Introduction

All dielectric materials have electroactivity, i.e., the capability of changing their dimension or shape in response to an applied electric field. [START_REF] Dang | Fundamentals, processes and applications of high-permittivity polymer matrix composites[END_REF] The electroactive polymer (EAP) and their polymer nanocomposites are particularly interesting for applications ranging from actuators, sensors, to generators, due to their low moduli, high strain capabilities, and the ease of processing at low cost and customizable electromechanical coupling properties.

Generally, EAPs have ability of inducing strains that are two orders of magnitude greater than the rigid and fragile electroactive ceramics. They show higher response speed as compared with the shape memory alloys and polymers. [START_REF] Yuan | Shape memory nanocomposite fibers for untethered high-energy microengines[END_REF] Because of these features, EAPs can be strongly comparable to biological muscles and have been long named "artificial muscles". [START_REF] Bar-Cohen | Electroactive polymers as artificial muscles: A review[END_REF] A challenge has been even posted to the community to develop a robotic arm actuated by artificial muscles to win an arm wrestling match with a human opponent.

In addition to actuators, EAPs have actually also revealed their potential in the sensory applications such as haptic sensing, blood pressure and pulse rate monitoring, and chemical sensing. [START_REF] Vijaya | Piezoelectric Materials and Devices: Applications in Engineering and Medical Sciences[END_REF] Furthermore, EAPs even act as key active materials in generators. With ever-increasing functionalities of portable electronics (e.g., wireless sensors and transmitters) and wireless micro-systems, their energy requirements have been dramatically elevated. While the use of batteries is troublesome due to the environmental concern and limited lifespan, necessitating their periodic replacement. The obvious solution to address this challenge is to develop self-powered systems that completely rely on the energy harvested from the human body or the ambient environment. EAPs have been demonstrated to be able to scavenge vibrational mechanical energy [START_REF] Lallart | Electrostrictive polymers for mechanical energy harvesting[END_REF] and ocean wave energy [START_REF] Jean | Standing Wave Tube Electro Active Polymer Wave Energy Converter[END_REF].

EAPs can be divided into different subgroups based on the crystal class (i.e., centrosymmetric or noncentrosymmetric) that they are belong to. As a dielectric material with a center of symmetry is stimulated by an electric field, the symmetry will cancel the movements of cations and anions and does not lead to a net deformation in the crystal. Nevertheless, the chemical bonds are not harmonic and the anharmonicity of the bonds gives rise to second-order effects resulting in small net deformation of the lattice. [START_REF] Vijaya | Piezoelectric Materials and Devices: Applications in Engineering and Medical Sciences[END_REF] The deformation is found to be proportional to the square of the electric field, and independent on the direction of the field. This effect is called electrostriction. Because such anharmonic effect exists in all dielectrics, all dielectrics are electrostrictive materials.

On the other hand, as a noncentrosymmetric dielectric material is subjected to an external electric field, the asymmetric movement of the neighboring ions results in much more significant deformation of the crystal. Such deformation dominates the net one induced by the anharmonicity of the chemical bonds. [START_REF] Vijaya | Piezoelectric Materials and Devices: Applications in Engineering and Medical Sciences[END_REF]. Moreover, it is linearly proportional to the magnitude of the electric field. Reversing the applied field will change the direction of deformation. These materials are called piezoelectrics. A detailed classification of electroactive (dielectric) materials is shown in Figure 7.1.

Figure7.1. Classification of dielectric materials.

In this chapter, we focus on electrostrictive polymeric materials. The fundamental issues and promising applications will be addressed. The recent progress toward giant-electrostriction EAPs in the field will be reviewed and some longstanding challenges for their practical applications will be also discussed.

Electrostriction Relations

Electrostriction represents the ability of materials to change their dimensions in response to an applied electric stimulus. As aforementioned, it is present in all materials, regardless of the symmetry. It is generally defined as a quadratic coupling between strain (𝑆 𝑖𝑗 ) and polarization (𝑃 𝑚 ) [START_REF] Damjanovic | Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics[END_REF][START_REF] Lallart | Electrostrictive polymers for mechanical energy harvesting[END_REF])

𝑆 𝑖𝑗 = 𝑆 𝑖𝑗𝑘𝑙 𝑃 𝑇 𝑘𝑙 + 𝑄 𝑖𝑗𝑚𝑛 𝑃 𝑚 𝑃 𝑛 [7.1] 𝐸 𝑚 = 𝜀′ 𝑚𝑛 𝑇 𝑃 𝑛 + 2𝑄 𝑘𝑙𝑚𝑛 𝑇 𝑘𝑙 𝑃 𝑛 [7.2]
Where 𝑄 𝑖𝑗𝑘𝑙 and 𝑄 𝑘𝑙𝑚𝑛 are the polarization-related electrostriction coefficients, 𝑆 𝑖𝑗𝑘𝑙 𝑃 is the elastic compliance, 𝑇 𝑘𝑙 is the stress, 𝐸 𝑚 is the electric field, and 𝜀′ 𝑚𝑛 𝑇 is the inverse of the linear dielectric permittivity.

The polarization 𝑃 that is induced in a polarizable material by an applied electric field vector 𝐸 is given by:

𝑃 𝑚 = 𝜒 𝑚𝑘 𝐸 𝑘 [7.3] 𝑃 𝑛 = 𝜒 𝑛𝑙 𝐸 𝑙 [7.4]
where 𝜒 is the second-rank tensor known as the dielectric susceptibility of the material. This relation is valid only for linear materials or in a linear limit for nonlinear materials. The dielectric susceptibility 𝜒 is a proportionality constant that describes the degree of polarization of a dielectric material in response to an applied electric field. The greater the dielectric susceptibility, the greater polarization of a material in response to a given field, and thereby more significant reduction of the total electric field inside the material. The dielectric susceptibility can be expressed as

𝜒 = 𝜀 0 (𝜀 𝑟 -1)) -;[7.5]
Where 𝜀 0 = 8.854 × 10 -12 F 𝑚 -1 is the dielectric permittivity of free space, and 𝜀 𝑟 is the relative dielectric permittivity of the material. Where 𝜀 𝑖𝑗 is the dielectric permittivity of the material, 𝛿 𝑖𝑗 is Kronecker's symbol.

As 𝑖 = 𝑗, 𝛿 𝑖𝑗 = 1

[7.10] As 𝑖 ≠ 𝑗, 𝛿 𝑖𝑗 = 0 [7.11]
For most of the ferroelectric materials, 𝜀 0 𝛿 𝑖𝑗 ≪ 𝜒 𝑖𝑗 , therefore 𝜀 𝑖𝑗 ≈ 𝜒 𝑖𝑗 . [START_REF] Damjanovic | Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics[END_REF] Bases on the above equations, we can obtain

𝐷 𝑖 = 𝜀 𝑖𝑘 𝑇 𝐸 𝑘 + 2𝑀 𝑖𝑗𝑘𝑙 𝑇 𝑘𝑙 𝐸 𝑙 [7.12]
Where 𝜀 𝑖𝑘 𝑇 is the linear dielectric permittivity of materials under constant mechanical stress.

Determination of the Electrostriction Coefficient

Experimental determination of the electrostriction coefficient M is typically performed in the actuator mode. The polymer samples are coated with conducting metallic or organic layers on the two opposite surface to form a sandwiched capacitor structure. The electrostriction coefficient is therefore characterized by measuring the induced thickness variation under certain magnitudes of high electrical field. In this case, the global strain S is the sum of two distinct contributions, i.e., SE and SMax. The former SE corresponds to the true electrostriction that originates from direct coupling between the polarization and mechanical response, while the latter one SMax is as a result of Maxwell stress that originates from the Coulomb interaction between oppositely charged compliant electrodes. Therefore the apparent electrostriction coefficient, 𝑀 𝑎𝑝𝑝 , is given by

𝑀 𝑎𝑝𝑝 = 𝑆/𝐸 2 [7.13]
Plenty of previous research have evaluated the electrostriction coefficients of various EAPs based on the actuator method. For example, Wongtimnoi et al. [START_REF] Wongtimnoi | Improvement of electrostrictive properties of a polyether-based polyurethane elastomer filled with conductive carbon black[END_REF] studied the electrostrictive properties of a polyether-based polyurethane (PU) elastomer and its corresponding composites filled with conductive carbon black (CB) by measuring the thickness strain induced by external electric fields (Figure 7.2a). Near the percolation threshold, the composite demonstrated an electrostriction coefficient as high as -1.2x10 -14 m 2 /V 2 . Park et al. [START_REF] Park | Actuating single wall carbon nanotube-polymer composites: Intrinsic unimorphs[END_REF] reported a novel electroactive single-walled carbon nanotube (SWNT)-polyimide composite. This composite intrinsically forms a unimorph during the fabrication process to actuate bending deformation in response to an applied electric field (Figure 7.2b). A strain of 2.6% can be obtained at low driving voltages (<1 MV/m). It was claimed that such strong actuation capabilities of the electroactive polymers were gained from the incorporation of SWNTs into these polymers. The electrostriction coefficients (M33) of the EAP composites were estimated from -3.8x10 -15 to -1.2x10 - 13 m 2 /V 2 . The latter represents the state of the art. [START_REF] Wongtimnoi | Polyurethanes électrostricitifs et nanocomposites : caractérisation et analyse des mécanismes de couplage électromécaniques[END_REF]) Copyright 2011, Elsevier. (b) Schematic structure of an intrinsic unimorph of SWNT composite film without and with an electric field. [START_REF] Park | Actuating single wall carbon nanotube-polymer composites: Intrinsic unimorphs[END_REF] The actuator approach can be easily implemented experimentally and highly efficient to characterize the global electrostriction. Nevertheless, it necessitates the application of a large voltage of about several hundred volts. At high voltage, assumptions are needed to determine whether the deformation is dominated by true electrostriction or by the Maxwell stress (Figure 7.3). Actually, it is very difficult to differentiate the true electrostriction from the Maxwell stress experimentally, particularly for heterogeneous dielectric systems. Because both of them induce bulk deformations that display a quadratic dependence with the electrical field. According to the literature, the actuation for elastomeric materials with low permittivity (𝜀 𝑟 ′ < 3), such as silicones, natural rubber latex and acrylates results mainly from Maxwell stress. On the other hand, for high permittivity (𝜀 𝑟 ′ > 5) polar polymers, such as polyvinylidene fluoride (PVDF) and PU, the actuation is dominated by the electrostriction mechanism and Maxwell stress as well.

The electrostriction actually also describes the ability of materials of changing their dielectric properties with deformation applied upon them, as show in Eq. [7.12]. By replacing the 𝐷 𝑖 with Eq. [7.9], one can obtain

𝑀 𝑖𝑗 ≈ ∆𝜀 𝑖𝑗 2𝑇 𝑖𝑗 ≈ ∆𝜀 𝑖𝑗 /2𝑌𝜑 𝑗 [7.16]
Where Y is the young's modulus and 𝜑 𝑗 is the mechanical strain. Being different from the actuator mode widely used in previous studies, we have developed a new method based on Eq.[7.16], i.e., measuring the permittivity variation with applied mechanical deformation. This method allows for direct measurements of the true electrostriction coefficients with a negligible contribution of the Maxwell stress. In our measurements, a voltage of only 1 V is applied, which corresponds to an electrical field below 10 -3 V/µm. The importance of Maxwell stress can be estimated by comparing the strain resultant from Maxwell stress and the strain mechanically applied by the tensile test mechanical instrument. The strain SMax resultant from Maxwell stress is given by

𝑆 𝑀𝑎𝑥 = 𝜀 𝑖𝑗 𝐸 2 𝑌 [7.17]
where E and Y respectively are the electrical field and the Young's modulus of the material. This strain (~10 -6 %) is totally negligible compared to the strain mechanically applied (~10%). In the case of an isotropic electrostrictive polymer film subjected to a stress 𝜏11 and to an electrical field along its thickness direction 𝐸3 (Figure 7.4a), the electrostriction coefficient can be expressed as

𝑀 31 = ∆𝜀 31 /2𝑌𝜑 11 [7.18]
While in an isotropic electrostrictive polymer film subjected to a stress 𝜏33 and to an electrical field along its thickness direction 𝐸3 (Figure 7.4b), the electrostriction is caculated as The mechanism by which a parallel plate capacitor changes its capacitance in response to a mechanical stretching. [START_REF] Yuan | Giant Electrostriction of Soft Nanocomposites Based on Liquid Crystalline Graphene[END_REF] Copyright 2018, American Chemical Society.

𝑀 33 = ∆𝜀 33 /2𝑌𝜑 11 [7.19]
The linear mechanics of the mechanical stretching process can be described as

𝜑 𝑥 = ∆𝑙 𝑙 [7.20] 𝜑 𝑦 = 𝜑 𝑧 = 𝜈𝜑 𝑥 [7.21] 𝑙 𝑠𝑡𝑟𝑒𝑡𝑐ℎ = 𝑙 + Δ𝑙 = 𝑙 + 𝜑 𝑥 𝑙 [7.22] 𝑤 𝑠𝑡𝑟𝑒𝑡𝑐ℎ = 𝑤 -Δ𝑤 = 𝑤 -𝜈𝜑 𝑥 𝑤 [7.23] ℎ 𝑠𝑡𝑟𝑒𝑡𝑐ℎ = ℎ -Δℎ = ℎ -𝜈𝜑 𝑥 ℎ [7.24]
Where 𝜈 is Poisson's ratio, 𝜑 𝑥 is the strain, and l, w, and h are the dimensions of the capacitor in a relaxed state. Considering the dimensions of the capacitor in its stretched state, one can deduce the value of the capacitance Cstretch:

𝐶 𝑠𝑡𝑟𝑒𝑡𝑐ℎ = 𝜀 0 𝜀 𝑠𝑡𝑟𝑒𝑡𝑐ℎ ( 𝑤 𝑠𝑡𝑟𝑒𝑡𝑐ℎ 𝑙 𝑠𝑡𝑟𝑒𝑡𝑐ℎ ℎ 𝑠𝑡𝑟𝑒𝑡𝑐ℎ ) = 𝜀 0 𝜀 𝑠𝑡𝑟𝑒𝑡𝑐ℎ ( (𝑤-𝜈𝜑 𝑥 𝑤)(𝑙+𝜑 𝑥 𝑙) ℎ-𝜈𝜑 𝑥 ℎ ) = 𝜀 0 𝜀 𝑠𝑡𝑟𝑒𝑡𝑐ℎ 𝑤𝑙 ℎ (1 + 𝜑 𝑥 ) [7.25]
where, 𝜀 0 is the permittivity of free space, and 𝜀 𝑠𝑡𝑟𝑒𝑡𝑐ℎ is the relative permittivity in the stretched state. This electromechanical equation directly relates the capacitance to the permittivity, applied strain and initial geometry of the capacitor. Despite that this equation is derived from Poisson contraction, it does not depend on the value of Poisson's ratio in the case of isotropic dielectric materials. Therefore, by measuring the capacitance at different strains, we can directly determine the permittivity variations (∆𝜀) of the material in response to mechanical deformations. This allows for calculation of true electrostrictive coefficients of a material directly through the aforementioned Eq. [7.18] or Eq.[7.19].

The Route to High Electrostriction Materials

Electrostrictive materials have been widely used to efficiently convert electrical and mechanical energy. [START_REF] Zhang | An all-organic composite actuator material with a high dielectric constant[END_REF] As aforementioned, all dielectric materials exhibit electrostriction. But certain materials, known as rexlor ferroelectrics have high electrostriction coefficient. [START_REF] Zhang | Giant electrostriction and relaxor ferroelectric behavior in electron-irradiated poly(vinylidene fluoridetrifluoroethylene) copolymer[END_REF] Today, the most used electrostrictive ceramics are lead based materials such as lead magnesium niobate (PMN), lead magnesium niobate-lead titanate (PMN-PT), and lead lanthanum zirconate titanate (PLZT). In spite of the high electrostrictive performances, they exibit several drawbacks, such as high sintering temperaure, brittleness, and lead-related environmental concern. Lead-free and flexible electrostrictive materials are always highly demanded, particularly for large-stroke actuation applications. In this regard, the polymeric materials, such as dielectric elastomers [START_REF] Brochu | Advances in Dielectric Elastomers for Actuators and Artificial Muscles[END_REF][START_REF] Lehmann | Giant lateral electrostriction in ferroelectric liquidcrystalline elastomers[END_REF][START_REF] Luna | Giant Electrostrictive Response and Piezoresistivity of Emulsion Templated Nanocomposites[END_REF][START_REF] Luna | Giant Permittivity Polymer Nanocomposites Obtained by Curing a Direct Emulsion[END_REF][START_REF] Pelrine | High-speed electrically actuated elastomers with strain greater than 100%[END_REF][START_REF] Yuan | Graphene liquid crystal retarded percolation for new high-k materials[END_REF], high-k elastomer composites, [START_REF] Clarke | Compatibilising effect of carbon black on morphology of NR-NBR blends[END_REF][START_REF] Huang | Electrical, thermophysical and micromechanical properties of ethylene-vinyl acetate elastomer composites with surface modified BaTiO3 nanoparticles[END_REF][START_REF] Kohlmeyer | Electrical and Dielectric Properties of Hydroxylated Carbon Nanotube-Elastomer Composites[END_REF][START_REF] Kuang | Fabrication of highly oriented hexagonal boron nitride nanosheet/elastomer nanocomposites with high thermal conductivity[END_REF][START_REF] Molberg | High Breakdown Field Dielectric Elastomer Actuators Using Encapsulated Polyaniline as High Dielectric Constant Filler[END_REF][START_REF] Romasanta | Increasing the performance of dielectric elastomer actuators: A review from the materials perspective[END_REF][START_REF] Wongtimnoi | Polyurethanes électrostricitifs et nanocomposites : caractérisation et analyse des mécanismes de couplage électromécaniques[END_REF] and PVDF and its copolymers [START_REF] Hagerman | E2I EPRI Specification: Guidelines for Preliminary Estimation of Power Production by Offshore Wave Energy Conversion Devices[END_REF][START_REF] Li | Exchange coupling in P(VDF-TrFE) copolymer based all-organic composites with giant electrostriction[END_REF][START_REF] Minh Quyen | All-organic electrostrictive polymer composites with low driving electrical voltages for micro-fluidic pump applications[END_REF][START_REF] Zhang | Giant electrostriction and relaxor ferroelectric behavior in electron-irradiated poly(vinylidene fluoridetrifluoroethylene) copolymer[END_REF], are more promising. However, most of the current electrostrictive polymers suffer from low electromechanical coupling coefficient (M~10 -18 m 2 /V 2 at 100 Hz) [START_REF] Guillot | Electrostrictive effect in polyurethanes[END_REF][START_REF] Hagerman | E2I EPRI Specification: Guidelines for Preliminary Estimation of Power Production by Offshore Wave Energy Conversion Devices[END_REF][START_REF] Zhang | An experimental investigation of electromechanical responses in a polyurethane elastomer[END_REF] due to their weak intrinsic polarization response, which severely restricts their applicability for practical devices. A big chanlenge that has been posed to the community is the developpment of electromechanically active polymeric materials with giant electrostriction. [START_REF] Yuan | Giant Electrostriction of Soft Nanocomposites Based on Liquid Crystalline Graphene[END_REF] As discussed in the previous section, the electrostriction is as a result of the changment of the polarization in materials in response to the external mechanical or electrical stimuli. The electromechanical properties can thus be optimized by improving the polarization response in polymers through designing structures on different scale levels.

NASA has developed a new electrostrictive graft elastomer, [START_REF] Su | Performance evaluation of bending actuators made from electrostrictive graft elastomers[END_REF][START_REF] Su | Electrostrictive graft elastomers and applications[END_REF] which is approximately 30 times stiffer than classic PU. These strong elasotmers have potential employment as an actuator that provides much higher mechanical energy and power output than soft neat elastomers. As illustrated in Figure 7.6, The graft elastomer contains flexible backbone chains and grafts that are the side chains of each backbone. The grafts physically cross-link with each other and eventually form crystal units. Both of the backbone chain and the crystal graft units have polarized monomers. These monomers contain atoms with electric partial charges, giving rise to dipole moments. These dipole moments will rotate to align with the external electric field and therefore stimulate backbone chain reorientation and induce a much larger electrostrictive strain than neat PU. [START_REF] Wang | Deformation mechanisms of electrostrictive graft elastomer[END_REF] Figure 7.6 Illustrative structure of the electrostrictive graft elastomer. [START_REF] Wang | Deformation mechanisms of electrostrictive graft elastomer[END_REF] Another example can be the dielectric elastomer gel. Kim and co-workers [START_REF] Kim | Electric Actuation of Nanostructured Thermoplastic Elastomer Gels with Ultralarge Electrostriction Coefficients[END_REF] introduced periodic nanoscale interfaces to dielectric elastomer gels and achieved an unexpected high electrostriction coefficient by creating numerous interfaces between dielectric mismatched nanostructured domains, resulting in a large electromechanical strain response at a relatively low electric field. Figure 7.7 shows a TEM picuture of the nanostructured domains in a thermoplastic elastomer gel composed of 20 wt% poly(styrene-block-ethylbutylene-block-styrene) (SEBS) and 80 wt% white mineral oil. The strong electrostrictive effect is attributed to the inhomogeneous electric field in the film thickness direction because of the high density of interfaces between dielectric mismatched periodic nanoscale domains. The demonstrated electrostrictive coefficient can be two orders of magnitude higher than that of neat PU. and 80 wt% white mineral oil. [START_REF] Kim | Electric Actuation of Nanostructured Thermoplastic Elastomer Gels with Ultralarge Electrostriction Coefficients[END_REF] Normal ferroelectric poly(vinylidene fluoride-trifluoroethylene) P(VDF-TrFE) polymers have excellent piezoelectric properteis. But after the introduction of nonpolar molecular structure in the ferroelectric phase [START_REF] Zhang | Giant electrostriction and relaxor ferroelectric behavior in electron-irradiated poly(vinylidene fluoridetrifluoroethylene) copolymer[END_REF] or the chemical copolymerization of a third bulky monomer [START_REF] Xia | High electromechanical responses in a poly(vinylidene fluoride-trifluoroethylene-chlorofluoroethylene) terpolymer[END_REF], it can be converted to relaxor ferroelectric polymers with large electrostriction. The improved electrostriction is due to the difference in the lattice constant between the polar and nonpolar phases and to the gradual increase of field-induced polarization in the relaxor P(VDF-TrFE) polymer.

Though improvements of electrostriction effect have been achieved based on the changement of polymer composition, a versatile strategy actually relies on the composite approach, i.e., mixing electroactive nanofillers with polymer matrices. It allows for a large tunability by varying the identity, shape, and content of nanoadditives.

Giant permittivity can actually be otained in electrically heterogeneous systems composed of conducting particles and insulating polymer matrices. [START_REF] Luna | Giant Electrostrictive Response and Piezoresistivity of Emulsion Templated Nanocomposites[END_REF][START_REF] Luna | Giant Permittivity Polymer Nanocomposites Obtained by Curing a Direct Emulsion[END_REF][START_REF] Yuan | Vertically Aligned Carbon Nanotube Arrays on SiC Microplatelets: A High Figure-of-Merit Strategy for Achieving Large Dielectric Constant and Low Loss in Polymer Composites[END_REF][START_REF] Yuan | Giant Dielectric Permittivity Nanocomposites: Realizing True Potential of Pristine Carbon Nanotubes in Polyvinylidene Fluoride Matrix through an Enhanced Interfacial Interaction[END_REF][START_REF] Yuan | Anisotropic Percolation of SiC-Carbon Nanotube Hybrids: A New Route Towards Thermally Conductive High-k Polymer Composites[END_REF] The permittivity of a composite material diverges at the percolation threshold. (Dang et al, 2013, In review.;[START_REF] Dang | Fundamentals, processes and applications of high-permittivity polymer matrix composites[END_REF][START_REF] Luna | Giant Electrostrictive Response and Piezoresistivity of Emulsion Templated Nanocomposites[END_REF][START_REF] Luna | Giant Permittivity Polymer Nanocomposites Obtained by Curing a Direct Emulsion[END_REF][START_REF] Yuan | Giant Electrostriction of Soft Nanocomposites Based on Liquid Crystalline Graphene[END_REF][START_REF] Yuan | Graphene liquid crystal retarded percolation for new high-k materials[END_REF][START_REF] Yuan | Anisotropic Percolation of SiC-Carbon Nanotube Hybrids: A New Route Towards Thermally Conductive High-k Polymer Composites[END_REF][START_REF] Yuan | Percolation of carbon nanomaterials for high-k polymer nanocomposites[END_REF] Systems at this point become critically sensitive, providing thereby the possibility for giant electrostriction. [START_REF] Luna | Giant Electrostrictive Response and Piezoresistivity of Emulsion Templated Nanocomposites[END_REF] A large amount of research have been conducted by randomly mixing conducting particles, such as metal nanoparticles and carbon based nanomaterials, with polymers to reach a high permittivity. The improved electromechanical performance can be associated to the variation of local electrical field as well as to the strong interfacial polarization as a result of charge accumulation. [START_REF] Lallart | Electrostrictive polymers for mechanical energy harvesting[END_REF] Typical solution methods were used to randomly mix the nanoparticles with polymer matrices. Recently, we reported a soft electrostrictive elastomer nanocomposite composed of reduced graphene oxide (rGO) in polydimethylsiloxane (PDMS) matrix. The advantage of using graphene is based on the essential feature of their large surface area, which is critical to achieve large polarization in composites. A diblock copolymer aminopropylmethylsiloxane-b-dimethylsiloxane (PAPMS-b-PDMS) was chosen to adsorb onto graphene oxide (GO) nanosheets and stabilize their dispersion in composite precursor solutions. Such interaction is based on the electrostatic attraction between negative charges of GO nanosheets and positive ones of the copolymer (Figure 7.9a). As such, the GO nanosheets continuously transferred from water phase into diethyl ether phase as this interaction at water/oil interface proceeded (Figure 7.9b). [START_REF] Yuan | Graphene liquid crystal retarded percolation for new high-k materials[END_REF] The obtained GO suspension is well suited to prepare PDMS matrix composites, without showing any reaggregation of GO nanosheets during the process. Moreover, with increasing concentration of such GO suspension, a nematic liquid crystal phase can form even with a low GO content (Figure 7.9c,d).

Experimentally we assessed the percolation transition in rGO/PDMS composites. The margin improvement of conductivity at 1.0 wt.% indicates that this loading is well below the percolation threshold. However, at this rGO-PDMS mass ratio, liquid crystal (LC) transition has already taken place (Figure 7.9d), indicating that the flakes with large anisotropy tend to self-assemble into nematic LCs before they form a percolated path along the direction of the conductivity measurement. This phenomenon would fundamentally change our views on the interest of graphene for conductive composites. Such competition can nevertheless be positively exploited to develop dielectric composites. The ordered liquid crystalline graphene structure is expected to promote the formation of effective microcapacitors to afford a very high permittivity (~2000 at 0.1 Hz). [START_REF] Yuan | Graphene liquid crystal retarded percolation for new high-k materials[END_REF]

Figure 7.9 (a) Schematic illustration of the electrostatic attraction between negative charges of GO nanosheets and positive ones of the PAPMS-b-PDMS copolymer. (b) Photograph shows the phase transfer of GO flakes from the aqueous to the organic phase. (c) Photograph of soft PDMS composites with GO (brown) and rGO (black)

nanosheets at a loading of 1 wt.%. (d) Optical micrographs between crossed polarizers of GO suspensions in PDMS monomer at a concentration of 1 wt.%. [START_REF] Yuan | Graphene liquid crystal retarded percolation for new high-k materials[END_REF] It is found that graphene doped PDMS exhibits very large variations of permittivity within a very small strain range due to the liquid crystalline alignment of the graphene sheets within the polymer. Such large variations yield ultralarge electrostriction coefficient (M~10 -14 m 2 /V 2 at 0.1 Hz), superior to those (from10 -18 m 2 /V 2 to 10 -15 m 2 /V 2 at 0.1 Hz) obtained in conventional polymer nanocomposites. [START_REF] Guiffard | Enhanced electric fieldinduced strain in non-percolative carbon nanopowder/polyurethane composites[END_REF][START_REF] Guyomar | The compressive electrical field electrostrictive coefficient M33 of electroactive polymer composites and its saturation versus electrical field, polymer thickness, frequency, and fillers[END_REF][START_REF] Jaaoh | Deformation on segment-structure of electrostrictive polyurethane/polyaniline blends[END_REF][START_REF] Jaaoh | Enhanced strain response and energy harvesting capabilities of electrostrictive polyurethane composites filled with conducting polyaniline[END_REF][START_REF] Lallart | Electrostrictive polymers for mechanical energy harvesting[END_REF][START_REF] Park | Actuating single wall carbon nanotube-polymer composites: Intrinsic unimorphs[END_REF][START_REF] Wongtimnoi | Electrostrictive thermoplastic polyurethane-based nanocomposites filled with carboxyl-functionalized multi-walled carbon nanotubes (MWCNT-COOH): Properties and improvement of electromechanical activity[END_REF][START_REF] Wongtimnoi | Improvement of electrostrictive properties of a polyether-based polyurethane elastomer filled with conductive carbon black[END_REF]

Applications of Electrostrictive Materials

Electrostrictive polymeric materials have been well known for their significant advantages of facile proccessibility, short response times (less than milliseconds), low hysteresis, and the capability of generating high strain levels (>7%) which is far above those of traditional inorganic electrostrictive materials or piezoceramics. Therefore, ever-increasing research attention has been attracted to the community of electrostrictive polymers during the past decades because of their promising applications ranging from functional actuators, capacitive sensors, to integrated microelectromechanical systems. Recently, electrostrictive polymers have even been shown to be potentially useful for harvesting energy from vibration sources to power electronic devices.

Actuators

Dielectric elastomer, as one type of EAPs, has been widely used to develop fuctional actuators, including artifical muscles. Dielectric elastomer deforms due to the electrostatic interaction between two electrodes with opposite electric charges. In earlier years, most of the focus was on actuator configurations. Electroded dielectric elastomer films can be rolled into a scroll, stretched over a frame, or laminated on a flexible substrate to form unimorphs and bimorphs, even formed into a tubular shape. A number of configurations have been fabricated and are shown [START_REF] Kornbluh | Electroelastomers: Applications of dielectric elastomer transducers for actuation, generation and smart structures[END_REF]

Capacitive Sensors

In addtion to actuators, the electrostrictive polymers can be sandwiched with a pair of compliant electrodes to fabricat a variable capacitor, which can be used to detect pressure or deformation in the change of capacitance. Such pressure sensors strategically place on a flexible synthetic skin working in the pressure range of 1 kPa to 1 MPa would be able to detect fingertip texture, hand grip, and in-shoe pressures.

According to Equation [1.25], for a sensor fabricated in three layers with a dielectric between the electrodes, the capacitance will be proportional to strain under a simple stretching. As we know, the small strain sensing can be also realized by electrical resistance measurement of elastomer composites embedded with conducting nanoparticles. However, for large strain sensing that involves filled elastomers, the measurement will be non-linear due to geometric effect and transient changes in resistivity of the electrodes as well.

Capacitance measurement offers an opportunity for reliable measurement that is nearly unaffected by transient changes in electrode resistance. The capacitive strain gauge factor 𝜅 can be written as

𝜅 = ∆𝐶/𝐶 0 𝜑 𝑥 = ∆𝜀/𝜀 𝜑 𝑥 + ∆𝜀 𝜀 + 1 [7.26]
Where ∆𝜀 is the permittivity variation. For nonpolar dielectrics like silicon rubber,∆𝜀 = 0, so the 𝜅 = 1. While for electrically heterogeneous dielectrics, ∆𝜀 ≠ 0, the capacitance variation ∆𝐶 is determined by the geometry effect coupled with the true electrostriction effect of the dielectrics, as the case of near percolated elastomer composites with conducting nanoparticles. Plenty of reported capacitive sensors are based on pure silicons sandwiched by diverse compliant electrodes. For instance, the Bao's group reported a capacitive sensor with transparant SWNT films as complient electrodes. It can detect a pressure up to 50 kPa and strain up to 50% (Figure 7.11).

The capacitance linearly increases with the strain, showing a slop near 1. In spite that is is not very sensitive to extremely low pressures, it is still able to detect pressures generated by a fly (20 mg).The transparency larely expands the application scope, such as touch screens and electronic skins with various options of skin color. [START_REF] Lipomi | Skin-like pressure and strain sensors based on transparent elastic films of carbon nanotubes[END_REF] In order to improve the sensitivity of capacitive sensors, the key is to employ soft materials with high electrostriction coefficient. According to Equation [1.26], it is promising to combine the geometry effect and electrostriction to induce a large capacitance variation for a given strian level. Based on the near percolated rGO composite demonstrated in Figure 7.9, we showed a highly efficient capacitive sensor in response to a cyclic mechanical tensile loading (Figure 7.12a). Such sensor device is stretched to a strain of 4%, after that the deformation was released with the same rate as stretching. Such a cycle was repeated 20 times. It was noticed that during the first four cycles, the capacitance variation slightly decreased from 2.4 nF to 2.2 nF yet hereafter tended to remain stable (Figure 7.12b). Recalling that the capacitance at relax state is 4.1 nF, a constant capacitive gauge factor of -13.6 was obtained.

Compared with the capacitive sensors sole based on the geometry effect (gauge factor <1), the electrostrictive composite sensors show a much higher sensitivity, which derives from their ultrahigh electrostriction. (b)Time dependence of the strain and relative permittivity at 0.1 Hz over the 20 stretching/releasing cycles. [START_REF] Yuan | Giant Electrostriction of Soft Nanocomposites Based on Liquid Crystalline Graphene[END_REF] Copyright 2018, American Chemical Society.

Mechanical Energy Harvesting

Dielectric elastomers have demonstrated great promise as actuator and sensory materials. Nevertheless, their potential in converting mechanical enegy into electricity in a generator mode has been less exploited. Actually dielectric elastomer acts electrically as variable capacitors. If a low voltage is placed on a pre-strained dielectric elastomer, the contraction works against the electrostatic field stress and raises the voltage of the charge, therefore generating electrical energy. Such simplicity, combined with demonstrated high energy density and high efficiency [START_REF] Lallart | Electrostrictive polymers for mechanical energy harvesting[END_REF][START_REF] Pelrine | Dielectric elastomers: Generator mode fundamentals and applications[END_REF] suggests that dielectric elastomers are promising for a wide range of energy harvesting applications. Comparing with competing energy harvesting technologies, such as electromagnetic or piezoelectric ones, dielectric elastomers show less stiffness so that they can be applied large deformations and operate even at frequency as low as 0.1 Hz (ocean waves). Their energy density can reach 0.4 J/g, which is one order of magnitude higher than those of the counterparts. Meanwhile their energy convertion efficiency can be higher than 50%, while that of electromagnetic one is usually below 20%.

Figure 7.13 shows the performance limits and also the basic operational cycles of a dielectric elastomer generator. In detail, there are four basic steps in the simplest operational cycles: (1) the film is stretched to its maximum stretch state; (2) a voltage or charge is applied to the film; (3) the film relaxes its internal elastic energy with constant charge, constant voltage or constant field; and (4) the charge is removed from the film to return it to its initial state. The horizontal axis of Figure 7.13 is a variable related to the change in capacitance, while the vertical axis is the square of the voltage or electric field across the film. Therefore the shaded blue area rerepsents the amount of extracted energy. The net amount of extracted energy per unit volume of film for the constant charge, constant voltage, and constant field cycles are respectively expressed as uQ, uV and uE. They can be caculated as [START_REF] Kornbluh | From boots to buoys: Promises and challenges of dielectric elastomer energy harvesting[END_REF]2012)

𝜇 𝑄 = 1 2 𝜀𝐸 𝑚𝑎𝑥 2 [ 𝛾 2 -1 𝛾 2 ] [7.27] 𝜇 𝑉 = 1 2 𝜀𝐸 𝑚𝑎𝑥 2 [ 𝛾 2 -1 𝛾 2 ] [7.28] 𝜇 𝐸 = 𝜀𝐸 𝑚𝑎𝑥 2 ln (𝛾) [7.29]
Where 𝐸 𝑚𝑎𝑥 and 𝜀 are respectively the maximum electric field and permittivity of the film samples. 𝛾 is the ratio of stretched area/unstretched area.

Figure 7.13

The performance limits and operational cycles of a dielectric elastomer generator. [START_REF] Kornbluh | Dielectric elastomers: Stretching the capabilities of energy harvesting[END_REF] Based on previous discussion, desirable dielectric elastomers for mechanical energy harvesting should have high breakdown strength, high permittivity, low dielectric losses. Meanwhile, they should sustain high flexibility to be deformed with small forces, low viscoelastic losses and low creep and stress relaxation effect. In the past decades, the most common candidate materials considered for generator applications are the ones based on commercial formulations of acrylics and silicones [START_REF] Carpi | Dielectric Elastomers as Electromechanical Transducers[END_REF] as well as polyurethane-based polymers [START_REF] Wongtimnoi | Electrostrictive thermoplastic polyurethane-based nanocomposites filled with carboxyl-functionalized multi-walled carbon nanotubes (MWCNT-COOH): Properties and improvement of electromechanical activity[END_REF][START_REF] Wongtimnoi | Improvement of electrostrictive properties of a polyether-based polyurethane elastomer filled with conductive carbon black[END_REF].

Pelrine et al. [START_REF] Pelrine | Dielectric elastomers: Generator mode fundamentals and applications[END_REF]) demonstrated the fundamentals and applications of neat silicone as generator. They used the constant charge cycle to harvest the stored mechanical energy as the prestrained silicone elastomer relaxed. The energy harvesting circuit is shown in Figure 7.14. The film was initially biased with a bias voltage supplied through a diode. A drain resistor in parallel exhibits a huge resistance so that the charge that is drained during contraction can be negligible. The dielectric elastomers have low enough leakeage current so that the decay back to the bias voltage can take tens of seconds. The extracted energy can be easily manifested by the boosting and decay of the voltage on the drain resistor. Dielectric elastomers are well suited to applications in which electrical power comes from relatively large motions, such as those produced by wind energy, waves and human activities. [START_REF] Kornbluh | From boots to buoys: Promises and challenges of dielectric elastomer energy harvesting[END_REF]2012) have developed a heel-strike generator using electrode-coated silicone polymers. Such generator can be installed in a normal shoe or boot, as shown in cross-section of the device with and without compression. [START_REF] Ashley | Artificial muscles[END_REF] The compression of the heel during normal walking can harvest power from human activity yet does not add any physical burden to the wearer. The device produced an electrical output of 0.8 J per step (the power is 1 W). The dielectric elastomer generator is an array of multilayer diaphragms that transfer the compression of the heel into deflection of the active layers (Figure 7.15b), that consist of 20 stacked layers of dielectric elastomer films. The device used prestrained VHB 4910 acrylic and performed with a maximum energy density of about 0.3 J/g.

Recently, SRI International has demonstrated the use of dielectric eleatomer for harvesting the energy of ocean waves. Ocean wave power has been called "blue energy", and has potential to produce clean, renewable energy in an eveiomentally sound manner. Harvesting such blue energy is critical as the world moves toward a more secure and sustainable energy future. It was shown that the ocean wave energy could potentially meet 10% of the total world electric demande. The U.S. Electric Power Research Institute (EPRI) estimated the viable resource in the United States at 255 TWh per year (6% of demand), comparable to the energy currently generated in the United States by conventional hydropower. [START_REF] Hagerman | E2I EPRI Specification: Guidelines for Preliminary Estimation of Power Production by Offshore Wave Energy Conversion Devices[END_REF] Based on the direct conversion of ocean wave to mechanical stretching and contract of the dielectric elastomer, Kornbluh et al. have developed a proof-of-principle generator system. [START_REF] Kornbluh | From boots to buoys: Promises and challenges of dielectric elastomer energy harvesting[END_REF] This system is shown in Figure 7.16 . The system was tested at sea in the Pacific Ocean near Santa Cruz, California. It produced an output of >25 J by using about 220 g of active dielectric elastomer material, giving rise to an energy density of more than 0.1 J/g. The efficiency of the energy-harvesting circuit used was 78%. The overall performance suggests that dielectric elastomers may indeed be practical for large-scale power generation. [START_REF] Kornbluh | From boots to buoys: Promises and challenges of dielectric elastomer energy harvesting[END_REF] By recognizing that high permittivity is desirable for mechanical energy harvesting, many researchers have focused on improving the energy conversion capabilities of elastomers upon incorperating diverse functional nanoparticles, such as carbon black, [START_REF] Wongtimnoi | Polyurethanes électrostricitifs et nanocomposites : caractérisation et analyse des mécanismes de couplage électromécaniques[END_REF] barium titanate, [START_REF] Yang | Enhanced energy conversion efficiency in the surface modified BaTiO3 nanoparticles/polyurethane nanocomposites for potential dielectric elastomer generators[END_REF][START_REF] Yin | Dielectric Elastomer Generator with Improved Energy Density and Conversion Efficiency Based on Polyurethane Composites[END_REF] and reduced graphene oxide. [START_REF] Nesser | All-organic microelectromechanical systems integrating electrostrictive nanocomposite for mechanical energy harvesting[END_REF] Yin et al. targeted generators with improved performance of mechanoelectric energy conversion by designing a ternary composite, which was composed of thermoplastic PU, barium titanate (BT) nanoparticles, and dibutyl phthalate (DBP) plasticizers. [START_REF] Yin | Dielectric Elastomer Generator with Improved Energy Density and Conversion Efficiency Based on Polyurethane Composites[END_REF] They simultaneously achieved an increase of 28% in permittivity and a decrease of 43% in Young's modulus by blending the PU with 25 phr of BT and 60 phr of DBP. Both of them are very desirable for improving the mechanical energy havesting capability. By simply stretching the sample to an area expansion of 80% and appling a bias voltage of 900 V (The circuit is shown in Figure 7.17a), the terpolymer with 15% of BT and 60% of DBP exhibited a harvested energy density up to 1.71 mJ/cm 3 and an efficiency of 0.45%. The harvested energy can light a LED array, consisting of 30 blue LEDs in series (Figure 7.17b). As a comparision, they tested the widely investigated dielectric elastomer, VHB4910, under the same stretching-releasing operations and only obtained an energy density of 0.21 mJ/cm 3 . These results clearly validate a principle that composite elastomers with tailored properties are indeed desirable for developing advanced generators. [START_REF] Yin | Dielectric Elastomer Generator with Improved Energy Density and Conversion Efficiency Based on Polyurethane Composites[END_REF] Copyright 2017, American Chemical Society.

In addition to the high-k nanoparticles, conductive nanofiller-based near percolative composites have also shown improved electrostrictive properteis. They are potentially promising to be exploited for developping generators. Very recently, [START_REF] Nesser | All-organic microelectromechanical systems integrating electrostrictive nanocomposite for mechanical energy harvesting[END_REF] integrated electrostrictive nanocomposites, composed of reduced graphene oxide dispersed in polydimethylsiloxane, into allorganic microcantilever resonators specifically designed for mechanical energy harvesting from ambient vibrations. Figure 7.18a and b presented the top and side views of the micro cantilever, consisting of electrostrictive PDMS/rGO composites coated by two layers of conducting polymer electrodes. The organic Micro-Electro-Mechanical Systems (MEMS) were fabricated at low-cost and in an all-organic approach by means of printing methods associated to xurography. The low resonant frequency (<100 Hz) of the resulting microcantilevers can be realized by simply incorporating a seismic mass at their tip. With the help of the constant voltage circuit (Figure 7.18c), the MEMS contilever provides a power of more than 6 μW/cm 3 for an acceleration amplitude of 1 g at the MEMS resonant frequency of 17 Hz. The demonstrated output power density is higher than that extracted from polyurethanebased electrostrictive materials [START_REF] Jaaoh | Enhanced strain response and energy harvesting capabilities of electrostrictive polyurethane composites filled with conducting polyaniline[END_REF], and is on the same level as the one generated by PVDF terpolymers. [START_REF] Cottinet | Analysis of ac-dc Conversion for Energy Harvesting Using an Electrostrictive Polymer P(VDF-TrFE-CFE)[END_REF] In spite of the tremendous progress of the generators based on the electrostrictive polymeric materials, there are still several challenges to be addressed in the future. The lifetime and reliabilty issues may be the greatest challenge to adoption of dielectric elastomers. Tradeoff performances could be made by entending the lifetimes yet has been seldom characterized in the literature. Reduction of the dielectric losses of electrostrictive mateials can be another key issue for improving the output and efficiency of generaotors. Another challenge relies on the design of energy harvesting circuits. In many cases, tradeoffs will be necessary between circuit complexity and cost. Usually, the former was prefered to have high efficiency yet increases the cost of the generators. Further, the energy harvesting cycles should be optimized according to the properties of mateials and the enviormental mechancial resources.

Conclusions and Perspectives

All dielectric materials have electrostriction effect. This chapter starts from the classification of dielectric mateials and then reviews the fundamental and application related issues for the electrostrictive materials. Particular attention is given on the means of characterizing the electrostriction coefficient as well as the routes to high electrostriction materials. Comparing with the changement of polymer composition, the composite approach can be a more versatile strategy. It allows for a large tunability of electrostriction of polymers by varying the identity, shape, and content of nanoadditives. At the end, actuation, sensory and energy harvesting applications of electrostrictive materials are summerized and examples for each application are given. Emphasis was put on generator appliation. Challenges for further improving the generator performances and large scale applications are discussed in detail. From a more general point of view, it is noted that fabrication of nanocomposites is usally guided by intuitive expectations and empirical approaches. Advanced modeling should be further developed in order to provide a more rational guidance of experimental efforts. It is expected that the development of new formulation and microfabrication technologies will help in the optimization and control of nanocomposite structures to achieve enhanced electrostrictive properties in the years to come.

Figure 7

 7 Figure 7.2 (a) A sandwiched structure used to evaluate the electrostrictive coefficient by measuring the thickness variation in response to the electric field. (Wongtimnoi et al, 2011) Copyright 2011, Elsevier. (b) Schematic structure of an intrinsic unimorph ofSWNT composite film without and with an electric field.[START_REF] Park | Actuating single wall carbon nanotube-polymer composites: Intrinsic unimorphs[END_REF]

Figure 7 . 3

 73 Figure 7.3 An illustration demonstrating the different mechanisms for the actuation of the materials in response to an applied electric field.

Figure 7 . 4

 74 Figure 7.4 Two modes used to characterize the electrostriction coefficient by measuring the permittivity variation with mechanical deformation. So the question arises. How to measure the variation of dielectric permittivity? We have combined the dielectric spectroscopy with tensile test to meet this end (Figure 7.5a). (Yuan et al, 2018) The change in capacitance of a parallel plate capacitor relies on Poisson contraction as well as permittivity variation (Figure 7.5b). Longitudinal stretching of the capacitor causes elongation in length yet shrinkage in width and thickness due to Poisson's effect.

Figure 7

 7 Figure 7.5 (a) Setups for the measurement of permittivity variation during mechanical stretching. (b)The mechanism by which a parallel plate capacitor changes its capacitance in response to a mechanical stretching.[START_REF] Yuan | Giant Electrostriction of Soft Nanocomposites Based on Liquid Crystalline Graphene[END_REF] Copyright 2018, American Chemical Society.

Figure 7 . 7

 77 Figure 7.7 TEM image of a thermoplastic elastomer gel composed of 20 wt% SEBSand 80 wt% white mineral oil.[START_REF] Kim | Electric Actuation of Nanostructured Thermoplastic Elastomer Gels with Ultralarge Electrostriction Coefficients[END_REF]

Figure 7 . 8

 78 demonstrates an example of CB/PU composites. Unfortunately, electrostriction coefficient experimentally achieved until now remain low. This is likely due to a lack of optimization of the morphology of the near-percolated networks formed by the particles.

Figure 7 . 8

 78 Figure 7.8 Typical solution mixing method to prepare electrostrictive CB/PU composites.[START_REF] Wongtimnoi | Polyurethanes électrostricitifs et nanocomposites : caractérisation et analyse des mécanismes de couplage électromécaniques[END_REF]

  Figure 7.10. In more recent years, the focus has turned to exploit material properties, particulaly the electrostriction coefficient (see previous section) to improve the actuating performances. O'Halloran et al.(O'Halloran et al, 2008) in a review paper summerized the operating principle and actuation mechanisms behind the actuator technology, highlighted its advantages over other existing actuator technologies, identified the challenges and examined the main research focus within the field.

Figure 7 .

 7 Figure 7.10 Actuator configurations based on dielectric elastomers.[START_REF] Kornbluh | Electroelastomers: Applications of dielectric elastomer transducers for actuation, generation and smart structures[END_REF]

Figure 7 .

 7 Figure 7.11 (a) Schematic illustration of a stretchable capacitor with SWNT electrodes (top). The capacitor dimension changes under pressure (bottom left) or in response to mechanical stretching (bottom right). (b,c) The capacitance changes as a function of pressure P (b) and strain ɛ (c). (d,e), ΔC/C0 versus time t over four cycles with applied pressure (d) and stretching (e).[START_REF] Lipomi | Skin-like pressure and strain sensors based on transparent elastic films of carbon nanotubes[END_REF]

Figure 7 .

 7 Figure 7.12 (a) Stress as a function of strain under a cyclic tensile loading-deloading.(b)Time dependence of the strain and relative permittivity at 0.1 Hz over the 20 stretching/releasing cycles.[START_REF] Yuan | Giant Electrostriction of Soft Nanocomposites Based on Liquid Crystalline Graphene[END_REF] Copyright 2018, American Chemical Society.

Figure 7 .

 7 Figure 7.14 The circuit used for mechanical energy harvesting and the measured voltage on the drain resistor as the dielectric elastomer (DE) relaxes.[START_REF] Pelrine | Dielectric elastomers: Generator mode fundamentals and applications[END_REF]

Figure

Figure 1 .

 1 Figure 1.15. (a)A photo of a dielectric elastomer based heel-strike generator, (b) across-section of the device with and without compression.[START_REF] Ashley | Artificial muscles[END_REF]

Figure 7 . 16 .

 716 Figure 7.16. Ocean wave power generator based on electrode coated dielectric elastomers.[START_REF] Kornbluh | From boots to buoys: Promises and challenges of dielectric elastomer energy harvesting[END_REF]

Figure 7 .

 7 Figure 7.17 (a) The electric circuit used for mechancial energy harvesting. (b) 30 blue LEDs in series can be lighted by the harvested energy during each cycle.[START_REF] Yin | Dielectric Elastomer Generator with Improved Energy Density and Conversion Efficiency Based on Polyurethane Composites[END_REF] Copyright 2017, American Chemical Society.

Figure 7 .

 7 Figure 7.18 Top (a) and side (b) view of the microelectromechanical systems (MEMS) cantilever, consisting of electrostrictive PDMS/rGO composites coated by two layers of electrodes.

  On the other hand, the total surface charge density that is induced in the materials by the applied field is given by the dielectric displacement vector 𝐷 𝑖 ,

	= 𝜀 𝑖𝑗 𝐸 𝑗	[7.9]
	Combining Eq [7.3], [7.4] with
	[7.1], one obtains
	𝑆 𝑖𝑗 = 𝑆 𝑖𝑗𝑘𝑙 𝐸 𝑇 𝑘𝑙 + 𝑀 𝑖𝑗𝑘𝑙 𝐸 𝑘 𝐸 𝑙	[7.6]
	Where 𝑆 𝑖𝑗𝑘𝑙
	𝑀 𝑖𝑗𝑘𝑙 = 𝜒 𝑚𝑘 𝜒 𝑛𝑙 𝑄 𝑖𝑗𝑚𝑛
	= 𝜀 0 2 (𝜀 𝑟 -1) 2 𝑄 𝑖𝑗𝑚𝑛	[7.7]
	𝐷 𝑖 = 𝜀 0 𝐸 𝑖 + 𝑃 𝑖	[7.8]
	Combining with Equation [7.3], one can deduce
	𝐷 𝑖 = 𝜀 0 𝐸 𝑖 + 𝜒 𝑖𝑗 𝐸 𝑗
	= 𝜀 0 𝛿 𝑖𝑗 𝐸 𝑗 + 𝜒 𝑖𝑗 𝐸 𝑗
	= (𝜀 0 𝛿 𝑖𝑗 + 𝜒 𝑖𝑗 )𝐸 𝑗

𝐸

is the elastic compliance under constant electric field, 𝑀 𝑖𝑗𝑘𝑙 is the electric field-related electrostriction coefficient, and it relates with 𝑄 𝑖𝑗𝑚𝑛 by


Guillaume Fayet
email: guillaume.fayet@ineris.fr

Patricia Rotureau

Patricia Vicot

On the Hazard Investigation of Polymerizing Substances through Experimental Methods and Theoretical Chemistry

Polymerizing substances can undergo strongly exothermic reactions under specific conditions such as transport. For this reason, they are considered under specific classification and safety recommendations. But their classification is mainly based on criteria initially intended for self-reactive substances and organic peroxides. So, considering their specific reactivity mechanisms, adaptations of the test methods and classification criteria could be needed and proposed. In this context, Ineris engaged efforts to evaluate the possible limits of experimental methods and the potential of theoretical chemistry tools to investigate the hazards and reaction mechanisms associated to polymerizing substances and to their self-polymerization.

Introduction

Among the chemical reactions involved in industrial accidents, uncontrolled polymerizations of substances like unsaturated hydrocarbons (ethylene, propylene, styrene, butadiene, etc.) and unsaturated nitriles (acrylonitrile, etc.) are found. In particular, accidentology highlights cases of polymerization runaway during the use of such substances in chemical industrial processes or their transport in tankers. [START_REF] Barpi | Analysis, Research and Information on Accidents (ARIA) Database[END_REF] For a long time, these substances were not specifically classified in regulations. Since only 2013, these materials are classified as "Polymerizing substances and mixtures (stabilized)" of Division 4.1 (with generic UN numbers 3531 to 3534) for the Transport of Dangerous Goods (TDG) (United Nations, 2019b) but they are not yet recognized in other regulations, such as the GHS (Globally Harmonized System of Classification and Labelling of Chemicals) (United Nations, 2019a) or the European CLP (Classification, Labelling and Packaging) (2008). Defined as substances which, without stabilization, are likely to undergo a strongly exothermic reaction leading to the formation of larger molecules or to the formation of polymers under specific conditions such as transport, their classification is based on several criteria initially intended for self-reactive substances and organic peroxides. However, these materials have specific reactivity mechanisms and adaptations of the tests and classification criteria may be necessary. After a summary of the accidentology, the regulatory framework applied to polymerizing substances is discussed with an analysis of the testing methods used in existing classification schemes. Their adequacy and practical drawbacks when characterizing the properties of polymerizing substances were analyzed. The completeness of the classification testing procedure was also criticized regarding the specificities of polymerizing substances and notably the efficiency of stabilizers and the prescription and sizing of decompression devices on packaging. At last, the use of quantum chemical methods for the theoretical study of self-polymerization mechanisms was investigated with the toluene and its derivatives as a reference case study.

Accidentology

The ARIA (Analysis, Research and Information on Accidents) database [START_REF] Barpi | Analysis, Research and Information on Accidents (ARIA) Database[END_REF], supported by the French Ministry in charge of the Environment, catalogues 85 accidents caused by or involving self-polymerizing substances between 1962 and 2019. 65 of them took place in France and 20 abroad. The distribution of the number of accidents per ten-year period is shown in Table 1 and demonstrates their stable occurrence over the last three decades. 1960-1969 1 1970-1979 1 1980-1989 7 1990-1999 26 2000-2009 30 2010-2019 20 Due to the multiple applications of polymerizing substances, accidents/incidents affect many economic areas like rubber and plastic manufacturing, chemical industry, textile manufacturing, transport or waste treatment.

The accident causes found in these 85 accidents were proposed as related to: The consequences of these events are very diverse from one accident to another, depending on its context and situation. They are often associated to increases in temperature and pressure, which can lead to fires, mechanical breakages and releases of products. The products released by these accidental polymerization events are often toxic and, depending on the quantities, can affect the company's staff as well as surrounding local residents. An example of event with significant consequences is an accident occurred in Himeji (Japan) on September 30 th 2012. The explosion of a tank containing acrylic acid was followed by a fire in a chemical plant resulting in 37 deaths, 5 seriously injured and 31 slightly injured people. From an economic point of view, the production losses and damages were estimated at 450 M€ and 15 M€, respectively.

Figure 1: Flow chart for classification of self-reactive and polymerizing substances

Critical analysis of testing methods to classify polymerizing materials

To be classified as "Polymerizing substances and mixtures (stabilized)" of Division 4.1 (with generic UN numbers 3531 to 3534) for the TDG, a substance must meet the criteria detailed in the flowchart presented in Figure 1.

In case where the substance presents hazards of classes 1 to 8, it will be classified according to those hazards which are predominant but with the word "STABILIZED" in the proper shipping name to indicate the risk of polymerization.

Polymerizing substances are distinguished from self-reactive ones by the knowledge of their subjacent mechanisms (polymerization rather than decomposition). Otherwise, as the classification criteria are very similar to those for self-reactive substances, a polymerizing substance could be classified as self-reactive. Indeed, the classification of polymerizing substances is based on test results recommended in the UN Manual of Tests and Criteria, developed for organic peroxides and self-reactive substances, which are subject to decomposition rather than polymerization. These tests are listed in Table 2. However, due to the specific behaviors of polymerizing substances, some difficulties may be encountered in their experimental implementation. For instance, the tests using single-use containers (UN H.1, UN H.3 and DSC) do not pose any problem with sample "solidification" during polymerization whereas, this is not the case for UN H.2 and H.4 tests, which use a fragile apparatus (Dewar vessel) that can be impossible to clean after the tests. For these lasts, alternative tests would deserve to be proposed and validated. Then, Model Regulation requires substances with a polymerization hazard (even those with additional hazards of classes 1 to 8) to be transported in specific packaging with different requirements depending on whether the substances are in solid or liquid form. Among these requirements, two parameters would require further analyzes:

• Check the effectiveness of the stabilizer to avoid uncontrolled polymerization during transport.

It depends on different factors and in particular on the type of stabilizer (e.g. thermal stabilizers, antioxidants). It also requires the consideration of criteria representative of the conditions of transport, whatever the mode used (road, rail, sea and air transport), including its duration and the climatic conditions.

• Check that the packaging is designed to allow gases/vapors releases in order to avoid overpressure or if it needs a pressure relief device and size it.

Upon their specific reaction mechanisms, polymerizing substances may release a certain amount of gas with a flow rate depending on the substance. So, it could be important to check that, in case of polymerization, the pressure relief devices are correctly sized to avoid mechanical rupture.

Unfortunately, no specific regulatory tests are nowadays implemented to guide the verification of these two safety points for polymerizing substances.

Theoretical study of self-polymerization reactions

Knowledge of reaction mechanisms (decomposition or polymerization) is a key information to well classify the substances as polymerizing or not (for instance as self-reactive). As a complement to experimental testing, quantum chemical methods have been successfully used to evidence the subjacent mechanisms of other hazardous reactions (like peroxidation [START_REF] Tommaso | Oxidation mechanism of diethyl ether: a complex process for a simple molecule[END_REF] or chemical incompatibilities [START_REF] Cagnina | Modeling Chemical Incompatibility: Ammonium Nitrate and Sodium Salt of Dichloroisocyanuric Acid as a Case Study[END_REF][START_REF] Menicacci | Toward the Mechanistic Understanding of the Additives' Role on Ammonium Nitrate Decomposition: Calcium Carbonate and Calcium Sulfate as Case Studies[END_REF]. So, works were engaged towards their use for the theoretical study of possible self-polymerization of these materials with the toluene and its derivatives as a reference case study [START_REF] Fayet | Theoretical chemistry investigation of polymerizing materials: styrene and its derivatives as a case study[END_REF].

Indeed, calorimetric data are available for the toluene, α-methylstyrene (AMS) and trans-β-methylstyrene (TBMS) demonstrating a thermal polymerization at an onset temperature of 103°C for toluene (with a heat of reaction ∆H of 602 J/g) compared to more than 350°C for both AMS and TBMS (with ∆H < 280 J/g) by differential scanning calorimetry at a 4°C/min temperature rate, as shown in Table 3 [START_REF] Lin | Comparison of thermal polymerization mechanisms for αmethylstyrene and trans -β-methylstyrene[END_REF]. It should be noticed that, for AMS and TBMS, a first exothermic pic of low heat (proposed as associated to a dimerization process) is observed prior to polymerization, at lower temperature. At first, the detailed reaction mechanisms involved in the self-polymerization of toluene were theoretically investigated by use of the density functional theory (DFT). These mechanisms were already investigated in literature using the B3LYP/6-31G* and BPW91/6-31G* methods [START_REF] Khuong | The Mechanism of the Self-Initiated Thermal Polymerization of Styrene. Theoretical Solution of a Classic Problem[END_REF] but this study presented some failures. In particular, the calculated diradical transition state of one main reaction of the mechanism (a Diels-Alder type reaction) was found as presenting unstable wave functions.

For this reason, these mechanisms have been analyzed again using a more recent functional, M05-2X, identified as more relevant for Diels-Alder reactions in benchmark studies [START_REF] Johnson | Delocalization errors in density functionals and implications for main-group thermochemistry[END_REF][START_REF] Zhao | Density Functional Theory for Reaction Energies: Test of Meta and Hybrid Meta Functionals, Range-Separated Functionals, and Other High-Performance Functionals[END_REF].

Finally, the new theoretical study (at M05-2X / 6-31+G(d,p) level) overcame the limits of the study of Khuong since the corresponding Transition States were found stable. Moreover, some deviations were found between the new method and the results of Khuong, in agreement with the observations made in the benchmark studies (highlighting an underestimation of reaction exothermicity using B3LYP). However, the global reaction mechanism and relative balance between the different reactions remained the same. As shown in Figure 3, the formation of DCB and the Diels-Alder reaction present similar activation energies (associated to their respective transition states TS1 and TSDA). Then, formation of the monoradicals (responsible for polymerization propagation) is slightly more difficult than trimerization, which agrees with experimental data evidencing an higher production of A-Sty trimers than polymers during the polymerization process [START_REF] Khuong | The Mechanism of the Self-Initiated Thermal Polymerization of Styrene. Theoretical Solution of a Classic Problem[END_REF]. The next step of the study now consists in computing these main mechanisms for AMS and TBMS. These calculations are under progress and may help to understand why self-polymerization occurs at higher temperature for these compounds than for styrene despite their similar structures. For instance, the relative activation energies could be influenced making the propagation less favorable compared to the two other mechanisms, or the presence of additional methyl groups in AMS and TBMS could favor other reactions compared to styrene.

Conclusions

The new classification framework proposed for the transport of polymerizing materials encourages some adaptations of the testing methods and criteria for their classification. They concern both the feasibility of the testing methods already implemented in the TDG regulation but also additional testing methods to fulfill the requirements of the classification (notably regarding the efficiency of stabilizers and the need and sizing of pressure relief devices). The use of theoretical chemistry also opens new perspectives towards an in-depth understanding of their thermal self-polymerization mechanisms and a better identification of compounds subject to possible hazardous polymerization issues. These theoretical studies could, for instance, be the basis of development of practical tools complementary to experimental methods, to support the identification of reaction types (e.g. selfpolymerization rather than self-decomposition) within the classification process.

Figure 2 :

 2 Figure2: Reaction mechanisms involved and in competition with the initiation of styrene thermal polymerization, as proposed by[START_REF] Khuong | The Mechanism of the Self-Initiated Thermal Polymerization of Styrene. Theoretical Solution of a Classic Problem[END_REF]

Figure 3 :

 3 Figure3: Energetic profile of the mechanisms involved and in competition with the initiation of styrene thermal polymerization, as calculated atM05-2X/6-31+G(d,p) level[START_REF] Fayet | Theoretical chemistry investigation of polymerizing materials: styrene and its derivatives as a case study[END_REF]

Table 1 :

 1 Distribution of number accidents by ten-year period

	Period	Number of accidents

Table 2 :

 2 UN test methods used for the classification of polymerizing substances

	Heat of reaction	DSC calorimeter tests
		Test Code	Name of test
	Self-Accelerated Polymerization Temperature (SAPT)	UN Test H.1 UN Test H.2 UN Test H.3	United States SADT test, Adiabatic storage test (AST) Isothermal Storage Test (IST)
		UN Test H.4	Heat accumulation storage test

Table 3 :

 3 Experimental onset and peak temperatures (To and Tp) and heat of reaction ∆H obtained from the DSC tests[START_REF] Lin | Comparison of thermal polymerization mechanisms for αmethylstyrene and trans -β-methylstyrene[END_REF]

		Styrene	AMS	TBMS
	To (°C)	103.3	136.5 356.3	74.6 385.4
	Tp (°C)	205.8	140.5 402.9	97.6 415.6
	∆H (J/g)	601.7	17.6 266.3	29.8 277.3

Acknowledgments

Quantum-chemical calculations were performed using HPC resources from GENCI-TGCC (Grant 2020-A0030810307).


Pierre Chot-Plassot

Alexis Vignes
email: alexis.vignes@ineris.fr

Carlos Murillo

Jean-Marc Lacome

Stephanie El-Zahlanieh

Nathalie Bardin-Monnier

Olivier Dufaud

Combining CFD and Experimental Approaches to Optimize a Spray Release in a 20 L Sphere

European regulations require that the risk of an explosive atmosphere (ATEX) associated with the production of a flammable mist be assessed. In order to properly investigate mist flammability, a dedicated experimental tool was developed based on the standardized 20L explosion sphere and an additional Venturi injection system. As part of its design, the conditions of mist generation must be fully controlled. To complement the experimental approach, a computational fluid dynamic modeling (CFD) has been developed with an Euler-Lagrange method. It aimed at determining the homogeneity of the mist and the influence of the turbulence on the droplet size distribution (coalescence / fragmentation dynamics) as it greatly influences the mechanism of combustion. Based on previous studies and on industrial considerations, ethanol and kerosene were chosen. Modeling results were compared to the droplet size distribution determined by laser in-situ measurements and to the flow characteristics measured by particle image velocimetry. The CFD modeling confirmed the significant effect of the turbulence on the droplets coalescence and enabled to improve the understanding of the phenomenology related to the dispersion of a flammable mist into a 20L sphere. These results are consistent with the droplet particle distributions measured experimentally and show that CFD modeling can support further experimental developments so that it can constitute the basis for a new standard 20L explosion sphere for mists testing.

Introduction

European regulations require an assessment of the explosive atmosphere (ATEX) risk associated with the production of a flammable aerosol (e.g. diesel, lube oil, gasoline mists). However, such a risk analysis is sometimes difficult due to the lack of tools to correlate dispersion conditions with their flammability and there is no standard for the experimental determination of the ignition sensitivity or explosion severity of a mist. The specifications for a new standard must include the availability of test equipment, its ease of use and both its specific and versatile nature. Therefore, a new experimental set-up was developed based on the classic 20L explosion sphere, already conventionally used to determine the explosion severity of gases and dusts. However, mists do not, of course, have the same properties and behavior as powders or gases: unlike gases, their dispersion in the air is not stable over time; unlike powders, droplets are subject to coalescence and can be captured onto the sphere wall (rain-out phenomenon). Their properties and specificities imply some modifications of the set-up and adaptation of the test procedures. In order to describe the dispersion of a flammable spray in the modified 20L explosion sphere and to determine the most conservative or representative test conditions, experimental characterizations have been performed using i) particle image velocimetry to study the turbulence of the mist, ii) laser in-situ measurements for the determination of the droplet size determination, iii) high-speed videos, to observe the homogeneity and stability of the suspension. Preliminary tests were performed on various hydrocarbons, such as kerosene JetA1, diesel, ethanol, biodiesel or light fuel oil. Nevertheless, as extensive as it may be, this experimental study cannot cover all possible configurations because the influential parameters (type of ejection nozzle, diameter, temperature, injection pressure, nature of the fluid, etc.) and their combinations are too numerous.

In this context, a computational fluid dynamic modeling (CFD) has been developed with an Euler-Lagrange approach to analyze the biphasic flow development inside the test apparatus and assess the flow conditions pertaining to the fragmentation and the coalescence of the dispersed phase. After adjusting the CFD simulation to the design specifications of the apparatus, CFD simulations were used to investigate the local concentration in the 20L explosion sphere beyond the experimental observations to predict the droplet size distribution, the mist homogeneity and the time-evolution of the local concentration near the ignition source.

Experimental set-up and CFD modeling

The experimental set-up will be briefly described as well as the models applied in the CFD approach.

The modified 20L sphere for mists testing

The experimental set-up was described by [START_REF] El -Zahlanieh | A step toward lifting the fog off mist explosions: Comparative study of three fuels[END_REF]. It consists of a standard 20L explosion sphere, whose dust container and outlet valve have been removed and replaced by a Venturi system connected at the bottom of the sphere. The mist is dispersed by the injection of an air-liquid mixture at a high pressure through an injection nozzle. Various injection sets of different diameters are available. Two electronic valves allow the control of the inlet flow rates and the air/liquid ratio. Before injecting the air/fuel mixture, a partial vacuum is created in the sphere at a calculated pressure depending on the mist concentration to be attained. When the mist has been completely injected, the pressure inside the sphere is equal to the atmospheric pressure. An ignition system (spark or chemical igniter) allows the combustion of the hydrocarbon mist. In addition to the nature of the fluid, the mass injected, the injection time and pressure, the nozzle type and diameter can be varied to change both the mist concentration and droplet size distribution. Tests were notably performed on ethanol, kerosene, isooctane, lube oil and biodiesel. This study will be limited to kerosene JetA1, with a single experimental configuration ('N1' nozzle) and an injection pressure of 3 bar.

Experimental characterization of JetA1 mists

Some variables were characterized experimentally in order to both feed and validate the numerical approach. The droplet size distribution (DSD) was determined in situ (at the center of the 20L sphere) using a laser sensor (Helos from Sympatec) on a 0.5 -175 µm range at a frequency of up to 2000 Hz. Unfortunately, the analysis duration is limited to 5 s due to the rain out phenomenon occurring on the glass walls. The droplet size distribution obtained with a 'N1' nozzle is rather stable over time and can be represented by the characteristic diameters shown in Table 1. It should be noted that, by using other injection nozzles, mean droplet diameter up to 80 µm can be obtained. An average flow rate of 2.5.10 -4 kg/s was measured for the liquid phase, which represents from 17 to 33 wt% of the biphasic flow. Particle image velocimetry (PIV) was implemented to study the time evolution of the droplet velocity and of the root-mean-square velocity (vrms) of the flow. Velocity magnitudes v ranging from 0.5 to 15 m/s were attained at the sphere center during the mist generation, with a maximum vrms of 2 m/s at the end of the liquid injection.

Dimensionless numbers related to mist generation and CFD theoretical approach

Using the experimental parameters presented in section 2.2 and the fluid properties, a set of dimensionless numbers of interest for this study was evaluated (Table 2). A study in the duct connecting the air tank and the nozzle was carried out on the PHAST software. This resulted in an outlet velocity at the orifice equal to 313 m/s which will be considered here. For these dimensionless calculations only, the droplet size distribution was assumed monomodal and equal to the mean diameter d50. According to the values of the dimensionless numbers represented in Table 2, the spray is qualified as fine (volume fraction lower than 1 %). However, the use of a four-way coupling is still necessary because the jet is confined, the flow is highly turbulent and dropletdroplet interactions cannot be neglected. As expected, Weber number is greater than 350 [START_REF] Pilch | Use of breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced breakup of a liquid drop[END_REF] and Ohnesorge number is less than 1 (Kadesh, 2020), which means that a catastrophic breakup of the droplets occurs when they exit the nozzle (fragmentation). The droplets follow the air trajectory (St < 1) and a collisional regime is assumed (Kn < 1) because the experiment showed collisions between the droplets. Jet was atomized at 95% at the orifice. An Euler-Lagrangian solver was then chosen to model the droplets and gas phase. The simulations were done using OpenFoam v1912. The dimensionless numbers detailed hereafter were used to select the most suitable solver: SprayFoam solver was chosen because it takes into account fragmentation and allows modeling the evolution of the droplet sizes distribution. It considers the fragmentation and the collision of the droplets. It should be noted that the wall-droplet interactions within the sphere was not considered in this preliminary modeling study. Wall-droplet interactions were then configured so that the droplets stick to the wall (Table 3).The turbulence models used are of the RANS type, i.e. only the large scales of the turbulence will be solved numerically while the small and medium scales will be modeled. Regarding the high Re and the confined environment, the k-ω sst model was implemented. It consists in applying the classical k-ε model far from the walls and the k-ω model close to the wall (Table 3). The turbulence models used are of the RANS type, i.e. only the large scales of the turbulence will be solved numerically while the small and medium scales will be modelled. Regarding the high Re and the confined environment, the k-ω sst model was implemented. It consists in applying the classical k-ε model far from the walls and the k-ω model close to the wall (Table 3). O'Rourke's model was chosen to represent the collision between two droplets (Table 3). It is based on a statistical approach and assumes that two droplets can only collide if they are in the same cell. Therefore, the result of a collision can be a coalescence, if the droplets collide head-on, or a rebound, for an oblique collision.

Setup of the CFD modeling

The kerosene spray is considered as a two-phase jet in a confined, compressible, unstable and turbulent environment. Modeling and meshing were done on Ansys v2021 (Design Modeler and Meshing) (Figure 1). An optimal mesh size is based on a good quality of orthogonality of the meshes. A cartesian cut-cell mesh (CutCell) was chosen and a refinement was made near the injection (Figure 1b) and at the center of the sphere (Figures 1.c) and d) in order to obtain accurate results. The quality of orthogonality is 98% and the number of meshes is set at 500,000. The dimensions of the 'N1' orifice are of the order of 1.8.10 -3 x 5.8.10 -4 m 2 . In order to obtain a sufficient accuracy, at least six meshes per dimension must be placed in this area, which implies a mesh size of approximately 1.10 -4 m. The simulation is done by a Crank-Nicholson semi-implicit time scheme. Spatial schemes were chosen according to the mesh orthogonality and to a lesser numerical diffusion. The time step dt is calculated with respect to the smallest mesh of the geometry. By definition, the stability criterion (CFL) is equal to 1; beyond that value, the stability and accuracy of the results are significantly reduced:

𝐶𝐶𝐶𝐶𝐶𝐶 = 𝑣𝑣. 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (1)
This implies that a time step equals 3.10 -7 , which is too costly in terms of computing time. It was decided to increase the time step to 1.5.10 -5 s, therefore the corresponding CFL value is much higher. Iterative procedure was achieved with solver factors to obtain the best compromise between stability and accuracy. The mist injection was represented by fluid parcels containing several droplets of the same physical size in order to reduce their number and keep the same mass of injected liquid constant. 10 5 parcels per second containing an average of 10000 droplets has been configured following a sensitivity analysis, which aimed at determining the adequate calculation time to represent liquid mass flowrate reliably. A system of ordinary differential equations is solved to predict mass, velocity and temperature evolutions. The droplets were assumed to be spherical. Mass flow rate of gas and liquid are set to 4.9 10 -4 kg/s and 2.5.10 -4 kg/s respectively.

Results and discussion

CFD simulations were validated using the experimental results presented in section 2.2. Then, the hydrodynamics of the mist and the time-evolution of its droplet size distribution (coalescence/fragmentation) was studied. It should be stressed that the knowledge of the DSD at the center of the sphere (ignitor location) is crucial to better understand the mist combustion mechanism.

Analysis of spray velocity iso-contour lines

Figure 2 shows the velocity vector during mist generation, from 300 to 1500 ms on a slice set up at the sphere center (velocity magnitude u in m/s). The fluid ejected from the nozzle disperses vertically in a cone and then hits the upper part of the sphere at around 400 ms (Figure 2b). Once the mist contacts the walls, recirculation zones are formed in the middle of the sphere (Figure 3). In the middle of the sphere, the turbulent kinetic energy k (velocity fluctuation) as well as the velocity is high. In order to implement this preliminary CFD modeling study, the initial DSD at the orifice was taken equal to the experimental DSD at the center of the sphere at t=1 s, keeping in mind that the DSD at the orifice cannot be observed experimentally. In the future, the initial DSD will be adapted to reach experimental DSD at the center of the sphere. Meanwhile, this current study is a first step to investigate the main phenomena and to validate the consistency of the approach. In order to observe by CFD modeling the evolution of the droplet size and velocity distribution, a block was modelled in Paraview at the center of the sphere to capture the droplets passing through this volume. The droplet size distribution at the center of the sphere is shown in Figure 5. The shape of the DSD obtained by CFD modeling looks like the experimental DSD obtained in the center of the sphere at t=1.5 s (Figure 5). However, the mean diameter of the numerical DSD (4.5 μm) is lower than the mean diameter of the experimental DSD (7.5 μm). The initial DSD at the orifice will then require to be modified by increasing the mean diameter of the droplets. It should be also noted that, due to the choice of the collision model (O'Rourke -Table 3,) c Further mesh refinements will be tested in order to ensure that coalescence collisions is well represented in the CFD modeling compared to the experimental reality (Pischke, 2014). Furthermore, it has been observed previously existence of recirculation zones (turbulence) in the center of the sphere. This aspect is all the more important that a strong correlation exists between turbulence and coalescence/fragmentation which influences the combustion mechanism.

s generation

The mesh size will be adapted in a near future to refine the results and better discriminate the phenomena of collision, evaporation and fragmentation.

Conclusions

The work carried out made it possible to study the phenomenology related to the dispersion of a non-flammable fog in a 20 L explosion sphere and to investigate the behavior of the fuel dispersion. In the course of this study, an overview of liquid and diphasic free air jets was presented to introduce the experimental study carried out. This study modelled and simulated a supersonic spray of air and kerosene in a 20L closed sphere using Ansys meshing and Openfoam. The flow was solved by an Eulerian-Lagrangian method and the equations were discretized temporally in semi-implicit by the Crank-Nicholson scheme. A dimensionless study of Reynolds, Stokes, Knudsen, Ohnesorge, Weber numbers based on the experimental observations at the centre of the device were used to set up the input data for the CFD simulation. A CFD calibration strategy on the experimental model and the development of the CFD configuration were described. The numerical results showed the impact of turbulence on the droplet size distribution (coalescence/fragmentation). Strong turbulence and recirculation zones in the middle of the 20L sphere were observed to influence droplet size through coalescence and fragmentation phenomena. These results will support experimental investigations on the combustion of mist droplets as it is directly under the influence of the DSD and turbulence. Further investigations are also foreseen to explore rain-out (droplets-wall interactions) phenomenon than can lead to the formation of a layer of liquid on the walls of the 20L sphere which may influence the explosion severity of the mist by changing locally vapor concentrations.

Figure 1 :

 1 Figure 1: 20 L explosion sphere modelled and meshed with the CutCell meshing algorithm. a) global view, b) focus on the injection zone, c) lateral view 1 and d) lateral view 2

Figure 2 :

 2 Figure 2: Time-evolution of the jet velocity in the sphere at a) 300 ms, b) 500 ms, c) 1100 ms and d) 1500 ms

Figure 3 :

 3 Figure 3: Representation of streamlines and iso-contour lines of a) velocity in lateral view 1, b) velocity in lateral view 2, c) turbulent kinetic energy in lateral view 1 and b) turbulent kinetic energy in lateral view 2.

Figure 4

 4 Figure4shows the temperature iso-contour lines at the orifice. It was found that a) the temperature decreases up to -75°C within the expansion zone and then increases to the initial temperature of 20°C ; b) the jet accelerates from 390 m/s at the nozzle exit to 510 m/s in the expansion zone and then slows down to a velocity close to 1 m/s. The cooling down effect and the speed up in velocity are explained by the expansion of the supersonic air jet. The velocities obtained by CFD and PHAST are of the same order of magnitude. The decrease of the CFL stability criterion could enable to obtain more accurate results.

Figure 4 :

 4 Figure 4: Representation of a) temperature and b) velocity iso-contour lines at the nozzle 3.2 Time and spatial evolution of the droplet size distribution

Figure 5 :

 5 Figure 5: Comparison between CFD results and the experimental kerosene DSD at the sphere center after 1.5 s generation

Table 1 :

 1 Characteristic diameters of JetA1 mists generated in the 20L sphere

	Diameters (µm)	d10	d50	d90	d99	d3,2
	Kerosene JetA1	5.5	7.3	10.6	13.7	7.3

Table 2 :

 2 Characteristic dimensionless numbers for the generation of a JetA1 mist in the 20L sphere at the injection orifice

	Dimensionless number	Vol. fraction Reair	We	Oh	St	Kn
	Value (-)	0.02 %	52000	21500	0.13	2.4.10 -4	< 1

Table 3 :

 3 Theoretical models used for CFD modeling of a JetA1 mist in the 20L sphere

	Phenomena	Forces	Turbulence	DSD	Fragmentatio	Heat transfer	Collision Interactions
					n
	Models	Gravity,	k-ε and	Rosin-	Reitz-	Ranz-Marshall O'Rourke Standard Wall
		drag force	k-ω sst	Rammler	Diwakar		Interaction

Acknowledgments

This work was supported financially by the French Ministry for the Ecological and Solidary Transition and The French Ministry for Higher Education and Research.