Adrian Villalba Weinberg

Dominique Goeuriot

Jacques Poirier
email: jacques.poirier@univ-orleans.fr

Cyrille Varona

Xavier Chaucherie

Mullite-zirconia composite for the bonding phase of refractory bricks in hazardous waste incineration rotary kiln

Keywords: mullite/zirconia, reaction sintering, phosphate, cyanite, microstructure

Although mullite-zirconia composites made from zircon, alumina, and andalusite meet the requirements for many refractory applications, little effort has been made to transfer these composites to the bonding phase (the 'matrix') of refractory bricks. In this paper, we investigate how this could be achieved through better control of secondary oxides. The high temperature phases were simulated with thermodynamic software and linked to the microstructures, mineralogy, and properties of the composites. The results revealed that the system is very sensitive to Na2O, which harmed the microstructure considerably. By contrast, TiO2 and P2O5 additions proved beneficial, allowing complete zircon decomposition at 1550 °C while providing the required green strength.

Decohesion between the matrix and aggregates due to high matrix shrinkage can be prevented by partially substituting andalusite with the volume-increasing mineral kyanite. Based on these findings, a novel refractory brick was developed and tested with success at an industrial scale.

Introduction

Reaction-sintered mullite-zirconia ceramics that use zircon (ZrSiO4) as the zirconia source have many beneficial qualities as refractory materials, including refractoriness, chemical stability in contact with slag [START_REF] Bouchetou | Synthesis of an innovative zirconia-mullite raw material sintered from andalusite and zircon precursors and an evaluation of its corrosion and thermal shock performance[END_REF],

high strength [START_REF] Boch | Preparation and properties of reaction-sintered mullite-ZrO2 ceramics[END_REF][START_REF] Barfi Sistani | Improving the microstructural and mechanical properties of in-situ zirconia-mullite composites by optimizing the simultaneous effect of mechanical activation and additives[END_REF], and outstanding thermal shock resistance [START_REF] Rendtorff | Thermal shock behavior of dense mullite-zirconia composites obtained by two processing routes[END_REF][START_REF] Hamidouche | Thermomechanical behaviour of mullite-zirconia composite[END_REF]. These properties make mullite-zirconia a promising candidate as bonding system of novel refractories for hazardous waste incinerators, because mullitezirconia exhibits excellent thermal shock resistance in comparison with bauxite, andalusite or alumina-chromia refractory bricks or castables used nowadays. The in-situ formation of 20 vol% well-dispersed zirconia grains raises the material's strength and fracture toughness by up to 50% [START_REF] Boch | Preparation and properties of reaction-sintered mullite-ZrO2 ceramics[END_REF] compared to that of plain mullite.

Nevertheless, these results are obtained when using high-purity ceramic powders; it is difficult to transfer them to refractory bricks, where the bonding phase (the 'matrix') is made from raw materials containing impurities. Therefore, despite the potential advantages of using a mullite-zirconia bonding phase, most refractory bricks are still mullite-bonded. Refractory manufacturers use graded particle size distributions, with particle sizes ranging from a few microns to a few millimetres. The finer part of this distribution (≤200 μm; typically 30-40% of the total raw material) forms the matrix; larger particles are considered aggregates. The matrix materials ought to be economically priced and accessible in large quantities.

Many studies in recent years have attempted to employ different aluminosilicates for the synthesis of mullite-zirconia composites. Minerals from the sillimanite group have attracted most attention [START_REF] Bouchetou | Synthesis of an innovative zirconia-mullite raw material sintered from andalusite and zircon precursors and an evaluation of its corrosion and thermal shock performance[END_REF][START_REF] Kumar | Synthesis and characterization of mullite-zirconia composites by reaction sintering of zircon flour and sillimanite beach sand[END_REF][START_REF] Majidian | Phase evolution, microstructure, and mechanical properties of alumina-mullite-zirconia composites prepared by iranian andalusite[END_REF][START_REF] Chotard | Thermomechanical characterisation of mullite zirconia composites sintered from andalusite for high temperature applications[END_REF]. These comprise sillimanite, andalusite, and kyanite, which are all polymorphs of Al2SiO5. In addition to the economic advantage, mullite-transformed andalusite can enhance the thermal shock resistance of the composite [START_REF] Bouchetou | Synthesis of an innovative zirconia-mullite raw material sintered from andalusite and zircon precursors and an evaluation of its corrosion and thermal shock performance[END_REF]. The starting materials ZrSiO4, Al2O3, and Al2SiO5 react in several steps to form mullite and zirconia, which, simplified, can be summarised to reaction [START_REF] Bouchetou | Synthesis of an innovative zirconia-mullite raw material sintered from andalusite and zircon precursors and an evaluation of its corrosion and thermal shock performance[END_REF], where 0 ≤ x ≤ 0.667.

2 -3 + + → 1 - + (1)
The sintering reaction is very sensitive to the presence of secondary oxides [START_REF] Pena | The zircon thermal behaviour: effect of impurities[END_REF], which are omnipresent in refractory-grade raw materials, either in the form of impurities or intentionally added as sintering aids, pressing aids, temporary binders, etc. In this respect, Na2O, TiO2, and P2O5 are critical. Na2O is a major impurity in Bayer alumina. TiO2 is a common impurity in many raw materials, but it is also a common sintering aid for mullite-zirconia composites [START_REF] Barfi Sistani | Improving the microstructural and mechanical properties of in-situ zirconia-mullite composites by optimizing the simultaneous effect of mechanical activation and additives[END_REF][START_REF] Melo | Multicomponent toughened ceramic materials obtained by reaction sintering, Part 3 System ZrO2-Al2O3-SiO2-TiO2[END_REF]. P2O5 is a prominent additive in the refractory industry as it provides the required green strength and plasticity [START_REF] Kingery | Fundamental study of phosphate bonding in refractories: II, Cold setting properties[END_REF], and reacts with alumina at high temperatures to form AlPO4, which reinforces the refractory bonding [START_REF] Toy | Phosphate bonding with several calcined aluminas[END_REF]. However, while the effects of Na2O and TiO2 on the sintering reaction have been studied in detail, the effect of phosphate is not well understood. Therefore, there is a need to investigate the effects of P2O5 additives and to compare them to the effects of Na2O and TiO2, for the refractories studied in this paper.

In this study, we apply a mullite-zirconia matrix made from zircon-alumina-andalusite/kyanite mixes to the bonding phase of refractory bricks and we investigate the effects of P2O5, Na2O, and TiO2 on the system. The findings offer an understanding of how to successfully control the reaction-sintering process in refractory bricks.

First, we focus on the matrix itself, regarding the phases that form at high temperature. Second, we shed light on the matrix-aggregate interdependence within a refractory brick. We also address the problem of matrix shrinkage and explain how kyanite can help to solve this problem.

Materials and methods

Fabrication of matrix samples

Matrix samples were synthesised through a laboratory processing route that aims to reproduce the industrial process, albeit at a smaller scale. The task was to transfer the matrix, elaborated in the laboratory, to an industrial refractory brick made of matrix (bonding phase) and aggregates. The main requirements with regard to the industrial process are short and dry mixing, uniaxial pressing, and sintering below 1600 °C.

Matrix samples were prepared by mixing the raw materials in a tumbler mixer (Turbula ® T2C) for 2 h. The absence of aggregates was compensated by adding alumina balls (Ø10 mm) to aid the mixing process, which were subsequently removed. The blended mixture was then pressed into pills (Ø20 × 10 mm) using a uniaxial pressure of 150 MPa (5584, Instron, US). The matrix pills were sintered at various temperatures between 1450 and 1600 °C. The heating and cooling rates were 2.5 °C/min and the holding time was 3 h. To analyse the effects of Na2O, TiO2, and P2O5 oxide additions on the matrix properties and sintering reactions, fine powders of Na2CO3, rutile (TiO2), and monoaluminum phosphate (Al(H2PO4)3) were respectively added to the raw materials, to give an equivalent oxide content of 1, 2, or 3 mol%.

Fabrication of brick samples

Brick samples (Wedges of 250 × 198 × 103/90 mm) were prepared by mixing the starting materials (aggregates, matrix components, water) in an Eirich-type mixer. The aggregate/matrix volume ratio was 7:3 for all samples. The amount of water was adjusted to yield 2 wt% moisture in every blend. The grain size distribution was designed following the modified Andreasen model developed by Dinger and Funk [START_REF] Dinger | Particle-Packing Phenomena and Their Application in Materials Processing[END_REF]. The maximum and minimum grain sizes of 6 mm and 2 µm, respectively, were the same for all formulations. After a total mixing time of 30 min, the batch was pressed to bricks using a uniaxial pressure of 150 MPa. Finally, the bricks were dried overnight at 110 °C and fired at 1500, 1550, or 1600 °C. The heating rate and dwell time were the same as for the sintering step of the matrix samples (2.5 °C/min and 3 h, respectively).

The brick formulations contained the same matrix, composed of 31.3 wt% zircon, 37.8 wt% alumina, 10.2 wt% andalusite, 15 wt% kyanite, 2.6 wt% (3 mol%) TiO2, and 3.1 wt% (2 mol%) P2O5. Various aggregate systems were evaluated, but only the two most suitable systems are presented in this paper. The first system used a combination of andalusite (0.1-1.6 mm) and high alumina chamotte (1-6 mm) aggregates, while the second system used fused zirconia-mullite (FZM; 0.1-6 mm) aggregates. The chemical compositions of the studied aggregates are summarised in Table 1.

Thermodynamic modelling

The equilibrium phases at 1000-1600 °C were calculated using FactSage® (version 6.4), a thermodynamic database computing software. The FactPS and FToxid databases (both updated in 2013) were used for pure substances and oxides, respectively. The ambient pressure was set to 1 atm. Due to missing/erroneous data of K2O compounds, the percentage of K2O was added to the Na2O content; since these alkali oxides are chemically alike, this should not considerably distort the resulting liquid portion.

Characterisation methods

The microstructures, mineral compositions, and properties (porosity, compressive strength) of the sintered samples were evaluated with regard to the thermodynamic calculation results. The microstructures were examined using scanning electron microscopy (SEM; JSM-6500F, JEOL, Japan) with polished and carbonmetallised samples. For better contrast between zircon, zirconia, and mullite, the backscattering mode was chosen. The crystalline phases in the matrix were determined for the matrix pills by X-ray diffraction (PANalytical X'Pert Pro, Netherlands) using Cu Kα radiation. The open porosity was measured via Archimedes' method according to DIN EN 993-1. The cold crushing strength (CCS) of the refractories was measured according to DIN EN 993-5 on cubes of 50 × 50 × 50 mm.

Results

Effects of secondary oxides on mineralogy and microstructure

The diffraction patterns in Fig. 1 of zircon-alumina-andalusite mixtures (without additives) sintered at different temperatures demonstrate that zircon dissociation actually takes place between 1450 and 1570 °C. For the P2O5-doped samples, the X-ray survey testifies a slight shift of both andalusite and zircon dissociation towards higher temperatures. After sintering at 1550°C, the zircon peaks are yet well visible on the diffractogram. Contrary to our expectations, crystalline AlPO4 was not detected by X-ray diffraction. AlPO4 might have been incorporated into the silica-rich amorphous phase, considering that AlPO4 and SiO2 are chemically similar [START_REF] Handke | Spectroscopic studies of SiO2-AlPO4 solid solutions[END_REF]. To verify this, a silica-free matrix was fabricated from alumina, zirconia, and Al(H2PO4)3. X-ray diffraction measurements confirmed that in this case, crystalline AlPO4 is indeed formed.

Therefore, when in contact with the silica-rich amorphous phase present in the mullite-zirconia composites, phosphate is likely to form an amorphous phase instead of crystalline AlPO4.

In order to observe how zircon starts to dissociate, the same area of an additive-free matrix sample was imaged sequentially after heating to 1450 and 1475 °C. The SEM images in Fig. 3 show two newly formed zirconia grains after the second heating step, which emerged from several adjacent zircon particles. Meanwhile, non-transformed zircon particle clusters did not coalesce. With the addition of TiO2 (Fig. 4 (c)), zircon is completely transformed after sintering at 1550 °C.

Nevertheless, the microstructure does not vitrify, contrary to the case for Na2O, although the porosity is still reduced somewhat compared to that in the additive-free matrix. The final zirconia grains are about 5 µm in size, similarly to the zirconia grains in the additive-free matrix with fully dissociated zircon (sintered at 1570 °C).

The impact of P2O5 on the zircon decomposition is marginal (Fig. 4 (d)). The morphology is similar to that of the additive-free matrix, with some zircon particles still remaining after sintering at 1550 °C. However, the porosity is reduced on account of the improved compaction during pressing.

Further experiments revealed that a combination of 2 mol% P2O5 and 3 mol% TiO2 provided optimal green properties and complete zircon dissociation at 1550 °C. Therefore, these combined additives were chosen for the brick samples.

The preparation of bricks

During the development of the bricks, there was a noticeable drop in mechanical strength when a certain sintering temperature was surpassed. SEM observations revealed that this was caused by high shrinkage of the matrix, as it let to decohesion between the aggregates and matrix. An example is illustrated in Fig. 5. It is possible to counter high matrix shrinkage by employing materials that increase in volume during sintering. Kyanite (Al2SiO5), a polymorph of andalusite, has such an effect. Thus, to counterbalance the shrinkage, the andalusite in the matrix was gradually replaced by kyanite. Fig. 6 shows that, in the additive-free matrix, approximately one third of the andalusite needs to be replaced by kyanite to obtain a matrix that neither shrinks nor expands at 1550 °C. Accordingly, the kyanite and andalusite contents of the matrix were adjusted to attain zero shrinkage when sintering at 1550 °C. Fig. 6. Impact of kyanite on the volume expansion and porosity of a reaction-sintered (1550 °C) mullitezirconia (20 wt% ZrO2) matrix made from andalusite/kyanite, alumina, and zircon (without additives).

Representative microstructures of the bricks with andalusite/Mulcoa 60 and FZM aggregates are depicted in Fig. 7 (a) and (b), respectively. In both cases, the zirconia particles are well distributed and there are no large gaps between the aggregates and matrix. However, fine cracks appeared at the matrix/aggregate interface when using andalusite/Mulcoa 60 aggregates, which were absent when using the FZM aggregates.

Properties of novel bricks and applications

The sintering temperature had a clear impact on the mechanical properties of the bricks (Fig. 8). Because the andalusite/kyanite ratio and additive content were respectively designed to give zero matrix shrinkage and complete zircon dissociation at 1550 °C, a maximum in strength was obtained when sintering at 1550 °C. If the sintering temperature is above or below this designed temperature, the desired matrix properties are no longer guaranteed. Fig. 9 compares the properties of bricks sintered at 1550 °C with and without P2O5 in the matrix. The phosphate addition lowered the porosity and greatly increased the strength. This effect was even more pronounced for the brick with FZM aggregates, because the phosphate bonds between the aggregates and matrix are intact (see Fig. 7). The typical compressive strengths obtained with FZM aggregates and P2O5 doping were >200 MPa, which is exceptional for refractory bricks. The mullite-zirconia-bonded brick with FZM aggregates was produced industrially and tested in two rotary kilns for hazardous waste incineration. The material had very good performance in the kiln inlet, where the refractory bricks were exposed to mechanical load caused by solid wastes and to thermal shock provoked by cold liquid wastes dropping on the hot refractory lining [START_REF] Villalba Weinberg | Extending refractory lifetime in rotary kilns for hazardous waste incineration[END_REF]. After 20 months of use, the lining thickness was reduced in the first kiln from 250 to 180 mm and in the second kiln from 250 to 200 mm. With these slow wear rates, a lifetime of five to seven years is expected to be attained, in comparison to only one to two years for the previously used bauxite, andalusite, and alumina-chromia refractory bricks or castables.

Discussion

Prior studies have showcased the benefits of mullite-zirconia composites made from zircon, alumina, and andalusite. In this study, we investigated the possibility of applying this material as the bonding matrix phase of refractory bricks, with a focus on the impacts of secondary oxides on the high temperature sintering reactions, microstructure, and resulting properties.

The sintering process of ZrSiO4-Al2O3-Al2SiO5 mixtures can be divided into four steps, represented by Reactions ( 2)-( 5).

Al SiO

!"##"$%&"'(, %&*%#+,"'(, -. /0%&"'( → Al Si O 1+##"'(-'.%&,3-.$(* ,"##"$%&"'(, %&*%#+,"'(,-. /0%&"'(

+ SiO

(2)

2SiO + 3Al O → Al Si O 4."$%.0 $+##"'( (3)
ZrSiO → ZrO + SiO (4) 2SiO + 3Al O → Al Si O !(7-&*%.0 $+##"'( (5)
The first two mullitisation steps (Reactions (2) and ( 3)) begin at about 1300 °C upon heating and conclude at around 1400 °C [START_REF] Bouchetou | Synthesis of an innovative zirconia-mullite raw material sintered from andalusite and zircon precursors and an evaluation of its corrosion and thermal shock performance[END_REF][START_REF] Bouchetou | Mullite grown from fired andalusite grains: the role of impurities and of the high temperature liquid phase on the kinetics of mullitization and consequences on thermal shocks resistance[END_REF]. From a technological point of view, the critical step is zircon decomposition (Reaction ( 4)), because it requires a high temperature. Pure zircon dissociates into zirconia and silica at 1673 °C [START_REF] Kaiser | Thermal stability of zircon (ZrSiO4)[END_REF]. In the presence of alumina, this temperature reduces to 1450-1600 °C, depending on the reactivity and purity of the starting powders [START_REF] Pena | The zircon thermal behaviour: effect of impurities[END_REF][START_REF] Di Rupo | Solid state reactions in the ZrO2 • SiO2 -αAl2O3 system[END_REF][START_REF] Lathabai | Reaction-Bonded Mullite/Zirconia Composites[END_REF]. The last step (Reaction ( 5)) is the reaction between the zircon-released silica and the remaining alumina to form secondary mullite.

Role of impurities and alkaline/TiO2 sintering aids

All four reactions (2) -( 5) are diffusion dependent, making them sensitive to the presence of a liquid phase.

This explains why impurities, in particular fluxing agents, are so critical to this system -in both a positive and negative sense [START_REF] Pena | The zircon thermal behaviour: effect of impurities[END_REF]. On one hand, they accelerate the reaction sintering processes, but on the other, they can harm the microstructure and deteriorate the final properties.

The thermodynamically stable phases of a theoretical 30% ZrSiO4-36% Al2O3-34% Al2SiO5 (wt%) mixture without impurities and those of the same mixture including impurities (0.17 wt% Na2O, 0.12 wt% TiO2, 0.19 wt% Fe2O3, 0.09 wt% MgO, and 0.02 wt% CaO) are shown in Fig. 10(a) and (b), respectively. In the theoretical composition without impurities, insignificant amounts (0.2 wt%) of liquid phase appear at 1556 °C.

However, when the impurities are considered, a liquid phase emerges at only 1074 °C. This phase contains some SiO2 and Al2O3 and all the accessible Na2O. By 1200 °C, the liquid phase comprises 1.3 wt% of the matrix. At 1300 °C and above, MgO, CaO, and Fe2O3 enter the liquid state and increasing amounts of SiO2, Al2O3, and some ZrO2 dissolve. At 1550 °C, 3.0 wt% liquid is formed. Moreover, between 1300 and 1600 °C, mullite transforms from 3Al2O3•2SiO2 (3:2 mullite) to 2Al2O3•SiO2 (2:1 mullite). This transformation is possibly related to the liquid phase, as it does not occur in the impurity-free calculation. It should be noted that zirconia is thermodynamically preferred over zircon (ZrSiO4) at 1008 °C. However, from a kinetic point of view, this temperature is too low for the diffusion processes, so zircon does not actually decompose at this low temperature. This also implies that ZrO2 does not transform back into ZrSiO4 during cooling. Fig. 10. Equilibrium phases of a 30% ZrSiO4-36% Al2O3-34% Al2SiO5 (wt%) mixture (a) excluding and (b)

including impurities (0.17 wt% Na2O, 0.12 wt% TiO2, 0.04 wt% P2O5, 0.19 wt% Fe2O3, 0.09 wt% MgO, and 0.02 wt% CaO). (c) Expanded view of oxides in liquid state for the impurity-containing matrix. m-and t-ZrO2 are monoclinic and tetragonal zirconia, respectively.

These thermodynamic calculations demonstrate that even a low impurity content in the raw materials generates non-negligible amounts of liquid phase, which reduce the zircon decomposition temperature. In our experiments, the zircon decomposition had begun by 1470 °C and was completed by 1570 °C, despite the relatively coarse-grained raw materials.

The effects of secondary oxide additions (Na2O, TiO2, and P2O5) are exhibited in the equilibrium phase diagrams in Fig. 11. The Na2O addition had a strong but negative impact. According to the thermodynamic calculations depicted in Fig. 11(a), an addition of 0.3 wt% Na2O would triple the liquid portion at 1200 °C (from 1.3 to 3.8 wt%) and more than double it at 1550 °C (from 3.0 to 6.6 wt%). Consequently, Na2O shifted the first two mullitisation steps to lower temperatures. This might explain the smaller zirconia grain sizes obtained when Na2O was added: the low temperature liquid could have promoted primary mullite growth (Reaction (3)), and the zirconia grains from zircon decomposition could have been trapped within the primary mullite network, hindering the zirconia grains from growing together. Unfortunately, while the small grain size is desirable, the final matrix exhibited very high amounts of a glassy phase. Thus, Na2O is highly undesirable for refractory purposes. generates 1.3 and 3.9 wt% liquid at 1200 and 1550 °C, respectively, which is slightly greater than that in the additive-free matrix. Moreover, the results indicate that TiO2 forms a solid solution with ZrO2, which stabilises the tetragonal zirconia (t-ZrO2) modification below 1170 °C. These results are in accordance with those of Melo et al. [START_REF] Moya | Microstructure and mechanical properties of mullite/ZrO2 composites[END_REF], who emphasised that TiO2 forms a transitory liquid phase at about 1400 °C and then diffuses into the ZrO2 and mullite crystals, which can both accommodate 4 wt% TiO2. Nevertheless, the stabilising effect of TiO2 on the t-ZrO2 phase suggested in the literature and our thermodynamic calculations was not observed experimentally in our TiO2-doped samples. It could be that TiO2 merely did not have enough time to diffuse from the transitory liquid phase into the zirconia grains. Furthermore, the rather high zirconia grain size (5 µm) and the high porosity both work against the tetragonal zirconia stabilisation. Nonetheless, zirconia stabilisation is a rather unimportant detail for refractory materials, because their operating temperatures are usually above the m↔t transformation temperature. Overall, TiO2 is a suitable sintering aid for mullite-zirconia-bonded refractory materials.

dissociation of the surrounding zircon particles and enables the coalescence of particles. This supports the theory of Di Rupo and Anseau [START_REF] Di Rupo | Solid state reactions in the ZrO2 • SiO2 -αAl2O3 system[END_REF]; the released silica first forms a viscous pre-mullite phase that gradually dissolves the neighbouring alumina before eventually crystallising. This viscous pre-mullite phase may enable the embedded zircon and zirconia particles to merge together, increasing the ZrO2 grain size. On the contrary, if the primary mullite crystals are well developed before zircon starts to dissociate, they could help to inhibit coalescence. Therefore, slow heating rates or a dwell at 1400 °C should be conducted to obtain smaller ZrO2 grains.

At high temperatures, 2:1 mullite is thermodynamically preferred over 3:2 mullite, especially when the liquid portion is high. It is likely that the 3:2 → 2:1 mullite transformation accelerates liquid formation even further because it releases free silica. In other words, the liquid itself promotes the formation of more liquid.

Schmücker et al. [START_REF] Schmücker | Mechanism of 2/1-to 3/2-mullite transformation at 1650 °C, 1[END_REF] investigated the mullite transformation experimentally, and found that the liquid phase is essential to enable the diffusion of Si and O out of the mullite crystal and Al from the surrounding alumina particles into the 2:1-mullite crystal. Conversely, this suggests that a slow cooling rate or even a dwell at 1300 °C is preferred to give the diffusion-controlled back reaction from 2:1 → 3:2 mullite time to occur, which would reduce the amount of glassy phase in the final composite.

Aggregate choice

To successfully transfer the good matrix properties to the brick, the choice of aggregate is important.

Mulcoa 60 (high alumina chamotte) had a loose interface with the matrix and therefore bestowed relatively poor strength. The loose bonding probably had two origins. First, there is a thermal expansion mismatch between the matrix and chamotte, which provoked cracking. Second, the TiO2 impurities in Mulcoa 60 caused higher matrix shrinkage. In contrast, FZM aggregates suited well; the chemical similarity of the matrix and aggregate materials led to strong cohesion at the matrix/aggregate interface.

Conclusions

(1) Effects of additives and secondary phases Phosphate precursors provide the necessary green strength and plasticity to the raw material mix. Moreover, although P2O5 slightly retards the zircon decomposition temperature, it effectively reduces the porosity and increases the final strength. TiO2 is well-adapted to stimulate zircon decomposition in refractory materials, as it does not create much liquid at high temperatures. Na2O has a highly undesirable effect on the sintering reaction because it produces excessive amounts of liquid phase at relatively low temperatures -this is even true for contents as low as 0.3 wt% Na2O. Therefore, the raw materials must be selected carefully with respect to alkali oxides.

(2) Preparation of bricks Matrix shrinkage should be avoided because it leads to decohesion at the matrix/aggregate interface.

Partially replacing andalusite with kyanite helps to reduce the net matrix shrinkage to zero. FZM aggregates are well-suited for use with a mullite-zirconia matrix, whereas high alumina chamotte and coarse andalusite aggregates lead to cracking at the matrix/aggregate interface.

The understanding of the high temperature phase transformations suggests to apply low heating and cooling rates, for the following reasons. Slow heating rates, or a dwell at 1400 °C, should help to develop primary mullite crystals before zircon decomposition. A well-developed primary mullite network may inhibit zirconia coalescence and eventually reduce the zirconia grain size. Slow cooling rates or, if possible, a dwell at 1300 °C are recommended to encourage the 3:2 → 2:1 mullite transformation and reduce the amorphous phase fraction.

(3) Applications A novel mullite-zirconia-bonded refractory material has been produced and field-tested in a rotary kiln incinerator, where the material was exposed to thermal shock and mechanical load. The material persisted much longer than mullite-and alumina-chromia-bonded refractories. We believe similar success could be achieved in other refractory areas, especially if alternative and more cost-effective aggregates could be found that are compatible with this matrix.

Fig. 1 .

 1 Fig. 1. X-ray diffraction patterns of a zircon-alumina-andalusite mixture (without additives) sintered at different temperatures. The mineralogical phases are C -corundum (Al2O3), M -mullite (Al6Si2O13), An -andalusite (Al2SiO5), ZS -zircon (ZrSiO4), Zm -monoclinic zirconia (m-ZrO2), and Zt -tetragonal zirconia (t-ZrO2).

Fig. 3 .

 3 Fig. 3. SEM images of a zircon-alumina-andalusite matrix sample (without additives) after heat treatments at (a) 1450 °C and (b) 1475 °C.

Fig. 5 .

 5 Fig. 5. SEM images of a mullite-zirconia-bonded brick sintered at (a) 1450 °C, (b) 1500 °C, (c) 1550 °C, and (d) 1600 °C, revealing decohesion between the matrix and aggregates (Mulcoa 60) caused by matrix shrinkage.

Fig. 7 .

 7 Fig. 7. SEM images of mullite-zirconia-bonded refractory bricks made with (a) andalusite/Mulcoa 60 aggregates and (b) fused zirconia-mullite aggregates (FZM). Both bricks used the same matrix (31.3 wt% zircon, 37.8 wt% alumina, 10.2 wt% andalusite, 15 wt% kyanite, 2.6 wt% (3 mol%) TiO2, and 3.1 wt% (2 mol%) P2O5) and sintering temperature (1550 °C).

Fig. 8 .

 8 Fig. 8. Cold crushing strength (CSS) vs. sintering temperature of a mullite-zirconia-bonded brick (Aggregates: andalusite/Mulcoa 60).

Fig. 9 .

 9 Fig. 9. Comparison of (a) open porosity and (b) cold crushing strength (CSS) of mullite-zirconia (MZ)-bonded bricks with and without 2 mol% P2O5 in the matrix. Andalusite/Mulcoa 60 (An/M60) and fused zirconia-mullite (FZM) aggregates were used. All bricks were sintered at 1550 °C.

Fig. 11 .

 11 Fig. 11. Impacts of 1 mol% oxide additions on the thermodynamic equilibrium of a 30% ZrSiO4-36% Al2O3-34% Al2SiO5 (wt%) mixture (including impurities): (a) +0.34 wt% Na2O, (b) +0.88 wt% TiO2, and (c) +1.55 wt% P2O5. m-and t-ZrO2 are monoclinic and tetragonal zirconia, respectively. ss: solid solution.

Table 1 .

 1 Raw material composition for matrix (without additives) and aggregate materials.

	Preliminary tests showed that using only very fine powders led to agglomeration during mixing and
	lamination during pressing. Both problems were resolved by designing a graded particle size distribution with
	fine zircon powder, intermediate alumina, and coarser andalusite or kyanite (Table 1). A portion of the
	andalusite was replaced with kyanite (percentage k in table 1) to mitigate matrix shrinkage (see Section 3.2).
	Matrix

component Grain size [μm] Composition [wt%] wt% of matrix Al2O3 SiO2 Fe2O3 TiO2 P2O5 Na2O K2O CaO MgO ZrO2

	Micronized zircon	≈2	0.7	34.0	0.10	0.25	0.12	0.00	0.00	0.00	0.20	64.0 3	31.3
	Calcined alumina	≈4	99.5	0.0	0.01	0.00	0.00	0.27	0.00	0.01	0.00	0.0	22.9
	Fused alumina	≈50	99.7	0.0	0.02	0.00	0.00	0.16	0.00	0.01	0.01	0.0	15.2
	Andalusite	<160	60.8	38.1	0.50	0.15	0.00	0.10	0.15	0.05	0.10	0.0	30.6-k
	Kyanite	<125	57.5	40.3	0.6	1.2	0.15	0.04	0.07	0.04	0.03	-	k
	Matrix	2-160	56.8	22.3	0.19	0.12	0.04	0.12	0.05	0.02	0.09	20.0 3	100.0
	Aggregate
	Andalusite	100-1600	58.7	38.5	1.1	0.2	-	0.1	0.4	0.2	0.2	-
	Mulcoa 60 1	1000-6000	60.0	35.8	1.2	2.4	-	0.1	0.1	0.1	0.1	-
	FZM 2	100-6000	45.8	17.1	0.1	0.1	-	-	-	0.1	-	36.5 3
	1 High alumina chamotte; 2 Fused zirconia-mullite; 3 ZrO2+HfO2

Acknowledgements and Support

This research was realised within a PhD thesis defended on April 2017 in Saint-Étienne, France. The thesis was financed by BONY SA and the French National Association for Research and Technology (ANRT).

Industrial tests were conducted in two hazardous waste incineration plants operated by SARPI-VEOLIA.

Role of phosphate

In mullite-bonded bricks, plasticity and green strength are provided by clay. However, owing to the high silica content of clay, it can only be used in reaction-sintered mullite-zirconia at the expense of reducing the zircon content or by accepting a portion of free silica in the sintered matrix. P2O5 precursors represent a more suitable alternative to clay. Just a few percent of a viscous monoaluminum phosphate solution assures sufficient green strength by delivering the required plasticity and forming chemical bonds at ambient temperature [START_REF] Kingery | Fundamental study of phosphate bonding in refractories: II, Cold setting properties[END_REF]. At high temperatures, AlPO4 is formed [START_REF] Toy | Phosphate bonding with several calcined aluminas[END_REF], which can reinforce the mullite-zirconia bonding.

In the investigated materials, P2O5 considerably reduced the porosity and increased the mechanical strength owing to the improved pressing behaviour and additional (amorphous) AlPO4 bonding. The impact of phosphate on the high temperature equilibrium was marginal (Fig. 11 (c)). The liquid phase accounts for 3.9 wt% of the matrix at 1550 °C, which is the same as that for the TiO2-containing matrix, and only 0.9 wt% more than that in the additive-free mixture. However, thermodynamic modelling suggested that P2O5 has a slight retarding effect on the liquid phase formation. In the P2O5-doped mixture, the calculations predict the formation of a liquid phase at approximately 1300 °C, i.e. 200 °C higher than that for the additive-free matrix. Consequently, the primary mullitisation step and the zircon decomposition were delayed. To enable full zircon dissociation at 1550 °C, the phosphate needed to be added in conjunction with TiO2. The matrix with 2 mol% P2O5 and 3 mol% TiO2 yielded the most promising results. A patent covering this bonding phase has been applied for.

Problem of and solution for matrix shrinkage

While shrinkage does not pose a substantial problem in classic ceramics, shrinkage is undesirable in refractory matrices. This is because the coarse and dense aggregates show practically no shrinkage during sintering. In fact, they form a rigid structure of closely packed aggregates which prevent the brick from shrinking. Hence, a shrinking matrix does not pull the aggregates closer together, but instead causes decohesion between the aggregates and matrix, as illustrated by the micrographs in Fig. 5. The high shrinkage associated with reaction-sintered ceramics therefore poses a serious problem for refractory bricks.

We have demonstrated that kyanite is a well-adapted solution to this problem. Kyanite has the same chemical composition as andalusite (Al2SiO5) but a higher density. Upon heat treatment, both kyanite and andalusite are analogously dissociated into mullite and a silica-rich amorphous phase [START_REF] Creig | Formation of mullite from Cyanite, Andalusite, and Sillimanite[END_REF]. The volume change of this mullitisation process can be calculated from the molar volumes, specified in [START_REF] Robie | Selected X-ray crystallographic data, molar volumes and densities of minerals and related substances[END_REF]. Assuming complete mullitisation according to Reactions (2) and (3), the resulting theoretical expansion is 18.3 vol% for kyanite, yet only 4.6 vol% for andalusite. By substituting a portion of the andalusite for kyanite, the net volume change of the matrix can be reduced to zero at a given sintering temperature. Evidently, kyanite's enormous volume change can cause cracking, cavities, and increased porosity if excessive kyanite is added; therefore, it should be dosed cautiously.

Implications for the sintering curve

By tailoring the matrix shrinkage to a specific sintering temperature, it is paramount to strictly comply with this exact temperature during fabrication. Otherwise, decohesion between the matrix and aggregates is likely to occur, weakening the material.

Slow heating and cooling rates are also suggested for the following reasons. Our SEM observations revealed that adjacent zircon particles dissociate simultaneously, and that the newly formed zirconia particles subsequently grow together. It can be deduced that the silica released upon zircon dissociation activates the


D Graeff

V Lespinet-Najib

T Letouze

C Vennin

J M André

ACDC : A new model for the Analysis of Communication in Dyad Cooperation validated by mixed methods

Keywords: Dyad cooperation, conversational exchanges, temporal dynamics, collaborative cognitics, mixed methods, knowledge eXchange 1

This article has two objectives. First is the presentation of an updated model of conversation analysis designed and tested in a specific cooperation situation. Secondly, its validation by mixed methods combining a qualitative study based on user-centered methods and a quantitative analysis using interrater agreements. The first highlights the importance of adapting the model to the cooperation task by adding or merging typology categories. The second reveals guidelines for the use and reliability of the model based on rater skills. Interrater agreements show that an expert could use our model alone whereas novices should use pair rating for satisfactory results.

Highlights :

• Formalization of a new model for conversation analysis during a cooperation situation.

• The nature of conversational exchanges and their temporal dynamics are both represented.

• Conversational exchange typology is adapted and validated by mixed methods.

• Interrater agreements indicate good reliability of the model.

• An expert could use our model alone whereas novices should pair rate a conversation.

Introduction

Social cognition refers to how we perceive, process and interpret social information [START_REF] Bertoux | Cognition sociale[END_REF]. The performance of a task (cognitive, motor, etc.) in a dyad situation can take place in a context of coordination, cooperation or collaboration [START_REF] Belkadi | Démarche de modélisation d'une situation de conception collaborative[END_REF] (Table 1). In the case of coordination, the goals are not necessarily the same for both people. Moreover, if there is a hierarchical relationship between them, the relationship of authority means that the tasks to be performed by each are often of a different nature (Gulati et al., 2012). Conversely, in the case of cooperation or collaboration, the goal is common. It is then the way of achieving this goal that will differentiate the two situations (Sébastien, 2001;Gulati et al., 2012;Henri and Lundgren-Cayrol, 1997). In a collaboration task, the responsibility is individual and we speak of associative interdependence. In a cooperation task, each person performs a part of the overall task. In this case, there is a strong interdependence between the two people and the success of the task is the responsibility of the group dynamics. Thus, contributions are meaningful only if understood and complemented by the work of others. The effectiveness of a collaboration or cooperation task depends on the quality of verbal communication between people [START_REF] Clark | Using Language[END_REF]Meier et al., 2007): request for advice, proposal, suggestion, consensus, strategy development, order, etc. [START_REF] Clark | Using Language[END_REF]Meier et al., 2007). In the present work, we focus on dyad cooperation situations. Even if the cooperation situation is based on a clear division of labor and thus on collective interdependence, the group dynamics that are created can lead to the emergence of social roles, such as leadership, followership or others (Demary, 2018, Dinh et al., 2014;Uhl-Bien et al., 2014;Oc and Bashshur, 2013;Tee et al., 2013). Thus, when two individuals perform a common cognitive task with a performance objective, they are led to implement one (or more) interaction strategies. Several authors have proposed semantic characterizations of these communications (Clark and Brennan, 1991;Meier et al., 2007). For example, Meier et al. (2007) proposed characterizing communications between two people based on 5 aspects: communication, informational process, coordination, interpersonal relations, and motivation. This classification is interesting, but limited: 1) it does not allow specifying the nature of the exchanges nor the roles of each person, 2) it does not consider the temporal dynamics of the exchanges. To analyze the data of an oral exchange, it is necessary to identify the parameters that must be observed during the interaction. They must also be transcribed and formalized into nonaudio data that can be more easily processed and compared. To do this, data must be classified and characterized. The case studied here is that of a cooperation dyad playing the same game of Tetris with complementary roles. The communicative interactions between the individuals were observed as short and repeated throughout the common task. The projective model of informational dialogue presented by Vernant (1992) is not suitable here since the modeling would have to be done for each exchange independently.

Our objective is to represent the entire dialogue and all the exchanges that make it up on the same graphical representation in order to account for the dynamics of the exchanges and their fluctuations. Since we want to identify the elements of the dialogue to characterize the dynamics of the exchanges, we are interested in the nature of these exchanges. Does the individual address the other person? Is he or she asking the other person to do something? Does he talk to himself? In the situation we are studying, individuals must make decisions quickly and perform actions to complete the cognitive task and comply with the instructions given to achieve the best possible score. We therefore rely on the collection of the exchanges expressed and the observation of the resulting decision making. It is a performative language, linked to an action, as defined by [START_REF] Austin | How to do things with words[END_REF].

Giving or accepting instructions is an illocutionary act that has an immediate effect on the game (Searle, 1972;[START_REF] Burger | Desire for Control and Interpersonal Interaction Styles[END_REF]. These instructions form the core of our model. Additionally, we add elements that characterize the interaction between individuals in terms of valency and intention, based on Bales' categories (1950;[START_REF] Bales | Rôles centrés sur la tâche et rôles sociaux dans des groupes ayant des problèmes à résoudre[END_REF]: the scope of the interactions (affective versus work-oriented), the value (positive versus negative), and the nature (intervention centered on oneself, on others, or on another intervention). This typology is interesting because it makes it possible to characterize the nature of exchanges according to their temporal dynamics.

Discourse analysis have been mainly qualitative or quantitative methods. But another alternative would be using mixed-methods and combining qualitative and quantitative approaches for data collection and analysis (Creamer, 2016). For example, it is now a common procedure to use statistical analysis on qualitative data to ensure the validity and reliability of discourse coding (Spooren, 2010). Sometimes, using qualitative and quantitative data in concert at the inference level is also the only way to draw conclusions [START_REF] Campbell | Understanding Inter-rater Disagreement: A Mixed Methods Approach[END_REF].

So we choose to use a mixed-methods approach to complete the objective of this work : 1-to formalize a model for the analysis of conversational exchanges within a dyad cooperation situation and 2-to consider the temporal dimension and propose a relevant, graphical representation of these exchanges.

Description of the design process

-Description of the cooperation task

There are many cooperation tasks already studied in dyad situations, including cooperative Tetris games (Mast and de Vries, 2016;Noah et al., 2015). This game is a playful, intuitive task with simple rules and it is easy to distribute the possible actions on the game pieces between the two people in an equitable way in terms of cognitive load (Noah et al., 2015). For example, participant 1 can move the pieces on the horizontal axis (i.e. from right to left) and lower them and participant 2 can rotate counterclockwise or clockwise. The success of this task, i.e. to build lines as quickly as possible, requires a very strong cooperation between the two participants. As illustrated in Fig. 1, the participants are placed side by side without seeing each other to force cooperation through verbal communication. They each have a screen in front of them that allows them to see the game and the actions performed. The experimental protocol takes place in two stages: 1) Participants play Tetris alone for 10 minutes to habituate to the controls; 2) they are then assembled in dyads and play cooperatively for 10 minutes. They are instructed to achieve the best possible score and are encouraged to communicate verbally as much as they wish, but they cannot see each other. In this situation, we record the conversational exchanges between the participants. This allows us to collect a corpus of data on which we develop and formalize our model for Analyzing Communication in Dyad Cooperation, called the ACDC model.

-Design process of the ACDC model

Formalizing the analysis model took place in several steps as illustrated in Fig. 2 •

-Mixed-method approach

A mixed approach is developed to design and validate the analysis model.

The development of this methodology is motivated by the observation that there is no satisfactory methodology dedicated to the modelling of conversational activity. Existing methods are not easily usable and their handling is complex. We present here a process for designing and validating a new analysis model adapted to our particular situation and accessible to non-expert experimenters.

The design process starts with the development of a model, first validated using user-centred qualitative methods. Then the model is validated by an experiment where qualitative data is collected, analyzed and interpreted. The transformation of the data is initially done through qualitative transcription, characterization and formalization of the conversations. This application of the ACDC model to a qualitative database is conducted by several judges and results in integrated qualitative and quantified data, more detailed below. Finally, this data transformation allows us to compare different points of view through inter-judge measurements and to quantitatively validate the reliability of our analysis model.

The integrated qualitative and quantified data are also used at the inference level, in order to allow a more solid understanding of the results of the experiment and to identify possible improvements to our analysis model.

The final formalization of our analysis model integrates the qualitative and quantified data into a single visual plan. This formalization is linked to a mixed methods' technique called "joint display", as suggested by Guetterman et al (2015). In our study, we use this joint display to capture and understand the dissonances and congruences between the formulations of the dyads and judges, and to suggest improvements to our ACDC model.

-Description of the analysis model

The objective of the model is to specify the nature of conversational exchanges in dyad cooperation situations and to propose a relevant graphical representation which considers temporal dynamics. Thus, our analysis model is based on two complementary phases:

• Determine the typology (nature and duration) of the exchanges between the two participants; • Formulate a relevant graphical representation of these exchanges and their temporal dynamics

-Determination of a typology for conversational exchanges

A conversation refers to the totality of dialogue between two participants during the experience. In this case, we model a conversation of 10-minutes. Conversations are composed of several exchanges, themselves made up of one or more successive speeches. Each time one of the two participants speaks, he or she makes a speech. Speaking is the smallest unit in our model. We consider an exchange to be all the successive speeches of the two participants that follow the same action statement. As soon as one of the participants starts talking about another action or another subject, a new exchange has begun. Thus, when transcribing the recordings according to the ACDC model, the entire verbal interaction is broken down into exchanges. For each defined exchange, several characteristics are specified:

1. The temporality: start and end time to the nearest second 2. The initiator of the exchange (Participant 1 or Participant 2) 3. The initiator of the action (Participant 1 or Participant 2) 4. The length of the exchange (discrete timing) 5. The presence of speech cuts (yes or no) 6. The type of exchange: directive actions (orders, instructions, proposals) or non-directive actions (comments, soliloquies, non-verbal actions) 7. The type of final decision made: assent or dissent 8. The qualifiers of the interaction: task-related interactions (information communication, questioning, negative game-oriented remark, positive game-oriented remark, control) and socio-emotional interactions (positive feedbacks to the partner, negative feedbacks to the partner)

Initiator of the exchange and initiator of the action

The initiator of the exchange is distinguished from the initiator of the action. It may happen that one participant elicits the participation of the other but does not state an action taken or to be taken. To initiate an action, a direction must be verbalized, a non-verbal action must be taken, or a comment or soliloquy must be made. Being proactive and engaging the other person is relevant. Therefore, to avoid any loss of information, the initiator of the exchange will also be considered and symbolized in the modeling. Thus, the initiator of the exchange and the initiator of the action are often the same person, but this is not systematic. For example, if an exchange starts with an invitation to give a directive: the participant giving the directive is the initiator of the action, but the other one is the initiator of the exchange.

Length of exchanges and speech cuts

The length of an exchange corresponds to the number of successive speeches by each person in the same exchange, so it is a numerical value. It thus gives information on the quantity of each person's speeches and on the tendency to respond to the other person or not. This variable is essential to have a global vision of the communication between two participants. Speech cuts make it possible to translate certain modalities of the conversational exchange, such as speaking at the same time as the other person or not letting him/her finish the sentence.

Types of exchanges

In order to identify the dynamics of verbal conversation between two participants cooperating on the same cognitive task, we chose to characterize the initiated actions. The exchanges are categorized according to syntactic distinctions (use of the indicative, imperative, conditional), recipient distinctions (speaking to oneself or to the other) and vocal tone distinctions (certain or uncertain tone, authoritative or non-authoritative tone). The classification of actions follows the taxonomy identified by Searle (1972Searle ( , 1990)), based on research by [START_REF] Austin | How to do things with words[END_REF]. Indeed, it is partly a matter of specifying so-called directive performative language acts. Performative means that the speaker expects the listener to perform an action, which is why we speak of an initiator of action to designate the issuer of this type of statement. In addition, they are directive, i.e. they ask explicitly the listener to perform a specific action. In Vernant (1992), this categorization corresponds to the 'directional initiator'. Searle (1990) and Vernant (1992) group orders, advice, questions or requests for undertakings under these directive performative language acts.

In order to validate the categories with respect to the task of the cooperative Tetris game, we conducted focus groups and card sorting sessions. Fourteen students aged 20 to 30, including 8 men and 6 women, were divided into three groups, each of which participated in a focus group organized around the characterization of the categories, followed by a card sorting session. Each group sorted 28 exchanges extracted from a 10-minute audio recording from a pre-experimental cooperative Tetris game. They were instructed to form groups of excerpts focusing only on the instructions given in the excerpts -and not on the answers -and to form as many groups as they wished, within the limit of 7. Once the groups were formed, they were asked to carry out labelling work and to give a name to each of the categories created. Based on the results, we formed three sub-categories under the heading of directive actions (i.e.

Directive Performatives -see Table 2):

• orders, which include injunctions, orders and safe orders,

• instructions, which can be related to the notion of uncertain order because one addresses the other directly but not as certain and authoritatively as with an order, • proposals, which include clear proposals awaiting confirmation, hesitant proposals requiring confirmation and suggestions.

Then our model considers non-directive actions:

• comments, of a consensual (and no longer performative) type, which are used for description, • soliloquies, which do not include the other and are therefore not illocutionary acts (the soliloquy being a discourse by a person who is talking to himself or thinking aloud),

• non-verbal actions, which are not acts of language.

Types of decisions

For the purpose of relevance, we are only interested in the final decision made by the original recipient. Thus, we note as a type of final decision the assent of the addressee to the initial directive with four sub-categories:

• agreement,

• acceptance,

• authorization,

• dealership, And if not, the recipient's dissent to the original directive, in other words, refusal.

Qualifiers of interactions

We sought to characterize the nature of the interactions between the two individuals. [START_REF] Bales | Interaction process analysis : a method for the study of small groups[END_REF][START_REF] Bales | Rôles centrés sur la tâche et rôles sociaux dans des groupes ayant des problèmes à résoudre[END_REF] distinguishes task-related interactions from socio-emotional interactions. Furthermore, he classifies these two broad categories into two groups: "in question" and "in response" for taskrelated interactions and "positive emotional domain" or "negative" for interactions within the socioemotional domain. Concerning the task-related interactions, we have collected those concerning:

• communication of information,

• questioning,

• negative game-oriented remarks,

• positive, game-oriented remarks,

• control.

Socio-emotional interactions are:

• negative feedbacks to the partner,

• positive feedbacks to the partner.

Events

Events allow transcription of context information (example: Tetris game finished, restart of a new game). This is information that is relevant for the analysis but that cannot be symbolized in the modeling.

-Development of a graphical representation of the exchanges and their temporal dynamics

Once the typology for characterizing conversational exchanges was defined, a graphical representation was proposed to account for the temporal dynamics. Several types of graphic proposals have been put forward (circular, frieze, etc.). A focus group enabled us to validate a symbology of the typology of exchanges (Table 3), then a representation in the form of a timeline (Fig. 3).

Categories Sub-categories and symbology

Initiator 3).

Fig. 3 shows the evolution of conversational exchanges observed for 5 minutes. It is characterized as follows:

• The time course is on a horizontal axis, the time unit is 10 seconds.

• The grey bars at the top and bottom of the timeline position each person of the dyad (participant 1 at the top and participant 2 at the bottom). • The middle part characterizes the exchanges according to the previously defined categories.

The graphical representation of the ACDC model (Fig. 3) corresponds to a joint display (Guetterman et al, 2015) and combines qualitative and quantitative data in the same visual plane. The qualitative data are accessible through the symbology of types of exchanges, types of decisions and qualifiers of interactions (Table 3). The quantitative data are also visualized at a glance in the graphical representation (Fig. 4) we can observe :

On the horizontal axis: the timing and distribution of exchanges during the conversation.

On the vertical axis: the distribution of the actions initiated between the participants, the duration of the exchanges and the distribution and quantity of the interactions. The difficulty of the game is progressive with the level increasing every 5 completed lines. After a short presentation of the context of the experiment, they are asked to play Tetris individually. All individuals started playing simultaneously and raised their hand when they lost so that observers could note the score and the level reached; the game then restarted. All participants played for a total of ten minutes, regardless of the number of games lost. The students then completed an anonymous Google Forms online questionnaire which included their assigned participant code. This questionnaire (i) collected personal data, (ii) self-assessed performance on Tetris, and (iii) had 5 questions on the practice of video games.

-Constitution of dyads for the task of cooperative Tetris.

The formation of the dyads for the experiment was carried out using the data collected during the pre-tests. Men and women were separated to form non-mixed dyads. In order to have subjects with homogeneous game performances (neither too weak nor too strong), we excluded subjects with performances in the first and last quartile. In total, we obtained a sample of 8 dyads: 3 male and 5 female.

-The Cooperative Tetris experience

The 8 dyads formed participated in the experiment as follows: The participants were informed individually of their schedule without knowing that they were in dyads. The observers received them in a room equipped with two computers with the version of Tetris developed on Matlab and used in pre-test (see Fig. 1). The two participants were placed at tables next to each other and separated by a screen. They each had a screen and a keyboard with which they could use two keys to play Tetris. The participant "P1" of each dyad was always in charge of the rotation of the pieces and their rapid descent, via the "up arrow" and "down arrow" keys of the keyboard. Participant "P2" was always in charge of horizontal movements using the "O" (left) and "P" (right) keys of the keyboard. The difficulty was not progressive and the level was set at 5 to simplify score comparison among the dyads. Participants were told that they could communicate verbally as much as they wished and that their goal was to achieve the best possible score in Tetris. As in the pre-tests, they were told that they must play for 10 minutes and that the game would be restarted if they lost. Two audio recordings were made for each participant throughout the experiment. In addition, observers took notes according to an observation grid. They observed the distribution of the exchanges in 2-minute increments and the general dynamics of the exchanges. They also noted events that would not appear in the audio recordings, such as intermediate scores in case of failure. The experiment was completed when the 10 minutes were up.

-The corpus

The results of 7 dyads (out of 8) were collected due to technical problems encountered during the experiment for the first dyad. All audio files were transcribed word-for-word and then a re-transcription of each, exchange by exchange, was carried out by a jury of two experts who had already mastered transcription as well as the model we propose.

The exchanges were treated according to our model of Analysis of Communication in Dyad Cooperation (ACDC) (examples in Table 4).

Example 1 1. Retranscription P2 : "How do you want it?" P1 : "Flat, I'll move it to the left" P2 : "There, like that ?" P1 : "Yeah" For an easier understanding, we will illustrate the application of our model through a single dyad, the one named BH3.

Once the transcription is done, we characterize and formalize the exchanges with a timeline (Fig. 5).

Fig. 5 : Completed paper timeline of the first 2 minutes of conversational exchanges for the dyad BH3

(jury alpha : 2 experts). Refer to the typology symbols of the ACDC model (Table 3).

From the timeline, a matrix of temporal data collects and characterizes all conversational exchanges. The matrix form is intended to facilitate statistical processing and interpretation (Table 5). 00:00 P1 0 3 comment null 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00:07 P1 0 3 instruction acceptance 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 00:13 P2 0 1 proposal acceptance 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 00:17 P2 0 4 comment null 0 2 0 0 0 0 0 1 1 1 0 0 0 0 0 0 00:25 P1 0 2 instruction undetermined 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 00:33 P1 0 2 proposal agreement 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00:38 P1 0 2 instruction acceptance 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00:46 P1 0 0 instruction null 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00:52 P2 0 1 comment null 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00:55 P2 0 2 order acceptance 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 01:05 P2 0 3 order acceptance 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 01:11 P1 0 0 instruction null 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01:16 P1 0 2 instruction agreement 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 01:22 P1 0 0 soliloquy null 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01:24 P1 0 0 instruction null 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 01:24 P2 0 1 instruction acceptance 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 01:29 P1 0 1 proposal refusal 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01:31 P2 0 1 instruction acceptance 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 01:34 P2 1 2 non-verbal agreement 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01:40 P1 1 4 order agreement 0 3 0 0 0 0 0 0 1 1 0 0 0 0 0 1 01:51 P2 0 3 order agreement 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 01:56 P1 0 0 comment null 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01:58 P2 0 3 proposal dealership 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 Table 5 : Time matrix of the first 2 minutes of conversational exchanges for the dyad BH3 (jury alpha : 2 experts).

Fig. 6 shows the ACDC model's final graphical representation of the exchanges of BH3 dyad, constructed from the data collected in the time matrix (Table 5).

Fig. 6 : Final graphical ACDC model representation for the first 2 minutes of conversational exchanges for the dyad BH3 (jury alpha : 2 experts). Refer to the typology symbols of the ACDC model (Table 3).

-Validation of the model

Human interrater agreement represents one of the gold standard against which we measure natural language processing performance [START_REF] Campbell | Understanding Inter-rater Disagreement: A Mixed Methods Approach[END_REF]. Therefore, we chose to calculate interrater agreements between judges and juries in order to validate the ACDC model and its reliability.

-Interrater agreement between judges

Initially, we compared decisions made individually. Two experts in the analysis model separately analyzed and modeled the conversational exchanges of the same dyad: the BF1 dyad. This dyad was chosen by the two experts as they judged the exchanges of this dyad to be dense, rather complex and exhaustive in terms of the representation of the different components of the model.

Results

Tables 6 and7 compare the analyses proposed by the two expert judges.

For dichotomous variables, we compared the responses of the two judges. We first identified the number of common exchanges with the same timing (+/-3 seconds). Then, for those exchanges, we noted the percent agreement between the two judges for the following scores:

• % of actors in common: percentage of times the judges identified the same actor (possible answer: participant 1 or participant 2) • % of initiators of the exchange in common: percentage of times the judges identified the same initiator of the conversational exchange (participant 1 or participant 2)

• % of speech cuts in common: percentage of times the judges identified the same speech cuts • % of interactions in common: percentage of times the judges identified the same qualifier of interaction (questioning, providing information, positive feedbacks to the partner, etc.). For qualitative variables, agreement between the two expert judges was analyzed using Cohen's (1960) kappa statistic: a coefficient to measure agreement between two qualitative variables, and commonly used to measure degree of agreement between two judges. The specificity of this coefficient is that it considers the theoretical proportion of "random" agreements. We used the Cohen's kappa coefficient interpretation grid proposed by Landis and Koch (1977): The specificity of Cohen's (1960) kappa coefficient is that it considers the theoretical proportion of "random" agreements. We calculated interrater agreement on length, type of interaction, type of directive, type of decision and type of other action. In the case of perfectly ordered sub-categories, serious disagreement is more likely to be sanctioned than slight disagreement between two contiguous sub-categories (Cohen 1968;Bakeman et al. 1997;[START_REF] Bakeman | Observing interaction: An introduction to sequential analysis[END_REF]. Thus, for the linear variable length of interaction, we used a linearly weighted Cohen's kappa. A quadratic-weighted measure of kappa was applied to directive actions with the following ordering: order, instruction, and proposal. We used an unweighted version of the kappa calculation for the other categories, which have subcategories that are too different to be ordered. Namely: the type of final decision and the type of non-directive action (non-verbal, soliloquy and comment).

• < 0: Disagreement • -

Inter-rater agreement

Agreement on the type of decision The results of interrater agreement between the two expert judges (percentage agreements and Cohen's kappa coefficients) show primarily strong and very strong, agreements, except for "type of decision" where agreement was moderate.

-Interrater agreement between juries

Previous interrater agreements were on decisions made individually. We also compared decisions made in pairs, as the confrontation of subjectivities could make it possible to partially neutralize them, and to conform more closely to the established grids. We compared the decisions by a pair of non-expert judges with those by a pair of expert judges on the same dyad : the BH3 dyad.

We set up 3 juries (X, Y and Z), made up of non-expert pairs and composed of people aged 21 to 25. Jury X was a pair of men. Jury Y a pair of women. Jury Z mixed. Jury (alpha) was made up of two judges expert in the model and constituted a reference.

Each jury was placed in a separate room and had unlimited time to re-transcribe the verbal conversations of the BH3 dyad using the audio recording, according to the previously described methods. They had the following resources available: a step-by-step guide to re-transcription, presentation and explanation grids for the different elements, decision support trees, detailed and didactic examples, a blank paper timeline and markers.

Participants were free to ask as many questions as they wished during the discovery phase. When they said they were ready, the observers started the audio recording and left the room. The sound was lateralized to improve voice distinction (the first participant BH3P1 was heard to the left of the sound output, and BH3P2 to the right). Juries could pause and leave the room at their discretion as long as they did not talk to others about the current experience.

After about 40 minutes of reading the transcription documents they had prepared, the juries took between 2h00 and 3h30 to model the exchange on paper. It should be noted that the BH3 dyad was chosen for modelling by the pair of experts (alpha jury) because it is dense, rather complex and exhaustive in terms of representation of the different components of the model.

Results

The results of the interrater agreement validation among juries are presented in Tables 8 and9. We compared the results obtained by the 3 non-expert juries (jury X, Y and Z) with the reference alpha expert jury. The percentages agreements and Cohen's kappa coefficients were calculated between each combination of two juries, i.e. a total of 3 interrater agreement comparisons :

• jury X versus jury alpha,

• jury Y versus jury alpha,

• jury Z versus jury alpha. Results show that for the type of decision and the length of the exchange there was little or no agreement between expert and non-expert juries. For length of the exchange, this disagreement does not have a decisive impact on exchange typology but may reflect a difference in precision e.g., judges did not end the exchange in the same place because they aggregated two instructions or separated a single exchange into several. Conversely, concerning the type of directive and the type of other action, interrater agreements between juries were mostly moderate with one strong agreement.

-Discussion

The objective of this work was the formalization of a model to analyze conversations in dyad cooperation situations. Two points are essential in this model: 1-determining a typology to characterize the nature of conversational exchanges (in terms of type of exchange, final decision, interaction, etc.) and 2-proposing a visual representation to illustrate the exchanges and their temporal dynamics.

In order to achieve this goal, we used different methods from the field of psychology, ranging from user-centred design to psychometric and an ecological approach to method validation. The design process (Fig. 2) begins with the development of a model and then with its validation using focus groups and card sorting sessions. Then the model is validated experimentally. The results are analyzed by several experts and the different points of view are compared by means of inter-rater measurements. This process allows the model to be validated in the analysis.

What motivated the development of this methodology is the observation that there is no satisfactory methodology for modelling conversational activity. Indeed, the existing methods are not easily usable and their manipulation is complex. Here we have a process for designing and validating a new analysis model adapted to a particular situation and accessible to non-expert experimenters.

To validate the model, we applied it to a dyad cooperation situation -the Tetris game -and choose to use a mixed methods approach. We performed two validation steps : a qualitative one to validate the categories of our typology and their wording, and a quantitative one to test the ease of use and reliability thanks to statistical analysis on our qualitative data.

The following characteristics of a conversation exchange were included in the model: performative, directive or non-directive type, type of decision, assent or dissent, the type of interaction, socioemotional or task-related. The model considered the most relevant theories from the literature in order to have a theoretical framework for each of the categories of analysis model typology.

For the first validation, we refined the typology using qualitative methods derived from usercentered design (ISO 9241-210:2019(ISO 9241-210: , 2019;;Zarour and Alharbi, 2017), namely focus group and card sorting sessions. These qualitative methods made it possible to measure the degree of usability of our model and to guarantee a consensus on the choice of categories and their wording. As illustrated in Table 2, this allowed defining the types of exchange precisely by clearly separating directive actions (order, instruction and proposal) from non-directive actions (soliloquy, commentary, non-verbal action). This first stage of validation also made it possible to enrich the typology by adapting theoretical names and categories to a concrete analysis situation. This showed that theoretical models may be far from field work and may need refinement by considering the users' point of view to adapt to a particular situation.

Concerning the second validation, we performed statistical analysis on our qualitative data to test reliability and degree of transferability of the analysis model, i.e. whether it could be used by people other than its designers and lead to similar results. We used Cohen's kappa coefficient as a concordance index, this allowed precise measurement of interrater agreement on qualitative variables. Since Cohen's kappa (1960) considers the theoretical proportion of "random" agreements, it is possible to measure the degree of true agreement between two raters based on the distribution of their responses (Santos, 2018).

We measured two types of interrater agreement: between two expert judges and between two juries, one composed of a pair of experts and the other of a pair of novices. These interrater agreement validations allowed us to specify the conditions of use of our analysis model according to the expertise of the raters. Thus, an expert judge will be able to use the model alone whereas for novice juries it is preferable to be in pairs to obtain satisfactory results. In all cases, a training and an understanding of the typology of the exchanges and the graphical representation was necessary before using the model.

Without this quantitative arm to the study, we would have been left with inconclusive results on the reliability and degree of transferability of our model.

Using the mixed-method technique called joint display, we combined qualitative and quantitative results to find possible improvements. We compared some final graphical representations to capture and understand the dissonance and congruence between disagreeing judges. Although the analysis model was validated, the typology could be further improved, especially concerning the length of the exchanges and the type of final decision. The disagreements seen here did not have a decisive impact on the typology of exchanges but may reflect a difference in precision: judges may aggregate two directives or, on the contrary, detail an exchange into several. It would be interesting to adjust the transcription grid, either by making the calculation of length of the exchange more objective, or by modifying the type of final decision variable so that it is no longer dependent on the length of the exchange.

The analysis model detailed here allowed us to objectively characterize the dynamics within a dyad in a simple and experimental cooperation situation, namely a cooperative Tetris game. The model could likely also be used as an analytical framework in characterizing the emergence of social roles in a group dynamic such as leadership, followership, etc.

One of the strengths of this analytical model is the consideration of temporality, which allows observation of the dynamics of leadership within a dyad. The fluctuations and the eventual transitions of decision-making power are important and could be lost in a more global approach that did not consider the time variable. Similarly, the possibility of adding events in this model contextualizes exchanges and their associated fluctuations making it possible to observe whether leadership changes hands when the situation becomes complicated or after a failure. The qualifiers of interaction (positive or negative remarks) and the type of final decision allow determination of type of leadership and followership associated with the observed dyad.

It is quite possible to associate our analysis model with other tools such as personality tests.

For example, the Big Five Inventory test developed by John et al. ( 1991) combined with our model would make it possible to study the emergence of leadership. Authors such as Judge et al. (2002), for example, have used the Big Five Inventory and shown the link between the notion of leadership and the personality traits of extroversion, openness and awareness, making this type of personality test an interesting and relevant analytical tool for studying the emergence of leadership. Similarly, the desire for control scale developed by [START_REF] Burger | Desire for Control and Interpersonal Interaction Styles[END_REF] could be combined with our model. [START_REF] Burger | Desire for Control and Interpersonal Interaction Styles[END_REF] explores the relationship between desire for control and interpersonal interaction style. In a study on the behavior of students with high and low desire for control during a structured interview, Dembroski et al. (1984) had already highlighted the tendency of individuals with a high desire for control to express themselves more strongly, with a more explosive and rapid intonation and to be more competitive in their verbal interactions. [START_REF] Burger | Desire for Control and Interpersonal Interaction Styles[END_REF] showed that people with a high desire for control can also adopt a reserved interaction style in order to control the information communicated. The link between the degree of desire for control and the style of interaction adopted seems relevant to an analysis of leadership dynamics.

-Conclusion and perspectives

Our model has so far only been applied to a dyad cooperation situation, in a controlled context and on a simple cognitive task. A next step is to model a cooperation situation on a more complex task with less clearly defined roles for each participant. Games designed as collaborative such as cooperative board games or escape games, could be a good support for experimenting with the application of the method to another task. If adaptation of the method to more complex interactions is possible, it could then be applied to other real-life situations.

Real situations of dyad cooperation are not lacking in teaching and learning contexts. Authors such as Johnson et al. (1990) and Slavin (1990) have shown that working in small groups with a common goal optimizes learning. In the area of learning team sports in school, authors such as Doise andMugny (1997) andGilly, Fraisse andRoux (1988) have studied the impact of exchanges between game sequences on the tactical skills developed. In these situations, two players must coordinate their actions in the game and cooperate in conversational exchanges in order to conceptualize their actions and develop tactical problem-solving strategies. It would be interesting to test our model with this type of real dyad cooperation.

The extension of the method to the modelling of conversational exchanges of more than two people seems more complex. The ACDC model would be suitable with some adjustments, such as e.g., adding categories for identity of the recipient(s). On the other hand, the proposed graphical representation is for the moment limited to the modelling of a dialogue within a dyad and will therefore have to be rethought if it is to be extended to larger groups.

Fig. 7: Diagram describing the ACDC method with steps that can benefit from automation or semiautomation

In the longer term, we can envisage automating all or part of the process as illustrated in Fig. 7. The temporal visualization in chronological form could be directly generated by a software program from the data matrices. We could automate / semi-automate the modelling process by inputting the sound file and asking an artificial intelligence software program to cut out and characterize the exchanges on the basis of the proposed symbolism and according to rules imposed beforehand.

Currently, the rules are understandable by a human, but there is probably still too much subjectivity to obtain a systematic result. The use of learning neural networks, supervised or not, could open interesting perspectives in the classification of exchanges. This could allow us to eventually optimize the model, but more importantly, it could perhaps shed light on variables common to certain leadership profiles that humans would not have identified.

In future research we will focus on two aspects, the transfer of the method from other situations and the automation of the different steps.

Fig. 1 :

 1 Fig. 1 : Illustration of the experimental set-up: Two participants separated by a screen playing the same game of Tetris.

  Step 1: Development of the model based on the literature; • Step 2: Validation of the categories used and their wording using a focus group and a card sorting session; • Step 3: Experimentation phase, dyads are formed and carry out the task of cooperative Tetris. The conversations are recorded and constitute our corpus of data; • Step 4: Application of the ACDC analysis model to the data; • Step 5: Calculate interrater agreements between judges and juries to evaluate the ease of use of the analysis model and its reliability; • Step 6: Final formalization of the model and its best practices guide.

Fig. 2 :

 2 Fig. 2: The stages of the design process.

Fig. 3 :

 3 Fig. 3 : Graphical representation of the ACDC model for the first 5 minutes of conversational exchanges for the dyad BH3 (jury alpha : 2 experts). Refer to the typology symbols of the ACDC model(Table3).

Fig. 4 :

 4 Fig. 4 : Illustration of quantitative data visible in an example of graphical representation of the ACDC model (Fig. 3). Vertical full lines indicate any type of exchange. Full squares represent any type of interaction. Refer to the typology symbols of the ACDC model (Table3).

  , but to involve the other participant -> questioning P1 P1 answer by a directive -> initiator of the action P1 give a directive, P2 answer, P1 answer -> length of the exchange = 2 P1 says what he/she would do -> proposal P2 does not conflict -> agreement 3. Formalization Example 2 1. Retranscription P1 : "You can put it on the edge, at the far right" P2 : " 'Fuck you're good !" 2. Characterization P1 initiates the exchange P1 speaks to P2 to tell him/her what to do -> instruction P1 gives a directive, P2 answers -> length of the exchange = 1 P2 compliments P1 -> positive feedback to P1 P2 answers positively to P1 -> agreement 3. Formalization Table 4 : Detail of the re-transcription of two exchanges : re-transcription, characterization and formalization according to our ACDC model 4.2 -Application of the model to our corpus: example of a dyad (BH3)

  0.20: Very low agreement • 0.21 -0.40: Weak agreement • 0.41 -0.60: Moderate agreement (*) • 0.61 -0.80: Strong agreement (**) • 0.81 -1.00: Almost perfect agreement (***)

Table 1 :

 1 Comparison of the 3 types of dyad situations : coordination, collaboration and cooperation.

		Common goal	Interdependant responsabilities
	Coordination	Not necessarily	No
	Collaboration	Yes	"No" rather than "Yes"
	Cooperation	Yes	Yes

Category Origin of the category Types of exchange : directive vs non-directive action Searle (1990), Vernant (1992)

		Order
			Coming from the Focus
	Directive action	Instruction	group and card sorting
			sessions
		Proposal
		Soliloquy	Coming from the Focus
	Non-directive action	Comment	group and card sorting
			sessions
		Non-verbal action
	Types of decision : assent vs dissent	Bales (1950; 1972)
		Agreement
	Assent	Acceptance Authorization	Adapted from Bales (decomposition cat. 3)
		Dealership
	Dissent	Refusal Undetermined	Adapted from Bales (decomposition cat. 10)

Qualifiers of interaction : task-related vs socio-emotional Bales (1950; 1972)

	Communication of information	Adapted from Bales (merging cat. 5 et 6)
	Questioning	Adapted from Bales (merging cat. 7, 8, 9)
	Task-related
	Control	Bales (cat. 4)
	Positive game-oriented remark Coming from the Focus
		group and card sorting
	Negative game-oriented remark	sessions
	Positive feedback to the partner	Adapted from Bales (merging cat. 1, 2)
	Socio-emotiional related
	Negative feedback to the partner	Adapted from Bales (merging cat. 11, 12)

Table 2 :

 2 Summary of the origin of the categories used in the ACDC model typology.

Table 3 :

 3 Typology symbols of the ACDC model.

Acknowledgement

We are grateful for the support of our interns Sarah Till and Clara Noël, who helped us to carry out the cooperative tetris experience and collect our corpus of data.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflict of interest

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.

• Clark, H. H., & Brennan, S. E. (1991). Grounding in communication. In L. B. Resnick, J. M. Levine, & S. D. Teasley (Eds.), Perspectives on socially shared cognition (p. 127-149). American Psychological Association. https://doi.org/10.1037/10096-006 • Cohen, J. (1960). A coefficient of agreement for nominal scales. • Gilly, M., Fraisse, J., Roux, J. P., & Roux, J. P. (1988). Résolution de problèmes en dyades et progrès cognitifs chez des enfants de 11 à 13 ans: dynamiques interactives et mécanismes sociocognitifs.

• Gulati, R., Wohlgezogen, F., & Zhelyazkov, P. ( 2012