Marie Mahé

Peter Reynders
email: peter.reynders@merck.de

Alain Demourgues

Jean-Marc Heintz

Marie Mahe´and

Influence of a Ceramic Substrate on Aqueous Precipitation and Structural Evolution of Alumina Nano-Crystalline Coatings

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Either boehmite (c-AlOOH) or gibbsite (c-Al(OH) 3 ) nanocrystalline thin films (h% 100 nm) can be precipitated from AlCl 3 solution at fixed pH and temperature onto different substrates. It depends on the nature of the substrate (mica flakes, SiO 2 flakes, or a-Al 2 O 3 flakes), on their crystallographic properties (crystalline or amorphous), and on some experimental parameters (agitation rate, addition rate). According to the surface charge of the substrates, different alumina species are involved in the precipitation process. When negative charges are present on the substrate, the [Al 3 O(OH) 3 (OH 2 ) 9 ] 41 polycation is promoted, leading to the formation of the (Al 4 ) tetramer ([Al 4 O (OH) 10 (OH 2 ) 5 ] o ) and then to the precipitation of bohemite. When positive charges are present, a ligand bridge containing complex ([Al 3 O(OH) 3 (O 2 H 3 ) 3 (OH 2 ) 9 ] 1 ) is likely favored, giving rise to hexagonal ring structures or amorphous solids that lead to the formation of gibbsite. Besides the surface effects, crystalline substrates can act as a template during precipitation of aluminum species as shown for the formation of gibbsite on muscovite. Finally, calcination at 8501C of boehmite samples leads to porous c-Al 2 O 3 layers, while calcination of gibbsite leads to d-Al 2 O 3 layers.

I. Introduction

M ANY studies have been devoted to alumina because of its importance in industrial applications. Transition alu mina and a Al 2 O 3 are notably used as catalytic carriers. [1][START_REF] Euzen | Alumina[END_REF] Alu mina is also found in coatings and cosmetics as additive and substrate. [START_REF] Teaney | New Effect Pigments Using Innovative Substrates. Silica and Alumina Flakes Extend the Product Variety and Performance Range of Pearlescent Pigments[END_REF][4][5][START_REF] Pfaff | Angle-Dependent Optical Effects Deriving from Submicron Structures of Films and Pigments[END_REF] The nomenclature and the relative stability of the different ''alumina'' species is now quite well understood. 1,[START_REF] Euzen | Alumina[END_REF][START_REF] Stumpf | Thermal Transformations of Aluminas and Alumina Hydrates[END_REF][START_REF] Deboer | The Dehydration of Alumina Hydrates.II[END_REF][START_REF] Tertian | Transformations Thermiques et Hydrothermiques de l ´Alumine[END_REF][START_REF] Rouquerol | Thermal Decomposition of Gibbsite Under Low Pressures[END_REF][START_REF] Wilson | A Kinetic Study of the System g-AlOOH/Al 2 O 3[END_REF][START_REF] Wefers | Oxides and Hydroxides of Aluminum[END_REF][START_REF] Levin | Metastable Alumina Polymorphs: Crystal Structures and Transitions Sequences[END_REF] Aluminum compounds can be divided into four classes. Gibbsite (g Al(OH) 3 ) and bayerite (a Al(OH) 3 ) belong to the aluminum trihydroxide class. The aluminum oxide hydroxides compose a second class including the boehmite (g AlOOH) and the diaspore (a AlOOH). Transition alumina, including the g , Z , d , y , k , w, and r alumina, form the third class. They are ob tained from the hydrated species through thermal decompos ition. The thermal evolution of these metastable transition alumina leads to the formation of the thermodynamically stable phase of aluminum oxide: corundum a Al 2 O 3 . The base of alu mina polymorphism has been established for a long time, [START_REF] Stumpf | Thermal Transformations of Aluminas and Alumina Hydrates[END_REF] but some additional phases could still be identified like the y 0 , y, 00 and l phases. [START_REF] Levin | Metastable Alumina Polymorphs: Crystal Structures and Transitions Sequences[END_REF] In contrast to these papers, the present study deals with alu mina as layers on ceramic substrates and will address how the chemical composition and the morphology of ''alumina'' species are influenced by the presence and chemical nature of the sub strate. This question is important for the controlled and repro ducible formation of substrate based ''alumina'' layers, which are used in a number of industrial applications such as catalyst supports [1][START_REF] Euzen | Alumina[END_REF] and optical effect pigments. [START_REF] Pfaff | Angle-Dependent Optical Effects Deriving from Submicron Structures of Films and Pigments[END_REF] The latter, which are also called nacreous pigments, have been manufactured to imi tate the fascinating optical effect of natural pearls. [START_REF] Pfaff | Angle-Dependent Optical Effects Deriving from Submicron Structures of Films and Pigments[END_REF][START_REF] Maile | Effect Pigments Past, Present, and Future[END_REF][START_REF] Glausch | Special Effect Pigments[END_REF] They are composed of submicron layers deposited on a ceramic sub strate. Many products have already been developed for func tional purposes, e.g., security printing and heat management via IR reflection, and decorative purposes, e.g., cosmetics and auto motive paints. Alumina is of two fold interest: first, as a dense layer with a refractive index in a medium range to design optical multiplayer pigments, and second as a porous layer to fix func tional additives upon.

Alumina species are precipitated onto different flaky sub strates, all of which are commercially relevant for optical effect pigments. [START_REF] Maile | Effect Pigments Past, Present, and Future[END_REF] The phases obtained are compared with each other and to the powders generated without any substrates. The in fluence of the substrate on the precipitation of hydrated alumina is examined and a mechanism of precipitation is proposed for the different cases. The structural evolution of precipitated alu mina through heat treatment is also examined in order to design dense, respectively, porous layers as required by the different pigment applications.

II. Experimental Procedures (1) Powder Preparation

The alumina layers were deposited onto different substrates via an aqueous liquid deposition process (LDP). It corresponds to the controlled precipitation of aluminum chloride within an aqueous suspension containing the substrates. The pH value was maintained at 6.5 by addition of an aqueous sodium hydroxide solution (16%, Merck, Darmstadt, Germany). The temperature was maintained at 751C. The influence of three parameters was studied: the solution addition rate (SAR; mL/min), the agitation rate (AR rpm), and the volume of the suspension containing the substrate (VSS L). SAR corresponds to the addition rate of the AlCl 3 solution in the reactor and was chosen equal to 3 or 6 mL/min. AR corresponds to the speed of the mechanical stir rer used to homogenize the suspension. In the following experi ments, AR is equal to 1000 or 1300 rpm. Finally, VSS represents the volume of the substrate suspension (10 wt%) in the reactor at the beginning of the reaction. VSS is equal to 1 or 2L. The thickness of the layers can be controlled by varying the amount of AlCl 3 solution used in the LDP. The precipitates were filtered off, washed completely with deionized water, and dried at 1101C for 12 h. For the present study, a 9 wt% AlCl 3 solution was added until the theoretical (''Al 2 O 3 ''/pigment) ratio reached 45 wt%. The AlCl 3 solution was prepared by dissolving the required amount of AlCl 3 , 6H 2 O (Merck, purity 497%) in deionized water. Using the same conditions, alumina powders were precipitated without any substrate. All dried pigments were then calcined at 8501C under air for 30 min. All performed ex periments are listed in Tables I, II, and III.

(2) Substrates Three different substrates were used in this study: mica flakes (Merck, diameter 10 50 mm, mean thickness 400 nm), SiO 2 flakes (Merck, diameter 10 50 mm, thickness 450710 nm), and Al 2 O 3 flakes (Merck, diameter 10 50 mm, thickness 200710 nm). The mica used is the dioctahedral muscovite KAl 2 [AlSi 3 O 10 ](OH) 2 . [START_REF] Hildenbrand | Untersuchungen an du¨nnen Titandioxid-und Eisenoxidschichten auf verschiedenen Substraten (Investigation of Thin Titanium Dioxide Films and Iron Oxide films Deposited on Different Substrates)[END_REF] The structure of mica muscovite is based on a three layer system that can cleave easily along the (001) plane. [START_REF] Hildenbrand | Untersuchungen an du¨nnen Titandioxid-und Eisenoxidschichten auf verschiedenen Substraten (Investigation of Thin Titanium Dioxide Films and Iron Oxide films Deposited on Different Substrates)[END_REF][START_REF] Kuwahara | Muscovite Surface Structure Imaged by Fluid Contact Mode AFM[END_REF][START_REF] Jolivet | Metal Oxide Chemistry and Synthesis From Solution to Solid State[END_REF] These planes are composed of arrangements of hexagonal rings of SiO 4 tetrahedra. The surfaces of the muscovite on which aluminum is precipitated are actually oriented according to these (001) planes. The synthetic SiO 2 flakes that were used are produced by a web coating process. [START_REF] Pfaff | Angle-Dependent Optical Effects Deriving from Submicron Structures of Films and Pigments[END_REF][START_REF] Maile | Effect Pigments Past, Present, and Future[END_REF] These substrates are amorphous and present quite smooth surfaces as observed from high resolution SEM analyses. The main component of Al 2 O 3 flakes is a Al 2 O 3 (99.1 wt%). A small amount of TiO 2 is also present (0.5wt.%) as an amorphous phase. The surface of the flakes is preferentially orientated according to the (001) plane of a Al 2 O 3 corundum.

(3) Powder Characterization X ray diffraction (XRD) patterns were taken from a y/y dif fractometer (D5000, Bru¨ker AXS GmbH, Karlsruhe, Ger many). The radiation used was CuKa. The detector was moved with a step size of 0.0201 in 2y and a step time of 8 s. Diffractograms were typically recorded in an angular range be tween 51 and 801 in 2y. The morphology of the particles was studied by scanning electron microscopy (SEM; Leo 1530 Gem ini, Carl Zeiss NTS GmbH, Oberkochen, Germany). SEM was also used to measure the layer thickness on micrographs of frac tured pigments. The pigments were mixed with a lacquer and dried. The lacquer, containing the pigments, was then broken in such a way that some of the particles were also broken, which allowed the thickness of the coated film to be measured. Light microscopy micrographs of the samples were taken using an optical microscope (Eclipse ME 600, Nikon GmbH, Du¨sseldorf, Germany) under reflection conditions.

III. Results

(

1) Aluminum Hydroxides and Oxide Hydroxides Precipitated Under Different Conditions

The dried pigments were observed using light microscopy under reflection (Fig. 1). The pigments appeared transparent, which shows macroscopically the good quality of these pigments and the homogeneity of the layers. Tables I, II, and III give the crystallographic nature of the precipitated aluminum compounds as a function of the exper imental conditions. It shows that boehmite (g AlOOH) was al ways synthesized when there was no substrate in the suspension. Boehmite was also systematically precipitated onto SiO 2 flakes. Gibbsite (g Al(OH) 3 ) was generally precipitated on mica. Never theless, it was also possible to obtain boehmite on mica by choosing the right parameters (Mica2). Tables I andII II andIII). Figure 2 shows the X ray diffractogram of the material resulting from the pre cipitation of boehmite (g AlOOH) onto SiO 2 flakes. The broad ness of the peaks, corresponding to the boehmite structure, clearly indicates the low crystallinity of this phase. Figure 3 presents the X ray diffractogram corresponding to the phase precipitated onto mica. The analysis of this X ray diffractogram is more difficult than the previous one due to the high crystal linity of mica. However, g Al(OH) 3 (gibbsite) has been identi fied. The same observations can be made for X ray diffractograms of products precipitated onto a Al 2 O 3 flakes.

SEM cross sections of Mica1 and Mica2 experiments are presented in Fig. 4. As can be seen on these micrographs, the alumina layers are precipitated quite regularly onto the mica flakes. The thickness of the dried film obtained for Mica1 pig ment (h Mica1 % 100 nm) appears lower than that observed for Mica2 pigment (h Mica2 % 250 nm). However, the thickness value of the Mica2 layer should be considered carefully. It could be related to its porous microstructure that could be filled by the lacquer used for the cross section micrograph. The morphology of the dried layers, obtained from SEM images, is presented in Figs. 5 7. Boehmite layers exhibit a very porous and fibrillar microstructure. Those characteristics may also explain the high er thickness observed for Mica2 pigment compared with Mica1. Gibbsite layers are more compact and constituted of platelets like grains with a hexagonal shape. Concerning precursors made of gibbsite boehmite/a Al 2 O 3 flakes, SEM observations show that the two precipitated phases were really separated: boehmite (g AlOOH) gives a porous layer, while gibbsite (g Al(OH) 3 ) leads to platelets like grains (Fig. 7). Figure 8 (2) Thermal Evolution of the Hydrated Alumina Considering the calcination process has been used, the tem perature (8501C) is not high enough to enable the formation of a alumina. So, transition alumina are obtained. XRD patterns show that cubic g Al 2 O 3 is formed when the initial precipitated layer is boehmite (g AlOOH). The example of SiO 2 1 pigment is given in Fig. 9. The same porous morphology than that of the dried layer is obtained as illustrated by the SEM images of the top surface of SiO 2 1 pigment (Fig. 10). When the precursor pig ment contains a gibbsite (g Al(OH) 3 ) layer, the calcination pro cess leads to the formation of the tetragonal d alumina. This is illustrated in Fig. 11, which presents the XRD pattern of calc inated Mica1 pigments. In that case, the calcinated alumina lay ers (d Al 2 O 3 ) are highly cracked as can be seen on the SEM images of the top surface of Mica1 pigment (Fig. 12).

IV. Discussion

(

1) Precipitation of Aluminum Cations without Substrate

The chemistry of Al(III) is known to be complex. Depending on the processing techniques used, different crystal structures and morphologies are generated. The kinetic and/or thermodynamic factors, as well as the solvent, are likely to affect the behavior of complexes in solution and the reaction mechanism. Different precipitation processing techniques are known [START_REF] Euzen | Alumina[END_REF][START_REF] Jolivet | Metal Oxide Chemistry and Synthesis From Solution to Solid State[END_REF][START_REF] Henry | Aqueous Chemistry of Metal Cations: Hydrolysis, Condensation, and Complexation[END_REF] : hydroxyla tion via the addition of a base at room temperature, hydroxy lation via addition of an aluminum salt in a base with controlled pH, acidification of a basic aluminate solution or thermohydro lysis by heating of an acid solution of Al(III) ions to about 801 1001C. In our case, an aluminum chloride solution is slowly added to an aqueous media maintained at pH 5 6.5 and T 5 751C, which does not correspond exactly to the above pro cesses. Now, according to the literature, [START_REF] Euzen | Alumina[END_REF][START_REF] Jolivet | Metal Oxide Chemistry and Synthesis From Solution to Solid State[END_REF][START_REF] Henry | Aqueous Chemistry of Metal Cations: Hydrolysis, Condensation, and Complexation[END_REF] Al 31 may form different types of complexes in a solution that lead to the for 71 . Then, addition of a base to a solution containing Al 13 leads to the formation of a translucent amorphous gel. This unstable gel gives either alu minum hydroxide (Al(OH) 3 ) when the pH is low and the tem perature is between 251 and 801C or boehmite (g AlOOH) when the temperature is higher than 801C. 1,[START_REF] Euzen | Alumina[END_REF] In a less acidic medium and with an increase of temperature, the compact Al 3 complex may evolve toward the formation of a aluminum tetramer, [Al 4 O(OH) 10 (OH 2 ) 5 ] o (Al 4 ). These Al 4 tetramers condense to form boehmite (g AlOOH) as Al 4 is a good structural model for the oxyhydroxide nuclei. [START_REF] Euzen | Alumina[END_REF] Considering that both polycations can exist under the present experimental conditions (pH 5 6.5 and T 5 751C), the most likely species could be determined by com paring their basicity character or their electronegativity. Based on the model of partial charges, [START_REF] Jolivet | Metal Oxide Chemistry and Synthesis From Solution to Solid State[END_REF] average electronegativity has been calculated. w(Al 13 ) 5 2.6 and w(Al 4 ) 5 2.48. The difference is not large but it confirms the more acidic character of the Al 13 polycation. As the pH of the suspension was maintained at 6.5, Al 4 is certainly more present under these experimental conditions, leading to the formation of g AlOOH nanoparticles. Now, even if Al 13 develops within the solution, it leads to the formation of amorphous particles that can grow and crystallize on the surrounding particles, i.e., boehmite. As a matter of fact, boehmite was obtained in all our experiments (Tables I III).

(2) Influence of the Substrates (a) Physicochemical Properties of the Oxide Surfaces:

The chemical properties of the flake surfaces have to be taken into account in order to explain why gibbsite could be precipi tated in some cases, depending on the substrates present in the reactor. When an oxide is dispersed at a given pH in water, the nature of the surface species can be estimated from the value of the point of zero charge (PZC). For pH4PZC, O groups are present, conferring a basic character to the surface, while at pH PZC, no given charge is formally present. The PZC values are quite different for the oxides constituting the substrates. In the literature, the PZC of SiO 2 is about two to four. The PZC of a Al 2 O 3 particles is around eight, [START_REF] Jolivet | Metal Oxide Chemistry and Synthesis From Solution to Solid State[END_REF] but recent studies also showed that the (001) basal plane of a Al 2 O 3 has lower PZC than alu mina particles. The PZC of the basal plane, which constitutes most of the surface area of a Al 2 O 3 platelets, is actually around five. [START_REF] Franks | The Isoelectric Points of Sapphire Crystals and Alpha-Alumina Powder[END_REF][START_REF] Kershner | Zeta Potential Orientation Dependence of Sapphire Substrates[END_REF][START_REF] Kosmulski | pH-Dependent Surface Charging and Points of Zero Charge III. Update[END_REF] Considering that in our experiments, the precipitation reactions are carried out at pH 5 6.5, it means that the surface charge of the SiO 2 flakes is negative. The surface charge of the Al 2 O 3 flakes is certainly slightly negative on the basal plane but slightly positive on the side of the Al 2 O 3 flakes. [START_REF] Kershner | Zeta Potential Orientation Dependence of Sapphire Substrates[END_REF] Finally, the surface charge of the mica is negative on the larger plane (SiO 2 ) but could be slightly positive on the side of the substrates due to the presence of Al 2 O 3 . [START_REF] Reynders | Titanium Dioxide: Precipitation and Crystal Growth[END_REF] As a first approach, the double layer model can be used to understand the possible influence of the surface charge on the precipitated products. Counter ions can not diffuse into the first layer (Stern layer). But other species can be attracted to the surface and are able to penetrate the Stern layer and bind chemically on surface sites. This is the case for the aluminum trimer species (Al 3 ), mentioned previously. In order to take into account the role of surface charges of the different substrates, the chemical pathway going from Al 3 to the precipi tates should be considered in a more detailed way. Under the experimental conditions of the precipitation, the Al 3 trimer undergoes an intramolecular condensation according to the fol lowing reaction (Eq. ( 1) and Fig. 14) [START_REF] Jolivet | Metal Oxide Chemistry and Synthesis From Solution to Solid State[END_REF] :

½Al 3 ðOHÞ 4 ðOH 2 Þ 9 5þ þ H 2 O ! ½Al 3 OðOHÞ 3 ðOH 2 Þ 9 4þ þ H 3 O þ (1)
The latter being in equilibrium with a complex containing [H 3 O 2 ] À ligand bridges (Eq. ( 2)):

½Al 3 OðOHÞ 3 ðOH 2 Þ 9 4þ þ 3H 2 O 2½Al 3 OðOHÞ 3 ðO 2 H 3 Þ 3 ðOH 2 Þ 9 þ þ 3H 3 O þ ð2Þ
The ligand bridge containing complex [Al 3 O(OH) 3 (O 2 H 3 ) 3 (OH 2 ) 9 ] 1 is known to be able to react as a nucleophile ligand (m 3 O bridge) with positive charged species, in order to form the Al 13 polycation [START_REF] Jolivet | Metal Oxide Chemistry and Synthesis From Solution to Solid State[END_REF] (like the [Al(OH ) 6 ] 31 monomer in acidic medium). It means that the surface can play a hydrolyzing role. Depending on its acidity/basicity character, it can affect the precipitation phenomena through the evolution of a reaction (Eq. ( 2)). The second point to be taken into account is the crys talline properties of the substrates that can promote the growth of the initial nucleus according to the underlying lattice.

(b) Adsorption of Aluminum Species on SiO 2 Flakes: The silica flakes show a basic character (negative charge sur face) at pH 5 6.5. Then, the formation of the ligand bridge con taining complex ([Al 3 O(OH) 3 (O 2 H 3 ) 3 (OH 2 ) 9 ] 1 ) is not promoted compared with [Al 3 (OH) 4 (OH 2 ) 9 ] 51 and [Al 3 O(OH) 3 (OH 2 ) 9 ] 41 polycations (Fig. 14). Interactions between silica surface and these aluminum trimers will result in a deprotonation as well as intramolecular condensation (like in Fig. 13).

½Al 3 ðOHÞ 4 ðOH 2 Þ 9 5þ ! ½Al 3 OðOHÞ 3 ðOH 2 Þ 9 4þ ! ½Al 4 OðOHÞ 10 ðOH 2 Þ 5 0
It gives rise on the silica surface to the formation of the smallest stable polycation, i.e., Al 4 tetramer. As Al 4 is the main species governing the condensation of boehmite, it should lead to the formation of this structure onto the silica flakes, which has been observed for all experiments (Tables I III). Moreover, the silica flakes are amorphous. So, they cannot induce any template effect and, in that case, only chemical effects drive the formation of the solid.

(c) Adsorption of Aluminum Species on Al 2 O 3 Flakes: At pH 5 6.5, the surface charge of Al 2 O 3 flakes is slightly nega tive on the basal plane (PZCB5). The slightly negative basal plane of a Al 2 O 3 should promote the formation of the Al 4 tet ramer, which should lead to the formation of boehmite. In fact, the formation of boehmite onto the surface of the Al 2 O 3 plate lets is largely observed in all experiments. Now, it was also possible to precipitate a mixture of boehmite and gibbsite on a Al 2 O 3 platelets. In this case, the AlOOH (boehmite) is largely covering the surface of a Al 2 O 3 platelets, whereas grains of Al(OH) 3 (gibbsite) cover the sides of a Al 2 O 3 platelets (Fig. 8). In fact, depending on their orientations, a Al 2 O 3 planes exhibit different acidic behavior [START_REF] Franks | The Isoelectric Points of Sapphire Crystals and Alpha-Alumina Powder[END_REF][START_REF] Kershner | Zeta Potential Orientation Dependence of Sapphire Substrates[END_REF] and it is likely that the sides of the a Al 2 O 3 platelets exhibit a higher PZC than the basal planes of a Al 2 O 3 platelets insofar as they should become closer to the acidic properties of Al 2 O 3 powders. Then, at pH 5 6.5, the sides of a Al 2 O 3 platelets would present OH [START_REF] Franks | The Isoelectric Points of Sapphire Crystals and Alpha-Alumina Powder[END_REF] external groups. These acid species could directly interact with the ligand bridge con taining complex ([Al 3 O(OH) 3 (O 2 H 3 ) 3 (OH 2 ) 9 ] 1 ). Therefore, equilibrium (Eq. ( 2)) is displaced toward the formation of [Al 3 O(OH) 3 (O 2 H 3 ) 3 (OH 2 ) 9 ] 1 , which react with the sides of the alumina flakes. Aggregation of [Al 3 O(OH) 3 (O 2 H 3 ) 3 (OH 2 ) 9 ] 1 complexes onto the sides of alumina flakes can lead to amorph ous solids (like in the condensation of Al 13 polycation) or can be structured in hexagonal rings that lead to a structure close to the lamellar structure of Al(OH) 3 gibbsite. Now, a few platelet like grains of gibbsite Al(OH) 3 could also be seen on the surface of a Al 2 O 3 platelets (Fig. 8). In fact, at pH 6.5, the basicity of a Al 2 O 3 basal plane (PZCB5) is limited and a few acidic sites can still be present at the surface of a Al 2 O 3 where Al(OH) 3 grains can develop.

Nevertheless, it can be seen from experimental results (Tables II andIII) that two parameters influence the crystallization of the aluminum species on Al 2 O 3 flakes: the SAR and the AR. Changing these parameters allows the formation of gibbsite to be hindered or promoted. When the agitation rate is high (like in Al 2 O 3 4 experiment), a better dispersion of the suspension is ob tained. As the difference between pH of the suspension and PZC of alumina flakes is not so large, it then helps the more acidic sites of the sides to be available for the formation of gibbsite. Conversely, at a low AR, the positive aluminum species present in the media can be preferentially adsorbed on the negative sites of the basal plane. It favors the formation of a boehmite nu cleus, which rapidly covers the a Al 2 O 3 platelets to the detriment of gibbsite. Concerning the SAR, an increase of this parameter leads to the formation of boehmite only. In fact, the increase of SAR leads to an increase of the metal salt concentration and of the sodium hydroxide concentration as a function of time. The ionic force of the suspension is then higher, which allows an in crease of the surface charge of the a Al 2 O 3 platelets. Then, the basicity of the large surfaces of a Al 2 O 3 platelets is enhanced, which can promote the formation of AlOOH on a Al 2 O 3 plate lets. Indeed, precipitation of alumina species onto alumina flakes under these conditions (pH 5 6.5 and T 5 751C) depends on complex chemical equilibrium (condensation of ligand bridge Al 3 against Al 4 ). Then, the nucleation/growth processes are af fected by slight experimental variations, which explains why ei ther the mixed phases or only boehmite are obtained.

(d) Adsorption of Aluminum Species on Mica: In the case of mica, there is a charge balance around the substrates. The larger mica surface appears negative and could favor the condensation of an Al 4 tetramer (like in the case of silica flakes) while the side surfaces appear positive and could preferentially imply [Al 3 O(OH) 3 (O 2 H 3 ) 3 (OH 2 ) 9 ] 1 polycations with a ligand bridge. However, for that substrate where the larger surface is oriented according to (001) planes, the a, b, and g parameters of mica are close to that of gibbsite (Table IV). Therefore, mica can act like a template during the condensation and the crystalliza tion of Al 4 polycations, which favors the growth of the gibbsite phase instead of boehmite. This means that the crystalline struc ture of the mica plays a more important role than its surface charge (toward the formation of Al 4 ) in the crystallization pro cess. Yet, one experiment showed that it was possible to pre cipitate some boehmite on mica (Mica2 sample: Tables I andII). In this case, the effect of the charge surface is more important than the effect of the crystalline structure. For experiment Mica2 (where VSS 5 1), the metal salt concentration, as a function of time, is higher than for other experiments, which corresponds to an increase of the ionic force of the suspension. For mica, the number of charged site on the flake surface is limited by their mutual repulsion. Attenuation of this repulsion by the counter ions present in the solvatation layer results in an increase of the negative surface charge of the substrates. In this case, the surface charge could really compete with the effect of the crystalline structure. Second, the alumina species remain longer in the aqueous solution without being in contact with the substrates. They have more time to transform into boehmite seeds, which are thermodynamically stable in the media. These seeds could then be physisorbed by the substrate afterwards. These experi ments show that the role of the substrates is of first importance to determine the nature of the precipitated phase. Moreover, in the case of mica, there is a competition between the crystalline effect, which promotes gibbsite, and the surface charge effect, which promotes boehmite. The cubic Al 2 O 3 was systematically obtained for all samples containing boehmite. These layers were porous and low crystallized, which corresponds well to other reports in the literature, [1][START_REF] Euzen | Alumina[END_REF][START_REF] Tertian | Transformations Thermiques et Hydrothermiques de l ´Alumine[END_REF][START_REF] Rouquerol | Thermal Decomposition of Gibbsite Under Low Pressures[END_REF][START_REF] Wilson | A Kinetic Study of the System g-AlOOH/Al 2 O 3[END_REF][START_REF] Wefers | Oxides and Hydroxides of Aluminum[END_REF] mentioning that calcination of a non well crys talline ''fibrillar'' boehmite leads to the formation of the cubic g alumina between 4001 and 8901C. One should note that a more crystalline boehmite would have followed another trans formation sequence: boehmitetetragonal g aluminatetragonal d alumina. Then, the tetragonal d alumina would have been the generated phase at 8501C, 1,[START_REF] Wefers | Oxides and Hydroxides of Aluminum[END_REF] which was not the case here. Conversely, thermal treatment of gibbsite layers led to the formation of tetragonal d alumina. According to the litera ture, 1,[START_REF] Euzen | Alumina[END_REF][START_REF] Stumpf | Thermal Transformations of Aluminas and Alumina Hydrates[END_REF][START_REF] Deboer | The Dehydration of Alumina Hydrates.II[END_REF][START_REF] Tertian | Transformations Thermiques et Hydrothermiques de l ´Alumine[END_REF][START_REF] Rouquerol | Thermal Decomposition of Gibbsite Under Low Pressures[END_REF][START_REF] Wilson | A Kinetic Study of the System g-AlOOH/Al 2 O 3[END_REF][START_REF] Wefers | Oxides and Hydroxides of Aluminum[END_REF][START_REF] Ollivier | Crystal Structure of k-Alumina: An X-ray Powder Diffraction, TEM and NMR Study[END_REF] several calcination paths starting from gibbsite exist to yield corundum a alumina. These paths depend on the starting material and the calcination conditions. On the one hand, calcination of gibbsite at atmospheric pressure can lead to the formation of the transition w and k , and then, a Al 2 O 3 . On the other hand, another path at atmospheric pressure is known to involve first the formation of boehmite, which leads to the formation of transition tetragonal g , tetragonal d , monoclinic y , and finally a Al 2 O 3 . The second path occurs only if local hydrothermal conditions depending on the grain size and on the heating rate can be achieved. These hydrothermal conditions enable the transformation of gibbsite into boehmite at the be ginning of the calcination process. [1][START_REF] Euzen | Alumina[END_REF][START_REF] Wefers | Oxides and Hydroxides of Aluminum[END_REF][START_REF] Ollivier | Crystal Structure of k-Alumina: An X-ray Powder Diffraction, TEM and NMR Study[END_REF] Our gibbsite layers are mostly constituted of nanosize gibbsite grains deposited on a rigid substrate, which can then behave locally like an autoclave and enable boehmite formation. [START_REF] Euzen | Alumina[END_REF][START_REF] Rouquerol | Thermal Decomposition of Gibbsite Under Low Pressures[END_REF] This calcination path is sup ported by the fact that no k Al 2 O 3 phase has been identified during heat treatment.

V. Conclusion

The synthesis of alumina precursors on platelets like substrates has been presented. By controlling the different precipitation parameters, it was possible to induce different alumina precur sors depending on the nature of the underlying substrate. Gibb site (g Al(OH) 3 ), boehmite (g AlOOH), or a mixture of gibbsite and boehmite have been synthesized. When calcined at 8501C, the dense pure gibbsite layers lead to a cracked tetragonal d alumina layer. In contrast, calcination at 8501C of the porous boehmite leads to a crack free and porous cubic g Al 2 O 3 . This synthesis method gives rise to the preparation of alumina layers that could be used in pigment material for functional materials.

  point out the importance of the volume of the suspension (VSS) and of the solution addition rate (SAR) on the precipitation process in the presence of mica flakes. Boehmite (g AlOOH) or a mixture of boehmite (g AlOOH) and gibbsite (g Al(OH) 3 ) were synthesized on Al 2 O 3 flakes. The SAR and the agitation rate (AR) param eters could determine the structure of the hydrated alumina de posited on Al 2 O 3 flakes (Tables

  shows a porous AlOOH layer (boehmite) largely covering the surface of a Al 2 O 3 platelets, whereas platelet like grains of Al(OH) 3 (gibbsite) cover the sides of a Al 2 O 3 platelets. Few platelet like grains of Al(OH) 3 could also be seen on the larger surface of the sub strate.

Fig. 1 .

 1 Fig. 1. Light microscopy images of (a) SiO 2 1 and (b) Mica1 particles (reflection stage dark field).

Fig. 2 .

 2 Fig. 2. X ray diffraction (XRD) pattern of SiO 2 1 (dried state). Vertical lines correspond to the XRD pattern of boehmite (g AlOOH).

Fig. 3 .

 3 Fig. 3. X ray diffraction pattern of Mica1 (dried state). Vertical lines correspond to the X ray diffraction pattern of gibbsite (g Al(OH) 3 ). Other peaks are attributed to the mica substrate.

Fig. 4 .

 4 Fig. 4. Scanning electron microscopy micrographs of sections of (a) Mica1 film and (b) Mica2 film (dried state).

Fig. 5 .

 5 Fig. 5. Scanning electron microscopy micrographs of the surface of SiO 2 1 particles (dried state).

Fig. 6 .Fig. 7 .

 67 Fig. 6. Scanning electron microscopy micrographs of the surface of Mica1 particles (dried state).

Fig. 8 .

 8 Fig. 8. Scanning electron microscopy micrograph of an alumina flake on which gibbsite grains develop from the edges.

Fig. 9 .

 9 Fig. 9. X ray diffraction pattern of SiO 2 1 after calcination at 8501C. Vertical lines correspond to the X ray diffraction pattern of the cubic g Al 2 O 3 phase.

Fig. 10 .

 10 Fig. 10. Scanning electron microscopy micrograph of the surface of two SiO 2 1 particles after calcination at 8501C.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. X ray diffraction pattern of Mica1 after calcination at 8501C. Vertical lines correspond to the X ray diffraction pattern of the tetrago nal d Al 2 O 3 phase. Other peaks are attributed to the mica substrate.

Fig. 13 . 19 Fig. 14 .

 131914 Fig.13. Aluminum species involved in the condensation of either alu minum hydroxide (Al(OH) 3 ) or boehmite (g AlOOH), depending on the experimental conditions.[START_REF] Euzen | Alumina[END_REF][START_REF] Jolivet | Metal Oxide Chemistry and Synthesis From Solution to Solid State[END_REF][START_REF] Henry | Aqueous Chemistry of Metal Cations: Hydrolysis, Condensation, and Complexation[END_REF]

Table I .

 I Influence of Volume of the Substrate Suspension (VSS) on the Alumina Species Precipitated (SAR 6 and AR 1000)

		No substrate	SiO 2 platelets	Mica	Al 2 O 3 platelets
		No Sub1	SiO 2 1	M i c a 1	A l 2 O 3 1
	VSS 2 L
	1101C	Boehmite	Boehmite	Gibbsite	Boehmite
	8501C	g-alumina (cub.)	g-alumina (cub.)	d-alumina (tetra.)	g-alumina (cub.)
		No Sub2	SiO 2 2	M i c a 2	A l 2 O 3 2
	VSS 1 L
	1101C	Boehmite	Boehmite	Boehmite	Boehmite
	8501C	g-alumina (cub.)	g-alumina (cub.)	g-alumina (cub.)	g-alumina (cub.)
		No substrate	SiO 2 platelets	Mica	Al 2 O 3 platelets
		No Sub2	SiO 2 2	M i c a 2	A l 2 O 3 2
	SAR 6 mL/min
	1101C	Boehmite	Boehmite	Boehmite	Boehmite
	8501C	g-alumina (cub.)	g-alumina (cub.)	g-alumina (cub.)	g-alumina (cub.)
		No Sub3	SiO 2 3	M i c a 3	A l 2 O 3 3
	SAR 3 mL/min
	1101C	Boehmite	Boehmite	Gibbsite	Boehmite/Gibbsite
	8501C	g-alumina	g-alumina	d-alumina
		(cub.)	(cub.)	(tetra.)
	Table III. Influence of Agitation Rate (AR) on the Alumina Species Precipitated (VSS 2 and SAR 6)
		No substrate	SiO 2 platelets	Mica	Al 2 O 3 platelets
		No Sub1	SiO 2 1	M i c a 1	A l 2 O 3 1
	AR 1000 rpm
	1101C	Boehmite	Boehmite	Gibbsite	Boehmite
	8501C
		No Sub4	SiO 2 4		A l 2 O 3 4
	AR 1300 rpm
	1101C	Boehmite	Boehmite		Boehmite/Gibbsite
	8501C	g-alumina
		(cub.)

g-alumina (cub.) g-alumina (cub.) d-alumina (tetra.) g-alumina (cub.) g-alumina (cub.) g-alumina (cub.)/ d-alumina (tetra.) Table II. Influence of Solution Addition Rate (SAR) on the Alumina Species Precipitated (VSS 1 and AR 1000) g-alumina (cub.)/ d-alumina (tetra.)

  mations of polycations which can then act as nuclei for the Considering acid or neutral solutions, the species that should be considered are: [Al(OH) 3 (OH 2 ) 3 ] o and [Al 3 (OH) 4 (OH 2 ) 9 ] 51 (Al 3 ). The first one, [Al(OH) 3 (OH 2 ) 3 ] o , is involved in the con densation of boehmite (g AlOOH) when the pH is increased but only at room temperature. Therefore, this complex will not be considered as the starting species for the precipitation in the following discussion. The second one, [Al 3 (OH) 4 (OH 2 ) 9 ] 51 , is the only trimer that has been clearly identified in solution.[START_REF] Brown | The Hydrolysis of Metal Ions. Part 8. Aluminium (III)[END_REF] Owing to its conformation, it constitutes an intermediate for the aluminum polycondensation processes. It can yield two different polycations: [Al 13 O 4 (OH) 24 (OH 2 ) 12 ] 71 (Al 13 ) and [Al 4 O(OH) 10 (OH 2 ) 5 ] o (Al 4 ), depending on the experimental conditions. This is schematically illustrated in Fig.13.In acidic medium, Al 3 reacts with [Al(OH 2 ) 6 ] 31 to give the Al 13 polycation, [Al 13 O 4 (OH) 24 (OH 2 ) 12 ]

Table IV .

 IV Elementary Cells Parameters Precipitated Aluminum Species and Substrates

	(3) Structural Evolution of Alumina Nanoparticles
	Synthesized on a Substrate
		a (A ˚)	b( A ˚)	c( A ˚)	g (1)
	c-AlO(OH)
	Orthorhombic
	JCPDS 83-2384 Literature 13	3.6936 12.214 2.861 3.696	2.8679 12.233
	g-Al(OH) 3
	Monoclinic
	JCPDS 01-070-2038 8.68400 5.07800 9.73600 94.54 Literature 13 8.62 5.06 9.7 94
	Calcined mica muscovite
	Monoclinic Literature 16	5.229	9.076	20.322	95.74
	a-Al 2 O 3
	Rhombohedral
	JCPDS 46-1212	4.75870		12.9929

Acknowledgments

We thank Antje Muermann (Merck KGaA) for the XRD analysis, Helmut Plamper (Merck KGaA) for the SEM micrographs, and Katsuhisa Nitta and Atsuko Nishimagi (Merck, Onahama) for the data on alumina flakes.


Quentin Malé
email: male@cerfacs.fr

Olivier Vermorel

Frédéric Ravet

Thierry Poinsot

Direct numerical simulations and models for hot burnt gases jet ignition

Keywords: turbulent jet ignition, pre-chamber ignition, internal combustion engine, flameproof enclosure

come   L'archive ouverte pluridisciplinaire

This work uses multiple three-dimensional Direct Numerical Simulations (DNSs) to i) investigate the ignition process of a cold lean premixed mixture at atmospheric conditions by a jet of hot burnt gases that may be cooled before injection ii) evaluate models able to predict the outcome of such a scenario in terms of ignition. Understanding and being able to model ignition of cold premixed mixtures by hot burnt gases is essential to design systems like engines (to ensure ignition) and flameproof enclosures (to prevent ignition). Limited work has focused on the combined effects of the jet injection speed and temperature on ignition. This is difficult to do by using experiments only and DNS is a natural approach to gain knowledge on that point. By varying the hot jet injection speed and temperature, the three-dimensional, kinetically detailed, DNSs allow a parametric study of the impact of these parameters on the ignition process and provide data to build and test models. Simulations prove that jet injection speed and temperature (usually less than the adiabatic flame temperature because of cooling effects through the injection hole) directly govern ignition.

Introduction

The process of ignition of cold streams by jets of hot burnt gases is critical in many applications including flameproof enclosures [START_REF] Phillips | A Summary of Research into the Basic Principles of Flameproof Enclosure[END_REF][START_REF] Grantham | Electrical safety in hazardous environments[END_REF], reciprocating engines [START_REF] Benajes | Evaluation of the passive pre-chamber ignition concept for future high compression ratio turbocharged spark-ignition engines[END_REF][START_REF] Alvarez | A review of prechamber ignition systems as lean combustion technology for SI engines[END_REF][START_REF] Toulson | A Review of Pre-Chamber Initiated Jet Ignition Combustion Systems[END_REF], wave rotor engines [START_REF] Wijeyakulasuriya | Mixing and Ignition Potential of Transient Confined Turbulent Jet in a Wave Rotor Combustor[END_REF] or pulse detonation engines [START_REF] Lieberman | Detonation Initiation by a Hot Turbulent Jet for use in Pulse Detonation Engines[END_REF]. Since flameproof enclosures may not be able to withstand an internal explosion, holes must be big enough so that pressure does not increase beyond the enclosure limit but small enough so that the exhausted hot burnt gases cannot ignite a surrounding flammable mixture. In Internal Combustion Engines (ICEs), on the contrary, hot burnt gases jets may be used to ignite a flammable mixture: in pre-chamber ignition systems, an auxiliary charge is ignited inside a small pre-chamber linked to the main chamber of the engine by multiple ducts in order to produce hot turbulent jets. In this application, the holes need to be big enough to ensure ignition in the main chamber but small enough to produce turbulent fluctuations which speed up the main charge consumption. In all cases, the ability to predict ignition success or failure of the flamable mixture in which the hot burnt gases jets penetrate is critical for holes design.

Knowing the critical size for flame quenching in a duct [START_REF] Poinsot | Theoretical and Numerical Combustion[END_REF] is not enough to establish the duct size limit that separates ignition success from ignition failure.

It is not necessary that an healthy flame escapes through a hole for ignition of an outer flammable mixture to be realized: the flame can quench in the hole and still push a jet of hot gases which are able to ignite the fresh mixture depending on the temperature, composition and dynamics of the hot jet even if it has stopped reacting [START_REF] Yamaguchi | Ignition and burning process in a divided chamber bomb[END_REF][START_REF] Phillips | On the transmission of an explosion through a gap smaller than the quenching distance[END_REF][START_REF] Wolfhard | The passage of explosions through narrow cylindrical channels[END_REF][START_REF] Biswas | On ignition mechanisms of premixed CH4/air and H2/air using a hot turbulent jet generated by prechamber combustion[END_REF][START_REF] Sadanandan | Detailed investigation of ignition by hot gas jets[END_REF]. Therefore, the true ignition limits are set by the aerothermochemical processes taking place in the outer atmosphere in which hot burnt gases penetrate.

Several studies have been carried out on pre-chamber/main chamber systems to investigate the effect of several parameters (ignition location, hole size, fuel type, equivalence ratio, etc.) on the ignition of a fresh mixture. Yamaguchi et al. [START_REF] Yamaguchi | Ignition and burning process in a divided chamber bomb[END_REF] experimentally studied the effect of hole size, charge stratification and volume ratio of a divided chamber bomb filled with propane/air mixtures. They showed that the ignition pattern was greatly influenced by the hole size and the volume ratio and classified it into four categories, depending on the amount of flame kernel at the nozzle exit: chemical chain ignition, composite ignition, flame kernel ignition and flame front ignition. More recently, Sadanandan et al. [START_REF] Sadanandan | Detailed investigation of ignition by hot gas jets[END_REF] performed experimental investigations using hydrogen/air mixtures to gain information about the spatial and temporal evolution of the ignition process. A combination of mixing reactor model/spectroscopic simulations was used to link the observed OH LIF signals with certain states (extinction, ignition, and combustion). The influence of the hot jet temperature and speed of mixing between the burnt and fresh gases on the ignition process was highlighted: quenching of the flame inside the duct was observed by the absence of significant amount of OH radicals at the nozzle exit. Biswas et al. [START_REF] Biswas | On ignition mechanisms of premixed CH4/air and H2/air using a hot turbulent jet generated by prechamber combustion[END_REF] used an experimental setup to study the effects of pressure, temperature, equivalence ratio along with geometric factors on the ignition mechanisms of hydrogen/air mixtures. They observed ignition even if the flame was quenched passing through the connecting duct. A global Damköhler number was proposed to evaluate the ignition probability. It was constructed using a chemical time scale based on the fresh premixed mixture thermochemical properties and a flow time scale based on the velocity fluctuations properties at the nozzle exit. Results showed that this number contains essential features as it successfully delineated the ignition modes and ignition limits. However, the potential heat losses to the wall of the connecting duct which lower the ignition capacity of the hot jet are a missing key parameter. Mastorakos et al. [START_REF] Mastorakos | Fundamental Aspects of Jet Ignition for Natural Gas Engines[END_REF] studied the pre-chamber combustion, jet injection and subsequent premixed flame initiation for ethylene and methane/air mixtures using an experimental test rig and Large Eddy Simulation (LES). They described several jet ignition phases, including the "outer flame ignition" phase that is of interest for the present work. During this phase, the main ignition sites were spotted at the tip of the transient jet. High velocities and stretch rates inhibit ignition at the sides of the jet. OH* and CH* emissions suggest quenching of the flame inside the duct, with subsequent reignition. Following this work, Allison et al. [START_REF] Allison | Pre-chamber ignition mechanism: Experiments and simulations on turbulent jet flame structure[END_REF] used a similar setup to further investigate fundamental turbulent jet dynamics, with a particular emphasis placed on the effect of fuel type, mixture composition, orifice size, and ignition location. Qin et al. [START_REF] Fei | Detailed numerical simulation of transient mixing and combustion of premixed methane/air mixtures in a pre-chamber/main-chamber system relevant to internal combustion engines[END_REF] investigated full ignition and flame propagation processes in a methane/air mixture using Direct Numerical Simulation (DNS) and detailed chemical kinetics. The effects of the jet on the main chamber have been categorized as chemical, thermal and potential enrichment effect due to mixture stratification. For their configuration which uses a rich mixture in the pre-chamber, the jet hot species OH, CH 2 O, and HO 2 were found to play an important role in the ignition and propagation of the main chamber flame. In a similar way, limiting the DNSs to two dimensions, Benekos et al. [START_REF] Benekos | Prechamber ignition: An exploratory 2-D DNS study of the effects of initial temperature and main chamber composition[END_REF] performed a parametric study to investigate the combustion phenomenology and the jet ignition process under different initial temperatures, main chamber compositions and wall boundary conditions. Several findings are relevant for the present work: the pre-chamber/main chamber interaction begins as soon as combustion develops in the pre-chamber, which generates a transient unburnt jet in the main chamber, which in turn generates strong turbulence in the region close to the outlet of the connecting duct. This first non-reactive jet has an important effect on the subsequent interaction with the hot burnt gases jet later exiting the pre-chamber. Furthermore, they showed that the hot burnt gases jet exits the pre-chamber at a temperature that can be significantly lower than the adiabatic flame temperature of the corresponding mixture due to wall heat losses. In the main chamber, the local flame structure differed strongly from that of a one-dimensional premixed laminar flame.

A large part of the ignition processes does not depend on the details of the pre-chamber design and many authors using simulations decided to study only the jet of hot burnt gases into a premixed charge. Iglesias et al. [START_REF] Iglesias | Numerical analyses of deflagration initiation by a hot jet[END_REF] used axisymmetric Navier-Stokes computations under low-Mach approximation associated with a simple one-step chemistry to investigate hot jet ignition. They proposed a critical Damköhler number whose parametric dependence is investigated. They found that the Reynolds number exerts a limited influence on this critical Damköhler number while the influence of reactant diffusivity is much more pronounced, facilitating ignition. Ghorbani et al. [START_REF] Ghorbani | Ignition by transient hot turbulent jets: An investigation of ignition mechanisms by means of a PDF/REDIM method[END_REF] numerically investigated jet ignition of a hydrogen/air coflow using a Probability Density Function (PDF) method to elucidate the mechanisms leading to ignition and explain the processes governing the ignition delay time and location. It was shown that mixing and chemical kinetics have a strong influence on the ignition process and that a realistic model for ignition has to account for these processes. Validi et al. [START_REF] Validi | Numerical Study of Turbulent Jet Ignition in a Lean Premixed Configuration[END_REF] carried out a DNS of a statistically steady jet of hot gases interacting with a coflow of a hydrogen/air mixture. It was found that turbulence wrinkles and alters the local flame structure which differs significantly from standard turbulent premixed flames. Sidey et al. [START_REF] Sidey | Pre-Chamber Ignition Mechanism: Simulations of Transient Autoignition in a Mixing Layer Between Reactants and Partially-Burnt Products[END_REF] proposed an original approach, similar to the Representative Interactive Flamelet (RIF) method of Peters et al. [START_REF] Peters | Laminar flamelet concepts in turbulent combustion[END_REF][START_REF] Felsch | An extended flamelet model for multiple injections in DI Diesel engines[END_REF][START_REF] Hergart | Modeling the Combustion in a Small-Bore Diesel Engine Using a Method Based on Representative Interactive Flamelets[END_REF], to study the ignition of a methane/air mixture by fully or partially burnt products: they used one-dimensional transient flamelets assuming that ignition occurs in thin layers between hot products and fresh mixture. It was shown that high scalar dissipation rates are able to freeze chemistry which is not fast enough to keep with mixing. When the degree of reaction completion in the hot products stream is decreased, ignition is harder to achieve. Application of the partially burnt hot stream flamelets to pre-chamber jet ignition cases is nevertheless disputable as the pre-chamber acts as a fully burnt gases, potentially cooled, reservoir. The present work uses an extension of this model in Section 3.1. Despite these contributions, the effects of the jet injection speed and temperature on ignition have been rarely discussed independently: most recent parametric studies investigated the effect of variables influencing multiple jet ignition key parameters at the same time. For example in pre-chamber/main chamber systems, decreasing the diameter of the connecting duct increases heat losses from the flowing mixture to the walls but also increases the jet injection speed because of higher overpressures in the pre-chamber. Decoupling and studying independently the effects of the jet injection speed and temperature is difficult to achieve by using experiments. Another approach is to use DNS to directly control these key parameters. Therefore, the present work uses a three-dimensional, kinetically detailed, DNS parametric study to:

-gain knowledge on the physics of ignition by a jet of hot burnt gases; -investigate the effects of the jet injection speed and temperature on flame initiation and development;

-build and test models to predict the outcome of an ignition sequence.

The configuration is a generic case where a ducted jet of hot burnt gases enters a quiescent atmosphere filled with a premixed charge (Fig. 1). For the present study, the injected hot burnt gases have the same equivalence ratio as the atmosphere. This corresponds to a situation where the partially enclosed volume generating burnt gases is previously filled with the mixture of the atmosphere either by piston compression in ICEs or naturally in industrial safety applications.

Most existing jet ignition investigations in similar configurations [START_REF] Feyz | Scalar predictors of premixed gas ignition by a suddenly-starting hot jet[END_REF][START_REF] Fischer | Numerical investigation of the ignition of diethyl ether/air and propane/air mixtures by hot jets[END_REF][START_REF] Ghorbani | Ignition by transient hot turbulent jets: An investigation of ignition mechanisms by means of a PDF/REDIM method[END_REF][START_REF] Iglesias | Numerical analyses of deflagration initiation by a hot jet[END_REF] have used at least one of the two simplifications that follow, which are relaxed here i) the injected jet composition corresponds to hot burnt gases at the initial time and ii) the injected gases are non-cooled: their temperature is the adiabatic flame temperature. Relaxing these simplifications is important for two reasons:

-even if injecting burnt gases right from the start of the DNSs is convenient for computation purposes, it does not correspond to reality and it affects results: combustion in an upstream partially enclosed volume first pushes out fresh gases before that burnt gases reach the inlet of the duct and are pushed out in turn, entrained by the prior flow (Fig. 2). The structure of the jet head is then different from that of a sudden starting jet because of the entrainment effects and the turbulence potentially generated by the fresh gases injection near the nozzle exit [START_REF] Benekos | Prechamber ignition: An exploratory 2-D DNS study of the effects of initial temperature and main chamber composition[END_REF]. The DNSs must therefore take into account the entire injection sequence so that the jet develops as it would in real applications.

-burnt gases lose temperature to the walls when they flow through the injection hole. The temperature of the jet burnt gases T inj is therefore smaller than the adiabatic flame temperature T ad . It will be shown that the difference between T inj and T ad , T drop = T ad -T inj is one of the key parameters controlling the result of an ignition sequence. In piston engines using a pre-chamber for ignition, T drop can reach 200 K [START_REF] Malé | Large Eddy Simulation of Pre-Chamber Ignition in an Internal Combustion Engine[END_REF]. Benekos et al. [START_REF] Benekos | Prechamber ignition: An exploratory 2-D DNS study of the effects of initial temperature and main chamber composition[END_REF] found even larger T drop , up to 585 K in their simplified divided chamber setup. fresh gases are pushed through the duct until 2) burnt gases reach the orifice at a time t = ttr.

Direct Numerical Simulations

Configuration

Numerical Setup

The solver used for this work is an explicit cell-vertex massively-parallel code solving the compressible multi-species Navier-Stokes equations called AVBP [START_REF] Schonfeld | Steady and Unsteady Flow Simulations Using the Hybrid Flow Solver AVBP[END_REF][START_REF] Gicquel | High performance parallel computing of flows in complex geometries[END_REF]. To solve the transport equations, a fully explicit two-step Taylor-Galerkin finite element numerical scheme is used [START_REF] Colin | Development of High-Order Taylor-Galerkin Schemes for LES[END_REF] which offers third order accuracy in space and time on irregular grids. Viscosity follows a power law function of temperature. Thermal diffusivity is computed from the viscosity using a constant Prandtl number: Pr = 0.71. Species diffusion fluxes rely on Hirschfelder and Curtis approximation [START_REF] Hirschfelder | Molecular Theory of Gases and Liquids[END_REF] where diffusion coefficients are computed using constant Schmidt numbers. Table 1 gives the Schmidt and Lewis numbers of the main species used for these DNSs.

Chemistry Description

Propane is used as fuel, air as oxidizer. Description of the chemical kinetics relies on Analytically Reduced Chemistry (ARC) whose application to reacting flows is widely described by Felden [START_REF] Felden | Development of Analytically Reduced Chemistries and Applications in Large Eddy Simulations of Turbulent Combustion[END_REF]. ARC mechanisms allow to accurately describe the chemistry and contain intermediate species involved in the process of ignition by hot burnt gases [START_REF] Fei | Detailed numerical simulation of transient mixing and combustion of premixed methane/air mixtures in a pre-chamber/main-chamber system relevant to internal combustion engines[END_REF][START_REF] Allison | Pre-chamber ignition mechanism: Experiments and simulations on turbulent jet flame structure[END_REF][START_REF] Yamaguchi | Ignition and burning process in a divided chamber bomb[END_REF][START_REF] Benekos | Prechamber ignition: An exploratory 2-D DNS study of the effects of initial temperature and main chamber composition[END_REF], keeping the computational cost at an affordable level for a three-dimensional code [START_REF] Jaravel | Large Eddy Simulation of an industrial gas turbine combustor using reduced chemistry with accurate pollutant prediction[END_REF][START_REF] Rochette | Influence of chemical schemes, numerical method and dynamic turbulent combustion modeling on LES of premixed turbulent flames[END_REF][START_REF] Felden | Impact of direct integration of Analytically Reduced Chemistry in LES of a sooting swirled non-premixed combustor[END_REF][START_REF] Felden | Including real fuel chemistry in LES of turbulent spray combustion[END_REF]. The ARC mechanism is specially constructed for this work using the YARC tool [START_REF] Pepiot-Desjardins | Automatic Strategies to Model Transportation Fuel Surrogates[END_REF]. Reduction starts from the San Diego mechanism [START_REF] Prince | Short chemical-kinetic mechanisms for lowtemperature ignition of propane and ethane[END_REF][START_REF] Saxena | Testing a small detailed chemical-kinetic mechanism for the combustion of hydrogen and carbon monoxide[END_REF][START_REF] Petrova | A small detailed chemical-kinetic mechanism for hydrocarbon combustion[END_REF]. The final ARC mechanism contains 35 species (14 of them in quasi-steady state) and 161 reactions. A full description of the ARC scheme and of its validation is included in Appendix A.

Computational Domain and Resolution

The computational domain consists of a constant pressure atmosphere in which a 4 mm diameter cylindrical jet injects the burnt gases (Fig. 1). Walls are used on all boundaries and the domain is large enough so that pressure remains almost constant during the simulations (the maximum rise in mean pressure is 0.2 Pa). The unstructured tetrahedral meshes comprise a zone of interest with a characteristic cell size of 80 µm and, thanks to the unstructured capacities of the code, much larger cell sizes are used elsewhere. To optimize the computational cost, the zone of interest is adjusted to the jet penetration length of the simulated cases. Depending on the jet injection speed and the simulation end time, different grid configurations M1 to M5 are used (Fig. 5)

whose number of cells n cells is summarized in Table 2.

To check that the flame is correctly resolved using 80 µm cells, a onedimensional premixed laminar flame was computed with this resolution and compared with Cantera [START_REF] Goodwin | Cantera: An Object-oriented Software Toolkit for Chemical Kinetics[END_REF] results (Appendix B.1). Furthermore, a grid independence study on the full three-dimensional DNSs was used to verify that the ignition (Appendix B.2).

Species k C 3 H 8 O 2 H 2 O CO 2 CO H 2 H OH Sc k 1.

Initial Conditions

The atmosphere is initially at rest, filled with a perfectly premixed propane/air lean mixture at an equivalence ratio Φ u = 0.8, a temperature T u = 298 K and a pressure P a = 1 atm. The lean equivalence ratio corresponds to one of our target applications: ICEs equipped with pre-chamber. Atmospheric conditions were selected to make the three-dimensional DNS parametric study computationally feasible. Although this does not cover the entire operating range of ICEs, main ignition mechanisms are not expected to differ from those at ICE relevant conditions.

Boundary Conditions

The walls of the computational domain are maintained at the initial temperature T u . A Navier-Stokes characteristic boundary condition [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous flows[END_REF][START_REF] Poinsot | Theoretical and Numerical Combustion[END_REF][START_REF] Guezennec | Acoustically Nonreflecting and Reflecting Boundary Conditions for Vortcity Injection in Compressible Solvers[END_REF] is used to handle the inlet injection and limit acoustic reflections. No arbitrary turbulence is injected into the jet so that potential turbulent structures develop naturally in the atmosphere. To mimic the ignition sequence produced by the jet issuing from a pre-chamber (Fig. 2), the DNS boundary condition at the jet inlet uses two phases in time:

1) first, fresh gases are injected from the start of injection time t si to the transition time t tr ;

2) then, burnt gases are injected until the end of injection time t ei .

The injection timing is chosen so as to correspond to a typical injection due to combustion in a small partially enclosed volume [START_REF] Malé | Large Eddy Simulation of Pre-Chamber Ignition in an Internal Combustion Engine[END_REF]. Mathematically, this results in several functions to describe the spatial and temporal evolution of the primitive variables at the inlet:

-the radial profile of the inlet axial velocity reads

U (r, t) = f U (t) Û (r), (1)
where r is the duct radial distance from its axis, t is the simulation time, Û (r) is the spatial velocity profile

Û (r) = U inj 1 - r d/2 N , (2)
where U inj is the target injection speed, d is the duct diameter, N is the dampening function coefficient, and f U is the temporal velocity function

f U (t) = 1 2 tanh t -t si K U -tanh t -t ei K U , (3)
where K U is a parameter which controls the transition width;

-the temperature profile reads

T (r, t) = f b (t)T b (r) + (1 -f b (t))T u , (4)
where T b (r) is the spatial burnt gases temperature profile

T b (r) = T u + (T inj -T u ) 1 - r d/2 N , (5)
where T inj is the target hot burnt gases temperature

T inj = T ad -T drop , (6)
and f b (t) is the temporal burnt gases function

f b (t) = 1 2 tanh t -t tr K tr + 1 , (7)
where K tr is a parameter which controls the transition width;

-the species mass fraction profile reads

Y k (r, t) = f b (t)Y k,eq (r) + (1 -f b (t))Y k,u , (8)
where the k subscript refers to the k th species, u subscript refers to the fresh gases and Y k,eq (r) are the species mass fraction at equilibrium at the temperature T b (r).

Figure 3 and 4 show the spatial and temporal profiles resulting from the parameters chosen for this work (Tab. 3). K U is chosen such that the temporal velocity transition width defined as

ep t f U = max(f U ) -min(f U ) max df U dt , (9)
is equal to 40 µs which allows relatively smooth transition from zero velocity to jet injection. K tr is adjusted to produce a spatial burnt gases transition width defined as

ep x f b = max(f b ) -min(f b ) max df b dx , (10)
equal to the laminar flame thermal thickness δ 0 L .

t si [ms] t tr [ms] t ei [ms] K U [-] K tr [-] N [-] 0.05 0.35 1.05 2 • 10 -5 δ 0 L /(2U inj ) 20
Table 3: Parameters used to describe the inlet injection.

duct axis

Variability

When the flow becomes turbulent, several realizations of an ignition sequence may differ. However, the conditions encountered along the jet interface exhibit all possible states for every simulation even when the realizations differ slightly.

The main outcome (ignition or no ignition) then does not change: ignition may not occur at the same point but if it must occur for this regime, it will. This was checked by repeating the same simulation including various perturbations in spatial discretization (e.g. Appendix B) or in injection timing (e.g. Appendix C and Appendix D): the overall result was unaffected.

Results

Ignition Domain

Ignition sequences have been simulated for multiple combinations of inlet injection speed U inj and temperature drop T drop . Corresponding jet Reynolds numbers Re x defined as

Re x = U inj d ν x , (11)
where ν is the kinematic viscosity of the injected gases and the x subscript refers to the nature of the injected gases (burnt, b or unburnt, u), are displayed in Fig. 5 along with ignition results. Ignition sequences are classified as successful if the initiated flame kernels are sufficiently strong to induce self-sustained flames after that burnt gases injection stops and as failed otherwise. Ignition is governed by the competition between the rate of heat production due to chemical reactions of the fresh mixture diffusing into the hot jet and the rate of heat losses due to thermal diffusion. When the jet velocity increases, the diffusion process responsible for the transport of reactants toward the hot burnt gases becomes stronger. However, thermal diffusion also increases. It actually increases faster than molecular diffusion does, for fuels having a Lewis number greater than unity as for propane (Table 1). Hence, chemistry may not be able to compensate for heat losses any more (black crosses in Fig. 5). More intuitively, a too large temperature drop T drop for the injected hot burnt gases weakens chemistry and leads to ignition failure. The T drop maximum value for ignition success decreases with increasing speed until ignition is no longer possible (here, U inj above 200 m/s).

DNS allows to analyse the local flame structure during the ignition sequence: Fig. 6 displays the maximum Heat Release Rate (HRR) reached in the whole domain versus time for multiple cases. This local maximum HRR overshoots the maximum HRR found in a one-dimensional premixed laminar flame max(HRR) 1DL , showing that some additional mixing may take place before combustion actually starts and/or that the flow alters the internal flame structure from its canonical laminar form.

Local HRR close to or higher than max(HRR) 1DL do not necessary imply that global ignition will eventually be reached (see cases [U inj = 200 m/s,

T drop = 100 K], [U inj = 250 m/s, T drop = 0 K] and [U inj = 300 m/s, T drop = 0 K]).
Even if small local ignition spots are formed, the reactive flow structure may be disorganized by high turbulence intensity and not be able to give rise to self-supported propagating flames because of too many heat losses to the surrounding: these zones may be below the critical radius for flame kernel propagation.

Analysis of Ignition Sequences

Low Injection Speed. For low injection speeds, no high-intensity turbulence structure is found (Fig. 7) and the flame elements are "flamelet-like". The burnt gases jet develops in a mushroom shape and detaches from the hole lips when injection stops. For T drop = 0, 100, 200 and 300 K, the rolled-up toroidal structure at the jet head is a favourable place for flame kernel development, driven by large scale engulfment of fresh gases into the toroidal vortex core resulting in large scale mixing of fresh gases and hot burnt gases. At this low speed, it is necessary to increase the injected burnt gas cooling to T drop = 400 K to obtain a failed ignition sequence. The failure is not due to local flame kernels which ignite but are too small to grow: it is due to the fact that chemical reactions are not excited by the cooled jet and ignition does not occur anywhere in the flow. al. [START_REF] Huang | A transition on minimum ignition energy for lean turbulent methane combustion in flamelet and distributed regimes[END_REF][START_REF] Shy | More on Minimum Ignition Energy Transition for Lean Premixed Turbulent Methane Combustion in Flamelet and Distributed Regimes[END_REF][START_REF] Shy | Ignition transition in turbulent premixed combustion[END_REF] that revealed turbulent-distributed flame kernels during the ignition process of a premixed charge by a spark discharge under intense turbulence regime. In this distributed ignition mode, significantly higher ignition energies must be provided to ensure the development of turbulent flame kernels because of increased heat losses due to kernels disruption and turbulence.

0 K 100 K 200 K 300 K 400 K 0 K 100 K 200 K 300 K 0 K 100 K 200 K 0 K 100 K 200 K 0 K 100 K 0 K 100 K U inj =
High Injection Speed. For U inj > 200 m/s, ignition never happens due to high levels of heat losses exerted by the intense turbulence (Fig. 9). Only small reactive kernels appear and they are quickly extinguished by the flow so that global ignition of the atmosphere is never reached.

Chemical Explosive Mode Analysis

Chemical Explosive Mode Analysis (CEMA) is a useful diagnostic tool developed by Lu et al. [START_REF] Lu | Three-dimensional direct numerical simulation of a turbulent lifted hydrogen jet flame in heated coflow: a chemical explosive mode analysis[END_REF] to identify flame and ignition structure in DNS of where y is the vector of the local thermochemical dependent variables including temperature and species mass fractions, ω is the chemical source term, s is the diffusion source term and J ω is the Jacobian matrix of the chemical source terms

J ω = ∂ω ∂y . (13)
A Chemical Explosive Mode (CEM) exists if J ω has an unstable eigenvalue characterized by a positive real part λ e , related to the chemical Jacobian J ω by

λ e = b e • J ω • a e , (14)
where

where the projected chemical (φ ω ) and diffusion (φ s ) source terms are defined as

φ ω ≡ b e • ω, (16)
φ s ≡ b e • s, (17)
and defining the local combustion mode indicator

α = φ s /φ ω . ( 18
)
This ratio compares the relative alignment of diffusion and chemical source terms contributions with the fastest CEM and can be interpreted as follows:

-if α > 1, diffusion dominates chemistry in the direction of the fastest CEM and promotes the ignition of the mixture; -if α < -1, diffusion dominates chemistry in the opposite direction of the fastest CEM and inhibits the ignition of the mixture;

-if |α| ≤ 1 chemistry dominates diffusion in the direction of the fastest CEM and promotes the ignition of the mixture.

In premixed flames, the local combustion mode indicator α discriminates between the preheat zone where α > 1 and the reaction zone where |α| ≤ 1 (Fig. 10).

Models

While the previous DNSs reveal much of the physics controlling hot jet ignition, they are too complicated to be used in practice for any design process.

However, they can be used to build much simpler models as shown now. Modelling the scenario of Fig. 1 requires assumptions on the mechanisms controlling reaction zone pre-heat zone hot products mix at a certain rate and ignite or not. In each of these models, the jet is assumed to act as an infinite source of ignition sites having the same aerothermochemical properties. As a consequence, a single ignition site can be used as a representation of the multiple ignition attempts. This implies that these models can only be used to model jets that have reached a quasi-steady state (as is the case in the DNSs of this work) where ignition no longer depends on the duration of the injection.

Unsteady Flamelet for Ignition Prediction (UFIP) Model

As established by Peters [START_REF] Peters | Laminar flamelet concepts in turbulent combustion[END_REF], if chemistry is fast compared to transport processes, combustion takes place within asymptotically thin layers embedded in the flow. Applying this concept to jet ignition, the boundary of the jet may be seen as a continuous interface strained and convected by the flow [START_REF] Sidey | Pre-Chamber Ignition Mechanism: Simulations of Transient Autoignition in a Mixing Layer Between Reactants and Partially-Burnt Products[END_REF] (Fig. 12). Along this interface, the motion of any fluid line segment may be decomposed into translation, rotation and strain (given by the relative change in length of the line segment). In the reference frame of the fluid line segment, the only effect of motion that will be apparent is that of strain [START_REF] Marzouk | Unsteady Strained Flames: Fundamentals and Numerical Modeling[END_REF]. Translation and rotation may change the mixture that the flame front encounters, but only strain will alter the internal flame structure. These line segments embedded

Hypothesis

Sketch

COR UFIP

The jet mixes at a certain rate while the mixture reacts simultaneously.

Chemistry is fast compared to transport processes such that combustion takes place within thin layers embedded in the flow. open-source software Ember [START_REF] Speth | Ember: a quasi-one-dimensional[END_REF] is used to compute this unsteady flamelet with high efficiency [START_REF] Long | Ember: An open-source, transient solver for 1D reacting flow using large kinetic models, applied to strained extinction[END_REF], assuming an infinite reservoir of burnt gases on the product side of the counterflow flame.

A flame in this configuration can never be completely extinguished because there will always be a region of contact between reactants and products: a threshold of HRR is needed to distinguish between flamelets able to globally ignite the fresh mixture or not. This threshold is arbitrarily set to half of the max(HRR) found in a one-dimensional premixed laminar flame at the same fresh gases conditions. As the temperature drop increases at the burnt gases side of the flamelets, chemistry is weakened and the time to reach this threshold increases (Fig. 13). For a fuel with Lewis number greater than unity, heat diffuses more quickly than mass, reducing the temperature in the reaction zone when strain increases and therefore the HRR decreases monotonically with the strain rate a [START_REF] Petrov | An unsteady strained flame model for turbulent combustion simulations[END_REF].

Hot burnt gases jet ignition exhibits different structural characteristics with increasing turbulence intensity starting from wrinkled flames to a more disorganized combustion pattern where small eddies broke/broaden the reaction zone (Section 2). Results from UFIP can only be compared to DNSs lying in the flamelet regime where the reaction zones are not distributed/broadened and the burnt gases are not highly fragmented in space. This limit the range of application of the UFIP model to the U inj = 50 m/s cases only.

The UFIP model requires a relation between the stretch to impose to the representative flamelet-which reduces to the strain in its zero-curvature planar configuration-and the aerodynamic features of the jet. In a three-dimensional flow field, at each point of an isosurface of progress variable c, the flame stretch is defined as

κ = -nn : ∇u + ∇ • u + S d (∇ • n), (19)
where n is the flame normal pointing toward the fresh gases

n = - ∇c |∇c| , (20)
u is the flow velocity and S d is the flame displacement speed

S d = 1 |∇c| Dc Dt . (21)
The DNSs data can be post processed to evaluate the level of stretch that the interface between burnt and fresh gases undergoes. However, there are several obstacles: first, the flamelets at the interface of the jet are subject to a wide range of local stretches (Fig. 14). Second, the strain rate of the interface changes in time as it is randomly disturbed by the flow. Constructing a single strain rate to apply to a flamelet supposed to represent what is happening at the whole interface of the jet becomes difficult: locally, a thin layer may undergo low strain rates and lead to ignition even if globally, the average strain rate is high.

Moreover, a flame may survive at higher unsteady strain rates than it would at steady strain rates [START_REF] Najm | Numerical and Experimental Investigation of Vortical Flow-Flame Interaction[END_REF]. These difficulties prevent from using the unsteady and heterogeneous flow field of the DNSs to establish a relation between the UFIP strain to be applied and the jet characteristics. Here, to bypass this problem, a simpler approach is used: an equivalent strain rate a e is defined as the steady strain applied to UFIP flamelets which would be equivalent to what happens to the unsteady thin layers featuring large strain variation. a e is supposed to scale as the inverse of a jet flow time scale defined as the ratio between the duct diameter and the injection speed such that

a e = K UFIP U inj d , ( 22
)
where K UFIP is a scaling constant. The UFIP model is in good agreement with the DNSs for K UFIP = 0.06 which results in a e = 750 s -1 for U inj = 50 m/s, close to all of the most probable stretch values encountered in the DNSs for this injection speed (Fig. 14). For this strain rate, UFIP predicts an ignition failure for T drop > 300 K using the arbitrary threshold 0.5max(HRR) 1DL to distinguish between ignition success and failure (Fig. 13).

Convected Open Reactor (COR) Model

The UFIP model is limited to flamelet-like thin reaction fronts, which restricts its range of application to low injection speeds. As an alternative, a more general modelling approach (the COR model) is explored now (Fig. 12).

It consists in following an open reactor-small enough to be homogeneous-, convected into the chamber: hot burnt gases mix at a certain rate with the cold fresh gases within the open reactor and chemistry inside the reactor can be tracked using a simple Ordinary Differential Equation (ODE) solver with complex chemistry. In this approach, the mixture fraction between the hot jet and the cold gases ξ in the open reactor changes according to a mixing rate: it starts at ξ = 1 in the injected burnt gases jet and goes to ξ = 0 as the reactor is convected inside the chamber and mixes with the fresh gases. The details of the convection movement are unimportant once the evolution of ξ is specified.

The balance equations are those describing the evolution of an open system at constant pressure. The mass balance equation reads where m is the reactor mass, ṁin is the mass flow entering the reactor (i.e. that comes from the flammable atmosphere) while ṁout is the mass flow leaving the reactor. The rate of entrainment of the cold mixture into the system θ is defined as

dm dt = ṁin -ṁout , (23)
θ = ṁin m . ( 24
)
The species balance equations for the convected reactor read

dY k dt = θ Y in k -Y k + rT P ωk , (25)
where Y k is the mass fraction of the k th species, T is the temperature, r is the specific gas constant, P is the pressure and ωk is the chemical source term of the k th species. The energy balance equation can be rearranged to describe the evolution of temperature

dT dt = 1 Cp θ k Y in k h in k -h k - rT P k h k ωk , (26)
where Cp and h are the specific heat capacity at constant pressure and the enthalpy per unit mass, respectively. The evolution of the mixture fraction ξ is described by a passive scalar balance equation similar to Eq. 25 without chemical source term

dξ dt = θ ξ in -ξ = -ξθ, (27)
since ξ in = 0 (the fresh gases entering the reactor have a zero mixture fraction).

Thus, using chain's rule, the system evolution can be described as a function of ξ by a set of two equations:

dY k dξ = dY k dt dt dξ = - 1 ξ Y in k -Y k - rT θξP ωk (28)
and

dT dξ = dT dt dt dξ = - 1 ξCp k Y in k h in k -h k + rT ξθCpP k h k ωk . ( 29
)
This model was used to aid the interpretation of experimental results for burnt gases jet ignition [START_REF] Sadanandan | Detailed investigation of ignition by hot gas jets[END_REF], namely the OH LIF signals. A similar concept was also used to examine the evolution of a hot air kernel ejected into a stratified coflow [START_REF] Sforzo | Ignition Probability in a Stratified Turbulent Flow With a Sunken Fire Igniter[END_REF]. Here, the aim is to predict ignition. Balance equations 28 and 29 are integrated thanks to a SciPy [START_REF] Jones | SciPy: Open source scientific tools for Python[END_REF] wrapped ODE solver. Thermodynamic and kinetic parameters needed in each equation are computed using Cantera [START_REF] Goodwin | Cantera: An Object-oriented Software Toolkit for Chemical Kinetics[END_REF].

The weak part of the COR model is that all mixing aspects are hidden in the mixing rate θ (Eq. 24) which becomes the most critical parameter of the model.

The mixing rate θ should be equal to zero in the fresh and in the burnt gases (i.e.

θ(ξ = 1) = θ(ξ = 0) = 0) and should be non-zero in the mixing zones. Since mixing occurs by removal of scalar variance and the scalar dissipation rate of ξ χ ξ represents the rate at which the fluctuation of ξ are destroyed, it is assumed here that the mixing rate θ is linked to the rate at which mixing occurs in the jet and therefore to χ ξ so that a linear relation between the mixing rate θ and the scalar dissipation rate of ξ is used to close the model. To finalize the model for θ, the distribution of χ ξ as a function of ξ in the DNSs (Fig. 15) can be used.

χ ξ = f (ξ) seems to follow a parabolic curve such that θ may be modelled using the function

θ = θ 0 4ξ(1 -ξ), (30)
where θ 0 is the maximum mixing rate supposed to scale as the inverse of a jet flow time scale such that

θ 0 = K COR U inj d , ( 31
)
where K COR is a scaling constant. Figure 16 shows an example of COR model results for T drop = 100 K using various values for the maximum mixing rate θ 0 . As expected, for large θ 0 , chemistry is not fast enough to balance the cooling effect of the entrainment of fresh gases. Temperature quickly falls to the fresh gases temperature: it is an ignition failure. For smaller θ 0 , the rate of combustion heating exceeds the rate of entrainment cooling and temperature rises to the burnt gases temperature: it is an ignition success. The COR model captures the essence of the phenomena at play here: ignition is controlled by the balance between the rate of combustion heating due to the combustion of the entrained fresh fuel and the rate of mixing cooling due to the entrainment of fresh gases. From this, it is possible to compute θ crit 0 , the critical mixing rate which separates ignition success from ignition failure. Using K COR = 0.3 in Eq. 31 the ignition limit established by the COR model is in fairly good agreement with all DNS results (Fig. 17). This suggests that, despite its simplicity, the COR model (which accounts for complex chemistry effects) captures ignition limits reasonably. Figure 18 shows guidelines for a practical use of the COR model.

Conclusion

Multiple DNSs of hot burnt gases jet ignition sequence have been performed, varying the jet injection speed and temperature. Increasing the heat energy supply by increasing the jet injection speed does not necessary favour ignition: higher turbulence generation due to higher jet injection speed may prevent ignition to be reached. This is a very important information for ICEs designers, seeking to increase turbulence to speed up the main charge consumption through flame wrinkling. This work has highlighted a jet injection speed limit beyond which ignition would no longer be possible. This jet injection speed limit decreases when the temperature drop that the burnt gases undergo through the duct increases. This highlights the importance of taking into account heat losses in the small gaps from which jets issue, too often overlooked in jet ignition studies.

For modelling perspectives, this work has brought some fundamental knowledge on hot burnt gases jet ignition. The flame structure was found to be

T [K]

T drop = 100 K equili brium m i x i n g to ξ = 0 as the system fills with fresh gases. The mixing line corresponds to an infinitely fast mixing case while the equilibrium line corresponds to an infinitely fast chemistry case. strongly impacted by the jet injection speed. Under low-intensity and largescale turbulence conditions, turbulence wrinkles the flames but is not able to penetrate their preheat and reaction zones. In this condition, segments of the wrinkled flames can be considered as laminar flamelets. For high-intensity and small-scale turbulence, the entire structure of the reaction zone is substantially distorted by the penetration of small eddies. Thus, the reaction zone is broadened and/or distributed. This results in a combustion mode fundamentally distinct from that of laminar flamelets. This broadened/distributed reaction zone regime-already revealed in ICE turbulent jet ignition [START_REF] Malé | Large Eddy Simulation of Pre-Chamber Ignition in an Internal Combustion Engine[END_REF]-is depicted through visual analysis of the HRR field and supported by the CEMA diagnostic tool.

UFIP limit COR limit

Two models have been evaluated using the DNSs as reference: the UFIP model where ignition takes place within thin layers embedded in the flow and the COR model where the system mixes at a certain rate while the mixture reacts simultaneously. These models are simple one-dimensional and zero-dimensional models but they both integrate a key element which is a precise description of chemistry through complex chemical kinetics. Major drawbacks of the UFIP model are i) the need of a representative steady strain rate and ii) the flamelet regime assumption. The COR model adopts an opposite modeling strategy by assuming perfect mixing and no flamelet structure as the UFIP model does.

The ignition limits given by the COR model are in good agreement with the DNSs showing that it is a good candidate to predict the ignition of a flammable atmosphere by jets of hot burnt gases for which flamelet approaches (such as UFIP) are not adequate as many regimes encountered during these ignition sequences, especially at large injection speeds, do not fall in the flamelet regime.

Future work will integrate this model in ICE turbulent jet ignition modelling to track misfires and optimize the design of this technology. Axial distance is relative to the inflexion point of the temperature profile.

Figure 1 :

 1 Figure 1: Left: hot burnt gases jet injection from an upstream chamber into a downstream chamber through a duct. Right: corresponding DNSs configuration. ξ is the mixture fraction of gases that stem from the hot burnt gases jet.

Figure 2 :

 2 Figure 2: Typical injection sequence for ignition of fresh gases by a jet of hot burnt gases: 1)

Figure 3 :

 3 Figure 3: Spatial inlet profile for axial velocity and temperature.

Figure 4 :

 4 Figure 4: Temporal inlet functions for injection speed (f U ) and burnt gases (f b , steers both temperature and mass fractions).

Figure 5 :

 5 Figure 5: DNS results: ignition map of the studied cases in a U inj -T drop diagram. The mesh configuration used for each case is specified above the markers. Jet Reynolds numbers are given on the right.

UFigure 6 :

 6 Figure 6: Instantaneous and moving average maximum HRR in the domain for various values of inlet injection speed U inj and temperature drop T drop .

Figure 7 :

 7 Figure 7: Planar cut in the jet axis coloured by the HRR with three isolines of temperature. U inj = 50 m/s.

complex flows [ 48 ,Figure 8 :Figure 9 :

 4889 Figure 8: Planar cut in the jet axis coloured by the HRR with three isolines of temperature. U inj = 150 m/s.

  a e and b e are right and left eigenvectors associated with λ e , respectively. A CEM indicates the propensity of a local mixture to ignite in an lossless environment where s ω. If multiple CEMs are present, λ e refers to the largest eigenvalue by recognizing that ignition is controlled by the fastest CEM. Validation and more details on the practical implementation are included in Appendix E. Local combustion mode. Xu et al. [50] introduced a local combustion mode indicator α by projecting Eq. 12 in the direction of the fastest CEM Dφ ω Dt = λ e φ ω + λ e φ s + Db e Dt • ω(y),

Fig. 11 shows

 11 the local combustion mode indicator α for explosive mixture (i.e. λ e > 0) for low, moderate and high injection speed cases without any T drop . For low injection speed, the reaction zone is thin and not much distorted by the flow. The internal structure of the flames is very similar to a one-dimensional premixed laminar flame as shown by the one-dimensional plot through the interface between the fresh gases and the burnt gases. On the other hand, α demonstrates the broadening of the reaction zone for higher injection speeds already depicted in Fig 8. |α| ≤ 1 zones become thickened and/or distributed. For U inj = 50 m/s, projected diffusion source terms mostly act in favour of the fastest CEM as in the preheat zone of a premixed laminar flame where heat and combustion products diffuse back into the unburnt mixture leading to self-supported flames. For higher speeds, α < -1 zones appear where diffusion acts against the fastest CEM. In these zones, ignition is inhibited by the flow. For U inj = 250 m/s (ignition failure), |α| ≤ 1 zones are quickly broken up and α < -1 zones dominate. More generally, as expected for ignition by hot burnt gases, diffusion plays an important role in the ignition process as shown by large |α| > 1 zones.

Figure 10 :

 10 Figure 10: Example of CEMA applied to a one-dimensional premixed laminar flame at 1 atm, unburnt temperature Tu and equivalence ratio Φu. Stared variables are normalized by their maximum absolute value, Θ is the reduced temperature profile such that Θ(T = Tu) = 0 and Θ(T = T ad ) = 1. t = 1.0 ms t = 1.2 ms

Figure 11 :

 11 Figure 11: Planar cut along the jet axis coloured by the local combustion mode indicator α with three isolines of temperature and one isoline of HRR. Upper right plot reveals a CEMA flame structure similar to a one-dimensional premixed laminar flame using a line cut through the three-dimensional flame. Stared variables are normalized by their maximum absolute value, Θ is the reduced temperature. T drop = 0 K.

Figure 12 :

 12 Figure 12: The two models developed from DNS to describe ignition by a hot burnt gases jet.

Figure 13 :Figure 14 :

 1314 Figure 13: Temporal evolution of the maximum HRR in one-dimensional strained unsteady flamelets for multiple strain rates a and temperature drops T drop . Dashed lines do not cross the 0.5max(HRR) 1DL horizontal line. It is assumed that the corresponding flamelets are too weak to lead to global ignition in jet ignition cases.

Figure 15 :

 15 Figure 15: Scatter plot of the scalar dissipation rate of the jet mixture fraction χ ξ versus the corresponding value of jet mixture fraction ξ for the DNSs alongside the θ modelling function (Eq. 30) with K COR = 0.3 (-). T drop = 0 K, t = 0.4 ms.

Figure 16 :

 16 Figure 16: Evolution of the temperature inside multiple COR model open reactors for different maximum mixing rates θ 0 . The trajectories start at ξ = 1 (in the hot burnt gases jet) and go

Figure 17 :

 17 Figure 17: Ignition map of the studied cases (DNS results) together with the UFIP model ignition limit for K UFIP = 0.06 and the COR model ignition limit for K COR = 0.3.

Figure 18 :

 18 Figure 18: Flowchart showing the practical use of the COR model.

Figure A. 19 :

 19 Figure A.19: Temporal evolution of the consumption speed Sc for strained flames at 1 atm, unburnt temperature Tu and equivalence ratio Φu. Original, skeletal and ARC mechanisms.

Figure A. 20 :

 20 Figure A.20: Auto-ignition times τ AI of homogeneous unburnt mixtures at 1 atm and equivalence ratio Φu. Original, skeletal and ARC mechanisms. Associated errors relative to the original mechanism.

Figure A. 21 :

 21 Figure A.21: Ignition times τ I of mixtures of unburnt and burnt gases at 1 atm, unburnt temperature Tu and equivalence ratio Φu. Burnt gases are product of fresh gases equilibrium.Original, skeletal and ARC mechanisms. Associated errors relative to the original mechanism.

Figure A. 22 :

 22 Figure A.22: Times to go from 10 to 90 % temperature increment for mixtures of unburnt and burnt gases at 1 atm, unburnt temperature Tu and equivalence ratio Φu. Burnt gases are product of fresh gases equilibrium. Original, skeletal and ARC mechanisms. Associated errors relative to the original mechanism.

Figure B. 23 :

 23 Figure B.23: One-dimensional premixed laminar flame computed using AVBP and Cantera at 1 atm, unburnt temperature Tu and equivalence ratio Φu. Vertical thin lines indicate the AVBP grid points. Axial distance is relative to the inflexion point of the temperature profile.

Figure B. 24 :

 24 Figure B.24: Instantaneous and moving average maximum HRR alongside volume mean HRR in the domain for U inj = 100 m/s, T drop = 0 K and two different grid cell sizes.

Figure E. 28 :

 28 Figure E.28: Application of CEMA to a one-dimensional premixed laminar flame at 1 atm, unburnt temperature Tu and equivalence ratio Φu. Multiple configurations are shown: skeletal mechanism and analytical generation of the Jacobian matrix (Skeletal, pyJac), skeletal mechanism and finite difference approximation of the Jacobian matrix (Skeletal, finite diff.), ARC mechanism and finite difference approximation of the Jacobian matrix (ARC, finite diff.).

Table 1 :

 1 Schmidt Sc k and Lewis Le k numbers used for some of the species used for the DNS

		22 0.74 0.55 0.94 0.75 0.20 0.12 0.49
	Le k	1.72 1.04 0.77 1.33 1.05 0.29 0.17 0.68

of propane/air flames.

Table 2 :

 2 Number of cells in the different grid configurations.t tr plays a limited role on the ignition results. Appendix D shows that the injection of burnt gases is long enough so that the jet reaches a quasi-steady state where the ignition outcome no longer depends on the duration of the burnt gases injection. Velocity and temperature profiles at the jet injection section are quasi flat profiles to correspond to a short injection tube from a pre-chamber where boundary layers do not have time to develop. The injected burnt gases are equilibrium combustion products from a cold mixture initially at Φ u , T u and P a which is burnt and then cooled by a fixed temperature drop T drop before entering the DNS domain. The chemical composition of the injected burnt and potentially cooled gases is also changed to correspond to equilibrium values at temperature T inj = T ad -T drop .

Acknowledgements

This work was granted access to the high performance computing resources of TGCC ("Très Grand Centre de calcul du Commissariat à l'énergie atomique et aux énergies alternatives") under the allocation number 2019204881 attributed by PRACE (Partnership for Advanced Computing in Europe) and the allocation number A0072B10157 attributed by GENCI ("Grand Équipement National de Calcul Intensif"). Quentin Malé is the recipient of the CIFRE ("Conventions Industrielles de Formation par la Recherche") Ph.D. research fellowship number 2017/0295 attributed by ANRT ("Association Nationale de la Recherche et de la Technologie").

Declaration of Interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Analytical Reduced Chemistry Mechanism for Propane Air Flames

A set of canonical zero-or one-dimensional configurations is used to steer the reduction process towards an accurate ARC mechanism. Here, the configurations include one-dimensional premixed laminar flames and zero-dimensional auto-igniting homogeneous reactors at the thermodynamic conditions of this work. Starting from the detailed San Diego mechanism [START_REF] Prince | Short chemical-kinetic mechanisms for lowtemperature ignition of propane and ethane[END_REF][START_REF] Saxena | Testing a small detailed chemical-kinetic mechanism for the combustion of hydrogen and carbon monoxide[END_REF][START_REF] Petrova | A small detailed chemical-kinetic mechanism for hydrocarbon combustion[END_REF], a skeletal reduction is performed first: unimportant species and reactions are removed from the detailed mechanism using the Directed Relation Graph with Error Propagation (DRGEP) method [START_REF] Pepiot-Desjardins | An efficient error-propagation-based reduction method for large chemical kinetic mechanisms[END_REF]. The resulting skeletal mechanism is comprised of 35 species and 161 reactions. Then, 14 species are identified as being Quasi Steady State (QSS) species by the Level Of Importance (LOI) criterion [START_REF] Løvås | Automatic reduction procedure for chemical mechanisms applied to premixed methane/air flames[END_REF]. The resulting ARC mechanism retains 21 transported species and 14 QSS species.

For the conditions of this work (pressure P a = 1 atm, unburnt temperature T u = 298 K and equivalence ratio Φ u = 0.8), the relative error on the laminar Appendix E. Chemical Explosive Mode Analysis Numerical Implementation CEMA requires the computation of the Jacobian of the local chemical source terms J ω (Eq. [START_REF] Sadanandan | Detailed investigation of ignition by hot gas jets[END_REF]. While the open-source software pyJac [64] offers optimized analytical generation of Jacobian matrices for classical chemical mechanisms [START_REF] Niemeyer | pyJac: Analytical Jacobian generator for chemical kinetics[END_REF], it is not directly applicable to ARC mechanisms where only transported species are included in the governing equations of the reacting flow (Eq. 12).

To bypass this obstacle, first order finite differences are used in this work to approximate the Jacobian matrix of the local ARC mechanism chemical source terms. Eigenvalues and eigenvectors are computed using LAPACK [START_REF] Anderson | LAPACK Users' Guide[END_REF].

To validate the method, CEMA is applied to a one-dimensional premixed laminar flame using both pyJac and finite differences for the skeletal mechanism and only finite differences for the ARC mechanism (Fig. λ e continues to grow in the reaction zone where φ ω φ s before disappearing, separating pre-and post-ignition zones [START_REF] Lu | Three-dimensional direct numerical simulation of a turbulent lifted hydrogen jet flame in heated coflow: a chemical explosive mode analysis[END_REF].