Gildas Diguet
email: gildas.diguet.d4@tohoku.ac.jp

Jean-Yves Cavaille

Gael Sebald

Toshiyuki Takagi

Hiroshi Yabu

Ai Suzuki

Ryuji Miura

Physical behavior of electrostrictive polymers. Part 1: Polarization forces

The electrostrictive response of soft elastomers with the same stiffness, strongly depends on their chemical nature and their typically multiphase microstructure. Moreover, some elastomers exhibit a strongly time dependent electrostriction over tens of minutes, and up to now, no theoretical approach has been proposed to analyze experimental data on local parameters like the dielectric constants, conductivities and viscoelastic moduli of these composite-like materials. We consider the phenomenon where the deformation of a polymeric sample between two electrodes is proportional to the square of the applied field, which is known as electrostriction. The electrostatic attraction of charged electrodes is Maxwell electrostriction. In cases, such as block co-polymers with phase separation, the observed electrostriction reaches magnitudes more than 10 times higher than those achieved via the Maxwell process. Phenomenological analyses of experimental data are usually performed but few physical have been proposed to explain the difference. Therefore, we analyze the electric forces inside a composite-like polymer and estimate the corresponding deformation. Using data sets for polyurethane-based materials that exhibit phase separation during their processing, we propose a microstructural model corresponding to a composite where spherical particles randomly fill a matrix. The particles and matrix exhibit different values of physical parameters such as the (i) dielectric constant and electrical conductivity, which determine the local electric field and (ii) viscoelastic modulus, which determine the local stiffness. Because the phases are different, the electric field is not homogenous and the field gradient generates forces around the interfaces. Developing a 2D model, we compare simulation results to experimental literature and other modeling approaches, and discuss them in detail. The polarization forces are found to be responsible for 20% of the deformation in a material with 35% inhomogeneity. Though the time constants are consistent with experimental data, their contribution is smaller than the Maxwell contribution, and therefore other mechanisms are involved in the large electromechanical activity of polymers like polyurethanes.

Introduction

When a soft material such as rubber is placed inside an applied electric field E between two electrodes, it also experiences to their electrostatic attraction. This leads in turn to a compressive strain, known as the Maxwell strain, given by [START_REF] Pelrine | Electrostriction of polymer dielectrics with compliant electrodes as a means of actuation[END_REF]

𝑆 𝑚 = -( 𝜀 𝑌 ) 𝐸 2 = -𝑀 𝑚 𝐸 2 (1)
where ε and Y are the dielectric constant and Young modulus of the material, and Mm the Maxwell electrostrictive coefficient.

However, some electro-active polymers (EAPs) exhibit much higher deformations, S>>Sm [START_REF] Jaaoh | Contribution of Electrostriction in polyurethane/polyaniline blends[END_REF][START_REF] Su | Effects of transitional phenomena on the electric field induced strainelectrostrictive response of a segmented polyurethane elastomer[END_REF][START_REF] Guillot | Electrostrictive effect in polyurethanes[END_REF][START_REF] Jomaa | Difference between electrostriction kinetics, and mechanical response of segmented polyurethane-based EAP[END_REF][START_REF] Diaconu | Electrostriction of a polyurethane elastomer-based polyester[END_REF]. These high deformations are still quadratic at low applied electric field according to

S = -M𝐸 2 ( 2
)
where M>Mm is the electrostriction coefficient. Electrostriction occurs in ionic crystals [START_REF] Li | Electrostrictive effect in ferroelectrics: An alternative approach to improve piezoelectricity[END_REF] and is responsible for piezoelectricity in ferroelectrics that have spontaneous polarization [START_REF] Li | Electrostrictive effect in ferroelectrics: An alternative approach to improve piezoelectricity[END_REF][START_REF] Katsouras | The negative piezoelectric effect of the ferroelectric polymer poly(vinylidene fluoride)[END_REF][START_REF] Fukukawa | Electrostriction as the origin of piezoelectricity in ferroelectric polymers[END_REF]. Anderson calculated the strain of a homogenous polarizable material in a cubic lattice hence providing a correction to the Maxwell coefficient [START_REF] Anderson | Mechanical stress in a dielectric solid from a uniform electric field[END_REF],

which Shkel later generalized to the case of amorphous material [START_REF] Shkel | Electrostriction of polarizable materials: Comparison of models with experimental data[END_REF] with corrections based on the dielectric (permittivities), mechanical (Young modulus and Poisson ratio) and structural (lattice) properties.

The electrostrictive response of soft elastomers with the same stiffness strongly depends on their chemical natures, and for multiphase polymers, on their microstructures (such as spherical domains dispersed in continuous matrixes and lamellar materials) [START_REF] Pelrine | Electrostriction of polymer dielectrics with compliant electrodes as a means of actuation[END_REF][START_REF] Jaaoh | Contribution of Electrostriction in polyurethane/polyaniline blends[END_REF][START_REF] Su | Effects of transitional phenomena on the electric field induced strainelectrostrictive response of a segmented polyurethane elastomer[END_REF][START_REF] Guillot | Electrostrictive effect in polyurethanes[END_REF][START_REF] Jomaa | Difference between electrostriction kinetics, and mechanical response of segmented polyurethane-based EAP[END_REF][START_REF] Diaconu | Electrostriction of a polyurethane elastomer-based polyester[END_REF].

Experimental tests of electrostriction can be tricky, and it is difficult to obtain relevant measurements. For example, the constraint induced by the electrodes limits the deformation, as shown by Guyomar et al. [START_REF] Guyomar | Focus on the electrical field-induced strain of electroactive polymers and the observed saturation[END_REF]. They measured the deformation of 70 µm polyurethane (PU) films (i) coated with a 50 nm-thick gold layer and (ii) uncoated. An interferometer was used to measure the electrostrictive coefficients of both coated and uncoated films. Without the gold layer the PU strain went up to 4%, whereas that of the coated film barely reached 1% for the same applied field of 10 MV/m. The resulting electrostriction coefficients for the coated and the uncoated films were 4.83×10 -16 m 2 /V 2 and 1×10 -16 m 2 /V 2 , respectively. Measurements of the low-field permittivity (ε=4ε0) and mechanical modulus (Y=32 MPa) lead to Mm=ε/Y = 1.10×10 -18 m 2 /V 2 . To avoid any measurement artifacts, Zhenyi [START_REF] Zhenyi | High field electrostrictive response of polymers[END_REF] evaluated the electrostriction coefficient of a PU material using three different experimental apparatuses; the first was an accelerometer, the second an interferometer and the third a capacitance sensor. All measured values were higher than the Maxwell coefficient (1.7×10 -18 m 2 /V 2 ), and the values corrected by removing the Maxwell effect were M= 2.4×10 -17 m 2 /V 2 (accelerometer), 3.3×10 -17 m 2 /V 2 (interferometer), and 1×63.10 -17 m 2 /V 2 (capacitance sensor).

Diaconu et al. [START_REF] Diaconu | Electrostriction of a polyurethane elastomer-based polyester[END_REF][START_REF] Diaconu | Electromechanical response of polyurethane films with different thickness[END_REF][START_REF] Diaconu | Properties of polyurethane thin films[END_REF][START_REF] Diaconu | An experimental investigation of electroactive polyurethane[END_REF] performed several experiments on a PU to study its strong behavior. For example [START_REF] Diaconu | Electrostriction of a polyurethane elastomer-based polyester[END_REF], they used a liquid between the metallic electrodes and the polymer film to reduce friction in a Michaelson interferometer when measuring the electrostrictive properties of a 0.05 mm thick sample with low frequency dielectric constant of ε=7.2ε0 and elastic modulus of Y=13.1 MPa. Their measurements yielded M=7×75.10 -16 m 2 /V 2 , which is much higher than the simple Maxwell response evaluated at Mm= 4.86×10 -18 m 2 /V 2 . They also obtained similar results, with different film thicknesses [START_REF] Diaconu | Electromechanical response of polyurethane films with different thickness[END_REF]. All these data show that additional contribution to electrostriction must be taken into This pure electrostriction is an intrinsic function of the material polarization P [START_REF] Guiffard | Enhanced electric field-induced strain in non-percolative carbon nanopowder/polyurethane composites[END_REF] and is expressed as [START_REF] Guiffard | Enhanced electric field-induced strain in non-percolative carbon nanopowder/polyurethane composites[END_REF][START_REF] Wongtimnoi | Thickness-dependent microstructural and electromechanical properties in polyurethane films obtained by polymer solution casting[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]. The quantity Q is proportional to 1/εY. Therefore [START_REF] Guillot | Electrostrictive effect in polyurethanes[END_REF][START_REF] Guiffard | Enhanced electric field-induced strain in non-percolative carbon nanopowder/polyurethane composites[END_REF][START_REF] Wongtimnoi | Thickness-dependent microstructural and electromechanical properties in polyurethane films obtained by polymer solution casting[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] Lebrun | The characterization of the harvesting capabilities of an electrostrictive polymer composite[END_REF][START_REF] Roussel | Electromechanical study of polyurethane films with carbon black nanoparticles for MEMS actuators[END_REF],

S = QP 2 . At low field, S = Q(ε-ε0) 2 E 2 , yielding ME = Q(ε-ε0) 2
𝑀 𝐸 ≈ (𝜀 -𝜀 0 )/𝜀𝑌 (3)
However, polymers exhibiting ME >> Mm have either been filled with conducting particles or have intrinsic heterogeneities [START_REF] Jaaoh | Contribution of Electrostriction in polyurethane/polyaniline blends[END_REF][START_REF] Roussel | Electromechanical study of polyurethane films with carbon black nanoparticles for MEMS actuators[END_REF][START_REF] Jomaa | Modeling of segmented pure polyurethane electrostriction behaviors based on their nanostructural properties[END_REF][START_REF] Jo | Polyurethane-based actuators with various polyols[END_REF]. Diblock-copolymers PU, have such intrinsic heterogeneities induced by a phase separation during the sample preparation resulting in a distribution of particles of one phase in a continuous matrix (second phase), for block compositions far from 50%-50%. In either case, added particles or a two-phase system, these materials can be viewed as composite materials (particles embedded in a matrix). The microstructural features of diblock-copolymer PUs have been characterized [START_REF] Jomaa | Modeling of segmented pure polyurethane electrostriction behaviors based on their nanostructural properties[END_REF].

Some researchers have tried to increase the electrostriction coefficient M by increasing the dielectric constant, such as in the experiments performed by Putson and al. [START_REF] Putson | Large electromechanical strain at low electric field of modified polyurethane composites for flexible actuators[END_REF] on PU containing polyaniline (PANI). They succeeded in increasing the permittivity ε with an almost constant modulus as the PANI content increased. Additionally, M for the four tested PU materials was around 10 -16 m 2 /V 2 and Mm around 10 -17 m 2 /V 2 .

Although little work has been done at the theoretical level to describe macroscopic behavior using local parameters, it is interesting to note the work of Lefevre and Lopez-Pamies [START_REF] Lefèvre | The overall elastic dielectric properties of a suspension of spherical particles in rubber: An exact explicit solution in the small-deformation limit[END_REF], or Stephen et al [START_REF] Spinelli | Dielectric elastomer composites: A general closed-form solution in the small-deformation limit[END_REF]. They have proposed very rigorous equations and simulations of composites made up of elastomeric matrices incorporating rigid spherical or ellipsoidal particles. In other works, Lefevre and Lopez-Pamies [START_REF] Lefèvre | Nonlinear electro-elastic deformations of dielectric elastomer composites: I -Ideal elastic dielectrics[END_REF][START_REF] Lefèvre | Nonlinear electro-elastic deformations of dielectric elastomer composites: II Non-Gaussian elastic dielectrics[END_REF] have developed a framework devoted to nonlinear elastic behaviour, which must be taken into account for deformations higher than about 1% for elastomeric polymers: this is the case for the best electroactive materials found in the literature. Such approaches have been developed using sophisticated and elegant homogenization techniques.

The main difference with this work is that they did not take into account electrical conductivity and only considered static conditions.

Diguet et al. [START_REF] Diguet | Physical modeling of the electromechanical behavior of polar heterogenous polymers[END_REF] examined the idea that these composite-like materials have heterogeneous dielectric properties, which, in turn, locally modify the electric field and create forces. They calculated the force created by a single particle (with dielectric constant ε2) embedded in a matrix (with ε1≠ ε2) and estimated the resulting deformation.

Interestingly, these polymers also exhibit a time-dependent electrostriction S(t), as seen in Figure 1. Possible reasons for this behavior are (i) their viscoelasticity and (ii) their dielectric relaxations. Previous works have studied both mechanical [START_REF] Jomaa | Difference between electrostriction kinetics, and mechanical response of segmented polyurethane-based EAP[END_REF] and dielectric [START_REF] Jomaa | Dielectric properties of segmented polyurethanes for electromechanical applications[END_REF] behaviors for different PU types, and clearly found corresponding relaxation times at room temperature that were lower by more than five orders of magnitude than the typical electrostriction time.

This means that, when a force is applied, the polymer deforms in less than a millisecond owing solely to mechanical and dielectric effects. Interestingly, the conductivity relaxation time is on the same order of magnitude observed for S(t). This means that the driving forces are also time-dependent on the order of tens of seconds. In fact, the electric field is governed by two material electrical parameters: permittivity (ε) and conductivity (σ). The EAP polarization state depends on the time scale because of charging effects at the electrodes arising from the effective conductivity [START_REF] Jomaa | Difference between electrostriction kinetics, and mechanical response of segmented polyurethane-based EAP[END_REF] and dielectric constant. This time-dependent dielectric force or dielectrophoretic (DEP) force used in cellseparator devices or in concentrator devices [START_REF] Zhang | Dielectrophoresis for manipulation of micro/nano particles in microfluidic systems[END_REF][START_REF] Simmons | The Development of Polymeric Devices as Dielectrophoretic Separators and Concentrators[END_REF]. This force arises usually from the differences in electric properties (ε, σ) between biological cells and a medium, and depends on the field gradient as well as its frequency.

The force experienced by cells is either positive or negative depending on these factors. Although the literature on this topic is abundant [START_REF] Zhang | Dielectrophoresis for manipulation of micro/nano particles in microfluidic systems[END_REF][START_REF] Simmons | The Development of Polymeric Devices as Dielectrophoretic Separators and Concentrators[END_REF], DEP forces have never been taken into account in elastic soft materials subjected to electric fields.

In this paper, we evaluate the effect of DEP forces as a physical mechanism for pure electrostriction in a compositelike structure. We then calculate the electric field and forces in a polymer with many particles, and obtain the resulting deformation. For diblock-copolymer PU, the phase separation (composition gradient) between domains might not be sharp, so we also evaluated how the profile might influence the macroscopic deformation.

Methods

Materials

The studied materials were PU materials, provided by Lubrizol Corporation. These PUs are block co-polymers with hard segments (HSs) and soft segments (SSs). HSs are composed of 4.4' methylene bis (phenyl isocyanate) -(MDI) and 1,4-Butanediol (BD), and SSs consist of poly(teramethylene oxide) -(PTMO). Table 1 lists the weight fractions of the three samples studied. The molecular weight of the SSs (PTMO) were 1000 g/mol for PU60 and 88, and 2000 g/mol for PU75.

Film samples were made via solution casting. The first step was to dissolve each PU in N,N-dimethylformamide (DMF) at 350 K. The solutions were spread onto glass plates using the Doctor Blade applicator (Elcometer®), dried at 340 K for 1 day and 400 K for 3 h. The thickness of the dried films was 100 ± 1 m.

Because HSs and SSs are thermodynamically incompatible, a phase separation occurs during their processing, specifically during solvent evaporation. This leads to a rich HS phase or "hard domain" (HD) and a rich SS phase or "soft domain" (SD). If the SS fraction predominates, the material topology is more likely a soft matrix with hard inclusion particles dispersed in it.

Au electrodes were sputtered onto a disk of radius 2 cm on and below the PU films with a thickness around 10-20 nm. Samples were then placed between the two electrodes and a voltage was applied. The lower electrode was fixed and the top electrode was free to move. The movement of the top electrode was recorded using a double-beam laser based interferometer. Figure 1 shows the strain obtained versus time as a step electric field E0 = 2 MVm -1 was applied.

These polymers exhibit several relaxations times (Figure 2), which were characterized either with mechanical spectroscopy (using the homemade dynamic mechanical analyzer described in [START_REF] Jomaa | Difference between electrostriction kinetics, and mechanical response of segmented polyurethane-based EAP[END_REF]) or through dielectric spectroscopy (Modulab Materials Test System; see [START_REF] Jomaa | Dielectric properties of segmented polyurethanes for electromechanical applications[END_REF]). Their relaxation times are compared with the characteristic times associated with electrostriction kinetics. The α-relaxation times result from polymer chain movements and impact the mechanical behavior of the PUs as well as their dielectric properties because the chains contain dipoles. The αrelaxation times were then determined via dielectric and mechanical measurements [START_REF] Jomaa | Dielectric properties of segmented polyurethanes for electromechanical applications[END_REF]; they are at least five orders of magnitude lower than the times associated with electrostriction at room temperature. For reference, we have also provided the β relaxation times in Table 1, although they are slightly weaker than those for α. The small PU conductivity, which might originate in ionic diffusion and therefore thermally activated processes, introduces an additional characteristic time on the order of minutes for the three PUs [START_REF] Jomaa | Difference between electrostriction kinetics, and mechanical response of segmented polyurethane-based EAP[END_REF]. These two characteristic times, αrelaxation (accounting for mechanical and dielectric time effects) and the characteristic time of the conductivity, highlight the importance of conductivity in the time dependence of electrostriction. Table 1. PU properties, with hard domain (HD) average diameter <DHD>~10 nm [START_REF] Jomaa | Modeling of segmented pure polyurethane electrostriction behaviors based on their nanostructural properties[END_REF]. Characteristic times τes, τα and τβ are for full electrostriction development, and the αand β-mechanical relaxations, respectively; TgSD is the glass transition temperature of the soft domains (SDs) of the PU; R.T. refers to room temperature. PU Commercial references wt% HS [START_REF] Jomaa | Difference between electrostriction kinetics, and mechanical response of segmented polyurethane-based EAP[END_REF] HD-HD distance (nm) [START_REF] Wongtimnoi | Thickness-dependent microstructural and electromechanical properties in polyurethane films obtained by polymer solution casting[END_REF] TgSD

(K) [22] Y (MPa) R.T, 0.2 Hz [32] εr (-) R.T, 0.2 Hz [32]
τes (s) R.T.

[5] τα (s) R.T. [START_REF] Jomaa | Dielectric properties of segmented polyurethanes for electromechanical applications[END_REF] τβ (s) R.T.

[30]

S/E

Simulation

In this paper, the aim is to propose a physical analysis of the mechanisms of mechanical deformation due to the application of electrical forces on the material, and in particular the effect of conductivity (section 3). In this spirit, a material containing a random distribution of identical particles was considered, and the effect of their density and the gradient of their composition at the matrix-particle interface was tested. Several authors have calculated the electrical [START_REF] Huang | Understanding the strain dependent dielectric behavior of carbon black reinforced natural rubber-An interfacial or bulk phenomenon[END_REF] and mechanical [START_REF] Wang | Modeling of viscoelastic properties of nonpermeable porous rocks saturated with high viscous fluid at seismic frequencies at the core scale[END_REF] macroscopic properties of similar composites. Such results from a 2D computation are of course different from those obtained from a 3D computation, but the trends can be discussed: the influence of the parameters can be compared, and of course, the results are obtained faster than for a 3D computation.

The 2D force calculations were carried out using the "Electric Currents" module of the FEM software COMSOL 5.3.

An electric potential was applied to the top electrode, while bottom electrode was grounded electrode, generating an

Electric forces

Maxwell stress

Maxwell forces cause the attraction of two electrodes with opposite electrical charges. When a dielectric material is placed between the two charged plates, the force experienced depends on the dielectric constant of the medium.

The electrode displacement is limited by the countering elastic force of the polymer. The potential ΔV applied to the electrodes induces a field E0 given by E0=ΔV/h (with h the distance between the electrodes). From theory, the attractive stress acting on each electrode is fs=εE0 2 /2, and the resulting strain is Sm =-(ε/Y)E0 2 = -MmE0 2 . A typical value of Mm, for a polymer with ε~10ε0 and Y~1 MPa would be Mm ~10 -17 m 2 /V 2 , which is a thousand times weaker when compared than experimental values (Table 1), M ~10 -14 m 2 /V 2 .

While the Maxwell effect originates as a stress fs acting on the surface electrodes, we aim to evaluate the contribution of electric body forces (electric forces within the volume).

Body forces

For a polarizable material, the overall force is

𝐹 = ∫ 𝑓 𝑣 𝑑𝑉 𝑉 + ∮ 𝑓 𝑆 𝑑𝑆 𝑆 ( 4
)
where the electric force per unit volume of a material, with a polarization density P can be expressed as

𝑓 𝑣 = (𝑃 ⃗ ⋅ 𝛻 ⃗ )𝐸 ⃗ ( 5
)
An alternative is to calculate the internal forces in dielectric and magnetic materials. For example, we could use the equivalent charge density to obtain 𝑓 𝑣 = -(𝛻 ⃗ ⋅ 𝑃 ⃗ )𝐸 ⃗ . First, the total force 𝐹 is zero if the external field 𝐸 ⃗ is homogenous. Second, the difference between the two force expressions can be written as a gradient of a scalar quantity, which does not change the deformation in an incompressible medium like a polymer well above its glassrubber transition temperature; more details on this topic are in [START_REF] Choi | General formulation of equivalent magnetic charge method for force density distribution on interface of different materials[END_REF][START_REF] Zahn | Derivation of the Korteweg-Helmholtz electric and magnetic force densities including electrostriction and magnetostriction from the quasistatic Poynting's theorem[END_REF][START_REF] Petit | Calculation and analysis of local magnetic forces in ferrofluids[END_REF].

For the PUs studied here, we use the model of a composite material with a host matrix (SD) and filling particles (HD) having different sets of properties (ε1, σ1, Y1) and (ε2, σ2, Y2), respectively; we generally take ε2 ≠ ε1, σ2 ≠ σ1, and Y2 ≠ Y1. Increasing the HS weight percentage slightly increases the SD glass transition temperature TgSD when the phase separation between the SD and HD is not complete, implying that TgSD is much lower than the HD glass transition temperature TgHD. The result is that TgSD<300 K<TgHD, and Y2>>Y1 at room temperature. As discussed above, the conductivity of these polymers arises most probably from charge diffusion. This directly implies that, as Y2>>Y1, diffusion is much slower in the HD and the conductivities obey σ2<<σ1. Increasing the HS weight percentage decreases the conductivity of the composite, which is consistent with the data in Table 1. Finally, the HD has a higher dielectric constant than the SD because of their chemical structures, and therefore ε2>ε1.

For the composite, we first calculate the forces exerted on the HD considered as a system of spherical particles, then the forces applied to the SD treated as the host matrix.

3.2.1

Body forces inside the HD. Equation ( 5) expresses the dielectric force density experienced by a polarized volume in an inhomogeneous electric field. A spherical particle in the HD with given electrical properties (ε2, σ2) embedded in the SD medium, with electrical properties (ε1, σ1) is polarized as a function of (i) the applied field E, (ii) the particle radius R, and (iii) the set of ε2, σ2, ε1, and σ1. The particle undergoes a DEP force from Eq. ( 5) as a function of time [START_REF] Benguigui | More about the dielectrophoric force[END_REF][START_REF] Jones | Dielectrophoretic force calculation[END_REF]:

𝐹 𝐷𝐸𝑃 (𝑡) = 2𝜋𝑅 3 𝜀 1 [𝐾 𝜎 (1 -𝑒 -𝑡 𝜏 ⁄ ) + 𝐾 𝜀 𝑒 -𝑡 𝜏 ⁄ ]𝛻𝐸 2 (6)
with

𝐾 𝜎 = 𝜎 2 -𝜎 1 𝜎 2 + 2𝜎 1 (7)
𝐾 𝜀 = 𝜀 2 -𝜀 1 𝜀 2 + 2𝜀 1 (8)
and

𝜏 = 𝜀 2 + 2𝜀 1 𝜎 2 + 2𝜎 1 (9)
The conductivities are contrasted by K and the permittivities by K, and range values from -0.5 (for 1>>2 and 1>>2, respectively) to 1 (for 1<<2 and 1<<2, respectively). As expected, when K and K are zero (1 = 2 and 1 = 2,), the polarization force tends towards zero as for homogeneous materials.

Because all materials have finite conductivities, it is important to compare the duration of the experiment with the characteristic time obtained from Eq. ( 6) [START_REF] Zahn | Derivation of the Korteweg-Helmholtz electric and magnetic force densities including electrostriction and magnetostriction from the quasistatic Poynting's theorem[END_REF]. Each of the two exponentially decreasing terms of Eq. ( 6) is dominated by the term with either Kσ or Kε. Hence, the force is dominated by the Kε term for times much shorter than τ, and we have

𝐹 𝐷𝐸𝑃 (𝑡 << 𝜏) = 2𝜋𝑅 3 𝜀 1 𝐾 𝜀 𝛻𝐸 2 (10)
In contrast, for times much longer than τ, the force is dominated by the Kσ term:

𝐹 𝐷𝐸𝑃 (𝑡 >> 𝜏) = 2𝜋𝑅 3 𝜀 1 𝐾 𝜎 𝛻𝐸 2 . ( 11
)
The signs and strengths of the forces change depending on the experimental time, compared with τ. Steady states are dominated by the conductivities, which are limited because Kσ limits their difference.

Body forces inside soft domains.

As pointed out by Diguet et al. [START_REF] Diguet | Physical modeling of the electromechanical behavior of polar heterogenous polymers[END_REF], a particle with ε2>ε1 (Kε >0) generates a locally inhomogeneous electric field around itself. Similar to Eq. ( 5), this inhomogeneous electric field induces a body force inside a polarizable medium. The estimate was close to data obtained by extrapolating similar polymer blends, meaning ε2>ε1. Some elastomeric polymers have relative permittivities of around 10 and are usually considered insulators but, in reality, have low conductivities. Thus, by including the difference in conductivities and permittivities via Kσ and Kε, we can take time dependence into account for forces such as DEP. For example, following the previous case, where ε2>ε1, σ2<σ1, Kε >0, and Kσ <0, a representative elementary volume of the composite material is a spherical particle as in Fig. 3a, and the matrix experiences a body force depending on its position and time (Fig. 3b andc). Figure 3 presents map of the electric field and the resulting body force around a spherical particle calculated using Eq. ( 5) assuming 2D axisymmetry. The forces at a short time, τ/5 in Fig. 3b, are towards the particle near its top surface and directed outwards at the equator. Moreover, the body forces are stronger near the poles than near the equator. At a longer time, t=2τ in fig. 3c, the opposite occurs: the forces are outwards near the top surface of the particle and directed inwards at the equator, and they are stronger near the equator than near the poles.

In the single-particle picture, the sign of the force experienced by the dipoles in both HD (particle-system) and SD (matrix) depends clearly on the properties of both the HD and SD. The analytical model of a DEP force acting on a sphere helps in understanding forces inside a composite composed of a spherical HD inside an SD: body forces at times t<τ are dominated by the permittivities, and those at t>τ are dominated by the conductivities. In the next section, we compute the effects of such time-dependent forces in steady state (t>>τ) with many particles inside the matrix.

Application to electroactive polymers and discussion

We next propose a 2D model in which all the above effects are taken into account, namely, (i) the Maxwell forces and (ii) the body forces calculated in the HD and SD. The calculation is performed assuming "linear" materials, meaning that the elastic constants, dielectric constants, and conductivities are independent of the electric field and mechanical strain. Because it is difficult to determine the permittivities of the parent polymers, a preliminary calculation of the electronic distribution of the PU was carried out using tight-binding-based accelerated quantum chemical molecular dynamics [START_REF] Ahmed | CO Oxidation and NO Reduction on a MgO(100) Supported Pd Cluster: A Quantum Chemical Molecular Dynamics Study[END_REF] at 300 K. This gives average dipole values of 1.18 Debye and 2.40 Debye for PTMO and MDI-BDO, respectively. There is also literature evidence of a larger dipole moment for MDI-BDO than for PTMO [START_REF] Law | Conformational Characteristics of Poly (tetramethylene oxide)[END_REF][START_REF] Zhang | Theoretical study of hydrogen bonding interactions on MDI-based polyurethane[END_REF][START_REF] Ramirez | Energetic, Structural, and Vibrational Properties of 4,4′-Methylenediphenyl Diisocyanate with Relevance for Adhesion[END_REF]. These calculations enable the HD and SD permittivities to be evaluated separately. More generally we choose simple parameters to roughly match the effective typical PU values with ε=5 ε0 to 10ε0 and Y=1 MPa to 10 MPa. Parameters for each domain are provided in Table 2. The polymer matrix Poisson ratio is set as 0.49

(very close to typical value of 0.5) to avoid volumetric locking during numerical calculation of the mechanical response. Poisson ratio υ 0.49 0.3

Lastly, because the contribution of body forces results from electric field gradients, it is important to test the effect of the chemical gradient, near the HD-SD interface. In block copolymers like the PUs used here, the phase separation does not lead to an abrupt change in chemical compositions and therefore of properties, but rather a smooth change in them [START_REF] Seefried | Thermoplastic urethane elastomers. II. Effects of variations in hard-segment concentration[END_REF]. This is hence a little different from standard particle-matrix composites and may change the profiles of both the permittivity and conductivity gradients. To account for this peculiar aspect and evaluate its contributions to the body forces, we modeled each particle as a core and boundary with different properties, with intermediate layers (to mimic the interphase) having properties that interpolate smoothly between the core and boundary. The chosen variation function is [START_REF] Merlin | Electrodynamics of soft multilayered particles dispersions: dielectric permittivity and dynamic mobility[END_REF] 𝜌

𝑖 (𝑟 ) = 1 2 [1 -𝑡𝑎𝑛ℎ ( (𝑟 -𝑟 𝑖 ) -𝑟 𝑝 𝛼 )] ( 12
)
where r is the position in the composite (with r the norm of r ), ri is the center of the i-th HD-layer, rp=1.5 μm is the HD radius, and α is a coefficient setting the particle/matrix interphase size. A small value of α yields a smoother profile, whereas a large value yields a sharper profile. Hence, the Young modulus Y, conductivity , and permittivity  are obtained as spatial functions of the forms

𝑌 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 = 𝑌 𝑚𝑎𝑡𝑟𝑖𝑥 + (𝑌 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 -𝑌 𝑚𝑎𝑡𝑟𝑖𝑥 ) ∑ 𝜌 𝑖 (𝑟 ) 𝑁 𝑖=1 𝜎 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 = 𝜎 𝑚𝑎𝑡𝑟𝑖𝑥 + (𝜎 𝑚𝑎𝑡𝑟𝑖𝑥 -𝜎 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 ) ∑ 𝜌 𝑖 (𝑟 ) 𝑁 𝑖=1 ( 13
)
𝜀 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 = 𝜀 𝑚𝑎𝑡𝑟𝑖𝑥 + (𝜀 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 -𝜀 𝑚𝑎𝑡𝑟𝑖𝑥 ) ∑ 𝜌 𝑖 (𝑟 ) 𝑁 𝑖=1
where N is the number of HD-layers. The properties are plotted in Figure 4. A 2D geometry was configured consisting of a rectangle (matrix) with a random distribution of N circular particlelike inclusions, with N=200 (Figure 5a). An electric potential was applied so that the field was directed from top to bottom with an average strength |E0| = 1 MV/m (y-component) in the steady state (Figure 5a). The electric field ycomponent exhibits some local variation. In particular, the steady state field inside the particles is governed by the conductivity contrast; for example, the field inside a spherical particle is set as |Eint| = 3|E0|σ1/(σ2+2σ1) ~ 1.5|E0| ~ 1.5 MV/m if σ1>>σ2. The field is then weaker than E0 around the particles

The maximum particles fraction of 35% corresponds to about half of the percolation threshold, which is around 60%

for 2D geometry (compared with 20% for 3D).

Electrostriction coefficients

Before analyzing the physical results in more details, it is important to check the influence of the parameters used in this FEM calculation, such as (i) the effect of the random draw of the position of the particles and (ii) their number (at constant volume fraction): in fact, if the results of the calculations depend on the number of particles, we should consider another approach, such as periodic boundary conditions that suppress the edge effect, in order to guarantee reliable results whatever the random draw of the particles dispersed in the matrix [START_REF] Gusev | Representative volume element size for elastic composites: a numerical study[END_REF].

A random dispersion of particles in the matrix leads to many possible distributions which can affect the results, however, for N=100 particles with 16 different trials, an average strain <S>=-8.63×10 -6 was obtained with a small a standard deviation of 3×10 -8 . A second test was performed with the fraction =7% was performed varying the particle number N (and their radius) from 50 to 450. Results of this calculation are shown in Figure 6. Beside, for N=75, the calculation was done 15 times with 15 random distributions of particles. Results seems concentrate in the same range with an average value as (Si-<S(0.07)>)/<S(0.07)>~5×10 -3 , it is then verified that the result of the simulation is mainly controlled by the filling fraction and not by the choice of distribution. This shows a consistency of the method to achieve repeatable results. We calculated the deformation at different applied fields (i) induced by the surface stress only, and (ii) induced by all the forces, as seen in in Figure 5b. For N=200 particles; the deformation Sm obtained from the Maxwell stress is indicated by the blue dots and curve, while the deformation S obtained from the Maxwell stress and the body force together is indicated by the red dots and curve. As expected, both curves exhibit quadratic behavior with E0.

Moreover, the deformations are such that S=-ME0 2 and Sm =-MmE0 2 with Mm ~7.84×10 -18 m 2 /V 2 and M ~8.21×10 -18 m 2 /V 2 , showing that the body forces slightly reinforce the strain; the composite material is more compressed. This calculation with 200 particles represents a 2D filling factor of nearly 0.07. In this Figure 7, the effect of particle number, on the pure electrostriction coefficient is presented. At N=0, there is only the matrix, and the deformation reaches S=-ME0 2 with M=Mm ~9×10 -18 m 2 /V 2 and is less than expected from theory with ε/Y =5ε0/10 6 ~45×10 -18 m 2 /V 2 . This might be an effect of the 2D geometry and/or of the clamping of the electrodes [START_REF] Guyomar | Focus on the electrical field-induced strain of electroactive polymers and the observed saturation[END_REF][START_REF] Zhenyi | High field electrostrictive response of polymers[END_REF][START_REF] Wang | Clamping effect on the piezoelectric properties of poly(vinylidene fluoride-trifluoroethylene) copolymer[END_REF]. The calculated values at N=0 are a reference for scaling the effect of particle number. When adding particles, we observe a divergence between Mm (Maxwell stress only) and M (Maxwell stress plus body forces), meaning that body forces increase in strength as particle number increases. As discussed above, the electrostriction coefficient M has two contributions: the Maxwell coefficient Mm and the pure coefficient ME related to body forces: We see in Figure 7 that for N=1000 particles, M~5.4×10 -18 m 2 /V 2 , Mm~4.2×10 -18 m 2 /V 2 , and ME~1.2×10 -18 m 2 /V 2 .

𝑆 = 𝑀𝐸 0 2 = (𝑀 𝑚 + 𝑀 𝐸 )𝐸 0 2 (14)
This means that the body forces are responsible for 22% of the total calculated deformation. The Maxwell stress is the dominant component. Comparing the value M~9.0×10 -18 m 2 /V 2 obtained with N=0 to M~5.4×10 -18 m 2 /V 2 obtained with N=1000 particles, the deformation is reduced by 50% and the percentage of ME in M is increased by 22%.

Effective properties

Both coefficients Mm and M decrease with increasing particle number, probably because the composite material stiffens as the HD volume fraction increases. Filling the matrix with particles affects the properties of the material such as its dielectric constant, conductivity, and Young modulus. Hence, we extracted these effective values as the particles number increased:

𝜀 𝑒𝑓𝑓 = 〈𝐷〉/𝐸 0 (15)
𝜎 𝑒𝑓𝑓 = 〈𝐽〉/𝐸 0 ,

where <q> denotes the average of quantity q. The effective mechanical modulus of the structure was obtained through numerical experiments by measuring the strain due to a relatively small pressure (Ymatrix/100) on the top electrode.

Figure 8 shows the effective dielectric constant, conductivity, and Young modulus. 2).

Clearly, the dielectric constant and Young modulus both increase with increasing N because the particles have higher ε and Y values than the matrix. In contrast, the effective conductivity decreases because particles have lower conductivities than the matrix. The increase in the Young modulus with particle fraction (HD) is larger than that for the dielectric constant which is experimentally measured in the PU [START_REF] Jo | Polyurethane-based actuators with various polyols[END_REF]. Moreover, its behavior appears to be nonlinear, whereas the electrical properties look almost linear, at least well below the percolation threshold of the particles.

The spatial distribution of spherical particles in a matrix is known for its electrical properties, which are usually modeled by the Maxwell-Garnett equation [START_REF] Torquato | Random Heterogeneous Materials: Microstructure and Macroscopic Properties[END_REF]:

𝜎 𝑒𝑓𝑓 -𝜎 1 𝜎 𝑒𝑓𝑓 -(𝑑 -1)𝜎 1 = 𝜑 𝜎 2 -𝜎 1 𝜎 2 + (𝑑 -1)𝜎 1 (17)
With d is the dimension of the system, which is the Maxwell-Garnett in 3D (d=3) [START_REF] Ruppin | Evaluation of extended Maxwell-Garnett theories[END_REF]. The dielectric constant and conductivity follow the Maxwell-Garnett equation. This is usually valid in the dilute regime. A linear behavior of permittivity and conductivity with the volume fraction of particles is then effectively obtained.

The Young modulus, however, follows the non-linear model [START_REF] Lopez-Pamies | An exact result for the macroscopic response of particle-reinforced Neo-Hookean solids[END_REF],

𝑌 𝑒𝑓𝑓 𝑌 𝑚𝑎𝑡𝑟𝑖𝑥 = 1 1 -𝜑 2 (18)
Being a non-linear equation with the filling factor, this explains the larger increase in the Young modulus as the particle number increases comparatively to the electric equations which are linear.

The strains were estimated for the composites. As previously mentioned, Mm=εeff/Yeff decreases as N increases because of the higher increase in the Young modulus with particle fraction than for the dielectric constant. A higher HD content induces a lower strain, which is confirmed by experiment [START_REF] Jo | Polyurethane-based actuators with various polyols[END_REF]. Moreover, the pure coefficient ME related to the body forces increases as N increases (Figure 7). To analyze the effect of electrostrictive body forces on the resulting strain, we estimated the equivalent stress by multiplying the strain coefficient ME by the effective Young's modulus and the square of the applied field: MEYeffE0 2 . We used the effective Young modulus because we are looking at the displacement of the top surface of the composite material. The result, plotted in Figure 9, shows the equivalent electrostrictive stress exhibiting clearly linear behavior with particle number. Thus, each particle produces an effective stress of 2×10 -3 N/m 2 . 1). This confirms that increasing the HD fraction reduces M for the reasons presented above. Figure 10 shows the experimental curves of M, Y, , and  as functions of HS content, where the HS content (wt%) sets the HD fraction inside the material. The experimental properties (Figure 10) behave according to the calculation based on an HD particle in a matrix (Figs. [START_REF] Li | Electrostrictive effect in ferroelectrics: An alternative approach to improve piezoelectricity[END_REF][START_REF] Katsouras | The negative piezoelectric effect of the ferroelectric polymer poly(vinylidene fluoride)[END_REF]. In [START_REF] Petcharoen | Electrostrictive properties of thermoplastic polyurethane elastomer: Effects of urethane type and soft-hard segment composition[END_REF][START_REF] Lim | High-Field Maxwell Stress Effect of Dielectric Actuator Based on Segmented Polyurethane[END_REF], two PUs with different HS contents show different experimental deformations for the same applied field of 550V/mm: a measured deformation of 2.11×10 -3 for HS fraction of 42.4 wt%, and a deformation of 1.39×10 -3 for an HS fraction of 55.2 %(wt). The corresponding mechanical moduli, and conductivities respectively increased and decreased with increasing HS, as in Fig. 10. The effect for an SS molecule with different HS contents (wt%=20, 25, 30, and 35) has been tested previously [START_REF] Jo | Polyurethane-based actuators with various polyols[END_REF]; the selected SS were: poly(tetramethylene glycol) (PTMG), poly(neopentyl glycol adipate) (PNAD), and poly(dimethyl siloxane) (PDMS).

The effect was still as previously found: deformations decreased with increasing HS content independently of the type of SS molecule.

It is interesting to consider other heterogeneous materials, like composites obtained in dispersing conductive particles in a polymeric matrix, and to analyze the role of body forces. For example, a PU filled with particles such as carbon black [START_REF] Wongtimnoi | Improvement of electrostrictive properties of a polyether-based polyurethane elastomer filled with conductive carbon black[END_REF] or carbon nanotube [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] has higher permittivity, conductivity, and mechanical modulus, than a pure PU.

As discussed above, the higher permittivity of the particle limits DEP forces. The increase in Young modulus due to the stiff particles causes the deformation first to increase with increasing conductive particle fraction and then decrease when approaching the particle percolation threshold. This is due to the drastic increases in the effective mechanical modulus and electrical conductivity. Thus, the gain expected from the high electrical property values of the fillers disappears.

Particle/matrix interface profile effect

Keeping the properties of the matrix and particles constant (Table 2), we changed the value of α, which affects the profile of the HD-SD interface. Figure 11 shows various profiles of the relative permittivity gradients, obtained with Eqs. [START_REF] Guyomar | Focus on the electrical field-induced strain of electroactive polymers and the observed saturation[END_REF][START_REF] Zhenyi | High field electrostrictive response of polymers[END_REF]. The higher α is, the stronger the dielectric, stiffness, and conductivity gradients are, but in a more confined space.

Because the body forces come from gradients of electrical properties (relative permittivity and conductivity) and the mechanical response depends on the effective Young modulus, we calculated the deformation to estimate the resulting electrostriction coefficient for different values of α.

Figure 12 shows the effect of the interface thickness on the electrostriction coefficients M, Mm, and ME along with the effective properties of a composite material with N=200 particles. The electrostriction coefficient M increases as α increases; the sharper the interface is, the higher M is. The Maxwell contribution Mm increases as α increases. Because Mm is proportional to the ratio of the effective permittivity to the effective Young modulus, these two parameters change as α changes. In fact, the effective permittivity increases while the Young modulus decreases, which globally explains the increase in the Maxwell coefficient (see Table 3). of the electrostriction to the strain does not seem to depend on the property gradients, which is consistent with the fact that increasing  decreases the interphase thickness. The increase in  and  gradients is the origin of the body forces.

Conclusion

The experimental data for some polyurethanes show a typical electrostriction coefficient on the order of 10 -14 m 2 /V 2 and Maxwell strain on the order of 10 -17 m 2 /V 2 . In this study, we calculated PU deformation with a polarization contribution from heterogeneities in the PU after a partial phase separation. The DEP forces, which are timedependent body forces, contribute additionally to the deformation; the Maxwell contribution is a surface force acting on the electrodes. The deformation is complete a few tens of seconds after the voltage is applied because the electrical conductivity agrees with experimental data. The pure electrostriction coefficient of these body forces is responsible for up to 22% of the total deformation in the range of our calculation (up to a maximal 2D filling of 35%). Increasing the particle volume fraction above 35% (2D) is ineffective because a strong stiffening occurs in the composite material. The profile modifications of the electrical permittivity, electrical conductivity, and mechanical stiffness between stiff particles (HD) and the matrix (SD) also induce a change in the electrostriction coefficient, but this change remains on the same order of magnitude. Furthermore, the DEP magnitude is limited at short times by the permittivity contrast factor K and at long times by the conductivity contrast factor Kσ. It is not possible to enhance the electrostriction coefficient by a factor higher than a few units just by changing the HD permittivity and/or conductivity in two-phase polymers such as a PU or by adding high permittivity/conductivity particles to that polymer.

Our numerical simulations were first performed in as simple a framework as possible, i.e. with a limited number of particles, no periodic boundary conditions, and considering only linear constitutive behaviors, and we obtained results much smaller than the experimental data, at equivalent particle volume fraction, but with a large variation in the number of particles. Consequently, we can assume that the contribution of the polarization body force is too low to explain the discrepancy between the deformation measurements and the Maxwell strain among the possible scenarios. More investigations are needed to understand the large electrostriction observed in various copolymer-like PUs. It is noteworthy that recent works are focused on time dependent phenomena at the origin of polymer electroactivity [START_REF] Putson | Interface Polarization Effect on Dielectric and Electrical properties of Polyurethane (PU)/Polyaniline (PANI) Polymer Composites[END_REF][START_REF] Ghosh | Homogenization of time-dependent dielectric composites containing space charges, with applications to polymer nanoparticulate composites[END_REF]. A very interesting approach by Gosh et al., consists in showing that the dielectric constant drastically increases at decreasing frequency, i.e. increases as a function of time when an electric field step is applied to the material [START_REF] Putson | Interface Polarization Effect on Dielectric and Electrical properties of Polyurethane (PU)/Polyaniline (PANI) Polymer Composites[END_REF]. On the other hand, the electrical conductivity observed in our materials indicates that electrical charges are free to diffuse inside elastomers, and therefore the space charges are to be taken into account as possible contributions to the deformation as shown by some recent works [START_REF] Lefèvre | Homogenization of elastic dielectric composites with rapidly oscillating passive and active source terms[END_REF]. Following such consideration, we are currently investigating the electrostatic interactions of more or less mobile electrical charges responsible for the above conductivity, and will present it in Part 2 of this work.

Figure 1 .

 1 Figure 1. Electro-deformation of polyurethane. The electric field is switched on at time t=0.

Figure 2 .

 2 Figure 2. Experimental relaxation times in PUs. Data from [5,30].

  electric field across the materials along the y-axis. The local electric field 𝐸 ⃗ , electric polarization field 𝑃 ⃗ , current density 𝐽 , and electric displacement field 𝐷 ⃗ ⃗ , were calculated directly (𝐸 ⃗ = -∇ ⃗ ⃗ 𝑉, 𝐽 = 𝜎𝐸 ⃗ , ∇ ⃗ ⃗ • 𝐽 = 0 and 𝐷 ⃗ ⃗ = 𝜀𝐸 ⃗ = 𝜀 0 𝐸 ⃗ + 𝑃 ⃗ ). From these electrical quantities, we derived both the body forces 𝑓 𝑣 ⃗⃗⃗ in the volume and Maxwell stress 𝑓 𝑠 ⃗ ⃗⃗ (see next section), which were then inputted to the "Solid Mechanics" module in COMSOL to obtain the mechanical stress tensor 𝑇 ̿ (∇ ⃗ ⃗ • 𝑇 ̿ + 𝑓 𝑣 ⃗⃗⃗ = 0 ⃗ in the volume and 𝑇 ̿ • 𝑛 ⃗ = 𝑓 𝑠 ⃗ ⃗⃗ at the electrodes) and the deformation. We also chose the displacement of the top electrode to be along the electric field direction (y-axis), keeping the bottom electrode fixed. The strain was estimated from the displacement of the top electrode. The geometry has been meshed by the COMSOL Solid Mechanics and is composed for a calculation that included 200 particles of 3771110 quadratic triangular elements for a number of degrees of freedom of 22628481.

Figure 3 .

 3 Figure 3. a) Heterogeneous polymer model. Maps of the body force (red) and electric field (blue) around a particle with properties (ε2, σ2) in a surrounding medium with properties (ε1, σ1) with ε2>ε1 and σ2<σ1 at b) t= τ/5 and c) t= 2τ.

Figure 4 .

 4 Figure 4. a). PU component and its microstructures. b) Properties along a cut line in the composite material. The two ends of the cut line lie in the matrix, and the line cuts diametrically through the particle.

Figure 5 .

 5 Figure 5. a) Composite-like elastomer: with height H=10 -4 m (100 μm) and length L=2×10 -4 m randomly filled with 200 particles (rp=1.5 μm). The color scale represents the field strength along the direction perpendicular to the electrodes. The applied voltage (top and bottom) was such that the average electric field strength |E0| = 10 6 V/m. b) Effect of the electric field on strain with both body forces and surface forces (red curve) or only surface forces (blue curve).

Figure 6 .

 6 Figure 6. Test of the effect of the number of particles (at constant volume fraction of =7%) on the calculated value of S, and test of different random draw of the position of the particles (for N=75).

Figure 7 .

 7 Figure 7. Comparison between the effects of the electrostriction coefficients due to surface and body forces together (M), due to surface force alone (Mm), and due to body force alone (ME)

Figure 8 .

 8 Figure 8. Effective composite properties versus particle number extracted from the 2D simulation. The values for N=0, are those of the matrix (see Table2).

Figure 9 .

 9 Figure 9. Equivalent electrostrictive stress as a function of the particle number in the matrix.

Figure 10 .

 10 Figure 10. Experimental properties versus HS content of the 3 PUs (seeTable1).

Figure 11 .

 11 Figure 11. Effect of α on the permittivity along the same cut line as in Fig. 4.

Figure 12 .

 12 Figure 12. Electrostriction coefficients and effective composite properties as functions of α with N=200 particles.

Table 2 .

 2 Parameter values used for the calculation are typical of the PUs used.

	Matrix (1)	Particle (2)	Contrast

Table 3 .

 3 Effect of α on electrostriction coefficients.

	α	ε (ε0)	Y (MPa)	Mm (10 -18 m 2 /V 2 )	ME (10 -19 m 2 /V 2 )	M (10 -18 m 2 /V 2 )
	6	5.43	1.57	7.37	3.44	7.72
	30	5.45	1.41	8.17	3.96	8.57
	Relative change	0.3%	-9.9%	10.8%	15.1%	11.2%

Table 3 ,

 3 the relative change in Mm with α is unsurprisingly about equal to the sum of the relative changes in the dielectrics constant and Young modulus. The relative change in ME with α is the largest, but the values are still lower than those of M. For α=30, ME contributes 4.6% to M, while for α=6, it is almost the same at 4.5%. The contribution

Acknowledgments

This work was partly supported by the Japan Society for the Promotion of Science (JSPS) Core-to-Core Program, A. Advanced Research Networks, "International research core on smart layered materials and structures for energy saving". We thank Mark Kurban, from Edanz Group (www.edanzediting.com/ac) for editing a draft of this manuscript.


Shaoyun Zhou

Chrystelle A Bernard
email: chrystelle.bernard@rift.mech.tohoku.ac.jp

Kesavan Ravi

Hiroki Saito

Yuji Ichikawa

Kazuhiro Ogawa

Shu Yin

Development and Characterization of Photocatalytic GaN Coatings by Cold Spray Process

Keywords: Cold spray, Gallium nitride (GaN), Photocatalysis, Agglomerated powder, Splat test, Ceramic powder

For the first time, the low-pressure cold spray (LPCS) process was used to manufacture Gallium Nitride (GaN) films to enhance its photocatalytic properties and decrease its manufacturing cost. The deposition behavior of the GaN powder on stainless steel substrates was investigated. Several specimens, with sparsely deposited agglomerated GaN particles, were prepared under different spray conditions. Quantitative analysis of the evolution of the coverage area, deposited particle count, and average sectional area shows that, upon impact, agglomerated GaN particles disintegrate, leading to large deposition of small particles. By analyzing the cross-sectional area of the deposited 2 particle, no discernible permanent deformation of the substrate was observed. In addition, X-ray diffraction analysis of the coatings and powder indicated that no phase transformation occurred during the process. Based on Williamson-Hall analysis, the broader peaks of the coatings were mainly attributed to the distortions in the GaN lattice rather than changes in the crystallite size. At 400°C and 0.6 MPa, the deposition efficiency reached 5.3%, and the photocatalytic activities of the coating were about 33% (+9% compared to the powder). It is attributed to the higher specific surface area and roughness that the coatings exhibit after the breakage of the particles during the cold spray.

Introduction

The world faces serious environmental problems related to air and water pollution. The NOx gases emitted into the air because of human activities, such as the combustion of fossil fuels in large amounts, contribute to acid rain, global warming, and are hazardous to human health (Ref 1). In recent years, significant research progress has been made to reduce pollution and its effects on the environment. Several techniques are used for the treatment of polluted air and water. Compared to other common methods used for depollution, such as precipitation, adsorption on active carbon, selective catalytic or non-catalytic reactions, photocatalysis has emerged as a green and sustainable technology for solar energy conversion and environmental remediation (Ref 2). The splitting of water into H2 and O2 over photocatalysts by harvesting solar energy as a source of renewable energy has been attracting significant attention worldwide (Ref 3).

In addition, because no oxidizing agents, such as ozone, or heat are necessary, photocatalysis appears to be more economical than other methods (Ref 4).

Among numerous photocatalyst materials, fine oxide powders, such as TiO2 or ZnO, have received tremendous attention for the degradation of several environmental contaminants (Ref 5). These powders can easily be prepared using low-cost methods (Ref 6-9), such as combustion, hydrothermal, precipitation, and sol-gel methods (Ref 5,[START_REF] Sanchez-Martinez | Study of the Preparation of TiO2 Powder by Different Synthesis Methods[END_REF]. Another interesting material is Gallium nitride (GaN), a nitride ceramic material, which exhibits high photoactivity and chemical stability against photo-corrosion (Ref Photocatalysis occurring on the external and internal surfaces of the structures, the high surface-to-volume (S/V) ratio that powdered or nano-porous materials exhibit is a great advantage for enhancing their photocatalytic properties (Ref 13). However, a major drawback lies in the recuperation and recycling of the fine particles suspended in water after use, mainly because of the formation of aggregates ( Ref 14). Therefore, the development of nano-porous photocatalytic films manufactured from the powders is of prime interest (Ref 15,[START_REF] Yang | Low Temperature Deposition and Characterization of TiO2 Photocatalytic Film through Cold Spray[END_REF].

GaN photocatalyst coatings are conventionally obtained through metal organic chemical vapor deposition (MOCVD) ( Ref 17), molecular beam epitaxy (MBE) (Ref 18) or hydride vapor phase epitaxy (HVPE) ( Ref 19). However, they still have certain disadvantages that need to be resolved. For example, cracks are frequently observed after MOCVD (Ref 20), MBE is time-consuming, and HVPE is an expensive and highskill technique. Besides, these three techniques require high temperature operation (>850 °C) which is known to degradate the material photocatalytic properties due to the changes in the surface microstructure (Ref 21). Therefore, a low temperature operating process seems more suitable for retaining the powder's photocatalytic properties within the films.

Thus, for the fabrication of photocatalytic GaN films, a new manufacturing process combining the advantages of rapidity, cost efficiency, and low temperature is essential.

Among the various technologies available, the cold spray (CS) process, also known as cold gas dynamic spray process, appears to be a suitable candidate. The CS process is a solid-state coating technique based on the high-speed impact and consolidation of finely divided powder particles on different substrates (Ref 22). The coating was manufactured at a temperature well below the melting point of the sprayed feedstock. In contrast to other thermal spray techniques, such as thermal spray, sol-gel, and sputtering, the CS process has no risk of oxidation, thermal degradation, grain growth, and low energy consumption. Therefore, the CS technique can be considered to be a safe, green, and economical technology because of the absence of high-temperature explosive gases and radiation ( Ref 23). Considering that the bonding process by cold spray predominantly involves the prior particle/substrate interactions and following particle/particle interactions (Ref 24-26), meanwhile, ceramic particles not easy to be plastically deformed and melt (Ref 27), the coating formation would be conceivably elucidated from the first layer and following layers, respectively (Ref 28).

In this study, for the first time, GaN coatings on stainless steel substrates using a coldspray process were manufactured. The bonding mechanisms occurring at the interface of particle/substrate and particle/particle were investigated using splat and multi-pass tests, respectively. These two deposition processes were investigated to understand the coating formation and build-up. Later, NOx decomposition tests were conducted to evaluate the photocatalytic properties of the cold-sprayed GaN coatings and compare them with those of the GaN powder.

Materials and Methods

1 Powders and Substrates

Commercially available GaN powder (Toshima Manufacturing Co., Ltd., Saitama, Japan) with an average particle diameter of 1.7 µm, ascertained from the particle size distribution results, was used as feedstock to develop photocatalytic active coatings via the low-pressure cold spray (LPCS) technique. The morphology of the feedstock powder was investigated using a field emission scanning electron microscope (FE-SEM, Hitachi SU-70, Tokyo, Japan) equipped with an energy dispersive X-ray (EDX) system for elemental analysis, which was operated at 10 kV. The material properties of the GaN feedstock are listed in Table 1.

GaN powder was cold-sprayed onto as-received and mirror-polished stainless steel substrates. Prior to the cold spray experiments, the surface roughness of the stainless steel substrates were measured using a surface roughness measuring instrument, that is, a surface profilometer (Surfcorder SE300, Kosaka Laboratory Ltd., Tokyo, Japan) (Ref 26). As illustrated in Table 1, the mean roughness (Ra) and the maximum height of the profile (Rz) of the as-received stainless steel substrates are 0.10 ± 0.012 μm and 0.83 ± 0.076 μm, respectively. After mirror polishing, the values of Ra and Rz decreased to 0.06 ± 0.012 μm and 0.39 ± 0.101 μm, respectively. Subsequently, the Vickers hardness of stainless steel was measured at 216 HV by microhardness testing system (Fischerscope HM2000, Helmut Fischer) (Ref 26).

2 Splat Test and Coating Build-up

Cold-spraying was performed using the DYMET-423 equipment (Obninsk Center for Powder Spraying, Obninsk, Kaluga, Russia), which comprises an internal gas heater in the spraying gun. Figure 1 reveals a schematic illustration of the downstream injection cold spray apparatus used for the experiment. The spraying gun was attached to a manipulator (BZT Maschinenbau GmbH, Leopoldshöhe, Germany) moving at the same traverse speed. To control the standoff distance between the nozzle and substrate surface, and the traverse parameters, the substrate was mounted on an XYZ stage and manipulated by a personal computer (Ref 30).

Coating formation by cold spray technique can be considered as a two-step process: development of the first layer followed by the coating build-up (Ref 28). In that regards, the splat test appears as an ideal experimental approach for the study of the singleparticle impact and particle/substrate interaction. Splat tests were performed on mirrorpolished stainless steel substrates to remove the effects of the mechanical interlocking (Ref 31).

In total, six spray conditions (three inlet gas temperatures and two inlet gas pressures) were investigated for splat tests to characterize the deposition quality of the GaN powder on the mirror-polished stainless steel substrate (Table 2). A low feeding rate of 0.019 g/s was used to impinge particles sparsely onto the substrate, resulting in no overlapping of the particles. After the test, the particle/substrate interface was investigated, highlighting the deformation and deposition mechanisms of GaN ceramic particle on a metallic substrate.

After understanding the development of the first layer and the deposition mechanisms between GaN particle and stainless steel substrate, coating build-up was performed. The same spray conditions as for splat tests were used (see Table 2). As it was proven than an initial substrate roughness promotes the mechanical interlocking and coating buildup (Ref 32), coating build-up experiments were performed on as-received substrates.

3 Characterization of the First Layer Deposition Behavior

Back-scattered electron (BSE) micrography (FE-SEM, Hitachi SU-70, Tokyo, Japan) was used to study the deposition behavior of the GaN particle after the splat tests.

Sixteen BSE images were taken from the center to the edge of the spray position, perpendicular to the spray direction, at a magnification of ×500 (Fig. 2), covering a surface of 3 mm×254 µm. Meanwhile, to investigate the deposition behavior of the deposited particles, higher magnification images (×2500) were also taken at the center of each BSE image (at a magnification of ×500) (Fig. 2). Each BSE image was processed using the open-source software ImageJ. The study mainly focused on the identification of the number of particles being deposited, their location of deposition, and the coverage area on the substrate for each spray condition. Therefore, the deposition behavior of the GaN particles onto a stainless steel substrate was investigated and analyzed quantitatively.

4 Coatings Build-up Characterization

Sixteen passes of GaN powder were cold-sprayed onto the as-received stainless steel substrates to form thick coatings. The influence of the spray conditions (gas temperature and pressure, see Table 2) on the coating build-up were investigated to understand their effect on the coating characteristics and the particle/particle bonding mechanisms.

During the spray tests, the deposition efficiency (DE) was concluded according to the ratio of the mass of the deposited powder to the mass of the sprayed powder (Ref 26).

Before the SEM characterization, the thick coating specimens were embedded in a water-free epoxy resin, ground with SiC sandpaper, polished initially with a 3 µm, and subsequently with a 1 µm diamond polishing compound with colloidal silica suspension before immersion in an ultrasonic bath. This procedure helped produce mirror-polished specimens for SEM analysis without any impurities generated from the polishing step.

Phase analysis of the GaN powder and coatings samples were carried out using X-ray diffraction (XRD-7000, Shimadzu Scientific Instruments, Tokyo, Japan) with CuKα radiation having a wavelength of 1.5406 Å, operated at an acceleration voltage of 40 kV and an emission current of 30 mA, with 2θ from 10-80°, using a step size of 0.02° and a scan speed of 2.4 s/step.

5 Material Coating Characterization for Photocatalytic Properties

The NOx photocatalytic decomposition process offers an effective way for the abatement of NOx gas, particularly under sufficient visible-light irradiation (Ref 33).

As illustrated in Fig. 3, the photocatalytic properties of the GaN powder and coldsprayed GaN coatings on the stainless steel substrates were evaluated using NOx decomposition tests. Before the test, the NO gas concentration was set at 1 ppm, and was prepared by mixing compressed air with NO/N2 (2 ppm). The mixed gas was continuously introduced into the reactor at a flow rate of 200 mL/min. Notably, the powder was poured in a 40×40 mm 2 glass sample holder to ensure the same surface area of the coated specimen. As for the coated specimen all under a deposition area of 40×40 mm 2 in stainless steel substrates, the sample holder was placed on the stage inside the reactor to perform photocatalytic test. Thereafter, the reactor was placed in the dark for 30 min under continuous NO gas flow of 1 ppm concentration, to reach a dynamic adsorption-desorption equilibrium of NO gas. After the pretreatment, the tests were carried out at room temperature (25 °C ) and a 450 W high-pressure mercury (Hg) lamp irradiation for 10 min (Ref 33) in different light wavelengths. During irradiation, filters were selected for controlling the light wavelength (λ) in the photocatalytic reaction, that is, Fuji triacetyl cellulose filter for λ>510 nm, Kenko L41 Super Pro (W) filter for λ>400 nm, and Pyrex glass for λ>290 nm (Ref 34). Besides, the Hg lamp was cooled by a circulating water flow generated using a Pyrex jacket. After each irradiation test under different wavelengths, the sample was placed in the dark for 10 min by blocking the light, to return to the initial NO concentration.

During the photocatalytic reaction, the gas was directly delivered to a NOx analyzer (Yanaco ECL-88A, Kyoto, Japan) to measure the NO concentration. Thus, the deNOx ratio is defined by:

deNO𝑥 𝑟𝑎𝑡𝑖𝑜 = 𝐶 0 -𝐶 𝐶 0 × 100%, ( Eq 1)
where 𝐶 0 and 𝐶 are the initial and actual concentrations of NOx gas, respectively, obtained after the photocatalytic reaction.

Results and Discussion

1 Characterization of the Feedstock Powder

The SEM images of the feedstock powders are displayed in Fig. 4. Nano-sized particles (Fig. 4 (a)), with an average sectional area of 0.026 µm 2 , were naturally agglomerated into micro-sized particles (Fig. 4 (b)). These micro-sized particles possessed an ellipsoidal shape. Based on the results of particle size distribution using the Rosin-Rammler model provided by Nikkiso Co. Ltd., (Tokyo, Japan) (Ref 35), the mean particle diameter was estimated to be 1.97 ± 0.013 µm, which can be considered as the average length of the agglomerated micro-sized particles. From the SEM images, the average width and sectional area were found to be 0.74 ± 0.086 µm and 1.50 ± 0.045 µm 2 , respectively. Owing to their small sizes and high surface areas, they are also strongly affected by the electrostatic surface charge and van der Waals forces, leading to the formation of super agglomerated particles (Fig. 4 (c)) (Ref 29). In particular, the super agglomerated particles exhibited a porous microstructure.

2 First Layer Formation

Splat tests were performed on mirror-polished stainless steel substrates to analyze the particle deformation behavior and its bonding mechanisms with the substrate in absence of mechanical interlocking (Ref 36). Like most ceramics, bulk GaN exhibits a brittle behavior (Ref 37 -38). However, it was demonstrated that when the material size is reduced to the nanoscale, even diamonds can undergo plastic deformation (Ref 39).

Considering the brittle-to-ductile transition occurring at the nanoscale, the yield/fracture strengths show a strong size effect, which increase with the decrease of the material size

(Ref 40).
During splat tests, the original super agglomerated GaN particles, composed of microsized particles, would exhibit "pseudo deformation", which attribute to the disintegration of its inherent component micro-sized particles. Therefore, resultant deagglomeration of the micro-sized and super agglomerated particles due to the breakage of van der Waals bonds (Fig. 4 (a-b)). At the same time, higher extent of the breakage of the deposited particles was attributed to the increasing changes of impact energy by gas temperature and pressure. Therefore, the splat formation is mainly derived from the rearrangement of the interlinked micro-sized particles within the super-agglomerates.

The strategy to quantify the deposition tendency by counting the deposited particles under each impinging position is demonstrated in Fig. 5 particles are deposited on the substrate. However, the surface area of these particles do not appear in the feedstock powder. Therefore, to account for the initial particle feedstock, it can be assumed that the van der Waals bonds and electrostatic forces are weakened by the impact of the particles on the substrate, leading to the deagglomeration of the particles. Afterward, an increase revealed in the nano-sized particles being deposited onto the substrate, while the scattered super-agglomerated particles can be deposited. Therefore, the original micro-sized particles (Fig. 4 (b)) disintegrated into nano-sized particles (at an average sectional area between 0.02 and 10 µm 2 ), while the super agglomerated particles (Fig. 4 (c)) exhibited pseudo-deformation (non-plastic deformation but deagglomeration into new shape) to micro-sized particles (at the average sectional area greater than 10 µm 2 ), which mainly attribute to the disintegration of its inherent component micro-sized particles After the splat tests, the evolution of the substrate coverage area along the half-width of the spray area for different spray conditions is reported in Fig. 6. At the same gas pressure, compared to the slope of the substrate coverage area, the slope of the coverage area increased with an increase in the gas temperature. Similarly, at the same gas temperature, a higher coverage area of the substrate was obtained on increasing the gas pressure. At the outskirts of the spray area, the saturation of the coverage area was observed. Therefore, at the center of the spray position, the deposition of a large number of nano and original micro-sized particles (Fig. 5 (c)) resulted in a larger coverage area on the substrate, in contrast to the few super agglomerated particles deposited at the outskirts of the spray area. As discussed earlier, under the same spray parameters of traverse speed and stand-off distance, the conditions of 400 °C and 0.6 MPa enabled the particles to cover 28% of the coverage area deposited on the substrate, as compared to nearly 0% at a temperature of 25 °C. Therefore, a higher gas pressure and temperature of the spray would induce higher efficiency for deposition.

Along the half-width of the spray area from the center to the edge, the evolution of the coverage area and the number of micro or nano-sized particles were evaluated under each spray condition, as illustrated in Fig. 7. For each figure, the initial slope at 400 °C was smaller than that at the other temperatures. Fig. 7 (c) and (d) display the saturation of the curves corresponding to the number of micro-sized particles getting deposited at the positions of 1500 µm and 2300 µm, respectively. However, the saturation of the curves in Fig. 7 (e) and (f) is not evident. Therefore, the deposited micro-sized particles are localized close to the center of the spray area, while the distribution of the nano-sized particles was more uniform.

At 0.4 MPa, no significant difference was observed between the three temperatures in terms of the normalized coverage area and the number of deposited micro-and nanosized particles. Nevertheless, at a pressure of 0.6 MPa, close to the center of the spray area for specimens sprayed at 25 °C, a higher normalized coverage area of the substrate and number of deposited nano-/micro-sized particles were observed as compared to those at the two other temperatures.

Considering the Pearson product-moment correlation coefficient r obtained by linear least squares regression analysis, when r 2 (coefficient of determination, varies between 0 and 1) is closer to 1, a more linear relationship is obtained between the investigated parameters and spray position (Ref 41). Therefore, when r 2 is close to 1, it represents a more constant deposition of the particles and splat coatings from the center to the edge of the half-width spray area. For the different spray conditions, the coefficient of determination r 2 was determined and reported in Table 3. At 400 °C and 0.6 MPa, the coefficient of determination was found to be relatively higher than that for all the other conditions.

On Fig. 8, for each spray condition, the relationship between the number of deposited particles, their coverage area, and their position along the spray area is plotted. For each condition, the number of deposited nano-sized particles always tends to decrease from the center position to the edge. Meanwhile, under the same spray position, higher temperature or pressure conditions always resulted in deposition of a higher number of nano-sized particles.

The average particle size (cross-sectional area) of disintegrated nano-sized particles, sprayed at 400 °C and 300 °C, decreased distinctly from the center to the edge of the impinging position, meaning that relatively smaller disintegrated nano-sized particles would be preferentially deposited than the bigger particles. Moreover, it is evident that a higher gas pressure and temperature could provide a higher energy for the particles to disintegrate; therefore, the average area size would also be influenced by a smaller size.

For the condition of 400 °C and 0.6 MPa, the evolution of the substrate coverage area exhibited a higher efficiency for the formation of the first layer. In addition, the evolution of the coverage area and the number of deposited particles proved a more constant deposition of the particles from the center to the edge of the spray area.

Therefore, 400 °C and 0.6 MPa appears to be the best spray conditions for deposition of GaN powder on stainless steel substrate.

3 Cross-section Observation of Single Deposited Particle Deformation

Behavior

To investigate the microstructure and deformation behavior of the particles and substrate after the splat tests, a single deposited particle (micro-size) at 400 °C and 0.6 MPa was cut by a (focused ion beam) FIB. The SEM images of the cross-sectional area morphology at a tilt angle of 45° are displayed in Fig. 9 (a). The deformation of the substrate was not observed after the splat tests. In addition, the interface between the substrate and particle did not exhibit any sign of delamination, cracks, or micro-size porosity (Fig. 9 (b)).

Based on the topographical morphologies in Fig. 9 disintegrated into nano-sized particles and then deposited. Therefore, the deformation is entirely localized within the particle of different extent of disintegrated for micro-sized and super agglomerated particles (Ref 42). Hence, after dissipation, for particle pseudodeformation, breakage, and deposition, the energy remaining was insufficient to lead to substrate deformation.

4 Coating Build-up Characterization

Considering the above discussions, 400 °C and 0.6 MPa proved to be the better spray conditions for particle deposition under splat tests. To optimize the properties of the thick coatings (coating build-up), multi-pass tests were carried out for the six spray conditions under 16 passes. The as-received substrates with initial roughness were used to enhance the mechanical interlocking and improve the deposition of the first layer.

Theoretically, multiple layer formation slightly influenced by the original substrate roughness which mainly in the first layer formation, at the same time, the deformation extent of the substrate will be improved with the passes number increase which attributes to energy obtained from bombardment particles successive impact.

SEM images of the coating build-up after 16 passes, for each spray conditions, are presented in Fig. 10. From these images, the thickness was measured using ImageJ software and reported in Fig. 10. At room temperature, no thick coating was obtained independently of the gas pressure used. Thus, the particles could not be completely deposited onto the substrates with low kinetic energy. By increasing the gas temperature and pressure, a thick GaN coating on stainless steel substrate was obtained. In addition, with the increase of the gas temperature or pressure, the average thickness of the coatings also increased. For the obtained coatings, the deposition efficiency was calculated and reported in Fig. 11. Even though it remains low, higher deposition efficiency was reported for the spray condition of 400°C and 0.6 MPa.

Most of the deformed particles seem to be interlocked with their counterparts, creating more stable interfaces. Higher gas temperature or pressure also increases the substrate deformation due to the continuous particles' bombardment on its surface, resulting in an increase of the deposition efficiency. Therefore, upon the impact of the particles, higher substrate roughness is induced, leading to the enhancement of the mechanical interlocking. It also suggests that the compressive stresses induced by the impact help to close the nano-micro pores within the particles resulting in a very dense coating.

Therefore, the particles are well packed and interlocked with each other in cold-spray coatings (Ref 42). According to this and the former results, 400 °C and 0.6 MPa appear to be the better spray conditions to obtain a thick coating on the stainless steel substrate.

5 Evolution of the Coating Crystallite Size

XRD tests were performed before measuring their photocatalytic properties to verify the crystalline structures of the sprayed GaN coatings and powder. The spray conditions, roughness, and thickness for each specimen are listed in Table 4. The different GaN coatings (G1-G4) were cold-sprayed under the same pass number, but different spray conditions (gas temperature and pressure). In addition, it should be noted that G4 corresponds to the specimen sprayed under the best conditions for particle deposition and coating build-up (400 °C and 0.6 MPa).

A comparison of the XRD patterns of the feedstock powders, stainless steel substrate, and cold-sprayed GaN coatings on stainless steel substrates (G1, G2, G3, and G4) is depicted in Fig. 12. The diffraction pattern of the GaN powder exhibited 11 peaks corresponding to the different orientations of the crystalline phase. Using the pdf cards of the hexagonal wurtzite structure GaN (JCPDS, No. 74-0243) (Ref 43), the crystalline orientation related to each peak was identified and presented in Fig. 12.

The XRD diffraction patterns of the GaN coatings on the stainless steel substrates exhibited the same crystalline phase of the hexagonal wurtzite structure of the GaN powder (diffraction peak at the same position). Therefore, the cold-spray process does not lead to anyphase transformation of the GaN powder. The photocatalytic properties of the powder should be retained within the coating.

Compared to the GaN powder, it can be observed from Fig. 12 that the XRD spectra of the coatings fabricated from the LPCS process exhibit broadened peaks. To understand the evolution of the width in the diffraction peak, a simplified integral breadth method, the Williamson-Hall method, is used. This method considers the effect of the straininduced XRD peak broadening and can be used for the calculation of the intrinsic strain along with the particle size (Ref 44).

The physical line broadening of the X-ray diffraction peak occurs because of the size and micro-strain of the nanocrystals, and the total broadening 𝛽 𝑡𝑜𝑡𝑎𝑙 can be expressed by size-induced peak broadening, 𝛽 𝑠𝑖𝑧𝑒 , and strain-induced peak broadening, 𝛽 𝑠𝑡𝑟𝑎𝑖𝑛 , (Ref 45):

𝛽 𝑡𝑜𝑡𝑎𝑙 = 𝛽 𝑠𝑖𝑧𝑒 + 𝛽 𝑠𝑡𝑟𝑎𝑖𝑛 . (Eq 2)
By considering the strain in all the crystallographic directions to be uniform, the material can be considered as isotropic, i.e., its properties are independent of the direction chosen for the measurement. Therefore, 𝛽 𝑠𝑡𝑟𝑎𝑖𝑛 is related to the residual strain ɛ by (Ref 45):

𝛽 𝑠𝑡𝑟𝑎𝑖𝑛 = 4𝜀 tan 𝜃. ( Eq 3)
The origin of Eq 3 may be understood by taking the differential of the Bragg equation with respect to the d-spacing and diffraction angle.

Sample-induced peak broadening is a convolution of the effects due to both the crystallite size and stress. The total sample broadening is usually approximated as the sum of the terms expressed in Eqs 2 and 3:

𝛽 ℎ𝑘𝑙 𝑐𝑜𝑠𝜃 = 𝑘λ 𝐷 + 4𝜀 𝑠𝑖𝑛𝜃. ( Eq 4)
where 𝛽 ℎ𝑘𝑙 is the total broadening due to the strain and size in a particular peak having the value ℎ𝑘𝑙, 𝐷 is the crystallite size, 𝑘 is the shape factor constant equal to 0.94, and λ is the wavelength of the X-ray (0.1542 nm) (Ref 46). Fig. 13 presents the plot of Eq 4 for the GaN coatings and powder, with 4𝜀 𝑠𝑖𝑛𝜃 along the X-axis and 𝛽 ℎ𝑘𝑙 𝑐𝑜𝑠𝜃 along the Y-axis. Only the four main diffraction peaks were represented, (100), (002), (101), and (110). In this case, the other peaks were not included in the analysis because their intensities were not sufficiently strong to allow the meaningful fitting of their peak shapes.

Based on the least-squares linear approximation of these four peaks, the crystallite size and strain were approximated (Ref 47). According to Eq 4, the slope of the linear approximation corresponds to the intrinsic strain ɛ, whereas the intercept allows the calculation of the crystallite size D. For each specimen, based on the values for the strain and crystallite size that reported in Table 5, tensile strains were generated under the GaN coatings after cold spray process (Ref 48). As the average crystallite size for the GaN powder and each coating is almost the same, the magnitude of the strain broadening greatly exceeded the broadening due to crystal size. Therefore, the distortions in the lattice of GaN are the main source of the broadening of the diffraction peaks, which arise from the gradients in the micro strains induced by tensile forces.

Because the broadening of the diffraction peaks due to the changes in the crystallite size is relatively low (Ref 45), the CS process does not involve any microstructural change to the particle when coated.

6 Photocatalytic Properties of Cold-sprayed GaN Coatings

After the identification of the crystalline structures of the prepared samples (as-sprayed

GaN coatings and powder), the photocatalytic activity was evaluated by the decomposition of NOx gas under visible light irradiation in a continuous gas-flow reactor. As listed in Table 4, the roughness and thickness values for each specimen were also measured.

NOx decomposition tests were performed to evaluate the photocatalytic properties of our specimens. As illustrated in Fig. 14, under UV light (10-400 nm), GaN proved to be effective for the photocatalytic degradation of dyes. Among them, the photocatalytic properties of the cold-sprayed GaN coatings were approximately 33%, which was 9% higher than the DeNOx ratio of the GaN powder (approximately 24%).

As discussed earlier, the cold-sprayed GaN coatings exhibited higher photocatalytic properties than the GaN powder. Because the coatings and powder exhibit the same phase (Fig. 11) and crystallite size (see Table 5), it can be assumed that the enhancement of the photocatalytic properties is related to the evolution of the coating surface morphology.

After the LPCS process, no significant change was observed in the crystallite size, and the coatings did not include any impurities. Only a large difference in the strain between the powder and coatings is observed, which is induced by the tensile forces. It has been reported that photocatalysis is a surface reaction; hence, the photocatalytic properties are affected by the surface morphology (Ref 15). Considering the deformation or breakdown of the feedstock powder and the average sectional area of the disintegrated nano-sized particles (Fig. 8), the agglomerated particles tend to break into smaller parts.

The same tendency is observed for multi-pass tests. Because of the severe breakage of the deposited particles, a decrease in the particle size and an increase in the surface area of the deposited particles were observed in accordance with the evolution of the gas temperature and pressure. Therefore, increasing the gas temperature and pressure led to better photocatalytic properties of the GaN coatings compared to those of the GaN powder.

As mentioned, photocatalysis is a surface reaction. Therefore, the DeNOx ratio is also evaluated as a function of coating thickness and roughness (see Fig. 15) (Ref 49). The decomposition NO conversion ratio increases with increasing thickness and roughness of the different samples. In particular, considering that G2 shown a relatively higher decomposition NO conversion ratio with higher roughness but lower thickness than G3.

Thus, a higher roughness of the coatings will be more helpful to increase the photocatalytic activity of the GaN films manufactured by the CS process. The enhancement of the S/V ratio of the coatings will be necessary to improve their photocatalytic properties.

Conclusion

In this study, for the first time, nano-porous and photocatalytic active GaN coatings were fabricated with GaN agglomerated powder using the LPCS system. Based on the splat test results and quantitative analyses under different spray temperatures and pressures, the best spray conditions were determined to be 400 °C and 0.6 MPa. Upon the impact on the substrate, the agglomerated micrometer-sized GaN particles disintegrated, with a large number of the deposited disintegrated particles observed at the nozzle axis of the traversed area. Furthermore, the critical impact energy was measured for particles subjected to the highest gas pressures and temperatures, thereby improving the fragmentation of the agglomerated particles and the deposition efficiency during the process. Meanwhile, no discernible permanent deformation of the substrate was observed according to the cross-sectional observation by FIB.

Multi-pass tests were performed in a stainless steel substrate, and changes in the average thickness were evaluated for different gas temperatures and pressures. From the XRD patterns of the coating, powder, and substrate, no phase transformation of the material was observed before and after spraying. In addition, the Williamson-Hall analysis exhibited no significant changes in the crystallite size after spraying. However, a large increase in the intrinsic strain after cold spray induced a broadening of the XRD peaks.

Finally, the NOx decomposition test was conducted to evaluate the photocatalytic properties of the cold-sprayed GaN coatings. Higher decomposition of NO gas was reported for the cold-sprayed specimen under UV light (approximately 33%) than for the GaN powder, mainly because of the surface area enhancement after cold spraying without any photocatalytic property degradation. Therefore, a higher thickness and roughness would be beneficial for improving the photocatalytic properties of the GaN coatings.

  11). Kida et al. (Ref 12) demonstrated the photocatalytic properties of powdered GaN, allowing him to decompose water into oxygen and dihydrogen when GaN is exposed to light irradiation.

  under the spray conditions of 25 °C and 0.6 MPa. From the BSE images (Figs.5 (a) and (b)), the number of particles deposited was counted, correlated to their cross-sectional area, and reported in the histogram (Fig.5 (c)). The histogram displays the results of the deposited micro-sized particles (with an average sectional area greater than 10 µm 2 ) exhibited in Fig.5(a) (at a magnification of ×500) and the nano-sized particles (with an average sectional area between 0.02-10 µm 2 ) exhibited in Fig.5(b) (at a magnification of ×2500) as a function of the coverage area. Considering the initial particle size distribution and the ellipsoidal shape of the particles, an average length to width ratio equal to 3 can be assumed. The estimated particle area of the feedstock was compared with the area of the deposited particles (Fig.5 (c)). As illustrated in the histogram, a large number of nano-sized

Figure 7 (

 7 a), (c), and (e) correspond to the results obtained with an initial gas pressure of 0.4 MPa, whereas Figure7 (b), (d), and(f) were obtained by analyzing the specimen cold-sprayed at 0.6 MPa. The three investigated gas temperatures are presented.

  (c) of deposited particles derived from Fig. 4 (c) super agglomerated particles, it is notable that the shape of the original super agglomerated particles in the starting feedstock has been transformed from approximate spherical to flat contour, which undergoes non-plastic deformation but deagglomeration into a new shape. It implied that the inherent components, adjacent micro-sized particles disintegrated into nano-sized and then deagglomerated together, result in compressive stress increase, thereby reducing the initial void within the super agglomerated particles. Simultaneously, as shown in Fig. 9 (d) surrounded the super agglomerated deposited particles, the original micro-sized particles in Fig. 4 (b)

Figure Captions:Fig. 1

 1 Figure Captions:

Fig. 2

 2 Fig. 2 Back scattered electron (BSE) observation protocol used to evaluate the relative

Fig. 3

 3 Fig. 3 Schematic diagram of the NOx decomposition test.

Fig. 4

 4 Fig. 4 SEM images of the GaN feedstock: (a) nano-sized particles (average sectional area: 0.026 µm 2 ), (b) original agglomerated micro-sized particles (average length: 1.97 ± 0.013 µm, width: 0.74 ± 0.086 µm and sectional area: 1.50 ± 0.045 µm 2 ), and (c)

Fig. 5

 5 Fig. 5 (a) Deposited micro-sized particles (with average sectional area >10 µm 2 ) at a

Fig. 6 Fig. 7

 67 Fig. 6 Evolution of the substrate coverage area after splat tests along the half-width of

Fig. 8 Fig. 9

 89 Fig. 8 Evolution of the number of nano-sized particles correlated with the coverage area

Fig. 10

 10 Fig. 10 Evolution of the average coating thickness by SEM images under each spray

Fig. 11

 11 Fig. 11 Deposition efficiency of the GaN coatings under the spray condition in two gas

Fig. 12 Fig. 13

 1213 Fig.12XRD spectra of the GaN powder, GaN coatings and stainless steel substrate.

Fig. 14

 14 Fig. 14 Photocatalytic properties of each cold-sprayed specimen and GaN powder

Fig. 15

 15 Fig. 15 Evolution of the deNOx ratio with the (a) thickness and (b) roughness of the

Fig. 1

 1 Fig. 1 Schematic representation of a typical downstream injection cold spray system.

Fig. 2

 2 Fig. 2 Back scattered electron (BSE) observation protocol used to evaluate the relative

Fig. 3

 3 Fig. 3 Schematic diagram of the NOx decomposition test.

Fig. 4

 4 Fig. 4 SEM images of the GaN feedstock: (a) nano-sized particles (average sectional area: 0.026 µm 2 ), (b) original agglomerated micro-sized particles (average length: 1.97 ± 0.013 µm, width: 0.74 ± 0.086 µm and sectional area: 1.50 ± 0.045 µm 2 ), and (c) original super

Fig. 5

 5 Fig. 5 (a) Deposited micro-sized particles (with average sectional area >10 µm 2 ) at a

Fig. 6

 6 Fig. 6 Evolution of the substrate coverage area after splat tests along the half-width of the spray

Fig. 7

 7 Fig. 7 Evolution of the (a)-(b) coverage area, (c)-(d) number of micro-size particles, and (e)-(f)

Fig. 8

 8 Fig. 8 Evolution of the number of nano-sized particles correlated with the coverage area of

Fig. 9

 9 Fig. 9 SEM images of (a) the cross-sectional area of the deposited particles at a tilt angle of 45°,

Fig. 10

 10 Fig. 10 Evolution of the average coating thickness by SEM images under each spray condition

Fig. 11

 11 Fig. 11 Deposition efficiency of the GaN coatings under the spray condition in two gas

Fig. 12

 12 Fig. 12 XRD spectra of the GaN powder, GaN coatings and stainless steel substrate.

Fig. 13

 13 Fig. 13 Evolution of the Williamson-Hall plot for the coatings and GaN powder.

Table 1

 1 Material properties of GaN powder and stainless steel substrates.

		GaN
	Parameter	Density	Melting temperature	Band gap
		(g/cm3)	(K)	(eV)
	Value	0.7	>1600	3.4
		Stainless steel substrate
	Parameter	Roughness, Ra	Roughness, Rz	Hardness
		(µm)	(µm)	(HV)
	As-received	0.10 ± 0.012	0.83 ± 0.076
				216
	Mirror polishing	0.06 ± 0.012	0.39 ± 0.101

Table 2

 2 Cold spray parameters for splat and multi-pass tests.

	Parameter	Values
	Gas type	Compressed air
	Powder feeding rate (g/s)	0.019
	Gas pressure (MPa)	0.4, 0.6
	Gas temperature (°C)	RT, 300, 400
	Traverse speed, TS (mm/s)	20.0
	Stand-off distance, SoD (mm)	20.0
	Pass number	1, 16

Table 3

 3 Evolution of r 2 (coefficient of determination) under each spray condition.

	Conditions	r 2 for substrate	r 2 for number of	r 2 for number of
		coverage area	micro-sized particles	nano-sized particles
	25 °C and 0.4 MPa	0.94	0.87	0.97
	25 °C and 0.6 MPa	0.87	0.76	0.94
	300 °C and 0.4 MPa	0.91	0.83	0.96
	300 °C and 0.6 MPa	0.92	0.84	0.99
	400 °C and 0.4 MPa	0.95	0.86	0.99
	400 °C and 0.6 MPa	0.96	0.89	0.99

Table 4

 4 Roughness and thickness of each coating under different spray conditions.

		Feedstock	Spray	Spray	Spray pass	Roughness,	Thickness
		type	temperature	pressure	number	Ra (µm)	(µm)
	G1	GaN	300 °C	0.4 MPa	16	2.15	6.9
	G2	GaN	300 °C	0.6 MPa	16	4.16	7.8
	G3	GaN	400 °C	0.4 MPa	16	2.82	9.4
	G4	GaN	400 °C	0.6 MPa	16	5.47	10.4

Table 5

 5 Evolution of the strain and average crystallite size of each specimen.

		G1	G2	G3	G4	GaN powder
	Strain (×10 -3 )	3.0	1.7	1.5	2.8	0.2
	Crystallite size D (nm)	22.7	18.2	18.8	18.3	19.9

Acknowledgements

This study was partly supported by the JSPS KAKENHI Grant-in-Aid for Scientific Research (A) 17H01235.