Julie Nabias

Aktham Asfour
email: aktham.asfour@g2elab.grenoble-inp.fr

Jean-Paul Yonnet
email: jean-paul.yonnet@g2elab.grenoble-inp.fr

Temperature effect on GMI sensor: comparison between diagonal and off-diagonal response

Keywords: GMI sensor, diagonal, off-diagonal, sensitivity, offset, temperature, parameter of influence

In this paper, the effect of temperature on Giant Magneto-Impedance (GMI) sensors is studied in amorphous wire Co-Fe-Si-B from Unitika. Both diagonal, Z 11 , and the off-diagonal, Z 21 , components of the impedance tensor are considered, with a primary focus on the changes of intrinsic sensitivity and offset, which are the most crucial relevant quantities for practical sensor implementation. Experiments are conducted in the almost industrial temperature range of [-20 °C, 120 °C]. For the component Z 11 , the results show a maximum change of the offset of -0.13 (A/m)/°C (170 nT/°C), at a 33 A/m bias field. The sensitivity at this field increases for temperatures between -20 °C and +40 °C, and then it decreases (a maximum change of -0.5 %/°C). For the off-diagonal component, Z 21 , the change of the offset near the zero-field is smaller (-0.015 (A/m)/°C or -19 nT/°C), whereas the sensitivity increases with temperature by about +0.4 %/°C.

Introduction

Giant Magneto-Impedance (GMI) is the significant change of the high-frequency impedance of some soft magnetic conductors (amorphous wires, ribbons, thin-film structures) when they are submitted to an external magnetic field [START_REF] Panina | Magneto-impedance effect in amorphous wires[END_REF]. This effect can be used to implemented magnetic sensors based on the measurement of the impedance of the conductor. This impedance is, in reality, a tensor, which has two main useful components, usually called diagonal, Z 11 , and off-diagonal, Z 21 [START_REF] Sandacci | Off-diagonal impedance in amorphous wires and its application to linear magnetic sensors[END_REF]- [START_REF] Ipatov | Magnetoimpedance sensitive to dc bias current in amorphous microwires[END_REF]. One or the other of these components can be used to implement a GMI sensor. GMI sensors are still not very widespread, especially for industrial applications. These applications may include-but are not limited to-the contactless measurement of electrical current and linear or angular positions as well as the nondestructive testing, etc. For these kinds of applications, different issues related to the use of the sensor in a real environment of measurement may be involved. In this context, the impact of the working temperature on the performance of the sensor is of major concern. The robustness of the sensor versus the temperature changes is a key feature for manufacturers and users. Therefore, the knowledge of the influence of temperature is crucial in order to be corrected by suitable strategies of sensor implementation.

By far, the studies dealing with the temperature effect on the GMI are not very numerous and they are generally nonsystematic, especially for the case of amorphous wires. Some examples of these works, directly related to the GMI, can be found in [START_REF] Chiriac | Temperature dependence of the magneto-impedance effect[END_REF]- [START_REF] Dzhumazodaa | Temperature-stable magnetoimpedance (MI) of current-annealed Co-based amorphous microwires[END_REF]. Other works have investigated the temperature dependence of the GMI in ribbons [START_REF] Tehranchi | Temperature dependence of magnetoimpedance in annealed Co-based ribbons[END_REF]- [START_REF] Malátek | Temperature offset drift of GMI sensors[END_REF] and in thin-film structures [START_REF] Chlenova | Permalloy-Based Thin Film Structures: Magnetic Properties and the Giant Magnetoimpedance Effect in the Temperature Range Important for Biomedical Applications[END_REF]. Temperature drift of a few GMI sensor prototypes was also reported in [START_REF] Malátek | Temperature offset drift of GMI sensors[END_REF], [START_REF] Honkura | Development of amorphous wire type MI sensors for automobile use[END_REF], [START_REF] Malátek | Double-core GMI current sensor[END_REF]. However, most of the works related to amorphous wires were focused on the temperature dependence of the diagonal component, Z 11 . To the best of our knowledge, there are only some rare recent works dealing with the temperature dependence of the off-diagonal component Z 21 [START_REF] Panina | Off-diagonal magnetoimpedance in amorphous microwires for lowfield magnetic sensors[END_REF]- [START_REF] Nabias | Effect of the temperature on the off-diagonal Giant Magneto-Impedance (GMI) sensors[END_REF]. In addition, the effect of different annealing temperatures on this component has been recently addressed [START_REF] Yudanov | Temperature dependence of the off-diagonal magnetoimpedance in sensor configuration utilizing Co-rich amorphous wires[END_REF]. In this context, it is very important to make a clear difference between the effect of treatments and annealing at relatively high temperatures, which is generally performed in most of the works, and the impact of the working temperature on the sensor response [START_REF] Chlenova | Permalloy-Based Thin Film Structures: Magnetic Properties and the Giant Magnetoimpedance Effect in the Temperature Range Important for Biomedical Applications[END_REF]. This impact, especially at the ambient and industrial temperature ranges, is by far poorly investigated in the literature.

In this paper, a study is carried out to firstly evaluate the temperature dependence of the off-diagonal component in an almost industrial temperature range [-20°C, 120 °C]. Secondly, another main underlying idea of the study is to provide a clear and a comprehensive comparison of the temperature dependence between the diagonal and the off-diagonal responses. Finally, in the present investigation, the primary attention is paid to the change of most relevant parameters for sensor implementation, namely the offset and of the intrinsic sensitivity of both components, near a suitable working (or bias) point of the sensor. Indeed, most of the previous works considered the change of GMI ratio with temperature as a factor of merit. Despite the fact that this ratio allows describing the behavior of the diagonal component, it is generally not the most relevant quantity to consider for the sensor implementation. For example, amorphous wires may exhibit an important GMI ratio, and in the same time a poor sensitivity at the working point [START_REF] Ménard | Perspectives in giant magnetoimpedance magnetometry[END_REF]. In addition, for some applications, especially the measurement of DC fields, the study of the change of the offset of the sensor with temperature may be more important than the evolution of the GMI ratio.

The section II of the paper presents the developed experimental setup, together with a description of the general approach of investigation as well as clear definitions of the considered quantities. The obtained results and discussions are provided throughout the section III.

Brief overview of the studied quantities and experimental setup

In typical implementations of GMI sensors, a high frequency current i ac of constant amplitude is supplied to the amorphous wire (Fig. 1a). A first method to implement the sensor makes use of the diagonal component, , which is defined as the ratio between the voltage v ac across the wire and the current i ac . This component exhibits a nonlinear and even characteristics versus the magnetic field H as illustrated in Fig. 1b. Therefore, the application of an axial bias field H bias is required to fix the working point of the sensor in the region where the response is almost linear and where the intrinsic sensitivity is high. This sensitivity, noted S 11 , is defined as the derivative of the curve at the bias point according to (1):

= ∂ = ∂ (1)
This magnetic bias induces an offset, which is related to the impedance, |Z 11 (H bias )|. Consequently, an offset cancelling circuit is necessary to obtain a zero output for a zero measured field.

A second promising method to implement the GMI sensor is based on the use of the off-diagonal component, . This component can be obtained through the measurement of the voltage induced in a pick-up coil wound around the GMI wire (Fig. 1c). The origin of this induced voltage is the appearance of an AC longitudinal magnetization of the wire. In amorphous wires with circumferential anisotropy, as the case of the wires used in the present experiments, a significant appearance of this longitudinal magnetization can be obtained by superimposing a DC static current I dc to the high frequency current i ac [START_REF] Sandacci | Off-diagonal impedance in amorphous wires and its application to linear magnetic sensors[END_REF]- [START_REF] Ipatov | Magnetoimpedance sensitive to dc bias current in amorphous microwires[END_REF]. One of key features of the off-diagonal configuration is that the response is almost linear and odd near the zero-field for both real Im{ } Z exhibit similar behaviors (Fig. 1). The sensor implementation may make use of one or the other of these parts. This is why, and without loss of generality, only the real part is considered in current study.

The temperature dependence of the offset and sensitivity may have influence on the choice of the method of sensor implementation for some applications. Indeed, the offset change would be of major concern when the sensor should be used for the measurement of DC magnetic fields or electrical currents. The intrinsic sensitivity, together with additional electronic amplification, determines the gain of the open-loop of the sensor. When the sensor operates in closed-loop, which is frequently the case, it is very important to guarantee a "high enough" open-loop gain for the proper functioning of closed-loop. The change of the intrinsic sensitivity with temperature may therefore affect this functioning.

In summary, the temperature dependence of the offset and sensitivity of GMI sensor is a major issue that by far is not intensively investigated. That is why a complete setup has been built for the characterization of the temperature dependence of the GMI. A simplified scheme of this setup is illustrated in Fig. 2. An amorphous GMI wire of Co-Fe-Si-B type with a 100 µm diameter and 5 cm long, manufactured by Unitika, is used in the experiments. The choice of a relatively long wire was actually related to the context in which the present studied was carried out. Indeed, the work aimed to develop a toroidal current sensor based on the GMI. In such application, the sensitive element forms a loop (of at least several centimeters) around the conductor carrying the current of interest. This current produces a circumferential magnetic field, which is measured by the sensitive element. It may be worth noting that in applications requiring high spatial resolution, the use of shorter wires is necessary. In this case, the wire from Unitika may exhibit limitations related to the presence of a large demagnetization field near the ends of the wire [START_REF] Zhukova | Length effect in Co-rich amorphous wire[END_REF]. This demagnetization field is due to the presence of large longitudinal magnetization associated with the uniformly magnetized core of such wires [START_REF] A S Antonov | Residual quenching stresses in amorphous ferromagnetic wires produced by in-rotating-water spinning process[END_REF]. Consequently, it is by far difficult to make a GMI sensor having length smaller than 1 cm. If the spatial resolution is an issue, glass-coated wires should be preferred since they may allow realizing sensitive elements with a length less than 0.5 cm [START_REF] Gudoshnikov | Highly sensitive magnetometer based on the off-diagonal GMI effect in Co-rich glass-coated microwire[END_REF]. This is related to the fact that longitudinal magnetization is close to zero, as the exchange core in the center of the wire is quite small [START_REF] A S Antonov | Residual quenching stresses in glass-coated amorphous ferromagnetic wires[END_REF].

A pick-up coil of about 800 turns was wound over the entire length of 5 cm of used wire. The excitation current, i ac , was provided by the combination of a sinusoidal voltage generator and the resistor R g . A DC current I dc , could also be superimposed using a DC voltage source and the resistor R b . A capacitor prevents the DC power from reaching the AC source.

For the off-diagonal operation, the frequency of the AC current i ac is limited by the resonance of the pick-up coil, due to parasitic capacitance between turns. For our device, the resonance frequency of the coil was about a few megahertz. It should be noted that the input capacitance of the lock-in detector can interfere with the measurement by reducing resonance frequency of the coil. This is why a relatively low excitation frequency of 1 MHz is chosen. Furthermore, a relatively low amplitude (5 mA) of the excitation current is used in the experiment in order to ensure a linear regime of the GMI [START_REF] Seddaoui | Nonlinear electromagnetic response of ferromagnetic metals: Magnetoimpedance in microwires[END_REF].

The measured field H is applied using a solenoid powered by a low-frequency AC source. This source could generate triangle-shaped current at 0.5 Hz, giving a linear "scan" of the field to be measured. The coil current is known by measuring the voltage across a shunt Rs, giving the field value. The maximum field available H max is about 1 kA/m. The GMI wire is placed in a thermal chamber. The temperature in this chamber was regulated. The components Z 11 and Z 21 are measured for the temperature range of -20 ° C to + 120 ° C.

Results and discussions

The diagonal component

The amorphous wire is supplied by a high-frequency current i ac having 1 MHz of frequency and 5 mA of amplitude. No DC current is added. Fig. 3(a) shows the impedance modulus |Z 11 | as a function of the magnetic field, H, for different temperatures.

To simplify the presentation of the curves, and without loss of generality, only the parts of curves corresponding to positive fields are shown.

It can be seen that this impedance exhibits a significant temperature dependence. For a first description of the general behavior of the GMI effect with temperature, the GMI ratio is used, despite the fact that it is not the most relevant parameter for sensor implementation as it was mentioned in Section 1. Indeed, this ratio is still useful for the comparison purposes because it is widely used in the existing literature on the GMI effect.

The GMI ratio, which is a relative change of the impedance with the magnetic field, is defined by ( 3)

∆| | | | ⁄ % = | | | | | | X 100 (3)
where is the maximum field available.

At each temperature, the GMI ratio has a maximum, denoted ∆| | | | ⁄ . This maximum is plotted as a function of temperature in Fig. 3(b). It increases with the temperature between -20 °C and +20 °C, then decreases between +20 °C and +120 °C. This observation is consistent with the results that have been reported in some works. For example, Chiriac et. al. have showed that the maximum GMI ratio ∆| | | | ⁄ increased with temperature up to slightly higher than the ambient temperature (between 20 ° C and 50 ° C), then decreased [START_REF] Chiriac | Temperature dependence of the magneto-impedance effect[END_REF]. The experiment was conducted on a Co-Fe-Si-B amorphous wire of 120 µm diameter, excited by a 1 MHz/15 mA high-frequency current.

As it is mentioned in the Section 1, the primary focus of the current study concerns the sensitivity and offset changes. The intrinsic sensitivity S 11 was obtained by calculating the derivative of the curves of Fig. 3(a) with respect to the magnetic field. The results are illustrated in Fig. 3(c), where the sensitivity at each temperature is plotted as a function of the magnetic field. The maximum of this sensitivity, denoted S 11-max , is represented versus temperature in Fig. 3(d). It can be seen that S 11--max increases with the temperatures up to about 40 °C, then decreases for temperatures between 40 ° C and 120 ° C. The relative change, between the lowest sensitivity (at 120 ° C) and the highest one (at 40 ° C), is about 0.9 Ω/(A/m). Furthermore, it is worth noting that the magnetic field corresponding to S 11-max decreases when the temperature increases.

For a GMI sensor based on the diagonal component, the use of an axial bias field H bias is necessary, as it was mentioned in Section 2. Usually, this bias field is chosen to maximize the sensitivity at room temperature (20 °C for example). At this temperature S 11-max is obtained at a field of about 33 A/m (H bias in Fig. 3(a)). One of the most important points for the sensor design is the gain of the open-loop, which is proportional to the value of the sensitivity at H bias . This sensitivity S 11 (H bias ) is shown in Fig. 3(e) as a function of temperature. It varies at most between 2.1 Ω/(A/m) at 20 ° C and about 1 Ω/(A/m) at 120 ° C. This yields a temperature coefficient of sensitivity of about -0.5% / °C. A value of the same order of magnitude (0.2 % /°C) was reported in amorphous ribbon samples [START_REF] Malátek | Double-core GMI current sensor[END_REF].

The bias field determines also the value of the offset of the sensor, which is proportional to |Z 11 (H bias )|. This quantity is represented in Fig. 3(f) as a function of temperature.| 11 ( bias )| varies at most between 43 Ω (at 20 °C) and 32 Ω (at -20 °C). The relative change is about -0.28 Ω/°C. This change could be converted into a measurement drift of the magnetic field by introducing the temperature coefficient of offset expressed in (A/m) /°C (or in T/°C). For an intrinsic sensitivity of 2.1 Ω/(A/m) at 20 °C, a change of -0.28 Ω/°C of the offset corresponds to a temperature coefficient of about -0.13 (A/m)/°C (or -170 nT/°C). A temperature coefficient of 340 nT/K was reported in [START_REF] Malátek | Temperature offset drift of GMI sensors[END_REF] for a sample of amorphous ribbon of composition Co-Fe-Cr-Si-B. When considering the origin of the GMI, this temperature dependence could be explained by the change of circumferential permeability. Indeed, as it is well-known, in the intermediate frequency regime of (roughly from 100 kHz to 100 MHz); the GMI results from the modification of the skin depth of the high frequency current in the wire. This skin depth depends on the resistivity and on the circumferential permeability. The modification of the impedance with the applied field is directly related to the significant change of this permeability.

The temperature dependence of the GMI may by far have two origins: the change of electrical resistivity and the change of the permeability. However, in the range of temperatures studied, the modification of the resistivity does not explain the observed temperature dependence of the diagonal component. This was confirmed by the measurement of the static resistance R DC of the wire in the temperature range of 20 °C to 80 °C as illustrated in Fig. 4. The relative change of this resistance is about 0.007 %/°C, which is very small when compared to the relative changes of sensitivity, offset and GMI ratio. Therefore, the most likely interpretation of this temperature dependence is the modification of the circumferential permeability [START_REF] Dzhumazodaa | Temperature-stable magnetoimpedance (MI) of current-annealed Co-based amorphous microwires[END_REF], [START_REF] Panina | Off-diagonal magnetoimpedance in amorphous microwires for lowfield magnetic sensors[END_REF]- [START_REF] Nabias | Effect of the temperature on the off-diagonal Giant Magneto-Impedance (GMI) sensors[END_REF], [START_REF] Chizhik | Heating influence on magnetic structure in Co and Fe rich amorphous microwires[END_REF]- [START_REF] Kurniawan | Interplay of stress, temperature, and giant magnetoimpedance in amorphous soft magnets[END_REF]. This interpretation is also consistent with the observed modification of the field of maximum impedance, which decreases with the temperature elevation (Fig. 3(a)). As this field of maximum impedance is close to the field of anisotropy, the temperature dependence of the diagonal component is directly related to the modification of anisotropy, which in turn modifies the sensitivity. This temperature instability of the anisotropy may originate from internal residual stresses in the wire. These stresses are due to the fabrication process by rapid solidification. The temperature elevation induces a partial release or a change of the stress distribution, resulting in the change of the anisotropy and consequently the GMI response and sensitivity.

The off-diagonal component

For this experiment, the same high-frequency current as in Section 3.1 is used. A DC current (I dc = 10 mA) is also superimposed in order to observe the off-diagonal component in wires with circumferential anisotropy [START_REF] Sandacci | Off-diagonal impedance in amorphous wires and its application to linear magnetic sensors[END_REF]- [START_REF] Ipatov | Magnetoimpedance sensitive to dc bias current in amorphous microwires[END_REF]. This current is also needed to stabilize the almost linear response of the off-diagonal component [START_REF] Usov | Giant magneto-impedance effect in amorphous ferromagnetic wire with a weak helical anisotropy: Theory and experiment[END_REF]. of this component at different temperatures as a function of the magnetic field. In Fig. 5 (a), the curves exhibit the almost odd symmetry of the curves with respect to the zero-field point. It can however be noticed a slight asymmetry of the responses for positive and negative fields due to the presence of a weak helical anisotropy in the used wires [START_REF] Usov | Giant magneto-impedance effect in amorphous ferromagnetic wire with a weak helical anisotropy: Theory and experiment[END_REF].

The maximum of sensitivity is obtained for a field close to zero for all the temperatures studied, as shown in Fig. 5(b). This field is almost the same, unlike the previous results obtained with diagonal component. The sensitivity at zero-field S 21 (0) increases with temperature as seen in Fig. 6 In the same way as for Z 11 , these modifications of the off-diagonal response could also be attributed to the change of the anisotropy and the circumferential permeability of the wire with temperature [START_REF] Dzhumazodaa | Temperature-stable magnetoimpedance (MI) of current-annealed Co-based amorphous microwires[END_REF], [START_REF] Panina | Off-diagonal magnetoimpedance in amorphous microwires for lowfield magnetic sensors[END_REF], [START_REF] Yudanov | Temperature dependence of the off-diagonal magnetoimpedance in sensor configuration utilizing Co-rich amorphous wires[END_REF]. In Reference [START_REF] Panina | Off-diagonal magnetoimpedance in amorphous microwires for lowfield magnetic sensors[END_REF], Panina et.al have investigated the temperature dependence of the off-diagonal component in glass-coated amorphous wires (CoFeNiMoSi) having a total diameter of 42 µm (36 µm of metallic core). The experiments were conducted for an excitation frequency of 20 MHz. The temperature dependence of the off-diagonal component was interpreted by a modification of the anisotropy, which is due to the change of the stress distribution. It is to note that, compared to the used wires from Unitika, the residual internal stress distribution in glass-coated wires is different. It can result from different contributions. Indeed, in addition to stress resulting from the fabrication by rapid solidification or inhomogeneous quenching, the different thermal coefficients of expansion of metal and glass contribute also to stress distribution and consequently to the temperature dependence of the GMI [START_REF] Dzhumazodaa | Temperature-stable magnetoimpedance (MI) of current-annealed Co-based amorphous microwires[END_REF], [START_REF] Yudanov | Temperature dependence of the off-diagonal magnetoimpedance in sensor configuration utilizing Co-rich amorphous wires[END_REF]. All of these contributions exhibit strong temperature dependence. Panina et al. have showed an increase of the sensitivity of the off-diagonal response in glass-coated wires for temperatures between 20 °C and 60 °C, then a decrease for temperatures up to 100 °C [START_REF] Panina | Off-diagonal magnetoimpedance in amorphous microwires for lowfield magnetic sensors[END_REF]. Globally, these results were consistent with those obtained in the present study, in the sense that the temperature changes induced a modification of the response and of the sensitivity. However, the behavior of the sensitivity with temperature was different between the two studies. In reference [START_REF] Panina | Off-diagonal magnetoimpedance in amorphous microwires for lowfield magnetic sensors[END_REF], the decrease of the sensitivity started from 60 °C, whereas in the present study this decrease was observed from a temperature of 120 °C. This may be due to the different stress distributions and different mechanisms of stress relaxation between the two types of wires.

Conclusions

This study showed a temperature dependence of the both diagonal and off-diagonal responses of the GMI in the studied amorphous wire. In the temperature range considered, the comparison between the sensitivities of the diagonal response at a given bias field and of the off-diagonal one at zero-field shows that the change is almost in the same order of magnitude (-0.5% / °C and +0.4 % / °C, respectively). The off-diagonal response exhibited better offset stability than the diagonal one (changes were almost ten times lower).

From practical point of view, the temperature dependence may determine the choice of either diagonal or off-diagonal configuration many applications of GMI sensors. Both configurations have a roughly equivalent change of the sensitivity in the considered temperature range. However, when DC field measurements are issues, the offset stability of the off-diagonal response is clearly a high advantage.

Fig. 1 .

 1 Fig. 1. Basic implementations of GMI sensor (a); and typical curves of the diagonal and off-diagonal responses (b) & (c).

Fig. 2 .

 2 Fig. 2. Experimental setup for the temperature characterization of the diagonal and off-diagonal responses.

Fig. 3 .

 3 Fig. 3. Temperature dependence of the diagonal component for an amorphous wire of Co-Fe-Si-B type (100 µm diameter, 5 cm long) excited by a high-frequency current iac of 5 mA amplitude and 1 MHz frequency. (a) Modulus |Z11(H)| at different temperatures; (b) GMI ratio versus temperature; (c) Sensitivity S11 versus magnetic field at different temperatures; (d) Maximum of sensitivity S11-max versus temperature; (e) Sensitivity S11(Hbias) at the bias field of about 33 A/m; (f) Offset|Z11(Hbias)| as a function of temperature.

Fig. 4 .

 4 Fig. 4. Static resistance, RDC, of the wire versus temperature.

Fig. 5

 5

Fig. 5 .Fig. 6 .

 56 Fig. 5. Temperature dependence of the off-diagonal component for an amorphous wire of Co-Fe-Si-B type (100 µm diameter, 5 cm long) supplied by a high-frequency current iac (amplitude 5 mA, frequency 1 MHz) and a DC current (IDC= 10 mA). (a) Re{Z21} versus magnetic field at different temperatures; (b) Sensitivity S21 versus magnetic field at different temperatures.


C Meille

B Olagnon

G Yrieix

Foray

Belynda Benane

Guilhem P Baeza
email: guilhem.baeza@insa-lyon.fr

Bruno Chal

Lucian Roiban

Sylvain Meille

Christian Olagnon

Bernard Yrieix

Geneviève Foray

Multiscale Structure of Super Insulation Nano-Fumed Silicas Studied by SAXS, Tomography and Porosimetry

Keywords: Silica, Insulation, Tomography, Fractal Aggregates, SAXS

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

Introduction

Ranked among multi-scale materials, nanostructured silica powders are composed of nanometer sized elementary particles, organized at higher scales as aggregates and agglomerates. [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF] While the former are believed to be indivisible gatherings of elementary particles, the latter consist of clusters of weakly bounded aggregates that can be "broken" by applying mechanical stress, e.g. in nanocomposites, by "internal" mixing [START_REF] Baeza | Mechanism of aggregate formation in simplified industrial silica styrene-butadiene nanocomposites: effect of chain mass and grafting on rheology and structure[END_REF][START_REF] Musino | Synergistic Effect of Small Molecules on Large-Scale Structure of Simplified Industrial Nanocomposites[END_REF][START_REF] Schneider | Correlation of mass fractal dimension and cluster size of silica in styrene butadiene rubber composites[END_REF] or in solution via stirring [START_REF] Suzuki | Effects of rubber/filler interactions on deformation behavior of silica filled SBR systems[END_REF] and sonication. [START_REF] Rahman | Synthesis of silica nanoparticles by sol-gel: size-dependent properties, surface modification, and applications in silica-polymer nanocomposites-a review[END_REF][START_REF] Forny | Storing water in powder form by self-assembling hydrophobic silica nanoparticles[END_REF] When loosely packed, these nano-architectured materials are essentially composed of air (>80%) [START_REF] Forrest | Long-range correlations in smoke-particle aggregates[END_REF][START_REF] Johnson | Ordered mesoporous polymers of tunable pore size from colloidal silica templates[END_REF][START_REF] Orts-Gil | Characterisation of silica nanoparticles prior to in vitro studies: from primary particles to agglomerates[END_REF][START_REF] De Temmerman | Quantitative characterization of agglomerates and aggregates of pyrogenic and precipitated amorphous silica nanomaterials by transmission electron microscopy[END_REF][START_REF] Singh | Experimental investigations into thermal transport phenomena in vacuum insulation panels (VIPs) using fumed silica cores[END_REF] with typical pore sizes not exceeding 200 nm, making them excellent candidates for thermal insulation applications. Indeed, their very open structure (low fractal dimension), combined with the nanometric size of their pores, drastically limits heat conduction through both the solid skeleton and the gas routes. [START_REF] Kennard | Kinetic theory of gases, with an introduction to statistical mechanics[END_REF][START_REF] Gesser | Aerogels and related porous materials[END_REF][START_REF] Reichenauer | Relationship between pore size and the gas pressure dependence of the gaseous thermal conductivity[END_REF][START_REF] Koebel | Aerogel-based thermal superinsulation: an overview[END_REF][START_REF] Fricke | Vacuum insulation panels-From research to market[END_REF] Taking advantage of such remarkable properties, a wide range of thermal insulating systems based on nanostructured silicas and aerogels [START_REF] Yrieix | Aerogel-based material that is super-insulating at atmospheric pressure[END_REF][START_REF] Nosrati | Hygrothermal characteristics of aerogel-enhanced insulating materials under different humidity and temperature conditions[END_REF][START_REF] Baetens | Aerogel insulation for building applications: a stateof-the-art review[END_REF] have emerged in recent years. Among them, vacuum insulating panels (VIP), made of a compacted silica core in a sealed envelope, have been identified as one of the most promising technologies. [START_REF] Fricke | Vacuum insulation panels-From research to market[END_REF][START_REF] Johansson | Retrofitting of a listed brick and wood building using vacuum insulation panels on the exterior of the facade: Measurements and simulations[END_REF][START_REF] Baetens | Vacuum insulation panels for building applications: A review and beyond[END_REF] While the thermal conductivity of conventional thermal insulation materials such as mineral fibers, polyurethane foams and expanded polystyrene is situated between 25 10 -3 and 40 10 -3 W.m -1 .K -1 , [START_REF] Asdrubali | A review of unconventional sustainable building insulation materials[END_REF][START_REF] Schiavoni | Insulation materials for the building sector: A review and comparative analysis[END_REF][START_REF] Aditya | A review on insulation materials for energy conservation in buildings[END_REF] VIPs allow dividing it by as much as a factor of 8, reaching 2-5 10 -3 W.m -1 .K -1 when their inner pressure is maintained around 100 Pa. [START_REF] Brunner | Vacuum insulation panels for building applications-Continuous challenges and developments[END_REF] They therefore offer major improvements in terms of building renovation, notably by reducing the insulating layer thickness to less than 30 mm to fulfill current building insulation standards.

Nevertheless, VIP panels are faced by a major barrier to their development and commercialization, namely their price. They remain confined to high added value applications, historical renovation, and demonstration sites. [START_REF] Johansson | Interior insulation retrofit of a historical brick wall using vacuum insulation panels: Hygrothermal numerical simulations and laboratory investigations[END_REF][START_REF] Caps | Evacuated insulation panels filled with pyrogenic silica powders: properties and applications[END_REF] In order to enlarge their field of use, many studies have been launched for more than a decade to develop competitive and efficient silica formulations. In particular, two types of nanopowders have been promoted: (i) fumed silicas (FS) synthesized by pyrolysis in a flame up to 1500 °C and characterized by a hydrophobic surface [START_REF] Heine | High concentration agglomerate dynamics at high temperatures[END_REF][START_REF] Schaefer | Growth and structure of combustion aerosols: fumed silica[END_REF][START_REF] Witten | Diffusion-limited aggregation[END_REF] and (ii) precipitated silicas (PS), obtained from aqueous hydrolysis, leading to a quite hydrophilic behavior. [START_REF] Rahman | An optimized sol-gel synthesis of stable primary equivalent silica particles[END_REF][START_REF] Iler | The chemistry of silica, Solubility, Polymerization[END_REF] The former are currently used in VIPs in spite of their higher cost, because they can be employed to assemble panels with sufficient compression strength for handling at lower density (140 -200 kg.m -3 , i.e., 6 -10 vol.% solid phase). In comparison, the latter exhibit higher densities (200 -250 kg.m -3 i.e., 10 -12 vol.% solid phase) for a similar compression stress, synonymous with lower insulation properties. [START_REF] Quenard | Long-term performance of super insulating materials in building components & systems-IEA-EBC Annex 65[END_REF] The real challenge therefore resides in identifying the origin of such different behaviors in view to adapting either synthesis routes or composite VIP processing to the industrial scale for the potential mass market. While it is commonly accepted that the macro-properties of a powder are mostly driven by its morphology [START_REF] Ramos | Physical and chemical characterisation of some silicas and silica derivatives[END_REF] together with its surface state, [START_REF] Krumpfer | Make it nano-Keep it nano[END_REF] a clear picture capable of explaining this structure-property relationship is still missing. As so often in materials science, the task is made difficult by the multi-scale nature of the problem and the lack of a technique able to provide information on the whole range of sizes.

We address this challenge here. In particular, by combining structural characterization in both the direct and Fourier space, we demonstrate that the origin of the high porosity present in fumed silica resides in its pronounced low dimensionality character at the 50-350 nm length scale. On the one hand, transmission electronic microscopy (TEM) and electronic tomography (ET) reveal the flat (oblate) or pearl necklace (prolate) shape of the aggregates. On the other hand, in-depth analysis of small angle X-ray scattering (SAXS) patterns allows generalizing this result by suggesting the stacking of elongated objects when the compression stress is increased in powder pellets. Furthermore, we provide a self-consistent model to estimate the volume fraction in aggregates that we subsequently validate using mercury intrusion porosimetry. This original parallel approach opens the way to more systematic structural characterization in this field.

Experimental Section

Nature of the fumed silica: The fumed silica studied (Konasil 200, OCI Korea) had a skeletal (intrinsic) density of 2.2 g.cm -3, a tapped density close 0.05 g.cm -3 and a purity higher than 99.8%. [38] Its typical untapped density is expected to be close to 0.03 g.cm -3 , in fair agreement with the volume fraction of the "powder" estimated from SAXS (Table 1).

Sample preparation:

Pellets were shaped by oedometric compression at a 10 mm.min -1 loading rate, up to either 0.15, 0.6 or 1.2 MPa using a Zwick Roell press equipped with a 500 N load cell. Sample unloading was performed at 5 mm.min -1 .

Pellet masses were always equal to 100 mg to ensure adequate thickness for the SAXS measurements (ca. 2 mm). The suspension (50 mL) was prepared by adding the silica powder in ethanol (0.1 vol.%) and subsequent sonication at 500W for 1 minute with a Vibra-cell 500 (Sonics, USA) allowing to ensure a good and stable dispersion. During sonication, the vessel containing the suspension was immerged into cold water and ice to limit the ethanol evaporation. The resulting sample was transferred into a 2 mm diameter capillary tube (specialglass, WJ M-Glass) and measured in SAXS ca. 5 minutes later.

Transmission electron microscopy: TEM analyses were performed on both Jeol 2010F (200 keV) and EFI TITAN environmental TEM (80 keV-300 keV) electron microscopes operating under high vacuum. The samples were prepared using a "dry method" consisting in the following. A small amount of silica (ca. 20 mg) was first crushed into an agate mortar and transferred into a 25 cm 3 closed vessel. The latter was then shook by hand for five minutes in order to generate a "silica fume dust" which was subsequently captured manually with a perforated carbon film cupper grid with a "300" mesh. The images were finally recorded using minimum illumination of the samples to prevent deteriorating the silica.

Electron tomography: Electron tomography [START_REF] Midgley | 3D electron microscopy in the physical sciences: the development of Z-contrast and EFTEM tomography[END_REF]40] consists in recording series of tilted images of an object in order to reconstruct its volume in 3D. [START_REF] Carenco | The core contribution of transmission electron microscopy to functional nanomaterials engineering[END_REF][START_REF] Ersen | Exploring nanomaterials with 3D electron microscopy[END_REF] In this study, the tilt series were recorded using a EFI TITAN environmental TEM (80 keV-300 keV). The sample was prepared dry, as explained in the TEM section, using a microscopy grid on which gold nanoparticles of 5 nm diameter were deposited previously as fiducial markers. The microscopy grid was then set on a Fischione high tilt holder. Because the silica was extremely sensitive to the beam, great care was given to the electron dose received by the sample in order to prevent any damage. [START_REF] Roiban | Advanced three dimensional characterization of silica-based ultraporous materials[END_REF] 75 images with a pixel size of 0.33 nm were recorded. Image alignment was performed using the Imod software, [START_REF] Kremer | Computer Visualization of Three-Dimensional Image Data Using IMOD[END_REF] the reconstruction was built using 15 iterations of the ART algorithm implemented using TomoJ software and ImageJ. [START_REF] Messaoudii | TomoJ: tomography software for three-dimensional reconstruction in transmission electron microscopy[END_REF] Finally, the data segmentation and quantification were performed by combining tools implemented in ImageJ [START_REF] Bolte | A guided tour into subcellular colocalization analysis in light microscopy[END_REF][START_REF] Tschumperle | Vector-valued image regularization with PDEs: A common framework for different applications[END_REF] and 3D Slicer. [START_REF] Fedorov | 3D Slicer as an image computing platform for the Quantitative Imaging Network[END_REF] SAXS: Ultra-SAXS and SAXS measurements were performed on the beam line ID02 at the ESRF, Grenoble, France. Two sample-detector distances were used, 1 m (SAXS) and 10 m (USAXS) with a beam wavelength of = 1.1 Å to measure the scattering intensity ( ) over three decades of scattering vector (2 10 nm -1 to 2 nm -1 ), roughly corresponding from 3 μm to 3 nm in real space. The beam was rectangular with an approximate size of 0.1 × 0.2 mm 2 . Each diffractogram was obtained by averaging ten measurements performed at different locations in the samples. The few diffractograms presenting singularities (samples heterogeneities) were eliminated.

The transmission was systematically found to be in the 0.57 -0.68 range.

Mercury porosimetry:

Porosimetry measurements were performed on an AUTOPORE IV 9500 (Micrometrics Instrument Corporation, USA). A powder cell equipped with a 1.131 cm 3 capillary and a 5.913 cm 3 cell volume was used. All the measurements were corrected with a blank performed on an empty cell. Samples were all 100 mg in mass; a 4 h void was performed before measurements. The maximum pressure was 200 MPa; more than 50 points were recorded during both the loading and unloading sample steps.

Results and Discussion

Direct visualization at the nano-scale

In Figure 1a we present an electron micrograph of a fumed silica agglomerate of ca. 400-500 nm in diameter obtained by transmission electron microscopy in bright field mode (BF-TEM). Figures 1b and1c are images at higher magnifications showing the agglomerate's inner-structure presented in Figure 1a. These micrographs clearly reveal the presence of large pores and denser zones, called aggregates, presenting either flat or necklace shapes at the 100 nm lengthscale (see e.g. the white ellipsoids in Figures 1e-f). The aggregates are themselves made of silica particles well visible in Figures 1b-c exhibiting elliptical geometries with major and minor radii estimated to be =13-17 nm and =5-7 nm, respectively, from image analysis. Careful examination of Figure 1f also reveals a gradient in terms of particle radius, with the long axes of the particles passing progressively from 6 nm (bottom left corner) to 17 nm (top-right corner). This effect has been documented in the literature and is known to be generated by the flame synthesis of fumed silica, causing heterogeneous conditions along the aggregates. [START_REF] Boldridge | Morphological characterization of fumed silica aggregates[END_REF] Then, "XY" cross-sections extracted from the reconstructed volume of an agglomerate investigated with electron tomography (Figure 2a-c Based on these initial investigations, it appears possible to rationalize from simple geometrical arguments the lower apparent density of the compacts of fumed silica in comparison with their precipitated counterparts (mostly made of "more spherical"

aggregates [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Roiban | Advanced three dimensional characterization of silica-based ultraporous materials[END_REF]). In order to support and deepen the above investigation performed on limited pieces of materials, we propose in the following section to use a systematic SAXS description of the fumed silica prepared from different conditions.

Small Angle X-ray Scattering

In Figure 3a, we present the scattering intensity ( ) measured from different samples made of fumed silica: (i) three pressed pellets prepared by applying 0.15, 0.6 and 1.2 MPa in an oedometer cell (20 mm of diameter), (ii) the corresponding unpressed powder, and (iii) a suspension in ethanol with a nominal volume fraction in silica of 0.1 vol.%. As expected, all the samples made of silica in air ((i) and (ii))

exhibit similar intensities at high-, while the silica in ethanol scatters much less due to a weaker electronic contrast between the two phases. Once normalized by the volume fraction of silica Φ , i.e., the ratio of the volume truly occupied by the solid matter over the volume of the sample, calculated from the pellet's dimension of ca.

650 mm 3 and its mass, the pellets' signals all collapsed above 10 -1 nm -1 . This allowed extracting subsequently the volume fraction of silica within the unpressed powder (≈11 vol.%) and the corresponding "true" value for the suspension (≈0.07 vol.%).

Note that in the latter case, the signal had to be multiplied beforehand by the contrast ratio ∆ /∆ ! " , with ∆ # being the SAXS contrast in a biphasic (i, j) medium [START_REF] Lindner | Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter[END_REF] (see Supplementary Material -Notes). These results are grouped in Figure 3b and must be read in the following way:

For q > 0.3 nm -1 : This -range (momentum transfer), known as the Porod regime, [START_REF] Lindner | Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter[END_REF] is indicative of the smallest objects within the material only, i.e., the elementary nanoparticles. Once the data are properly normalized (by Φ and ∆ # in cm -4 ), they all overlap regardless of the structure at higher lengthscales. The assumption of perfect sphericity allowed estimating the radius of these "buildingblocks" through a simple power-law fit of the data such that ( ) = $ % (see Figure 3a-b). Here, exponent 4 stands for a perfectly smooth interface, in quite good agreement with Figure 1, while pre-factor $ can be used to calculate the Porod radius through the following expression:

& = 6( Φ ∆ $ (1)
Based on the estimation of ∆ , (see Supplementary Material -Notes), we For q <0.1 nm -1 : This -range concerns the powder structure at higher lengthscales, namely the aggregates and agglomerates -spanning almost two decades in direct space from ≈ 50 nm to 2 μm (see Figures 1 and2). Immediately below =0.1 nm -1 , the different signals start to separate, with the denser systems exhibiting lower intensity.

This trend indicates the presence of a deeper structure factor "correlation hole" in more compressed pellets rather than the decreasing size of the aggregates (related to their form factor). [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Genix | Determination of the local density of polydisperse nanoparticle assemblies[END_REF] To eliminate the structure factor from the measurement, we thus measured a very diluted and sonicated silica suspension (≈0.07 vol.%) in ethanol providing us with the apparent form factor of the agglomerates 0 9 //+ ( ) (grey signal called "Suspension" in Figure 3a-c; see also the schematic representation in Figure 5a). As can be seen, the derivative of that signal with respect to is monotonic -a strong argument suggesting, in fact, that most of the contribution of the interaggregate structure factor has been eliminated. [START_REF] Baeza | Revealing nanocomposite filler structures by swelling and small-angle X-ray scattering[END_REF] This important data then allows extracting the most probable gyration radius of the aggregates and the agglomerates from the breaks in slope positions observed in Here, we want to insist on the fact that the fractal dimension of the agglomerates should be extracted only from such "structure factor free" signals ("Suspension" in Figure 3), contrary to what is frequently proposed in the literature. [START_REF] Schneider | Correlation of mass fractal dimension and cluster size of silica in styrene butadiene rubber composites[END_REF][START_REF] Saint-Michel | Fractal behavior and scaling law of hydrophobic silica in polyol[END_REF][START_REF] Schneider | Correlation of mass fractal dimension and asymmetry[END_REF] In such cases, the scattering intensity at intermediate (10 -2 -10 -1 nm -1 , i.e., * * < < * ) originates from the product of the aggregate form factor and the inter-aggregate structure factor, leading to the erroneous quantification of the fractal dimension of bigger objects when simply taken as the slope of the (log-log) signal. We believe that this confusion comes from the absence of a clear maximum on ( ) and that the authors would not have performed such an analysis in the presence of well-defined structure peaks, like e.g., in colloidal systems. In comparison, this method would have led in our case to ? //+ @ varying from 1.0 to 2.0 for the 1.2 MPa pellet and the suspension in ethanol, respectively. In other words, it would have suggested that the density (or compacity) of the agglomerates is much higher in a sonicated suspension than in a pellet formed under high compression.

In addition, the utilization of the Kratky representation to transform the inflexion points of the multiscale form factor into a "trapezoid" or "peak" signal, presents the same restrictions. As can be seen clearly in Figure 3c, the Kratky peak position dramatically shifts to higher , passing from the suspension to the unpressed powder and pellets. Here again, this is unambiguously due to the correlation hole of the interaggregate structure factor, which pulls the signal down at low-for high Φ values, and certainly not to a dramatic decrease of the aggregate radius. (Indeed, we may have expected the opposite trend, i.e., smaller aggregates in the dispersed state).

Returning to the analysis of the fumed silica, measuring 0 9 //+ ( ) also makes it possible to calculate the apparent structure factor G && ( ) in the unpressed powder and pellets (Figure 4a) such that:

G && ( ) = ( ) 0 9 //+ ( ) (2)
As expected, all the apparent structure factors merge at high-q (same elementary particles) and show a clear correlation hole (destructive interferences) from 0.1 nm -1 and below. Interestingly, no major constructive interferences (e.g. structure peaks) appear in these functions, in line with what was observed by Baeza et al. [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Baeza | Revealing nanocomposite filler structures by swelling and small-angle X-ray scattering[END_REF] and Genix et al. [START_REF] Genix | Determination of the local density of polydisperse nanoparticle assemblies[END_REF] for similar powders but in contrast with experiments performed on colloidal silicas. [START_REF] Montes | Particles in model filled rubber: dispersion and mechanical properties[END_REF][START_REF] Musino | Aggregate Formation of Surface-Modified Nanoparticles in Solvents and Polymer Nanocomposites[END_REF][START_REF] Oberdisse | Aggregation of colloidal nanoparticles in polymer matrices[END_REF] This qualitative difference is usually explained by the considerable polydispersity in both the size (H > 0.3) and shape of the fumed/precipitated powders, as shown by simulations. [START_REF] Genix | Determination of the local density of polydisperse nanoparticle assemblies[END_REF] On a more quantitative level, the minimum of G && ( ), which is related to the isothermal compressibility [START_REF] Lindner | Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter[END_REF] and denoted G hereafter, was used to estimate the volume fraction in silica aggregates within nanocomposites through simulations and the approximation of Percus-Yevick (PY). [START_REF] Percus | Analysis of classical statistical mechanics by means of collective coordinates[END_REF] In our case, the G values can be obtained directly through the experimental determination of 9 //+ ( ) (Figure 4a) to estimate the volume fraction in aggregates Φ // (see Figure 4b) as shown in equation 3. (Note that Φ // is here formally defined as the ratio of the volume occupied by spheres of radius // containing both silica and air over the volume of the whole sample).

G ≡ G KL ( → 0) = (1 -NΦ // ) % (1 + 2NΦ // ) (3)
This function was calculated analytically for monodisperse spheres with no interactions, [START_REF] Percus | Analysis of classical statistical mechanics by means of collective coordinates[END_REF][START_REF] Ashcroft | Structure of binary liquid mixtures. I[END_REF] and recently extended empirically for polydisperse spheres with moderate attraction. [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF] Here, N is related to the polydispersity (standard deviation) H by N = 1 -1.13H as defined in ref. [START_REF] Genix | Determination of the local density of polydisperse nanoparticle assemblies[END_REF] for an infinite assembly of particles.

By using a log-normal distribution and H = 0.5, i.e. a geometrical standard deviation PG? = QRS (H) close to 1.65 as estimated from refs, [START_REF] Boldridge | Morphological characterization of fumed silica aggregates[END_REF], [START_REF] Oh | Comparison of nanoparticle exposures between fumed and solgel nano-silica manufacturing facilities[END_REF] it is therefore possible to estimate the volume fraction in aggregates for the different samples. We find respectively Φ // =0.17, 0.28, 0.35 and 0.40 for the unpressed powder, and the pellets pressed at 0.15, 0.6 and 1.2 MPa. From these values, we can then determine the compacity of the aggregates, i.e., their volume fraction in silica particles through , where e 0 is called the "fractal prefactor" or lacunarity, [START_REF] Boldridge | Morphological characterization of fumed silica aggregates[END_REF] likely to be related with the degree of overlap between the elementary beads. While its value is often taken as 1 in the literature, [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Schneider | Correlation of mass fractal dimension and asymmetry[END_REF] Oh and Sorensen showed that higher values were more sensible for DLCA morphologies such as fumed silica. [START_REF] Oh | The Effect of Overlap between Monomers on the Determination of Fractal Cluster Morphology[END_REF] In consequence,

TU = Φ Φ // ⁄ .
we fixed e 0 =1.3 in our analysis. The corresponding outputs (Φ // , TU and ? // @ ) are regrouped in Table 1 with the rest of the samples characteristics. For the sake of comparison, we also report apparent value of ? // @ for the pellets in the legend ( * ) under the symbol ? // @ && .

Importantly, while in the suspension the appellation "aggregate" is referring to noninterpenetrated sub-domains of isolated agglomerates, it rather refers to spheres of radius // in which the latter objects become interpenetrated, resulting in a higher apparent compacity in the untapped powder and the pellets (Figure 5). Also, it is worth to note that the calculation of the aggregates compacity TU is independent of any object size. On the contrary, its transformation into an (apparent) aggregate fractal dimension ? // @ (or ? // @ && ) is calculated from & =7.0 nm and // =40 nm (see above), assuming an identical compacity for all the aggregates. are measured experimentally by using SAXS.

(1) Φ // and TU are calculated from the SAXS measurements assuming H=0.5 (2) ? // @ and d // are calcuculated from TU with // =40 nm, & =7.0 nm and e 0 =1.3.

( * ) ? // @ && = 2.13, 2.13 and 2.11 for the pellets pressed at 0.15, 0.6 and 1.2 MPa respectively.

This approach allows highlighting the iso-structure of the aggregates in all the pellets regardless of the pressure applied in the 0.15 MPa -1.2 MPa range. As a matter of fact, they systematically appear to have a compacity around TU = 0.29, confirming that the silica structure is not impacted significantly by the compressive stress from the aggregate lengthscale and below. This trend, as well as the values of ? // @ (and eventually ? // @ && ), are found to be in good/fair agreement with other works. [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Boldridge | Morphological characterization of fumed silica aggregates[END_REF][START_REF] Wengeler | Turbulent hydrodynamic stress induced dispersion and fragmentation of nanoscale agglomerates[END_REF][START_REF] Kammler | Structure of flame-made silica nanoparticles by ultra-small-angle X-ray scattering[END_REF] Interestingly, in the untapped powder, in which the interpenetration of the agglomerates is presumably modest, the fractal dimension of the aggregates (seen as spheres of radius // ) is found to be ? // @ ≈ 1.85, close to the fractal dimension of the agglomerates observed in the suspension (? //+ @ ≈ 1.80). This result suggests therefore that "native" aggregates and agglomerates could be seen as self-similar objects, as one could have hypothesized from the almost constant exponent of the ( ) power law up to (/ & nm -1 in Figure 3a-b. Although this result is supported by Boldridge (? // @ ≈ 1.86), [START_REF] Boldridge | Morphological characterization of fumed silica aggregates[END_REF] we want to point here that higher values, closer to ? // @ ≈ 2.31, could have been obtained by taking H and e 0 respectively to 0.3 and 1.0 as proposed by Baeza et al. [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF] This would have meant that the compacity of the structure tend to decrease significantly with increasing the lengthscale, a sensible result often discussed in the literature.

To summarize our SAXS analysis we refer the reader to the Figure 5. First, one should consider the situation in the suspension (Figure 5a) where silica agglomerates are well separated, giving the opportunity to measure their form factor since at low volume fraction in silica (Φ ≈0.1 vol.%) one obtains ( ) = 0 9 //+ ( ). In this case, the aggregates interact with their neighbors which belong to the same agglomerate only. The fractal dimension of the agglomerates can be then extracted from the power law between * * and * , i.e. ? //+ @ = 1.80. The corresponding value for the "native"

aggregates can be roughly extrapolated to the same value through our model by considering a high polydispersity of the aggregates (H=0.5) and a lacunarity higher than 1 (e 0 = 1.3).

Passing then to the untapped powder (Figure 5b) is synonymous of increasing substantially the volume fraction of silica (Φ ≈3 vol.%). In this situation, the agglomerates are in contact and are likely to be interpenetrated considering their very open structure. This results therefore in additional interactions between the aggregates, making emerge the inter-aggregates structure factor which pulls down the scattering intensity at < * . From this stage, a sphere of radius // is thus containing more silica than in the previous case, making in consequence the apparent fractal dimension of the aggregates slightly higher ? // @ = 1.85. (That extra-silica comes either from different interpenetrated agglomerates or from the same one, once compressed).

Finally, the compression of the powder into pellets is leading to an even denser material in which the agglomerates are more interpenetrated (Figure 5c). In this context, the aggregates are in close contact with plenty others, particularly if they were originally elongated, resulting in a much lower isothermal compressibility quantifiable in SAXS through the fall of G (Figure 4). Following this logic, one can therefore rationalize the results reported in the Table 1, notably the significant raise of TU from 0.18 to 0.28 when passing from the powder to the compressed pellets.

Beyond the strong impact of the elongated nature of the aggregates on the structure of densely packed samples, it is also playing a crucial role on their mechanical properties. As a matter of fact, the fumed silica made of such low-dimensionality objects was found to have a lower percolation threshold in comparison with its precipitated counterpart, leading to enhanced mechanical properties at higher porosity content. [START_REF] Garboczi | Geometrical percolation threshold of overlapping ellipsoids[END_REF][START_REF] Meille | Linear elastic properties of 2D and 3D models of porous materials made from elongated objects[END_REF][START_REF] Sanahuja | Micromechanical explanation of elasticity and strength of gypsum: from elongated anisotropic crystals to isotropic porous polycrystals[END_REF] (The systematic comparison of both FS and PS mechanical properties will be dealt with in our next article).

Porosimetry

In order to compare the above results with macroscopic measurements, we performed mercury porosimetry experiments in a similar way as what was proposed by Smith et al., [START_REF] Smith | Structural studies of vapor-phase aggregates via mercury porosimetry[END_REF] to extract the cumulated pores volume contained in the powder at different lengthscales. [START_REF] Xu | Multiscale structures to describe porous media part I: theoretical background and invasion by fluids[END_REF][START_REF] Johnston | Compression effects in mercury porosimetry[END_REF] For fumed silica, the fraction in macropores (in this case corresponding to inter-aggregate pores) was measured as they buckled or bent under low mercury pressure. Also, the fraction in mesopores (in this case corresponding to intra-aggregate pores) was measured as mercury infiltrated the material. [START_REF] Julve | Analysis of mercury porosimetry curves of precipitated silica, as an example of compressible porous solids[END_REF] In Figure 6, we present the typical relative pore volume content vs.

pressure curves which clearly illustrate this duality, with the macro-and meso-pores fraction linked to the pinkish and blue zones, respectively. From the quantitative angle, these results show that in a pellet containing Φ = 11 vol.% (pressed at 1.2 MPa), 71 % of the pores are identified as "macro" while only 29% are identified as "meso" (see the orange and blue arrows in Figure 6 for a schematic representation of these two pores' categories). For further details on less compacted pellets with a higher macropores content, we refer the reader to the Supplementary Material - Therefore, by taking into account the solid fraction in the sample (Φ ), we obtain in a straightforward way the volume fraction in the macro-and meso-pores within the pellet, i.e. 9 l = 63 vol.% and 9 * = 26 vol.%, respectively. Remarkably, adding the former value to the volume fraction of aggregates obtained from the above SAXS analysis (Φ // =40 vol.%), results in 9 l + Φ // ≈ 1, providing therefore the quantitative confirmation that the material is composed of hollow aggregates and macropores when observed at a lengthscale larger than 100 nm. 1) and further supports the initial tomography estimations, i.e., ≈0.28), highlighting the robustness of our study. As a matter of fact, the connections between these three parameters obtained from experiments probing the material at different lengthscales represents, in our opinion, a significant progress in terms of structural description.

(The aggregate compacity obtained from tomography, SAXS and porosimetry, is actually "averaged" in the present work at the 100 nm, 1 mm, and 1 cm lengthscales respectively).

Conclusion

To summarize, we used a complementary set of experimental techniques in both direct and Fourier spaces to fully quantify the multiscale structure (5 -500 nm) of an industrially relevant fumed silica. We showed in particular that its high porosity, responsible for its outstanding thermal insulation properties, originated from the anisotropic nature of its constituents at the sub-micronic lengthscale. Since the latter characteristic was undoubtedly attributable to the synthesis route, it therefore appears to be of prime importance for the development of innovative powders with reduced costs and higher energetic efficiency.

From a more fundamental point of view, we adapted in this work a clear methodology for the treatment of complex SAXS data measured on highly polydisperse systems. In particular, this approach offers the possibility of extracting basic geometrical parameters such as elementary particle radius, mean aggregates size and compacity, all of which being highly relevant as inputs for computer simulations. Last but not least, and for the first time to our knowledge, we were able to establish a strong correlation between SAXS and mercury porosimetry experiments at two different lengthscales (intra-and inter-aggregates). volume as a function of the mercury pressure. Each level of grey stands for an independent measurement giving an idea of the error bar. The inflexion point around 59 MPa stands for the limit between the inter-aggregate ("macro") and intraaggregate ("meso") pores of radius ? l and ? * respectively.

  ) confirm the very open nature of such silica powders, regardless of the Z-coordinate. Indeed, removing the volume occupied by the largest pores makes it possible to estimate the compacity of the denser zones within the agglomerates, resulting in ca. ≈ 28 %. (Details of the data analysis are given in Supplementary Material, Figures SM.1 and SM.2, as well as in the related video SI1_silica_volume.avi). Furthermore, in addition to characterizing the high porosity content, reconstructing the agglomerates in 3D (Figures 2d-f) allows selectively scanning sub-parts of the objects, corroborating the elongated character of the aggregates at the 100 nm lengthscale. Indeed, Figures 2e and 2f show the presence of flat and necklace-shaped aggregates, respectively, confirming the previous TEM observations.

.

  obtain & ≈7.0 nm, a characteristic size resulting in a nanobead volume of ) & = % ( & =1.4 10 3 nm 3 , in comparison with the ellipsoid geometry defined in the previous section ) *++ & , = % ( = 2.2 10 3 nm 3 . A perfect match would have been obtained for & = 8.1 nm, giving an idea of the error bar. For 0.1>q >0.3 nm -1 : This short transition zone presents a change in slope, corresponding to the inflexion point of the nanoparticle form factor. The latter is usually described from the Guinier approximation used to extract the radius of gyration / of the scattering objects with the expression ( ) = 0 exp (-Note that in Figure 3b, a calculation based on the silica-air Xray contrast (∆ ) and the Porod radius ( & ) with the empirical Guinier-Porod model[51] (assuming / = & ), perfectly matches the experimental data, highlighting the self-consistency of the two descriptions.

Figure≈ 1 . 8

 18 Figure3b, respectively // = (/ * and //+ = (/ * * . Alternatively, one can use the Kratky representation (i.e. ( ) = ;( ))[START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Baeza | Effect of Grafting on Rheology and Structure of a Simplified Industrial Nanocomposite Silica/SBR[END_REF] to make emerge more clearly these two characteristic sizes through the formation of a trapeze-shaped signal in which the power law and the breaks in slope in Figure3bare transformed respectively into a plateau and its two extremities in Figure3c. Based on the latter, we estimated the following values: * =8 10 -2 ± 1 10 -2 nm -1 and * * =9 10 -3 ± 1 10 -3 nm -1 , leading to // = 40 ± 5 nm and //+ = 350 ± 50 nm, in agreement with the micrographs presented in Figure1. Note that such a plateau appears only if the fractal dimension of the agglomerates, readable from ? //+ @

Figure SM. 3

 3 Figure SM.3 and Notes.

-FiguresFigure 1 :Figure 2 :Figure 3 :

 123 Figures

Figure 4 :

 4 Figure 4: Analysis of the apparent structure factor calculated from SAXS experiments. a) Apparent structure factor obtained from G && ( ) = ( ) \ 0 9 //+ ( )_ ⁄ for the neat powder and the corresponding pellets compressed at 0.15, 0.6 and 1.2 MPa respectively. G values are reported in Table 1. b) Minimum value of the apparent structure factor G as a function of the volume fraction in silica aggregates Φ // . The solid line stands for the prediction calculated from the Percus-Yevick theory assuming a polydispersity of H=0.3 (equation 3).

Figure 5 :Figure 6 :

 56 Figure 5: Schematic representation of the agglomerates and aggregates in the different samples. a) the ethanol suspension, b) the (uncompressed) powder and c) the compressed pellets. The red circle stands for the size of the aggregates and the arrow represents their radius. The shaded disk highlights the number of neighbor increasing with the volume fraction in silica, explaining the raise of the aggregates apparent fractal dimension estimated from the SAXS data analysis (its radius is set to 2 // ).

Table 1 :

 1 Structural characteristics of the fumed powder from the SAXS analysis.

	Sample	Φ (0) ±0.01	G (0) ±0.02	Φ // ±0.05 (j)	TU ±0.1 (j)	@ ( ) ? // ±0.05	d // ±30 ( )
	Powder	0.02	0.38	0.17	0.18	1.85	33
	0.15 MPa	0.08	0.20	0.28	0.29	*	53
	0.6 MPa	0.10	0.14	0.35	0.29	*	53
	1.2 MPa	0.11	0.10	0.40	0.28

* 51 (0) Φ and G

  =0.30, is again in excellent agreement with the SAXS predictions regarding aggregates compacity ( TU =0.28 in Table

	and the denominator by	t m usu	(d is the number of aggregates in the pellet of volume
	) ), resulting in &	=	t m DEnEon]88 /m usu t \m DEnEon]88 pm qrDs _/m usu	or more simply, &	=	Φ DE Φ DE pK qrDs	. The
	result of this calculation, leading to &
	At a smaller lengthscale, considering the mesopores as intra-aggregate cavities allows
	calculating the compacity of the aggregates from the porosimetry outputs such that
	&	=	m DEnEon]88 m ]88	=	m DEnEon]88 m DEnEon]88 pm qrDs	, where )	// and ) * are the volume of silica
	and mesopores, respectively, within an aggregate of volume ) // . From this simple
	definition, we can then describe the whole sample by multiplying both the numerator

Table 2 :

 2 Parameters and variables used in the article.

	Full Name

Acknowledgements

All the authors are grateful to Michael Sztucki (ESRF Grenoble) for his technical support with the SAXS experiments and to CLYM for access to the microscopes. They also gratefully acknowledge funding from CIFRE grant 2014-0903, ADEME grant 1604C0019, and MATeB (joint MATEIS EDF R&D laboratory). All the authors are sincerely grateful to the Acta Materialia reviewers for their great contributions to this article.

Supplementary Material

Supplementary Material is available from the Acta Materialia website or from the author.

Conflict of Interest

The authors declare no conflict of interest