. Variations of the homogenized conductivity 𝐾 ℎ𝑜𝑚 as functions of the grid refinement for the Dirichlet problem (at the left) and the Neumann problem (at the right). Case of a composite with a population of inclusions and local anisotropic phases.

Fig. 13 .

 13 Fig. 13. Variations of the homogenized conductivity 𝐾 ℎ𝑜𝑚 22 as function of the grid refinement for the Dirichlet problem (at the left) and the Neumann problem (at the right). Case of an composite with a population of inclusions and local anisotropic phases.

Table 1

 1 Homogenized conductivities of the anisotropic composite. Comparison of the primal, dual and polarization based solutions (at 𝑁 = +∞) and the FEM one.

		Dirichlet			Neumann
		𝐾 ℎ𝑜𝑚 11	𝐾 ℎ𝑜𝑚 12	𝐾 ℎ𝑜𝑚 22	𝐾 ℎ𝑜𝑚 11	𝐾 ℎ𝑜𝑚 12	𝐾 ℎ𝑜𝑚 22
	FFT primal	4.4974	1.8904	4.0813	4.2808	1.8652	3.8221
		(0.080%) (0.026%) (0.010%) (0.096%) (0.016%) (0.034%)
	FFT dual	4.4927	1.8893	4.0804	4.2860	1.8653	3.8190
		(0.024%) (0.032%) (0.012%) (0.026%) (0.011%) (0.047%)
	FFT polarization 4.4932	1.8881	4.0820	4.2820	1.8656	3.8198
		(0.013%) (0.095%) (0.027%) (0.068%) (0.005%) (0.026%)

the flux: 𝜟 0 * (𝑯 ′ .𝝈 ′ -𝑬 ′ ) = 0 (A.5)

The following iterative process can then be used to solve the Dirichlet problem:

This iterative scheme is initialized with 𝝈 ′ 1 = 0. At convergence, the macroscopic flux is determined with Eq. (25).

A faster method for computing the solution of the problem consists to consider the polarization method that uses both Green tensors and the following definition for the polarization tensor:

The polarization based iterative scheme reads:

The iteration is initialized with 𝝉 ′ 1 = 0. In this equation 𝛼 and 𝛽 are two coefficients, their values must be chosen such that 𝛼 ∈ ]0, 2] and 𝛽 ∈ [-2, 0[ to ensure the convergence following [START_REF] Monchiet | Combining FFT methods and standard variational principles to compute bounds and estimates for the properties of elastic composites[END_REF]Bonnet (2012, 2013). In our applications, we use 𝛼 = 1 and 𝛽 = -1.

A.2. Neumann problem

By duality, one can easily derive the FFT based iterative scheme for the Neumann problem. The strain based iterative scheme is:

The stress based iterative scheme is:

And the dual version of the polarization based iterative scheme uses the field: .11) and reads;

The strain, stress and polarization based iterative schemes are initialized with 𝜺 ′ 1 = 0, 𝝈 ′ 1 = 𝜮 ′ and 𝝉 ′ 1 = 0.


Shian Guan

Aline Rougier

Oudomsack Viraphong

Dominique Denux

Nicolas Penin

Manuel Gaudon

Two-step synthesis of VO2(M

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

More than 10 kinds of vanadium dioxide (VO 2 ) crystalline phases have been reported in the literature. Tremendous research efforts have been devoted to the various VO 2 polymorphs syntheses. [START_REF] Yu | Phase evolution and crystal growth of VO 2 nanostructures under hydrothermal reactions[END_REF][START_REF] Alie | Direct synthesis of thermochromic VO 2 through hydrothermal reaction[END_REF][START_REF] Mutta | Facile hydrothermal synthesis of economically viable VO 2 (M1) counter electrode for dye sensitized solar cells[END_REF][START_REF] Batista | Synthesis and characterization of VO 2 -based thermochromic thin films for energyefficient windows[END_REF][START_REF] Cao | Solution-based fabrication of VO 2 (M) nanoparticles via lyophilisation[END_REF][START_REF] Cao | A simple and low-cost combustion method to prepare monoclinic VO 2 with superior thermochromic properties[END_REF][START_REF] Chen | An ethylene glycol reduction approach to metastable VO 2 nanowire arrays[END_REF][START_REF] Dahlman | Electrochemically Induced Transformations of Vanadium Dioxide Nanocrystals[END_REF][START_REF] Li | Low-Valent Vanadium Oxide Nanostructures with Controlled Crystal Structures and Morphologies[END_REF][START_REF] Liu | Metastable Vanadium Dioxide Nanobelts: Hydrothermal Synthesis, ElectricalTransport, and Magnetic Properties[END_REF][START_REF] Zhang | Manipulations from oxygen partial pressure on the higher energy electronic transition and dielectric function of VO 2 films during a metal-insulator transition process[END_REF] At low temperature, monoclinic VO 2 (M) is the most stable one and undergoes a first-order metal-insulator transition (MIT) to rutile phase VO 2 (R) at 68 °C associated with a drastic change in optical properties in the near-infrared range. [START_REF] Batista | Synthesis and characterization of VO 2 -based thermochromic thin films for energyefficient windows[END_REF][START_REF] Cao | A simple and low-cost combustion method to prepare monoclinic VO 2 with superior thermochromic properties[END_REF] Owing to these properties, VO 2 (M) has been considered in a wide range of applications including optical switching, optical sensing, and in particular as a strong candidate for smart windows because of its significant effect in energy savings. [START_REF] Batista | Synthesis and characterization of VO 2 -based thermochromic thin films for energyefficient windows[END_REF][START_REF] Chen | VO 2 -based double-layered films for smart windows: Optical design, all-solution preparation and improved properties[END_REF][START_REF] Goncalves | Smart optically active VO2 nanostructured layers applied in roof-type ceramic tiles for energy efficiency[END_REF][START_REF] Melnik | Low-temperature method for thermochromic high ordered VO 2 phase formation[END_REF][START_REF] Zhou | VO 2 thermochromic smart window for energy savings and generation[END_REF] Indeed, windows are known as one of the most energy-inefficient components of buildings. VO 2based thermochromic smart windows exhibit a self-regulating ability of optical transmission/reflection with temperature changes near room temperature, and so, they offer smart functions to decrease building energy consumption.

In close relation with smart window applications, the VO 2 (M) phase is mainly directly elaborated as thin films on glass substrates by physical deposition processes as magnetron sputtering, pulsed laser deposition (PLD), or epitaxial growth. [START_REF] Nag | Synthesis of vanadium dioxide thin films and nanoparticles[END_REF] More recently, several methods have been reported to prepare VO 2 nanoparticles, such as the sol-gel synthesis, [START_REF] Wang | sol-gel process and one-step annealing of vanadium dioxide thin films: Synthesis and thermochromic properties[END_REF] hydrothermal synthesis, [START_REF] Alie | Direct synthesis of thermochromic VO 2 through hydrothermal reaction[END_REF][START_REF] Hou | Facile synthesize VO 2 (M1) nanorods for a low-cost infrared photodetector Application[END_REF] and pyrolysis method. [START_REF] Wang | sol-gel process and one-step annealing of vanadium dioxide thin films: Synthesis and thermochromic properties[END_REF][START_REF] Zheng | Preparation and Characterization of VO 2 Nanopowders[END_REF] In comparison with physical deposition processes, the use of nanoparticles presents some advantages in energy efficiency owing to their small size, large surface area, great chemical reactivity, and mechanical strength associated with a low cost coating process at room temperature and at atmospheric pressure. The processes based on the pyrolysis of a vanadium oxide precursor have gained popularity, [START_REF] Wang | sol-gel process and one-step annealing of vanadium dioxide thin films: Synthesis and thermochromic properties[END_REF][START_REF] Zheng | Preparation and Characterization of VO 2 Nanopowders[END_REF] since (i) the VO 2 (M) composition (the homogeneous doping with various metallic elements is shown and the V/O ratio can be tuned) can be controlled from the parameters of the pyrolysis step (oxygen partial pressure, temperature), (ii) this synthesis route is really easy to set up (in comparison to hydrothermal processes, for illustration), and (iii) it shows a prospect of mass production. As a commonly used vanadium precursor, vanadyl ethylene glycolate (VEG) has been investigated through the pyrolysis method to synthesize VO 2 (M) submicronic powder. [START_REF] Choi | Synthesis of colloidal VO 2 nanoparticles for thermochromic applications[END_REF][START_REF] Mjejri | Low-Cost and Facile Synthesis of the Vanadium Oxides V 2 O 3 , VO 2 , and V 2 O 5 and Their Magnetic, Thermochromic and Electrochromic Properties[END_REF] In our group, Mjejri et al. reported the synthesis of VO 2 powder from a single polyol route by annealing the VEG directly under very low oxygen pressure (∼10 -5 Pa). [START_REF] Mjejri | Low-Cost and Facile Synthesis of the Vanadium Oxides V 2 O 3 , VO 2 , and V 2 O 5 and Their Magnetic, Thermochromic and Electrochromic Properties[END_REF] In order to eliminate or weaken the effect of energy surplus during VEG decomposition, Zhang et al. prepared VO 2 by dividing the VEG pyrolysis process into two steps. [START_REF] Zhang | A cost-effective method to fabricate VO 2 (M) nanoparticles and films with excellent thermochromic properties[END_REF] Zou et al. obtained VO 2 powder via the thermolysis of VEG precursor in air atmosphere. [START_REF] Zou | A low-temperature synthesis of monoclinic VO 2 in an atmosphere of air[END_REF] However, most of the studies in the literature focused on the thermochromic properties of synthesized VO 2 powders, while the study of the decomposition of VEG with different annealing temperatures has not yet been investigated. Moreover, the versatility of the oxidation number of vanadium leads to the difficultly to propose a robust flow-chart for a reproducible synthesis. Furthermore, even if all the studies point out the need for high crystalline quality for VO 2 powder, as far as we know, this aspect was not deeply investigated in previous studies by following the crystalline fraction from accurate X-ray investigations. This is a key issue as the occurrence of undesired amorphous VO 2 without any thermochromic properties would act drastically to the detriment of the smart optical/electrical effect of the asprepared powders. Thus, the calculation of VO 2 (M) crystallinity appears necessary and could greatly help us to evaluate the particles' thermochromic performances.

Herein, pure VO 2 (M) powder with high crystallinity is synthesized through a two-step thermolysis and postannealing process. The VEG is first decomposed in air, and a poor crystallized VO 2 powder is obtained after 3 min heating, prior to further annealing at higher temperatures in an oxygen-free environment for a full-crystallization. The crystallinity of three different VO 2 powders is carefully determined in parallel with the VO 2 particles morphological evolution. The VEG decomposition in the first step is investigated carefully, and it is found the carbon plays an important role in the second step of VO 2 crystallization. Moreover, novel and particular microstructures for V 2 O 3 and V 2 O 5 compounds are also synthesized through this method changing the atmosphere of the second step annealing. Step Thermolysis for the VEG Decomposition. A total of 1.42 g of NH 4 VO 3 powder was mixed with a certain amount of C 2 H 6 O 2 in a 250 mL three-neck flask and heated to 160 °C under vigorous stirring as reported. [START_REF] Mjejri | Low-Cost and Facile Synthesis of the Vanadium Oxides V 2 O 3 , VO 2 , and V 2 O 5 and Their Magnetic, Thermochromic and Electrochromic Properties[END_REF] The precipitate was collected by centrifugation, washed with ethanol three times, and dried overnight in oven at 80 °C to remove excess of ethanol. Then the obtained vanadyl ethylene glycolate (VEG) precursor was heated in the furnace (air) at 300 °C for durations from 1 to 90 min.

EXPERIMENTAL SECTION

2.2.2. Post Annealing Treatment of Poorly Crystallized VO 2 Powder. The resulting powders from first step were further annealed in a nearly oxygen-free atmosphere (PO 2 < 10 -4 bar) to enhance their crystallinity of VO 2 powders. After the decomposition of VEG for 3 min at 300 °C, the powder (VO x -3 min) was chosen as the starting material for the post annealing for 5 h at temperatures of 500 and 700 °C in the second step, as illustrated by Scheme 1. The annealing temperature was controlled by a temperature controller with a heating rate of about 10 °C/min. Furthermore, V 2 O 3 and V 2 O 5 with new and particular morphologies were prepared by annealing VO x -3 min powder under different conditions: 800 °C for 0.5 h under Ar/H 2 atmosphere (95:5 mol % ratio) for the preparation of vanadium sesquioxide, and 600 °C for 0.5 h under air for the preparation of vanadium pentoxide.

2.3. Characterizations. The crystal structures of the as-prepared samples were determined by X-ray diffraction (XRD) analyses (Philips PW1820, PANalytical X' Pert PRO MPD diffractometer) performed using a Cu-Kα1 radiation source (λ = 1.54056 Å), with a divergent slit of 1°and receiving slit size of 0.1 mm in a 2θ range from 8°to 80°. The identification of compounds was made by comparing the experimental XRD patterns to standards compiled by the Joint Committee on Powder Diffraction and Standards. Moreover, Rietveld refinements were performed using FullProf software. The morphology of the as-prepared particles was observed by SEM using a TESCAN Vega II SBH microscope and JEOL JSM-6700F, and also by high resolution transmission electron microscopy (HRTEM, JEOL 2200FS, operating at 200 kV). Carbon content of each sample was determined by CHNS elementary analyzer (Thermo Fisher Scientific). Thermogravimetric analysis (TGA) was carried out on a Setaram TGA instrument. Differential scanning calorimetry (DSC) experiment of as-obtained VO 2 powders was performed using PerkinElmer DSC. The reflectance spectra were measured using a Varian Cary UV-vis-NIR spectrophotometer in a wavelength region from 200 to 2500 nm.

RESULTS AND DISCUSSION

3.1. Decomposition of VEG. Vanadyl ethylene glycolate (VEG: VC 2 H 4 O 3 ) is prepared by polyol synthesis to be used as vanadium precursor. During its preparation, V(V) in NH 4 VO 3 is reduced to V(IV) by ethylene glycol (EG) with the consequent formation of vanadyl complex. EG is not only a well-known solvent used in the polyol method to prepare highly divided metal particles, but also a cross-linking reagent coordinating to the central metal ion to form a metal glycolate further leading to subsequent oligomerization.

To fully decompose VEG in a short time, referring to our own TGA measurements as well as the one reported with DSC data by Zhang et al., [START_REF] Zhang | A cost-effective method to fabricate VO 2 (M) nanoparticles and films with excellent thermochromic properties[END_REF] the VEG is annealed in air at 300 °C for various times (1-5 min, 8 and 90 min). As shown in Figure 1, VEG gradually transforms into VO 2 and then V 2 O 5 after air annealing. During the pyrolysis process of VEG, the effect of the annealing treatment can be separated into two stages.

In the first stage (0-3 min), the VEG precursor starts to decompose and forms a poorly crystallized VO 2 phase, consecutively to carbon chain combustion, as described in eq 1.

The intensity of the peak located at 2θ = 13.46°c orresponding to a preferred (110) Miller orientation of the VEG compound decreases gradually during heating at 300 °C in the first 2 min with simultaneous rising of VO 2 (M) peaks (the main peak being located at 2θ = 27.95°).

Two phases VO 2 (M) as a major phase and VO 2 (B) as a minor one coexist after the heating treatment for 3 min, named VO x -3 min, while no more VEG peaks are detected. With continuous heating, V(V) containing phases start to emerge due to the further oxidation in the second stage (4-90 min), as shown in eq 2.

XRD results show that traces of vanadium(V) oxide powders (V 2 O 5 and V 6 O 13 ) start to form at 4-5 min, resulting from the partial energy surplus due to the exothermic process of VEG decomposition. [START_REF] Zhang | A cost-effective method to fabricate VO 2 (M) nanoparticles and films with excellent thermochromic properties[END_REF] After 8 min annealing, some new peaks (like 2θ = 20.26°and 26.26°) (Figure 1), which belong to V 2 O 5 (space group: Pmmn, JCPDS No. 41-1426), appear, and VO 2 coexists with V 2 O 5 . Finally, VO 2 is entirely converted to V 2 O 5 after VEG being heated for 90 min.

+ → + + 2VO(OCH CH O) 5O 2VO 4H O 4CO 2 2 2 2 2 2
(1)

+ → 4VO O 2V O 2 2 2 5
(2)

Figure 2a insets displays the photos of the VO x obtained powders after decomposition of the VEG precursor. For deeper comparison of the optical properties of the decomposition products, the reflection spectra measured in the 200-2500 nm and in the UV-visible region (200-800 nm) are shown in Figure 2b, respectively. Undecomposed VEG shows a bluish gray color, corresponding to a reflection of visible radiation at around 470 nm. The color changes to black after being heated for 3 min with a rather flat reflectance spectra of less than 5% in the visible region, corresponding to the color of VO 2 powder. After longer heating, an orange color starts to emerge due to the presence of the V 2 O 5 phase. At 8 min, the orange mixes with the remaining black color from VO 2 , and the color of the obtained powder becomes dark green (VO x -8 min), and the reflection gap shifts to a higher wavelength (i.e., 560 nm). Finally, a full orange color appears after 90 min, which indicates the complete oxidation in V 2 O 5 , with a clear charge transfer band gap located at about 600 nm. The color of each powder is consistent with the evolution of the phases previously detected by XRD data in Figure 1.

The corresponding morphologies from low-magnification SEM (Figure 3) consist of elongated cuboids that remain basically unchanged after heating at 300 °C for 90 min. Because of the large surface energy, these cuboids easily aggregate, forming spherical aggregates about 5 μm in size.

In order to gain further insight into the decomposition process of VEG, VEG, and VO x -3 min samples were characterized by high resolution SEM. As shown in Figure 3a, VEG nanostructures exhibit an elongated cuboid morphology and a typical length of several micrometers (∼1.6 μm) and an average length of ∼350 nm (in its rectangular cross section). After 3 min heating in air and removal of the organic components (C and H), the cuboid shape remains, while the detailed morphology as cuboid surface changes from smooth to rough aspect (Figure 3a,b). Combined with its TEM image (inset in Figure 3b), the structure of cuboid after decomposition could be seen more clearly and composed of numerous small particles, which can be attributed to the removal of the organic part in the form of CO 2 and H 2 O during the VEG annealing (eq 1); i.e., the small particles can be attributed to the appearance of the VO 2 crystallites. Hence, VEG crystallites are shown to be the "reservoirs" of the germination and growth of the VO 2 phase. The morphology of the nanometric VO 2 issued from VEG decomposition is so "templated" by the VEG crystallite shape.

A powder architecture at two different scales is observed: isotropic VO 2 crystallites with growing size versus calcination duration are aggregated in a way that the polycrystalline assemblies retain the initial VEG crystallite shape and size.

3.2. Postannealing Treatment of VO x -3 min. 3.2.1. Synthesis of VO 2 . From the combination of the XRD data and morphology of decomposed products, the optimal time of heating VEG is around 3 min. The VO 2 powder (VO x -3 min) obtained from VEG decomposition is of relatively poor crystallinity due to the short heating time (3 min) at a rather low temperature (300 °C), as confirmed by the wide XRD peaks (Figure 1). Thus, a postannealing treatment at higher temperature for the VO x -3 min sample is necessary to prepare VO 2 powder with a higher crystallinity. During the second annealing step, it was found that the decomposition time for the VEG in the first step plays a critical role in the formation of VO 2 .

Four samples with different decomposition times in the first step (VEG, VO x -2 min, VO x -3 min, and VO x -5 min) were chosen for a higher temperature postannealing treatment. These samples were heat treated at 500 °C for 5 h in an environment free of oxygen (PO 2 < 10 -4 bar) in order to avoid VO 2 from being oxidized into V 2 O 5 .

As organic components, carbon materials are usually used as a reducing agent, like hydrazine reported by Cao et al., [START_REF] Cao | Solution-based fabrication of VO 2 (M) nanoparticles via lyophilisation[END_REF] and thus the carbon content of each sample is measured through CHNS before annealing, and its evolution vs heating time is shown in Figure 4a. Overall, the carbon content decreases during the decomposition (eq 1). The initial VEG contains 12.42% of carbon. After a decomposition time of 2 min, 7.89% of carbon still exists in the VO x -2 min sample, which can be attributed to undecomposed VEG remaining in the powder. After VEG is heated for 3 min (VO x -3 min), the carbon content decreases to 0.98%, indicating the nearly full decomposition of VEG. The residual carbon content remains unchanged for longer annealing times (VO x -4 min and VO x -5 min). The carbon contents of most samples before and after annealing at 500 °C are gathered in Table S1 in the Supporting Information.

The XRD patterns of postannealing samples are shown in Figure 4b. The 500 °C postannealing of VO x -3 min leads to the so-called S-3 min sample. Pure V 2 O 3 (S-0 min) is obtained when VEG is annealed at 500 °C in an oxygen-free atmosphere, in agreement with previously reports showing that only V 2 O 3 can be obtained under inert atmospheres. [START_REF] Zhang | A cost-effective method to fabricate VO 2 (M) nanoparticles and films with excellent thermochromic properties[END_REF][START_REF] Zou | A low-temperature synthesis of monoclinic VO 2 in an atmosphere of air[END_REF] Unexpectedly, the XRD of S-2 min sample also shows a pure V 2 O 3 phase (Figure S1). The lack of VO 2 peaks after post annealing treatment indicates that the V(IV) is probably reduced to V(III) by reducing agents for the VO x -2 min sample that is in fact a combination of VEG and VO 2 (Figure 1) and that contains quite a lot of carbon (7.89%). The existence of carbon probably causes the reduction of VO 2 to V 2 O 3 . When the VEG is heated for longer times (3 and 5 min), the carbon content further decreases to a quite low level (0.98%) (Figure 4a), which might avoid the reduction from VO 2 to V 2 O 3 . The XRD pattern of the S-3 min and the S-5 min samples showing mainly VO 2 (M) phase (Figure 4b) confirms our hypothesis that there is no V(III) after the postannealing treatment. However, in the S-3 min sample especially in the S-5 min sample, peaks of undesired V 6 O 13 phase, known as a V(IV) and V(V) mixed valence oxide, [START_REF] Choi | Synthesis of colloidal VO 2 nanoparticles for thermochromic applications[END_REF][START_REF] Zhang | Template-free synthesis of highly porous V 2 O 5 cuboids with enhanced performance for lithium ion batteries[END_REF] are visible. Its presence probably results from excessive oxidation during the VEG decomposition in the first step (Figure 1 and eq 2).

To summarize, a relatively pure VO 2 phase (S-3 min, called later S-500) with high crystallinity is synthesized by our twostep thermal process. First, a thermolysis step allows the preparation of poorly crystallized VO 2 (VO x -3 min), by decomposing the VEG in a strict heating time (∼3 min). The organic part of undecomposed VEG remaining in the powder when the heating time is shorter than 3 min will contribute to reduce the VO 2 to V 2 O 3 in the follow-up postannealing treatment. On the contrary, V(V) coexists with V(IV) in the powder when the decomposition process is longer than 3 min.

As V(V) is the most stable vanadium state, [START_REF] Zhang | Template-free synthesis of highly porous V 2 O 5 cuboids with enhanced performance for lithium ion batteries[END_REF] it is difficult to remove once it is formed. Those results emphasize that the decomposition time for VEG in the first thermolysis needs to be controlled very accurately.

To study the influences of crystallized temperature on VO 2 powders, [START_REF] Xiao | A cost-effective process to prepare VO 2 (M) powder and films with superior thermochromic properties[END_REF] VO x -3 min sample is chosen as the starting material for the postannealing treatment at a higher temperature (700 °C) in an oxygen-free environment (S-700). In an attempt to clarify structural analysis, XRD data were processed by the software program Fullprof, and the Rietveld structure refinement is shown in Figure 5a and Table 1. With satisfactory convergence factors (R Bragg = 4.85, R f = 4.2), the diffraction pattern is refined as the monoclinic phase of VO 2 (M) (space group P2 1 /c), with lattice parameters a = 5.7550(1) Å, b = 4.5291(7) Å, c = 5.3854(8) Å, and β = 122.6030(4)°. The corresponding structure deduced from unit cell parameters and V, O atomic positions extracted from Rietveld refinement is shown in Figure 5b. We confirm that the structure and atomic positions match with JCPDS data card 00-044-0252 from the reported structure of monoclinic VO 2 . [START_REF] Melnik | Low-temperature method for thermochromic high ordered VO 2 phase formation[END_REF] No characteristic peaks of any other phases are detected, indicating that the sample is highly pure. Traces of V 2 O 5 and V 6 O 13 , which were detected in the VO x -3 min powder before postannealing, are probably reduced to VO 2 during annealing at high temperature. [START_REF] Akande | Phase evolution of vanadium oxides obtained through temperature programmed calcinations of ammonium vanadate in hydrogen atmosphere and their humidity sensing properties[END_REF] This reduction is helped by a mechanism of dilution of the minor phases into the VO 2 main phase in order to decrease the surface energies (cosintering phenomenon).

The morphologies of VO 2 powders synthesized by VEG decomposition (VO x -3 min), and postannealing of VO x -3 min sample at 500 °C (S-500) and 700 °C (S-700) for 5 h, are displayed in Figure 6. The VO x -3 min sample shows a cuboid structure composed of uniform nanoparticles (Figure 6a). After postannealing at 500 °C, the elongated shape of the cuboids of VO 2 starts to break down and becomes a rod-like structure (Figure 6b) composed of larger nanoparticles. When the postannealing temperature is increased to 700 °C, the rodlike structure is further broken down, and the isolated VO 2 particles become more regular with clear grain boundaries (Figure 6c). To be more specific, the polyhedron (octahedron deformation: i.e., A and B) shape indicates a crystallization through a preferred growth direction (111) ensuring a lower surface energy. Meanwhile, the corresponding particles size distribution (Figure 6d) shows that the size slightly increases from 32 to 46 nm after the postannealing at 500 °C, and greatly increases to 745 nm when the temperature is 700 °C. In the literature reported by Xiao et al., [START_REF] Xiao | A cost-effective process to prepare VO 2 (M) powder and films with superior thermochromic properties[END_REF] a large sintering effect was observed for vanadium dioxide particles in the same temperature ranges. Indeed, a similar large increase of VO 2 particle size is correlated to the huge endothermal peak at 675 °C from DSC data. Those authors also discussed that the aggregation effect decreased drastically because of the low particles surface energy, and well-isolated VO 2 particles were observed, especially when the particle size was larger than 500 nm. The evolution of the morphology from VEG to S-700, is schematized in Figure 6e.

To determine the nature and the proportion of the various components present in the three kinds of VO 2 powders, powder X-ray diffraction was used (Figure 7). The quantification analysis, using Rietveld refinement with the Fullprof program, [START_REF] Pawar | Spin glass behavior and enhanced but frustrated magnetization in Ho 3+ substituted Co-Zn ferrite interacting nanoparticles[END_REF] was deduced from the scale factor of the different phases either amorphous or crystalline. [START_REF] Masudi | A General Method for Quantifying the Amorphous Phase in Nano Polycrystalline Materials[END_REF] An amount of calibration salt: lanthanum hexaboride (LaB 6 ), mixed as 10 wt % with the three VO 2 samples (VO x -3 min, S-500 and S-700), is used as reference for calculation of the components proportions. The Rietveld refinement results are displayed in Figure 7a-c, showing a good-quality fit. The diffraction peaks at 2θ = 27.95°(Figure 7d), corresponding to the preferred (011) orientation of VO 2 (M), are thinner and of higher intensity for the annealed VO 2 powders (S-500 and S-700). The decrease of full width at halfmaximum (fwhm) from 0.554°to 0.292°and 0.162°after the postannealing treatment at 500 and 700 °C, confirms the trend of the increase in average crystallite size in agreement with the TEM and SEM results in Figure 6. In addition to the larger crystallite sizes after being postannealed, the crystallinity of VO 2 particles also increases significantly. Deduced from the XRD patterns and analysis with the Fullprof program (Figure 7a-c), the weight percent of each component (including amorphous VO 2 ) in VO 2 powders is plotted in Figure 7e. The weight percent of crystalline VO 2 increases from 44% to 79% after postannealing at 500 °C (S-500). Furthermore, it reaches 100% for a postannealing temperature 700 °C (S-700). The proportion of crystallized VO 2 is improved clearly after postannealing in oxygen-free atmosphere. The enhanced crystallinity of VO 2 (M) should promote its thermochromic effect and make it more stable in the air environment.

It is well-known that the first order phase transition of VO 2 (M) occurs at around 68 °C with a large variation of latent heat. The DSC analysis of the three synthesized VO 2 samples (VO x -3 min, S-500 and S-700) was measured using the same conditions (heating rate: 5 °C/min, atmosphere: Ar, and in between -30 and 110 °C) for both heating and cooling cycles (Figure 8). The endothermic and exothermic peaks during heating and cooling cycles are clearly visible. After postannealing, a more obvious phase transition of S-500 and S-700 samples compared with the initial VO 2 (M) powder (VO x -3 min sample) is observed. To be more specific, the phase change enthalpy (ΔH) during the heating cycle increases greatly from 3.824 J/g to 40.837 J/g and 43.656 J/g, which is considerably high, and similar to the highest latent 43 J/g of the fine crystalline VO 2 nanoparticles reported by Chen et al. [START_REF] Chen | Fine crystalline VO 2 nanoparticles: synthesis, abnormal phase transition temperatures and excellent optical properties of a derived VO 2 nanocomposite foil[END_REF] and close to 51 J/g of bulk VO 2 . The high latent could be attributed to the increase of crystalline VO 2 percentage in the powder. As for the two or three convoluted exothermic peaks detected in the cooling period, they probably result from a series of effects combination, including defects, lattice expansion, size effects, and nonstoichiometry of synthesized VO 2 powders. [START_REF] Chen | Fine crystalline VO 2 nanoparticles: synthesis, abnormal phase transition temperatures and excellent optical properties of a derived VO 2 nanocomposite foil[END_REF] 3.2.2. Synthesis of V 2 O 5 and V 2 O 3 . In addition to the synthesis of VO 2 (M), this two-step thermolysis and annealing method is also suitable for the preparation of V 2 O 3 and V 2 O 5 using VO x -3 min as the starting material annealed under different conditions. The formation temperature of V 2 O 3 and V 2 O 5 is deduced from the thermogravimetric curve of VO x -3 min in Figure 9a, revealing that the sample mass is stable when VO x -3 min is annealed up to 800 °C in Ar flow with 5% H 2 or up to 600 °C in air. XRD shows that V 2 O 3 and V 2 O 5 are well obtained after annealing (Figure 9b). Interestingly, SEM results (Figure 9c) reveal that the V 2 O 3 powder shows an urchin-like morphology with a high specific surface area, which can be of great interest for the use of vanadium sesquioxide as battery materials; i.e., such morphology is able to provide a large number of active sites between the active materials and electrolytes. [START_REF] Liu | Flowerlike Vanadium Sesquioxide: Solvothermal Preparation and Electrochemical Properties[END_REF][START_REF] Xu | New-Phased Metastable V 2 O 3 Porous Urchinlike Micronanostructures: Facile Synthesis and Application in Aqueous Lithium Ion Batteries[END_REF] Moreover, the morphology of as-prepared V 2 O 5 exhibits regular plates, which can be of great advantage in terms of the lithium storage property for the use as positive electrode in batteries or electrochemical layer. [START_REF] Zhang | Template-free synthesis of highly porous V 2 O 5 cuboids with enhanced performance for lithium ion batteries[END_REF] Their morphological evolution is schematized in Figure 9d. Thanks to this new synthesis protocol, we are able to produce different morphologies, as V 2 O 3 and V 2 O 5 synthesized with this twostep thermolysis and annealing differ from the ones obtained by direct annealing of VEG as earlier reported. [START_REF] Mjejri | Low-Cost and Facile Synthesis of the Vanadium Oxides V 2 O 3 , VO 2 , and V 2 O 5 and Their Magnetic, Thermochromic and Electrochromic Properties[END_REF] Further investigations on their physical and chemical properties are under progress.

CONCLUSION

Herein, the decomposition of vanadyl ethylene glycolate (VEG), leading to VO 2 (M) particles with tunable crystallinity and morphology, through a two-step thermal treatment, a firstthermolysis step under air followed by postannealing under an oxygen-free atmosphere, has been investigated. Careful control of the time and the temperature parameters for the VEG thermolysis (first step) is required to obtain a pure VO 2 phase after the postannealing (second step). The use of shorter or longer time than ∼3 min of heating at 300 °C for VEG decomposition results, after the postannealing, in the formation V(III) or V(V) compounds, respectively. While the thermolysis step is performed with a strict time of 3 min at 300 °C, a pure VO 2 (M) compound is synthesized with various crystallite sizes depending on the temperature of the postannealing treatment under an oxygen-free atmosphere (500 or 700 °C). SEM and TEM observations show an increase in the crystallite sizes of the VO 2 (M) powders with temperature up to few microns for 700 °C postannealing temperature, while the crystallite shapes become regular polyhedra. Moreover, thanks to XRD analyses (Rietveld refinements), the weight percent of the crystalline VO 2 (M) phase in the powders obtained after the 300 °C thermolysis, after 500 °C postannealing, or after 700 °C postannealing is calculated. It increases from 44% to 79% reaching finally 100%. Hence, our synthesis protocol, a low cost and easy to implement synthesis route, allows us to obtain a pure and 100% crystalline VO 2 (M) phase using optimal conditions. Additionally, V 2 O 3 and V 2 O 5 oxides with particular morphologies were synthesized from a similar two-step process by only modifying the atmosphere of the postannealing treatment, namely, reductive H 2 -Ar atmosphere for the former with a lower V(III) oxidation state and air atmosphere for the latter with a higher V(V) oxidation state.

■ ASSOCIATED CONTENT S* Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.8b00753. Page S1, Table S1: CHNS analysis results on VO 2 compounds versus their thermal history. Page S2, Figure S1: Morphological analyses of the V 2 O 3 sample: SEM, TEM, and HRTEM investigations (PDF)

■ AUTHOR INFORMATION

Corresponding Author *E-mail: manuel.gaudon@icmcb.cnrs.fr.

ORCID

Aline Rougier: 0000-0002-1340-734X Manuel Gaudon: 0000-0002-6918-2004

Scheme 1 .

 1 Scheme 1. Illustrations of the Preparation of VO 2 , V 2 O 3 , and V 2 O 5

Figure 1 .

 1 Figure 1. XRD patterns of samples synthesized by thermolysis of VEG precursor at 300 °C in air with different annealing times (1-5 min, 8 and 90 min). Corresponding PDF numbers provided at the bottom.

Figure 2 .

 2 Figure 2. (a) The reflection spectra of the VEG, VO x -3 min, VO x -8 min, and VO x -90 min samples in the wavelength region of 200-2500 nm; insets show photos of the VEG and VO x samples; and (b) their reflection spectra in the visible region of 200-800 nm.

Figure 3 .

 3 Figure 3. SEM images of (a) VEG, and its decomposed products at 300 °C for different times: (b) VO x -3 min (inset shows the evident porosity from its corresponding TEM), (c) VO x -8 min, and (d) VO x -90 min.

Figure 4 .

 4 Figure 4. (a) Carbon content of samples obtained through heating VEG with different times at 300 °C in air (1-5 min). (b) The XRD patterns of samples (S-0 min, S-2 min, S-3 min, and S-5 min) synthesized after a postannealing treatment at 500 °C for 5 h in an oxygen-free atmosphere from VEG, VO x -2 min, VO x -3 min, and VO x -5 min, respectively (S-3 min was further named S-500).

Figure 5 .

 5 Figure 5. (a) Rietveld refinement of X-ray diffraction pattern for VO 2 (M) (S-700) synthesized at 700 °C for 5 h through post annealing of VO x -3 min powder. The observed and calculated intensities are represented by red cross and black solid line, respectively. The bottom blue line shows the fitting residual difference. The Bragg positions are represented by dark cyan ticks. (b) The refined structure of monoclinic VO 2 (M). The octahedron unit consists of one vanadium atom and six oxygen atoms, which are colored orange and red, respectively.

Figure 6 .

 6 Figure 6. TEM and SEM images of the as-prepared VO 2 (M) particles synthesized with conditions (a) VO x -3 min: 300 °C for 3 min in air through decomposition of VEG; (b) S-500: post annealing treatment of VO x -3 min at 500 °C for 5 h in oxygen-free atmosphere; (c) S-700: post annealing of VO x -3 min at 700 °C for 5 h in oxygenfree atmosphere. (d) Size distribution of the as-prepared VO 2 particles. (e) Schematic illustration of the morphological evolution for VO 2 particles. (A and B designate two kinds of deformed octahedra).

Figure 7 .

 7 Figure 7. Typical Rietveld refinements of XRD data from (a) VO x -3 min; (b) S-500; (c) S-700; (d) comparison of XRD peak at 2θ = 27.95°preferred (011) orientation for VO 2 (M); (e) and the corresponding weight percentage of each component.

Figure 8 .

 8 Figure 8. DSC curves of VO 2 samples of VO x -3 min (inset shows its enlargement of DSC data), S-500, and S-700.

Figure 9 .

 9 Figure 9. (a) TG curves of VO x -3 min samples annealing under Ar/ H 2 flow and air atmosphere; insets show the synthesis route of V 2 O 3 and V 2 O 5 from VO x -3 min; (b) XRD pattern of V 2 O 3 and V 2 O 5 samples after postannealing of VO x -3 min, (c) its corresponding SEM images, and (d) schematic illustration of the morphological evolution.

Table 1 .

 1 Lattice Parameters and Atomic Positions of VO 2 (M) (S-700)

	space group	P1 21/c 1 (14) monoclinic
	cell parameters	a	b	c	β
		5.7550(1) Å	4.5291(7) Å	5.3854(8) Å	122.6030(4)°a
	tom	Wyck position	x	y	z
	V1	4e	0.2393(6)	0.408(2)	0.0266(6)
	O1	4e	0.095(2)	0.216(2)	0.198(2)
	O2	4e	0.408(2)	0.720(2)	0.308(2)

min, respectively (S-3 min was further named S-500).

Funding

The Ph.D. grant of S.G. was supported by the China Scholarship Council.

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The authors would like to thank Eric Lebraud, Sonia Buffiere, and Laetitia Etienne for their assistance during XRD, SEM & TEM, CHNS and DSC measurements, and Issam Mjejri for helpful discussion. All authors contributed to the discussions and revisions of the manuscript.