M Mailard

J Chevalier
email: jerome.chevalier@insa-lyon.fr

L Gremillard

G P Baeza

E-J Courtial

S Marion

V Garnier

Keywords: Additive manufacturing, Robocasting, Rheology, Strength, Fracture

Robocasting, or Direct Ink Writing, can be used to create dense mono-or multi-materials ceramic parts using micro-extrusion of ceramic pastes through needles, whose position is controlled in 3D. Rheological properties of the ceramic pastes, printing parameters and thermal post processes (drying, debinding and sintering) are key parameters to control the quality of the printed parts. In this work, the rheological properties (including yield stress, shear-thinning behavior, storage/loss moduli and recovery time) of alumina pastes were characterized. Correlations were established between on one side the rheological properties and the printing conditions and on the other side the extrudability, the shape fidelity and the mechanical performance of the final parts. This paper thus defines an extended definition of printability, which includes functional requirements of the final parts in addition to the more classical processability criteria.

Introduction

Owing to its development in many domains, Additive Manufacturing (AM) of ceramics is becoming a key technology with a constantly growing market share [START_REF] Olhero | Conventional versus additive manufacturing in the structural performance of dense alumina-zirconia ceramics: 20 years of research, challenges and future perspectives[END_REF]. Robocasting, also called Direct Ink Writing (DIW), is one of the AM technologies enabling to build mono or multi-material ceramics layer by layer [START_REF] Chen | 3D printing of ceramics: A review[END_REF][START_REF]Women in 3D Printing, From Bones to Bridges and Everything in Between[END_REF][START_REF] Peng | Ceramic robocasting: recent achievements, potential, and future developments[END_REF][START_REF] Zocca | Additive manufacturing of ceramics: issues, potentialities, and opportunities[END_REF][START_REF] Chen | Dense ceramics with complex shape fabricated by 3D printing: A review[END_REF][START_REF] Lakhdar | Additive manufacturing of advanced ceramic materials[END_REF]. It consists in the continuous extrusion of a ceramic paste through a thin nozzle (generally 0.2-0.9 mm), the position of which is controlled in 3D. The extruded material is designated by different names in the literature (paste, ink, feedstock…); the term 'paste' will mostly be used throughout the present article. The robocasting process starts with the definition of a Computer Assisted Design (CAD) model. The design is then sliced, this step providing a definition of the nozzle path. The choice of an adequate slicing software is very important in order to minimize printing defects specially at the surface of printed part [START_REF] Smay | Directed Colloidal Assembly of 3D Periodic Structures[END_REF], as evidenced by Gleadall et al. who have shown the importance and the link between the success of the printing process and the software itself [START_REF] Gleadall | FullControl GCode Designer: Open-source software for unconstrained design in additive manufacturing[END_REF][START_REF] Gleadall | VOLCO: A predictive model for 3D printed microarchitecture[END_REF].

Although its general concept is rather simple, the success of robocasting requires ceramic pastes with specific rheological properties and appropriate processing parameters. In simple words, a ceramic paste is a mixture of a ceramic powder, organic components (binder, dispersant, and gelling agent) and a liquid carrier medium (mostly water). Non-Newtonian shear-thinning behavior is required in order to let the material, originally 'solid' in the syringe, become 'liquid' under shear stresses (generated by the shape of the nozzle and the printing speed) to flow through the nozzle [START_REF] Chan | Direct ink writing of hierarchical porous aluminastabilized emulsions: Rheology and printability[END_REF][START_REF] Barki | A. 3D Printed boehmite based objects[END_REF]. This physical phenomena is possible due to the reorganization of particles and polymers under shear stress. To describe this behavior, the Ostwald-de Waele's relation is often proposed [START_REF] Ovarlez | Caractérisation rhéologique des fluides à seuil[END_REF]:

𝜏 = 𝑘(𝛾) ̇𝑛 (Eq. 1)
𝜏 and𝛾̇ are the shear stress and strain rate respectively, where k is constant and n represents the difference with a Newtonian behavior (n = 1 for a Newtonian fluid). The apparent viscosity is then defined as 𝜂 = 𝑘𝛾̇𝑛 -1 . Shear-thinning is observed when n < 1, due to the decrease of viscosity under rising stress. A refinement of Eq. ( 1) is generally proposed on the basis of the Herschel-Bulkley equation (Eq. ( 2)), which includes a yield stress [START_REF] Ovarlez | Caractérisation rhéologique des fluides à seuil[END_REF]:

𝜏 = 𝜏 𝑦 + 𝑘(𝛾̇) 𝑛 (Eq. 2)
Indeed, the process of extrusion requires an elastic shape retention of the paste after extrusion [START_REF] Del-Mazo-Barbara | Rheological characterisation of ceramic inks for 3D direct ink writing: A review[END_REF]. This translates into sufficient yield stress and elastic modulus (often called stiffness) to avoid layers collapse, guarantee a good stability and thus the desired shape and dimensions [START_REF] Courtial | Silicone rheological behavior modification for 3D printing: Evaluation of yield stress impact on printed object properties[END_REF].

Defining viscosity and yield stress are first steps to characterize ceramic pastes, but a necessary step further consists of defining the overall viscoelastic behavior, often characterized through oscillatory measurements and resulting into the definition of a complex shear modulus (G*) such as [START_REF] Larson | A review of thixotropy and its rheological modelling[END_REF]:

𝐺 * (𝜔) = 𝐺′(𝜔) + 𝑖𝐺′′(𝜔) (Eq. 3)

In the above expression, 𝐺'and 𝐺'' are called the storage and loss moduli, respectively assigned to the in-phase (𝜑 = 0) and out-of-phase (𝜑 =

) responses of the viscoelastic material, ω is the angular frequency. The complex viscosity η* can be derived from these data using:

|𝜂 * | = |𝜂 ′ + 𝑖𝜂′′| = √𝐺′ 2 +𝐺′′ 2 𝜔 (Eq. 4)
While G' represents the elasticity of the material, i.e., its 'solid-like' behavior, G'' represents the dissipated energy, i.e., its 'liquid-like' behavior. Therefore, the G''/G' ratio, often expressed as a function of the phase shift between the stress and the strain, tan (), gives a relative indication on the fluidity of the material: G''/G' < 1 characterizes a 'solid-like' behavior, whereas G''/G'> 1 characterizes a 'liquid-like' material [START_REF] Tabard | Robocasting of highly porous ceramics scaffolds with hierarchized porosity[END_REF]. The stress for which G'=G'' is often called flow stress, f.

To determine these moduli, oscillatory rheology is performed. This rheological evaluation leads to the evolution of G' and G'' as a function of several variables such as the strain amplitude, the frequency of the solicitation, the time or even the temperature. Amplitude tests (or dynamic strain sweep) give access to the stiffness of the paste, which is defined as the constant G' value observed in the Linear Viscoeslatic Region (LVR), before it decreases with strain amplitude. This value is often called G'eq [START_REF] Instruments | A Basic Introduction to Rheology[END_REF].

All these rheological studies are widely used in bioprinting literature [START_REF] Champeau | 4D printing of hydrogels: a review[END_REF][START_REF] Fu | Printability in extrusion bioprinting[END_REF][START_REF] Lemarié | Rheology, simulation and data analysis toward bioprinting cell viability awareness[END_REF][START_REF] Lopez | FingerMap: a new approach to predict soft material 3D objects printability[END_REF] and start to pervade in the field of ceramics [START_REF] M'barki | Linking Rheology and Printability for Dense and Strong Ceramics by Direct Ink Writing[END_REF]. If aims of rheological studies in bioprinting are for example to study cells survival in the hydrogel during the printing process [START_REF] Naghieh | Printability-A key issue in extrusion-based bioprinting[END_REF][START_REF] Schwab | Printability and Shape Fidelity of Bioinks in 3D Bioprinting[END_REF], for ceramic science, attention must be given to the absence of printed defects to reach sufficient mechanical properties [START_REF] Feilden | 3D printing bioinspired ceramic composites[END_REF][START_REF] Feilden | Robocasting of structural ceramic parts with hydrogel inks[END_REF][START_REF] Neto | Synthetic and marine-derived porous scaffolds for bone tissue engineering[END_REF].The concept of 'printability' is something very complex and strongly related to the additive manufacturing process itself and the final properties targeted. Lot of authors gave some criteria in order to optimize entire printing processes from formulation to printing parameters. Those criteria are still discussed and need to be completed [START_REF] Corker | 3D printing with 2D colloids: designing rheology protocols to predict 'printability' of soft-materials[END_REF].

At this stage, it is important to define exactly how the yield stress is precisely obtained. Generally, it is described as the stress needed to make the fluid start or stop moving and it usually defines the highest stress where no flow is detected by the rheometer [START_REF] Dinkgreve | On different ways of measuring "the" yield stress[END_REF][START_REF] Coussot | Avalanche behavior in yield stress fluids[END_REF][START_REF] Chen | Rheological Techniques for Yield Stress Analysis[END_REF]. In fact, according to rheological definitions and general literature, three different yield stresses could be defined:

1. the static yield stress (𝜏 𝛾 𝑆𝑇𝐴𝑇 or 𝜎 𝛾 𝑆𝑇𝐴𝑇 ), which corresponds to the shear stress necessary to initiate a flow when increasing the shear stress. 2. the dynamic yield stress (𝜏 𝛾 𝐷𝑌𝑁 𝑜𝑟 𝜎 𝛾 𝐷𝑌𝑁 ), which is the minimal stress required to maintain the flow when decreasing the shear stress from the 'liquid-like state'.

3. and, to finish, the flow stress (f or f ), which is the stress for which G'=G''.

In addition to the conventionally used yield stress(es)  and viscosity η, the use of extended rheology measurements and complex shear modulus enables the definition of other criteria of interest as 'figures of merit' of ceramic pastes. For example, to ensure a minimal deformation of overhanging structures and a suitable extrudability, Lewis et al. proposed that the ratio ф (G'eq/ ) should be higher than 20. Another quantitative criterion from amplitude sweep corresponding to the ratio of yield stress and the flow stress, called 'FTI' for Flow Transition Index (it should be lower than 20), is commonly used to described 'printability' of paste [START_REF] Lewis | Direct Ink Writing of 3D Functional Materials[END_REF], [START_REF] Lewis | Colloidal processing of ceramics[END_REF]. This criterion, though, could not display a clear trend according to Corker et al. [START_REF] Corker | 3D printing with 2D colloids: designing rheology protocols to predict 'printability' of soft-materials[END_REF]. To account for possible slumping of the printing structures due to both gravity and surface tension, Smay et al. determined that the dynamic yield stress should be higher than the sum of the gravity forces, leading to the criterion ℎ 𝑚𝑎𝑥 = 𝜏 𝛾 𝐷𝑌𝑁 𝜌𝑔 [START_REF] Champeau | 4D printing of hydrogels: a review[END_REF][START_REF] Smay | Colloidal Inks for Directed Assembly of 3-D Periodic Structures[END_REF] (hmax is thus the maximum possible printable height with a given paste) [START_REF] Smay | Colloidal Inks for Directed Assembly of 3-D Periodic Structures[END_REF]. This obvious and physically grounded criterion can also be improved by taking into account the surface tension of the printed material, thus becoming: Ξ = 𝜏 𝛾 𝐷𝑌𝑁 𝜌𝑔ℎ+𝛾𝑅 -1 (Ξ> 1 is necessary to ensure shape retention after printing).

Another important, although somewhat 'under-emphasized', effect in viscoelastic material is thixotropy, which corresponds to a time-dependent shear-thinning property [START_REF] Mewis | Thixotropy -a general review[END_REF][START_REF] Mewis | Thixotropy[END_REF]. In practical terms, the viscosity of a thixotropic material not only depends on stress and strain rate, but also on time and shear history. Due to this time-dependence, when loading a paste first at high strain rate, then at lower (e.g. zero) strain rate, a stress relaxation will occur. This relaxation is related to timedependent microstructural evolutions ('healing') of the paste to come back to its original state. The time required to obtain a new viscosity similar to the original one is called 'recovery time' (or healing time) and has to be considered in order to choose the size of the printed part.

All the rheological requirements presented above are summarized in Fig. 1. Although these rheological properties are essential to ensure a good extrusion of the paste and no collapsing of the printed structure during the printing process, they do not fully ensure good performances of the final part (porosity, inter-filaments defects, surface state, and thus mechanical properties are not addressed by the above-mentioned criteria). Indeed, the quality of the printed part will depend on three families of parameters. First, similarly to all other ceramic processes, composition of pastes and their homogeneity are important factors to guarantee the least possible defects. For example, increasing the volume fraction of ceramic particles will lead to rheological modifications which could be harmful for a safe homogeneity during all the process. Second, environmental parameters, such as temperature and humidity, are also essential to avoid cracks during the printing process. The homogeneity of the filament can be mainly maintained under high humidity to avoid cracks due to too rapid drying [START_REF] Feilden | Additive Manufacturing of Ceramics and Ceramic Composites via Robocasting[END_REF][START_REF] Tabard | Elaboration de céramiques architecturées pour le stockage d'énergie thermique[END_REF][START_REF] Lorenz | Influence of platelet content on the fabrication of colloidal gels for robocasting: Experimental analysis and numerical simulation[END_REF]. Third, the choice of printing parameters will influence shape fidelity and final geometries, so indirectly, final properties: for example, printing speed will influence the overall aspect of the printing filament and its shape [START_REF] Trincat | Influence of the printing process on the traces produced by the discharge of 3Dprinted Liberators[END_REF][START_REF] Allum | Interlayer bonding has bulk-material strength in extrusion additive manufacturing: New understanding of anisotropy[END_REF]; the degree of overlap and size of the nozzle will influence the shape fidelity of ceramics due to the topography of the printed part related to layer stacking.

Figure 1: Rheological requirements in DIW process [START_REF] Zocca | Additive manufacturing of ceramics: issues, potentialities, and opportunities[END_REF][START_REF] Lopez | FingerMap: a new approach to predict soft material 3D objects printability[END_REF][START_REF] M'barki | Linking Rheology and Printability for Dense and Strong Ceramics by Direct Ink Writing[END_REF][START_REF] Corker | 3D printing with 2D colloids: designing rheology protocols to predict 'printability' of soft-materials[END_REF][START_REF] Lewis | Direct Ink Writing of 3D Functional Materials[END_REF][START_REF] Franchin | Optimization and Characterization of Preceramic Inks for Direct InkWriting of Ceramic Matrix Composite Structures[END_REF].

Finally, following printing, thermal treatments are necessary, first to dry the printed parts, then to debind and sinter them. These treatments will also influence the final properties, since they can induce defects such as drying cracks or distortions due to non-homogeneous shrinkage. All this post-printing step will thus influence the final functional (mechanical) properties [START_REF] Garnault | Frittage par micro-ondes de céramiques mises en forme par impression 3D. Génie des procédés[END_REF][START_REF] Hugonnet | Frittage et évolution de la microstructure au cours des traitements thermiques d' aimants NdFeB: influence sur les propriétés magnétiques[END_REF][START_REF] Li | Effects of Ceramic Density and Sintering Temperature on the Mechanical Properties of a Novel Polymer-Infiltrated Ceramic-Network Zirconia Dental Restorative (Filling) Material[END_REF][START_REF] Camposilvan | Surface stability and small-scale testing of zirconia[END_REF].

In this article, a multi-criteria approach to obtain printed ceramics with good shape and mechanical properties is emphasized. DIW technology was used to fabricate dense alumina pieces with two geometries (dense bars and hollow tubes). The influence of the rheology of the pastes at all steps of the process was evaluated. The rheology of the pastes was modified by varying the solids loading, nature and amount of the gelling agent and amount of the dispersing agent. Finally, mechanical strength of printed parts was measured. This work examines the correlations between all these parameters in order to extend the definition of "printability" beyond rheological properties only.

Materials and methods

Raw materials

Pluronic® F-127 (polyethylene oxide -polypropylene oxide -polyethylene oxide triblock copolymer, referred to as PF127) and Poly-Ethylene Glycol (PEG with molecular weight 4,000 g.mol -1 and 20,000 g.mol -1 respectively called hereafter PEG4000 and PEG20000) were purchased from Sigma Aldrich (Saint-Quentin-Fallavier, France) and used without further purification. A sodium polyacrylate (Darvan® 811, RT Vanderbilt, Norwalk, USA) was used as dispersing agent. The Alphaalumina ceramic powder was provided by Baikowski (Poisy, France); it was a Baikalox ® High Purity Alumina WA15 powder, with a specific surface area of 23.6 m 2 /g and a D50 of 0.15 µm. Milling balls (mixture of 2 mm diameter alumina balls and zirconia (Mg-PSZ) balls) were used to facilitate the homogenization of the pastes.

Ceramic pastes preparation

Pastes based on 2 different gelling agents (PF127 +PEG4000 or PEG20000) were prepared following identical steps. The first step was the preparation of the polymeric solution.

In the case of the PF127-based pastes, PF127 solutions at 20% w/v (20 g in 100 cm -3 of water) were prepared by mixing adequate amounts of PF127 powder and deionized water at 4 °C with a magnetic stirrer. The solution was then mixed with cycle at 2000 rpm for 2 min using a Speedmixer (Hauschild GmbH & Co.KG, Hamm, Germany) and stored at 0 °C for 24 h. In parallel, a PEG 4000 aqueous solution at 3% w/v was prepared by stirring PEG4000 powder with water using a magnetic stirrer, at 55-60 °C. After cooling, the PEG4000 solution was added to PF127 solution.

In the case of the PEG20000-based pastes, PEG solutions at 20% w/v and 10% w/v (with the PEG20000) were prepared by stirring PEG powder with water at 55-60 °C. Solutions were then mixed at 2000 rpm for 2 min using the Speedmixer and stored at 0 °C for 24 h.

Then the preparation steps were common for all pastes: 1. Addition of the dispersing agent (Darvan ® 811) 2. Addition of the mixing balls (same weight as the ceramic powder) 3. Addition of the alumina powder, in two steps in order to optimize the dispersion of the powder and the homogenization of the paste. After each addition the mixture was homogenized in the Speedmixer (4 cycles of 1 min at 1500-1850 rpm, with 2 min cooling in an ice bath between each cycle).

Four pastes were selected from a prior preliminary study to focus on the most demonstrative effects. Two of them were based on PF127 20% w/v solution and contain 38 or 42 vol% alumina powder. They are labeled respectively ADP38 and ADP42. A third paste was based on a 10% w/v PEG20000 solution and contained 38 vol% alumina. It is labeled ADPeg38. The last paste (labeled ADPeg38_20) was based on a 20% w/v PEG20000 solution and also contained 38 vol% alumina. The compositions of the pastes are detailed in Table [START_REF] Olhero | Conventional versus additive manufacturing in the structural performance of dense alumina-zirconia ceramics: 20 years of research, challenges and future perspectives[END_REF].

Rheological properties

For each ceramic paste, the following rheological tests were carried out using an ARES straincontrolled rheometer (Rheometrics instruments, USA) equipped with 2kFRTN1 transducer. The geometry used was a 25 mm cone-plane geometry with a cone angle of 0.1 rad and 0.05 mm gap. Measurement was performed keeping a constant temperature of 21 °C. The different rheological tests performed could be segmented into 3 categories:

-Oscillatory strain sweeps (controlled shear deformation) allow to determine the (nondestructive) deformation of the pastes at small strains and study the behavior in the Linear Visco-Elastic Region (LVER), through the evolution of storage and loss moduli (G' and G'') versus strain amplitude. They were conducted at a constant angular frequency of 10 rad/s, on three decades of deformation (0.1-100%). Value of the flow point 𝜏 𝑓 , static yield stress 𝜏 𝛾 and limits of the LVER were obtained. -Oscillatory frequency sweeps allow to describe the viscoelastic behavior of the pastes as a function of the frequency. High frequency values are used to mimic fast deformation. Low frequency values will mimic the 'steady' behavior in the syringe of the paste. Frequency sweep tests are ideal to study the internal microstructure and have an idea about the stability of the paste with time. These tests involved a frequency ramp from 0.01 to 100 rad/s, at a constant stress of 0.1 kPa (selected in agreement with the previous results, to remain on the plateau zone of G') in order to determine tan() and the viscoelastic behavior of the pastes. -Start-Up tests were conducted in order to determine the recovery time and the thixotropic behavior of formulation. They consisted in the measurement of the evolutions of the storage modulus and viscosity with time under three successive conditions: (1) a pre-shear of 3 s -1 for 120 s (to characterize the paste before extrusion), ( 2) a high shear-strain rate of 60 s -1 for 10 s (mimicking the extrusion) and ( 3) a shear rate of 3 s -1 during 500 s. The recovery time corresponds to the time needed for the paste to recover a stiffness or a viscosity similar (for example 90%) to its initial value [START_REF] Chen | A novel thixotropic magnesium phosphate-based bioink with excellent printability for application in 3D printing[END_REF]. -All rheological parameters obtained from these different tests are summarized in Table [START_REF] Chen | 3D printing of ceramics: A review[END_REF]. Stress for which G'=G'', which is considered as the transition between a 'liquid-like' and a 'solid-like' paste tG' Time needed to recover "high" viscosity or initial rigidity after the extrusion. Here tG' was defined as the time necessary to recover 90% of viscosity at rest (without shear).

Printing conditions

A Robocasting system (3D-Inks, Tulsa, USA) was used for all printings, controlled by the Robocad software, a graphical user interface designed to produce G-code interpretable by the printer. All the samples were printed with 0.410 mm diameter nozzles and the printing speed was fixed to 10 mm/s to obtain continuous filaments during the printing process. A layer height of 0.33 mm (corresponding to an overlap in the z-axis of 20% of the nozzle diameter) was chosen to allow good interpenetration of layers (and thus avoid large defects between filaments). With the same goal, a pitch of 0.4 mm was chosen (the pitch is the distance between the centers of two adjacent filaments).

Two types of samples were printed: 1. Bending bars with dimensions (l x w x h) of 36 ✕4✕ 5 mm 3 (dimensions after printing) consisted of 15 layers. Each layer was printed lengthwise, and was shifted widthwise by a filament half-diameter with respect to the previous layer to avoid longitudinal empty channels between each filament and obtain best final properties, in agreement with Feilden et al. [START_REF] Feilden | Robocasting of structural ceramic parts with hydrogel inks[END_REF]. 2. Hollow cylinders (outer diameter, height, thickness after printing: 20 ✕ 20✕ 1.2 mm3) were processed with a stack of 37 layers consisting of 3 filaments by layer. Each layer was printed directly on top of the previous one, without horizontal shift contrary to the bending bars, due to the small thickness of the structures.

Drying, debinding and sintering processes

Ceramics parts were dried in a climatic chamber (WKL 150 Weiss Technik, France). At first the samples were put in the climate chamber (25 °C and 98% HR) and the relative humidity kept constant at 98% HR while temperature was increased up to 60 °C during 1.5 h. Then the relative humidity was decreased to 30% in about 4 days while the temperature was maintained at 60 °C.

The debinding process consisted in a slow heating (0.5 °C/min) followed by a 30 min dwell at 400 °C, then a 1 °C/min heating to 600 °C with a 30 min dwell, followed by natural cooling. Sintering was conducted at 1500 °C with a 1-hour dwell (5 °C/min heating rate, natural cooling).

Solid state mechanical properties

Mechanical tests were performed with an INSTRON 8800 universal testing machine (Norwood, USA) with a 5 kN load-cell, at room temperature.

Four-point bending tests were conducted on as-sintered samples (i.e. without surface grinding nor polishing, in order to characterize the real surface state of components after 3D printing), at a crosshead speed of 0.5 mm/min with a support span L (mm) of 21 mm and inner span L' of 10 mm.

The flexural strength f (MPa) was obtained by:

𝜎 𝑓 = 3 2 𝐹 𝑀 ( 𝐿-𝐿′ 𝐵𝑊 2 ) (Eq. 5)
where FM is the load to failure, B and W the width and the height of sample respectively (measured after sintering on each sample).

Tubes were tested using a diametral flexural strength configuration. They were positioned with the printing origin at 45° with respect to the testing direction, so that the printing origin is not on a location where the stress is maximal. The strength was derived from the load to failure by the equation [START_REF] De With | Note on the use of the diametral compression test for the strength measurement of ceramics[END_REF]:

𝜎 𝑓 = 2𝐹 𝑀 𝑟 ( 6𝑟-𝑡 𝜋𝑙𝑡 2 (2𝑟-𝑡) ) (Eq. 6)
Where t is the wall thickness, r the mean radius, l is the length and FM is the fracture load.

Finally, Weibull moduli were deduced from the 4-point bending strengths, using two-parameters Weibull expression:

𝑃 𝑓 = 1 -𝑒𝑥𝑝 (-( 𝜎 𝜎 0 ) 𝑚 ) (Eq. 7)
where Pf is the probability of failure at applied stress , 0 the characteristic strength for which the probability of failure is 63% and m is the Weibull modulus, which gives an indication of failure distribution (low m value meaning a large variability). A careful and precise Weibull modulus analysis requires numerous specimens to be tested (typically more than 30, according to the ISO standards and according to literature [START_REF] Davies | Confidence limits for Weibull parameters estimated using linear least squares analysis[END_REF]). As we were limited to 10 (hollow structures) to 18 samples (bars), Weibull moduli obtained are given as indications only and will be associated to a confidence interval. A 95% confidence interval was given for both the characteristic strength and the Weibull modulus, following the work of Davies [START_REF] Davies | Confidence limits for Weibull parameters estimated using linear least squares analysis[END_REF].

Microstructural characterization

Densities of printed part were evaluated after sintering by the Archimedes method. Macroporosity assessments were also conducted by SEM analysis. Microstructures were characterized by Scanning Electron Microscopy (SEM) using a Supra40VP microscope (Zeiss, Oberkochen, Germany). Fracture surfaces of selected samples were analysed according to Morrell [START_REF] Morrell | Fractography and fracture toughness measurement[END_REF] in order to determine the fracture origin.

Results

Viscoelastic properties and Linear domains of the ceramic pastes

Results of oscillatory strain sweep tests are illustrated in Fig. [START_REF] Chen | 3D printing of ceramics: A review[END_REF]. They confirm the influence of solids loadings in four formulations: ADP38, ADP42, ADPeg38 and ADPeg38_20 and show their rheological behavior. PF127 (20%w/v) itself was also evaluated and showed much lower stiffness. All pastes, and PF127 alone are, as expected, (typical) viscoelastic materials with a decrease of G' with strain and a crossover between G' and G'' (from 'solid-like' to 'liquid-like'). Due to its small molecular weight, the addition of PEG4000 into PF127 solution did not lead to a significant modification of rheological properties which explains the choice to not show this result.

Figure 2. Viscoelastic properties (storage modulus G' and loss modulus G'') of four formulations and PF127 20% w/v by oscillatory rheology (strain sweep).

Fig. 2 also shows the presence of a LVER (areas where G' is a constant before starting to decrease) for all pastes. These areas are more or less extended according to the composition of the paste. ADPeg38_20 showed the highest G'eq, larger than 10 6 MPa. This is a too high stiffness for DIW, thus ADPeg38_20 was not studied further. ADP42 paste showed the second largest LVER. It also exhibited a high stiffness, with G'eq around 110 kPa, while ADP38 and ADPeg38 were around 30 kPa and 20 kPa respectively, which (if recovered after extrusion) would be sufficient to ensure shape retention.

The stiffness of a paste logically increases with the addition of ceramic particles. However, G'-G'' crossover is always visible, which confirms that the viscoelastic behavior required for DIW was obtained for all pastes. Under oscillatory strain sweep tests, ADPeg38 shows a behavior quite similar to ADP38 paste, while containing less organic compounds (to release during debinding) and being less sensitive to temperature.

Flow behavior

Fig. 3 represents the evolution of tan(δ) and of the complex viscosity (*) of the three remaining pastes as a function of the angular frequency, when staying in the LVER. Logically, tan(δ) is close to zero for low frequency, indicating a large value of G' compared to G''. Over the whole frequency domain scanned here and for the selected shear stress of 0.1 kPa, tan(δ) < 1 for the three pastes, which characterizes a solid-like behavior and could be assimilated to a tri-dimensional network. The 3 pastes show complex viscosity decreasing with frequency which confirms the shear-thinning behavior. Interestingly, the evolutions of complex viscosity with angular frequency of ADP38 and ADPeg38 are quite similar.

Yield stress and flow points

Solid state mechanical properties 4.1. Optimization of printing parameters

Defects which may appear in dense printed parts were studied before going towards final samples for testing. Typical defects that can be observed by SEM are illustrated in Fig. 5. Fig. 5A illustrates a bad homogeneity during the formulation process for ADP42 paste, due to a deficient amount of dispersing agent in relation to the volume fraction of alumina powder. Fig. 5B represents printed parts without enough overlap and pitch for ADP38 paste, thus without enough interpenetration of the neighboring filaments. Fig. 5C gives an example of a technology printing issue, with a lack of a layer in the G-Code which creates an hollow space after the first layer for ADP42 optimized paste. Fig. 5D illustrates an acceptable sample, with the optimized printing parameters but still a few remaining pores, due to a bad homogeneity during the mixing process of materials. These few examples highlight the fact that optimal rheology of the paste is mandatory, but not enough to obtain samples with a low amount of (small) defects.

Strength of printed parts

Parts printed with ADP38 and ADPeg38 exhibited very similar relative densities after sintering (respectively 91 ± 4% and 90 ± 5%) but with a quite high variability (density ranging from 90% to 98% for ADP38 for example). However the average density of parts printed with ADP42 was 85 ± 11%: the paste with the highest amount of alumina resulted (by far) in the lowest density after sintering, with the highest variability. Thus, pastes with a too high stiffness and yield stress are 'printable' but give rise to porous structures with insufficient density.

Fig. 6 illustrates results from fracture tests performed on alumina (unpolished) printed parts with the 3 formulations, in the form of fracture probability versus applied stress curves. In the 4-points bending configuration, the ADP42 bars (printed from the past with the highest solids content) show the lowest strengths values (from 100 to 300 MPa), while the two other modalities show non-significant differences. This is in agreement with the low density of pieces obtained from the highly-loaded paste.

The characteristic strength and Weibull modulus of the 4 types of samples are given in Table 3. The low Weibull moduli are also in agreement with the large density variability mentioned above. Overall, it is shown that the pastes should not be too stiff to obtain acceptable strength. It also shows that PEG can be a good substitute to Pluronic, since it gives, for half the amount of organics, a quite similar extrusion behavior and self-standing ability of the filaments, thus in turns similar strength properties. Last, strengths obtained in this work (σ0 of ADP38, ADPeg38 and ADP42 are respectively 350, 325 and 250 MPa) are significantly improved when compared to existing literature on DIW [START_REF] Cai | 3d-Printing of Ceramic Filaments with Ductile Metallic Cores[END_REF] and even comparable to the strength commercially available alumina ceramics, and this without post-process treatments like polishing. Interestingly, the diametral flexion fracture tests conducted on ADP38 tubes lead to σ0 values very similar to the ones obtained in 4-points bending. This highlights the adequacy of this paste to print structures with complex shapes with thin walls, since the mechanical properties were independent of the testing configuration (which would need confirmation on a larger set of testing configurations). It is important to notice that the low Weibull moduli indicate a quite large heterogeneity in our result. In fact, these parts were printed during summertime which lead to thermosensitive issue due to the presence of PF127. PF127 pastes based become quickly tough during the printing which should have been created random defects. To finish, high strength values (like 500-600 MPa) could be explained by the absence of big defect due to the AM technology and the good homogeneity of ceramic paste. ISO 6872 standard [START_REF]Dentistry -ceramic materials[END_REF] on ceramics for dentistry requires for example a flexural strength of 300 MPa for single crowns, which could thus be reachable with DIW and optimization of printing parameters and formulations.

Fracture surfaces of printed part

In order to determine the origin of the failure, fracture surfaces of samples tested in four-point bending were studied by SEM. Two examples of fracture surfaces for low strengths are illustrated in Fig. 7 in the green section. The yellow rectangle corresponds to first layers of each printed parts. At the top, a large crack, approximately 80 µm length, may have occurred during drying and debinding processes.

The pictures at the bottom show another type of defects, which may be caused by a contamination and lead to generation of elongated Beta-alumina grains, likely linked to the presence of calcium or sodium [START_REF] Altay | Microstructural evolution of calcium-doped α-alumina[END_REF][START_REF] Heavens | Structural transformation during sintering and annealing of beta alumina[END_REF]. These defects are between 80 and 100 µm in length and can be detrimental. They were rarely observed, but systematically led to low strength results.

Bars ADP38 Bars ADP42 Bars ADPeg38 Hollow (ADP38)

Figure 7. Analysis of defects in fracture surfaces of printed parts with lower strength (red) and higher strength (green).

Fracture surface of samples with higher strengths are also shown for comparison. The tensile faces show no distinguishable defect at low magnification. At higher magnification, a shell of 10 µm with a different surface roughness is observed. More important, the wavy surface of robocast samples is underlined, which by itself certainly plays on the strength of even the best samples.

Linking rheology and mechanical properties

Both the absence of major defects inside the parts and the fracture origins identified in Fig. 7 hint to the surface being one of the major limitations to obtain good mechanical properties. More detailed observations of the surface of printed parts, shown in Fig. 8, reveal a rougher surface for the samples printed with ADP42 (that presents the lowest strength and as a reminder the highest paste stiffness).

Interestingly the two pastes with 38% solid content, although exhibiting quite similar values of stiffness and yield stress, lead to different surface states after printing. To evaluate how to smoothen the surface prior (or without) any polishing step, one could adjust the recovery time. Indeed, Fig. 9 shows that a shorter recovery time is measured for ADP38 than for ADPeg38 (few seconds for ADP38 to recover 90% of the initial viscosity and ∼80 s for ADPeg38 to recover only 40% of the original viscosity). During this period, the lower viscosity allows the just-extruded filaments to flow slightly and thus present a smoother surface.

Discussion

The rheology of the paste is a key property to ensure the printability of the paste. Since numerous rheological parameters are available, they can be combined in printability criteria. Fig. 4D summarizes the rheological characteristics of the 3 alumina pastes selected and two calculated criteria: Ф (G'eq/𝜏 𝛾 ) and FTI (Flow Transition Index). According to the literature, both these criteria could delimit a printability domain defined on a storage modulus versus apparent yield stress space.

In the present work, all pastes are in the printability domain (FTI < 20 and Ф > 20). However, the printed parts exhibit different strength, illustrating the limitation of Ф and FTI: they will help to predict printability but not the final quality of the parts. Besides the printability criteria, successful printing of a ceramic part implies knowing the necessary conditions of the process to obtain the part with adequate shape, both internally (architecture) or externally. Some of the printer-based parameters are essential for good shape fidelity. The pitch, overlap and printing speed parameters must be optimized to allow good overlap of the filaments in order to obtain a completely filled body (when dense parts are desired as it is the case here). Rheological parameters can also help in obtaining a good shape fidelity. For example, an adequate recovery time may allow the filaments to better overlap each other prior to the re-augmentation of the storage. The recovery time is thus significant to print parts with smooth surface and dense body, so consequently with high mechanical properties. However, the recovery time must be adapted to the size of the part and the printing speed in order to minimize the time needed to print a part. In our work, with a printing speed of 10 mm/s and a part size of 36 mm in length and 4 mm in width, it takes about 36 s to print a whole layer (10 filaments). As the recovery time of the ADPeg38 paste is longer than this value (it requires more than 100 s to recover 50% of the initial viscosity), its viscosity will still be low when the next layer is printed. Having a too long recovery time might lead to a collapse of the structure and thus a lack of geometrical fidelity. Finally, a successfully printed ceramic part should present good functional properties (i.e. high strength in the present article). In order to obtain a functional part with homogeneous microstructure and good mechanical characteristics, it is necessary to avoid critical defects that could lead to failure. Thus, all the processing steps where the defects could originate from must be under control. Fig. 5A illustrates issues coming from the paste homogeneity: modifications of the mixing process described in the Ceramics paste preparation section, such as insufficient cooling and mixing time, may induce some aggregation of the PF127 and residual bubbles. Even with optimized paste preparation, although some printed parts may show no visible defects inside their body after a critical failure (Fig. 7), others show identifiable defects. Among them, the most critical seem to be the closest to the tensile surface, but also the large internal defects resulting from a lack of fusion between filaments. This kind of defects is found in the ADP42 bars (Fig. 8) (although in this case the surface morphology is more probably the origin of the failure). It is also observed by other authors (see for instance the Fig. 5 of Baltazar et al. [START_REF] Baltazar | Influence of filament patterning in structural properties of dense alumina ceramics printed by robocasting[END_REF]). These defects hardly impact the density, but they may be sufficiently large and acute to act as critical defects for the fracture. They may be avoided by fine-tuning the rheology as has been shown in the present article, together with optimizing the nozzle path.

To complement, one should also consider that the possibility of defects linked to debinding also points to the necessity of reducing the amounts of organics in the ceramic green body. Printing with PEG instead of the commonly used Pluronic is a good approach, since it allowed us to work with twice less organics. However, it should be possible to reduce even more the amount of organic materials, for instance by replacing PEG by gelatin or alginate (widely used in bioprinting by DIW [START_REF] Axpe | Applications of Alginate-Based Bioinks in 3D Bioprinting[END_REF][START_REF] Gao | Advanced Strategies for 3D Bioprinting of Tissue and Organ Analogs Using Alginate Hydrogel Bioinks[END_REF][START_REF] Gao | Optimization of gelatin-alginate composite bioink printability using rheological parameters: a systematic approach[END_REF]), or by using a completely different approach to get the right rheological properties, such as the dispersantbinder-coagulant approach, that uses minimal amounts of organics.

According to the previous paragraphs, Fig. 10 summarizes all the relevant parameters involved in the printing process of a ceramic part that this work has highlighted. This figure is not exhaustive and could be linked with the Fig. 9 of the relevant review from Lamnini et al. [START_REF] Lamnini | Robocasting of advanced ceramics: ink optimization and protocol to predict the printing parameters -A review[END_REF]. In our work, we consider that three successive levels of printability are required to achieve the desired part objectives: extrudability, shape fidelity and final performance. The first level, which indicates the ability of a paste to continuously exit the nozzle, depends on the printer's force, the rheology of the paste and the external environment (mainly humidity and temperature in this case). This level actually provides the prerequisites for proper paste extrusion. The shape fidelity, which corresponds to the second level of printability, defines the fidelity of the desired shape of the part, i.e. the external shape and the internal design (dense or empty zone). To be correctly achieved, this level must take into account the complexity of the part. The parameters of the printer and the rheology of the paste must be adapted together. This optimization work, which involves many variables, requires the user's experience and many experiments to be carried out. As explained in our study, special attention must be paid to the recovery time, which influences both the external appearance and the correct connection of the filaments inside the body of the part. The last printability level concerns the final performance. Indeed, a part cannot be considered as 'printable' if it does not reach the required mechanical properties after the whole process. Therefore, sufficient mechanical properties are desired as final performance in our study. Thus, defects must be reduced, both on the surface of the part and inside the part. Defects, which usually originate from small, local initial heterogeneities, can then develop during all the processing steps of the ceramic, from drying to sintering, depending on their optimization. Thus, having a good shape fidelity, i.e. a smooth surface and a dense and homogeneous microstructure, is also of primary importance for good mechanical properties. One should keep in mind that the definition of printability criteria is more advanced in the bioprinting field than in the ceramics field, and can encompass this idea of including the final function in the definition of the criteria. It is thus important to remember that the requisites for both fields are not all identical. Indeed, in particular the paramount functional requirement in bioprinting is to keep alive the cells contained in the paste, which leads to severe limitations in terms of the mechanical stresses to which cells are submitted, hence in terms of rheology of the paste. These demands cannot be translated in the ceramics field without adaptation. However, the idea of considering the final, desired properties as a part of the definition of printability criteria is common to the bioprinting field and to this article. The approach shown in the present paper enabled reaching good mechanical properties as compared to the literature: the strongest alumina shown here reach 350 MPa characteristic strength, with some samples resisting to 600 MPa, without polishing. This is 50% larger than previous results in the literature [START_REF] Feilden | Robocasting of structural ceramic parts with hydrogel inks[END_REF] and shows that further optimization may lead to robocast alumina parts with strength comparable to conventional processes. There are still issues with Weibull moduli obtained on ceramics made by additive manufacturing in general, showing a larger distribution of defects and thus a lower robustness than other more mature ceramic technologies. Weibull moduli obtained in this work, in line with previous literature [START_REF] Feilden | Robocasting of structural ceramic parts with hydrogel inks[END_REF], are a real point of vigilance for structural applications requiring low risk of failure. Reproducibility of additive manufactured ceramics should require further efforts. To go ahead in predicting the overall final performance of ceramic parts, machine learning, already applied to predict the mechanical properties of bio-ink [START_REF] Goh | A review on machine learning in 3D printing: applications, potential, and challenges[END_REF][START_REF] Lee | Machine learning-based design strategy for 3D printable bioink: elastic modulus and yield stress determine printability[END_REF], could be of great help in developing the pastes and printing conditions by reducing the time-consuming iterative process optimization.

Conclusion

Alumina printed parts were obtained by robocasting and exhibited in the best case characteristic flexural strength of about 350 MPa, which may be sufficient for several technical applications. It was necessary to optimize both rheological properties of the paste and printing parameters before reaching such fracture strength. In particular, and that might be counterintuitive for ceramists, pastes with the lower solid content reached the highest strength: this is due to their higher ability to flow and fill larger defects. Further optimization was obtained by exploiting the thixotropy: increasing the recovery time was proven to be beneficial to both surface roughness after printing and mechanical properties. Note that these results are linked to the rheology, not directly to the solid content: a better optimum might be reached for similar rheologies but higher solid content.

The optimization also concerned the formulation of the paste: even if Pluronic F127 is still among the most widely used gels to process ceramic pastes for direct ink writing, PEG can be an excellent substitute, enabling to fabricate pastes with similar rheological behavior, leading to a better shape fidelity, while facilitating debinding through the use of less organic material as compared to PF127.

Finally, during this work, three different kinds of rheological tests were performed to well define the printability, in order to obtain sufficient final mechanical properties without machining. The large set of rheological parameters illustrates why the ceramic community must be very precise in their definition and measurements, if only to avoid misunderstandings.

Figure 3 .

 3 Figure 3. Solid-like and shear-thinning behavior of alumina pastes: Evaluation of tan (δ) and viscosity in function of angular frequency in dynamic frequency sweep. Shear viscosity curves.

Fig. 4

 4 shows static yield stresses (τγ) and the flow stresses (τf) of the 3 pastes, obtained in dynamic strain sweep (G',G" versus oscillation stress) curves. Yield stresses values are determined when storage modulus start to drop at the end of the linear domain. Values obtained for the ADP38, ADPeg38 and ADP42 are respectively 100 Pa, 200 Pa and 500 Pa. Flow stresses which are calculated at the crossover G'=G'' are 250 Pa, 300 Pa and 1000 Pa. The 'breaking' area (between yield point and yield stress) is lower for ADPeg38 and ADP42, which can be explained by the lower homogeneity in ADP42 and the difficulty to break the network. Fig.4Dsummarizes the different rheological parameters obtained for the 3 pastes and the 'figures of merit' defined in the literature. All three pastes can be considered as 'printable' according to these criteria. ADP38 and ADPeg38 are quite close, which is interesting in order to use a paste with less organic compound.

Figure 4 .

 4 Figure 4. Flow stress (τf) and yield stress (τγ) of 3 pastes in dynamic strain sweep A) ADP38. B) ADPeg38. C) ADP42. D) summary of the measured values.

Figure 5 .

 5 Figure 5. Illustrations of defects found with non-optimized conditions related to A) Bad homogeneity of the paste (ADP42 with a deficiency in Darvan 811 dispersant: the paste was so stiff that the bubbles could not be eliminated), B) Bad choices of overlap in y and z, C) First layr missing in the G-Code (optimized ADP42), D) Air bubbles trapped.

Figure 6 . 4 -

 64 Figure 6. 4-point bending of alumina unpolished printed parts (bars and tubes) with 3 pastes: probability of failure versus stress.

Figure 8 .

 8 Figure 8. Topography analysis of three different formulations in relation to strength and rheological properties. ADP38 in blue andADPeg38 in green show lower stiffness and higher strength, when ADP42 in red shows higher stiffness and lower strength. Note the surface state of ADP42 with a larger roughness.

Figure 9 .

 9 Figure 9. Start-up tests, performed on ADP38 (in blue) and ADPeg38 (in green). ADPeg38, with a longer healing time, shows a smoother surface.

Figure 10 .

 10 Figure 10. All parameters involved in the success of DIW of dense ceramic parts with satisfying mechanical properties.

Table 1 .

 1 Composition of the alumina pastes.

	Name of the	Alumina	Solution of	Solution of	Solution of	Solution of	Darvan
	paste	(g)	PF127 20%	PEG 4000 at	PEG 20,000	PEG 20,000	811
			w/v (g)	3% w/v (g)	10% w/v (g)	20% w/v (g)	(g)
	ADP38	40	16.06	1.200			0.200
	ADP42	40	13.34	1.200			0.360
	ADPeg38	40			19.55		0.300
	ADPeg38-20	40				19.55	0.300

Table 2 .

 2 Definition of rheological properties.

	G'	Elastic (or storage) modulus
	G''	Viscous (or loss) modulus
	η*	Complex viscosity
	Tan()	Loss factor (indication of the fluidity of the material)
	LVER (G'eq) often called	Linear Visco-Elastic Region at small strain (and related storage modulus) sometimes
	G'LVER	called stiffness
	𝜏 𝛾 (yield stress) often called 𝜎 𝛾	Stress needed to leave the LVER
	𝜏 𝑓 (flow point or flow stress)
	often called 𝜎 𝑓

Acknowledgments

This work was funded by the Agence Nationale de la Recherche (ANR) under ID ANR-17-CE08-0021, in the frame of CERAPIDE project. This work was performed in parallel with the work of Dr. Tristan Garnault, who we wish also to thank for his help in the assessment of the macroporosity.

Declaration of Competing 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.


Mathieu Muratet
email: mathieu.muratet@lip6.fr

Amel Yessad

Thibault Carron
email: thibault.carron@lip6.fr

How to assist designers model learning games with Petri nets?

Keywords: Learning Game, Behavioral Model, Petri Net, Design Pattern, DSL

In previous research, we presented a methodological framework that provides players with adaptive feedback. The core of this framework relies on modeling the learning game with a Petri net. However, this modeling is a challenging task. Indeed, Petri nets are well adapted to model dynamic and complex systems but require a mastery of the underlying mathematical formalism to build them manually. In particular, when the learning game is characterized by a large freedom of action. In this paper, we present an authoring tool and its domain-specific language to assist designers to model learning games with Petri nets. We carried out a case study where our contribution was implemented. Results show that our contribution helps designers to build the Petri net in combination with classical Petri net editors which are still useful to visualize, to check and to validate the Petri nets built.

INTRODUCTION

Adaptive feedback in Technology-Enhanced Learning (TEL) is important to help learners acquiring competencies, progressing in problem solving and improving the learning process [START_REF] Thomas Bimba | Adaptive feedback in computerbased learning environments: a review[END_REF][START_REF] Valerie | Focus on formative feedback[END_REF][START_REF] Vanlehn | The Andes physics tutoring system: Lessons learned[END_REF]. In learning games, the adaptive feedback remains a challenging issue because the analysis of the learner traces is complex. Indeed, in learning games with a large amount of available actions, the learners are free to interact with the game objects and to explore the game environment. This makes it hard to understand and analyze the learner's traces and to answer questions like: Does the learner progress to the solution? What is the player's next action to perform in order to progress towards the solution? Which feedback is relevant to help the learner solve the problem, acquire a competency or break through erroneous cognitive reasoning?

To provide automatically hints adapted to the learner in learning games and help the player to progress in problem solving, we need to model the resolution process of the learning game. In previous research [START_REF] Muratet | Framework for Learner Assessment in Learning Games[END_REF][START_REF] Thomas | Petri Nets and Ontologies: Tools for the "Learning Player" Assessment in Serious Games[END_REF][START_REF] Yessad | Using the Petri Nets for the Learner Assessment in Serious Games[END_REF], we proposed a methodological framework to tackle this challenge by means of Petri nets (see Fig. 1). The framework is divided in two workflows: the first deals with the use of Petri nets to generate feedback adapted to the player difficulties (the feedback loop), and the second is the workflow to model the game with Petri nets.

We proposed a solution in [START_REF] Muratet | Understanding Learners' Behaviors in Serious Games[END_REF] for the first workflow (the feedback loop) and detailed the algorithm that exploits Petri nets properties to analyze the learners' actions, to characterize them with pedagogical labels, and to calculate a label-based score. These pedagogical labels provide users, and particularly teachers, with information about the learner's behavior such as, correct actions, erroneous actions, actions taking place too late or too early in the learner's solving process.

In this paper we focus on the second workflow that aims to assist designers to build the Petri net used by the feedback loop (see FilteredPn in Fig. 1). When designers develop the learning game and describe the different monitored interactions, as detailed in this paper, they build the learning game and export automatically a first Petri net called a full Petri net (see FullPn in Fig. 1). Afterward, an expert plays the game (several times if there is more than one solution available) and the traces are used to filter the full Petri net in order to build the filtered Petri net automatically. The full Petri net includes all monitored actions the learner can perform and the filtered Petri net is a sub-part of the full Petri net including only actions used by an expert to solve the game. Thus, it becomes possible to compare between the experts' solution and the player solution in order to provide relevant feedback to the player.

A critical part of this second workflow is the building of the FullPn. Indeed, modeling a complex system, like a game, with a Petri net is not a trivial task [START_REF] Naedele | Design Patterns in Petri Net System Modeling[END_REF].

Thus our main issue is: How to assist designers to model complex environments like learning games with Petri nets? We make the hypothesis that providing small patterns of Petri nets and hiding mathematical formalism of Petri nets with domain-specific language will help to model links between game objects. These links are processed to automatically build the full Petri net that models the entire learning game and thus avoids designers having to build it manually.

In the two next sections, we introduce the Petri net formalism and we give an overview of the existing research and related work. Section 4 deals with the contribution of this paper: the domainspecific language that helps designers to model links between game objects. Section 5 presents a case study in which our contribution was implemented and discusses the results. Finally, we conclude the paper and present some future works identified as useful to pursue this research. The behavior of many systems can be described in terms of system states and their changes. To simulate the dynamic behavior of a system, a state (or a marking) is changed according to the following transition (firing) rule:

(1) A transition 𝑡 is said to be enabled if each input place 𝑝 of 𝑡 is marked with at least 𝑤 (𝑝, 𝑡) tokens, where 𝑤 (𝑝, 𝑡) is the weight of the arc from 𝑝 to 𝑡. (2) An enabled transition may fire (depending on whether or not the corresponding event takes place in the simulated system). (3) The firing of an enabled transition 𝑡 removes 𝑤 (𝑝, 𝑡) tokens from each input place 𝑝 of 𝑡, and adds 𝑤 (𝑡, 𝑝) tokens to each output place 𝑝 of 𝑡, where 𝑤 (𝑡, 𝑝) is the weight of the arc from 𝑡 to 𝑝.

The set of all markings (states) that are reachable from the initial marking of a Petri net by firing the transitions represents the range of states of the system and is called the reachability graph. This graph is the core model we use to analyze the players' actions. In case the number of game states is infinite, particularly for unbounded Petri nets (some places become unbounded because of the increasing number of their tokens), the reachability graph cannot be calculated and is replaced by another graph called the coverability graph. More details about the Petri net formalism can be found in [START_REF] Murata | Petri nets: Properties, analysis and applications[END_REF][START_REF] Lyle | Petri Net Theory and the Modeling of Systems[END_REF].

In the context of this work, the places represent the states of the game objects and the transitions the actions that the player can perform in the game objects. Then, performing game actions can be simulated with a Petri net by triggering (firing) the transition associated with the game action according to the previous rule. This trigger will fire transition in the Petri net and change the current Petri net marking to keep it synchronized with the game state.

In the Petri nets we use to model games, we include "inhibitor" arcs that have a different pattern of operation compared to the standard arcs. In the case of an "inhibitor" arc, the transition 𝑡 is enabled if each input place 𝑝 of 𝑡 is marked with less than 𝑤 (𝑝, 𝑡) tokens. This kind of arc is interesting to model games, especially to model actions that are not possible if the game state exceeds a threshold. For instance, considering a simple game where players can be attacked (they lose 1 health point for any damage) and have the power to heal themselves (they get 10 health points back). Players can heal themselves only if their health points are less than or equal to 90 (to avoid exceeding the maximum of 100 health points). In this example, an inhibitor arc avoids creating an artificial place to store the number of health points lost (see Fig. 2). Saving this artificial place is crucial in our case because it avoids mixing the healing action and another game action, such as the attack in this example. In this context, using inhibitor arcs enables designers to simplify their work. Indeed, in this example, the designers will be able to express a constraint on the healing action without considering side effects to other game actions (see section 4 for details on how to express these constraints).

In sum, we use the Petri net formalism to model learning games and we use the coverability graph to check properties about the player's game solving. Thus, for example, we can answer questions like: is the game end reachable from the current state of the game (there is a sequence of enabled transitions from the current state to the final state)? Is the player stuck and can no longer progress in the game (no transition is reachable from the current game state)? See [START_REF] Muratet | Understanding Learners' Behaviors in Serious Games[END_REF] for more details.

POSITIONING

As we introduced in section 2, Petri net is a powerful tool to simulate the dynamics of complex systems. In his thesis, Dormans [START_REF] Dormans | Engineering emergence: applied theory for game design[END_REF] reviews several approaches to assist game designers and shows that Petri net is one of the most promising simulation tool but is less accessible to designers. This has been a known difficulty for quite some time, Naedele and Janneck [START_REF] Naedele | Design Patterns in Petri Net System Modeling[END_REF] identify in 1998 that "it is easy to grasp the basic concepts of places, transitions, and concurrent sequences, and that one is very quickly able to apply this to the modeling of systems that have an internal structure that only consists of applications of those basic concepts. The situation changes when one tries to model complex technical systems". Their contribution to help in this field was to identify and describe several design patterns such as removing all tokens stored in a particular place or synchronizing two concurrent sub-nets. These design patterns are very interesting but no hints are provided to help designers to connect patterns together. Gomes and Barros [START_REF] Gomes | Structuring and composability issues in Petri nets modeling[END_REF] have the same comment "Petri nets are often difficult to use in practice due to the problem of rapid model growth". Authors present a condensed view of the main structuring mechanisms for general system modeling with Petri nets. They introduce composition and refinement/abstraction strategies. Composition strategy includes fusion of Petri nets (called horizontal composition, as the nets are glued together at specific points, places or transitions) and folding that consists in identifying structural symmetries inside Petri nets. Refinement/abstraction strategy includes static macros (a place or a transition model a sub-net) and dynamic invocations (when a transition fires, a new net is dynamically created). One of their conclusions is that work on simple and intuitive notations for set modification would be interesting for further research.

Abstraction strategy was also studied with hierarchical Petri nets. Balas et al. [START_REF] Balas | Hierarchical Petri Nets for Story Plots Featuring Virtual Humans[END_REF] explain how they use hierarchical Petri nets to model plots and control their evolution in games emphasizing a story and featuring large worlds inhabited by virtual characters. Araújo and Roque [START_REF] Araújo | Modeling Games with Petri Nets[END_REF] discuss the applicability of Petri nets to model game systems and game flows compared with other languages such as UML and show that it is possible to model game systems with hierarchical Petri nets. A lot of tools enable designers to edit and analyze Petri nets [START_REF] Jie | A Survey of Petri Net Tools[END_REF], but all of them require mastering Petri net mathematical formalism.

Dormans [START_REF] Dormans | Simulating Mechanics to Study Emergence in Games[END_REF], inspired by Petri nets, proposes the Machination framework that allows designers to model and simulate games in an early stage of development. The author indicates that this All these research inspired us. We used the design pattern approach of Naedele and Janneck [START_REF] Naedele | Design Patterns in Petri Net System Modeling[END_REF] to provide small and reusable Petri nets to designers rather than asking them to model the game from scratch. We also use hierarchical Petri nets to model the game at various levels, especially the macro-transition approach [START_REF] Gomes | Structuring and composability issues in Petri nets modeling[END_REF] that abstracts sub-nets by transition inside the super-net. Finally, we propose a domain-specific language to help designers to update Petri nets. This domain-specific language implements the AND-split, AND-join, and OR-split introduced by Sun et al. [START_REF] Sun | Incremental Workflow Mining with Optional Patterns and Its Application to Production Printing Process[END_REF].

CONTRIBUTION

In section 1 we introduced the global architecture of the framework and we noticed that a key part of this framework was the building of a Petri net that models the learning game. In this section, we detail the method and tools we designed to tackle this challenge.

Our approach consists of asking designers to build manually small Petri nets. Each one of these small Petri nets models a behavior pattern that can be linked to a game object. We assume that this effort is much less than if we ask them to build a full Petri nets manually (which can be composed of several hundred places and transitions). We designed a set of generic Petri nets: activable, pickable, activationCount, exclusiveChoice, unique action, etc. The objective is to capitalize on all these small Petri nets to avoid to create them from scratch for each new game level. Fig. 3 depicts the two first examples previously cited. A game object can be linked with several behavior patterns and a behavior pattern can be linked with several game objects.

A domain-specific language to describe links between game objects

Each game object the designer decides to include in the Petri net is called a "monitored object". If a game object is monitored, all learner interactions with this object are considered in the feedback loop (see Fig. 1). We propose connecting the small Petri nets by formally explaining the links between the monitored entities with a domain-specific language.

To illustrate the process of using the full Petri net building we focus on an example (see Fig. 5), which presents a game level where the player has to open a locked frozen door. A key and a boiler are available in the scene to use in opening the door. The player has to pick up the key and turn on the boiler to melt the ice on the lock then use the key to unlock the door. The player can grab the key and turn on the boiler in any order, but opening the door has to be done last. In this example, we have three monitored game objects: the key, the boiler and the door.

Fig. 4 models relations to express links between MonitoredObjects. An object, like the door, is considered "monitored" if it contains at least one MonitoredComponent. Each MonitoredComponent contains a small Petri net (the "Activable behavior pattern" to the door, see Fig. 3) and a list of MonitoredActions. Each Mon-itoredAction has a reference to the monitored Transition (the gameAction) in the small Petri net (here the action "open" is associated to the transition "Enable" in the Activable behavior pattern) and a list of links. Links define constraints with other monitored objects (the key and the boiler). A Link is composed of four main attributes: a Verb to express the type of the constraint (REQUIRE AT LEAST, REQUIRE LESS THAN, GET or PRODUCE), a weight to value the constraint, a gameState that references a place of the Petri net in a monitored component and a label to rename the link. A monitored action can include several links with other monitored actions. The logicRel inside MonitoredAction enables to combine links with an AND/OR expressions. A logicRel has to follow a simple formal grammar where 𝐿 is the set of available labels: Then logic relation enables to link automatically small Petri nets together with basic Petri nets patterns (AND-split, AND-join, and OR-split [START_REF] Sun | Incremental Workflow Mining with Optional Patterns and Its Application to Production Printing Process[END_REF]). An operator * connects all links to the gameAction of the MonitoredAction, whereas an operator + duplicates the gameAction of the MonitoredAction and applies links to each copy.

Fig. 6 depicts several scenarios depending on AND/OR occurrences inside logicRel. The first scenario (S1) is the basic Petri net that models a simple counter that adds one token inside Counter place each time the GameAction is fired. The second (S2) is the adapted Petri net automatically built with a AND constraint: "require at least 1 token in State1 AND 15 tokens in State3" (𝑙𝑖𝑛𝑘1 * 𝑙𝑖𝑛𝑘2), in the resulting Petri net two arcs was added to link the GameAction with its input states. The third (S3) is the same as the second, but with an OR constraint: "require at least 1 token in State1 OR 15 tokens in State3" (𝑙𝑖𝑛𝑘1 +𝑙𝑖𝑛𝑘2), in this case the Game-Action was copied and each copy was linked with its appropriate input state. The fourth scenario (S4) illustrates a more complex constraint that combines AND and OR operators: "Require at least 1 token in State1 OR (15 tokens in State3 AND 2 tokens in State4)" (𝑙𝑖𝑛𝑘1 + (𝑙𝑖𝑛𝑘2 * 𝑙𝑖𝑛𝑘3)), in this case the output Petri net contains a copied GameAction (+ operator), on the first copy one arc was linked with State1 (𝑙𝑖𝑛𝑘1) and on the second copy two arcs was linked with State3 and State4 (𝑙𝑖𝑛𝑘2 * 𝑙𝑖𝑛𝑘3). The last scenario shows simply all kind of verbs to see their instance in Petri net formalism.

In our example of the locked frozen door (see Fig. 5), first, we link the key to the pickable behavior pattern and the door and the boiler with the actionable behavior pattern. Then, we define the opening of the door with the following constraint: "the boiler must be enabled AND the key must be in the avatar's hand". The output Petri net of this game level automatically built based on the behavior patterns selected and the links defined is shown in Fig. 7 (a).

Expressing a link obeys the following outlines:

(1) Selecting the action to apply a constraint on. This action refers to a transition included in one of the behavior patterns attached to a first game object. (2) Selecting the state linked to this constraint. This state refers to a place included in one of the behavior patterns attached to a second game object. (3) Selecting the type of link and its weight. We defined three kinds of links:

• The "Get" link refers, in the Petri net formalism, to an arc that from the input place consumes tokens equal to the weight of the arc. • The "Require" link offers two possibilities:

-The "at least" constraint refers to a "read" arc (in Petri net formalism) that does not consume tokens, but enables the output transition if the number of tokens inside the input place is greater than or equal to the weight of the arc. -The "less than" constraint refers to an "inhibitor" arc (in Petri net formalism) that does not consume tokens but disables the output transition if the number of tokens inside the input place is greater than or equal to the weight of the arc. • The "Produce" link refers, in the Petri net formalism, to an arc that produces tokens equal to its weight inside its output place. To summarize, a link enables to create arcs between a gameAction and a gameState, the Verb defines the type of arc, and the weight specify the amount of tokens in play. Several links define several constraints on a same gameAction. The logicRel structures the links and how to connect the gameAction with all linked gameStates. In this screenshot we implemented previous example and we changed the constraint of the previous example (the frozen door): "opening the door requires the boiler enabled OR the key in the avatar's hand". Formally, first we select the game object to apply the constraint on, the Door (1). Then we select one of its action, the turnOn action, renamed "Open" (2) and we add two constraints, the first is linked with the Key and "requires at least 1 (Key) inHand" (3) (this link is noted "𝑙0"), the second is linked with the Boiler and "requires at least 1 (Boiler) on" (4) (this link is noted "𝑙1"). At least one of these two links is required to open the door, so the logic expression is: "𝑙0 + 𝑙1" (5). We can translate this expression "𝑙0 + 𝑙1" like: "Open the frozen door Requires at least 1 Key inHand OR Requires at least 1 Boiler on". Then the resulting full Petri net with an OR operator is quite different in comparison with the AND version (see Fig. 7). Thus, the monitoring module integrated into Unity provides the following functionalities:

A plugin for Unity to monitor game entities and build the Petri net model

• Selecting game objects to bind with the monitoring system.

• Associating small Petri nets to each monitored game object.

• Defining links between monitored elements.

• Building the Petri net with all links previously defined.

CASE STUDY

We have experimented with our proposition during the development of an open source virtual escape game called E-LearningScape2 (see fig. 9). E-LearningScape is a numerical adaptation of a physical escape game 3 . In this numerical adaptation, the participants (in teams of 2 to 5 players) play the role of a sandman immersed in the dream of Camille. Their challenges will be to help Camille structure her thought in her dream by solving enigmas. The team members gather around a computer. One player controls the game and moves inside the virtual universe in a first-person navigation to discover interactive game objects and fragments of dreams giving access to clues in the real world. All members of the team solve enigmas inside and outside the game, these two facets feeding each other. E-LearningScape has two objectives, the main one is to introduce learners to concepts in one of the three knowledge domains considered in the game (pedagogy, accessibility or computer science), and the second objective is to promote team work and cohesion. E-LearningScape is made up of 17 enigmas that require the players to combine more than 40 game objects found into the game. Without being exhaustive, here are some examples of enigmas: finding a wire to connect words on a pin panel; deciphering a message with a cylinder mirror; revealing hidden clues with a black light; etc.

As in classic escape games, several enigmas can be performed in parallel and the players have to associate found clues to resolve them. The monitoring module we work on aims to generate in-game hints to help players to solve enigmas. As we explain in previous sections, the monitoring module require Petri nets and we focus this case study on the enigmas modeling.

GameAction

Enigmas modeling

The developer, who has coded the game, masters Unity and knows the Petri nets principle (he is not an expert of Petri nets). He had to model the 17 enigmas with the help of our Unity plugin and Tina editor [START_REF] Bernard Berthomieu | The tool TINA -construction of abstract state spaces for Petri nets and time Petri nets[END_REF]. The current model of the game is composed of 23 full Petri nets with a sum of 297 places, 231 transitions, and 558 arcs. All these full Petri nets were built from eight small Petri nets. From these eight small Petri nets, the developer reused four generic Petri nets we introduced in section 4 ("activable", "activationCount", "exclu-siveChoice", and "unique action") and designed with Tina four new small Petri nets. The first, "activableAndCount" is a merge between "activable" and "activationCount" patterns, it allows to count how many time a game object has been activated. The second, "answerQuestion" models question validation options (right and wrong answer). The last two are a simple place and a simple transition used to complete specific behavior. Fig. 10 shows these eight small Petri nets used to model E-LearningScape.

We do not show full Petri nets of all enigmas in this paper due to place limitation, but the reader can download and open each enigma's PNML files available at [START_REF] Mocah Team | Full Petri nets of E-LearningScape[END_REF] with Tina tool. We show just one example in Fig. 11 that illustrates what a full Petri net looks like, this full Petri net models the pin panel enigma that we mentioned in section 5.

Yet we present some interesting parts of Petri nets that were built. The game was not modeled with a unique Petri net for two main reasons: the first is a technical reason, computing the coverability graph of this Petri net would take too much time; the second is a usability reason, checking the validity of such model seemed very complex to the developer. Then, the developer chose to model enigmas' sequence with a super-net with only "unique action" patterns. For example, a gears enigma (the players have to select the right gear to unlock a mechanism) is available after resolving three previous enigmas. To model this constraint, the developer attached the "unique action" pattern to the four enigmas and add links to the "perform" action of the gears enigma (see Fig. 12). Each transition of this super-net models an entire enigma. The details to resolve each enigma are defined in dedicated full Petri nets. The full Petri net presented in Fig. 11 is one example. We discuss the conception of these full Petri nets in the next section.

Discussion

First of all, we notice 17 enigmas and 23 Petri net (6 Petri nets in addition). Five enigmas were modeled with two Petri nets. The last Figure 9: Screenshot of "E-Learning Scape", a virtual escape game on teaching skills additional Petri net is the super-net that structures which enigma is available depending on the others.

Among the enigmas that require two Petri nets, we found the last four enigmas of the game. For these enigmas the answers can be entered freely in four input fields, but when an answer was excepted in one input field it will no longer be accepted in the others. The case of these last four enigmas is interesting because the monitoring module has to know which enigma is validated. Then the developer chose to model the heart of each enigma in its own Petri net and the validation mechanism in different Petri nets. We focus on this validation mechanism. The developer used the "exclusiveChoice" and "answerQuestion" patterns. We present in Fig. 13 a simplified example including only two choices instead of four. The two "exclusiveChoice" patterns model the two possible input fields and the "answerQuestion" pattern the answer of one enigma. "choice1" of each "exclusiveChoice" models that the associated input field was chosen to validate the answer of this enigma and "choice2" models that the associated input field was chosen to answer to another enigma. Selecting "choice1" in one enigma has to inhibit others "choice1". To integrate this constraint, the developer added the following links on each "choice1": "Require less than 1 answer in answering AND produce 1 answer in answering". Then, "choice1" will be enabled if the "answering" place does not contain tokens. We notice that this modeling is not perfect because once the "correct" action will be fired, the "answering" place will be emptied and unlock the second "choice1". Developer would have to add another constraint on "answered" place like: "Require less than 1 answer in answered". In this case this mistake is not a problem because when the enigma is validated it is saved inside the super-net, then full Petri nets of this enigma will not be used anymore.

Another interesting modeling choice is the collecting behavior. Developer did not choose to use the "pickable" pattern we introduced in Fig. 3. He preferred using "uniqueAction" and "activable" patterns because, in E-LearningScape, a pickable game object can't be discarded and principally because it was implemented with two game objects, the "world object" and the "inventory object". Once the player clicks on the world object, it is disabled and inventory object is enabled. Then player believes he grabbed the object. Then developer attached the "unique action" pattern to the world object because it can be clicked once and the "actionable" pattern to inventory object because it can be selected and unselected. To model the constraint that the inventory game object can be used only when the word object was clicked he added a simple link to the two "actionable" actions (turnOn and turnOff). For instance, for a wire available in the game: "turnOn InventoryWire requires at least 1 WordWire picked". Fig. 14 shows how this constraint is defined in Unity and dotted area in Fig. 11 shows output result in full Petri net.

Finally, developer used the Unity plugin to associate small Petri nets with game objects and to define links between game objects to reflect game simulation. He exported regularly the Petri net he worked on with the Unity plugin and checked it with the simulator functionality of Tina. When he detected some inconsistencies with game simulation, he did not edit the Petri net with Tina and preferred to make modifications with the Unity plugin. He explained this choice because he was not sure how to update the Petri nets in particular for cases that contain complex relations. In this case, he preferred using the AND/OR expressions of the plugin.

We also identified situations not easy to model with current tool: (1) managing game objects that could appear dynamically in the game (not specified in the initial Petri net) and (2) simplifying the constraints defined on a minimum and maximum game object state, for instance to constrain a game action if the state of a game object is included between two values. These perspectives will enable us to improve current framework.

CONCLUSION AND FUTURE RESEARCH

In this paper, we presented a contribution to help learning game designers to model learning games. We based our work on Petri nets and we defined a domain-specific language to describe the links between small and reusable Petri net patterns that can be easily built. Based on this logical description we built automatically a full Petri net that models the player interactions in the game.

The tool we designed makes our conceptual approach operational. We experimented with our contribution by reviewing usage in a professional context. This case study shown a loop between the Unity plugin and Tina. The game developer designed 23 full Petri nets with a sum of 297 places, 231 transitions, and 558 arcs with only 8 small Petri net patterns. The plugin helps developer to build the Petri net thanks to the domain-specific language and Tina enables them to visualize, to check and to validate the resulting Petri nets.

The work done with E-LearningScape is a first step to implement a feedback loop inside the game. Currently the 17 enigmas Figure 12: Part of the super-net to show constraints added (bold arcs) to Gears enigma (Gears_perform_147) that will be available when enigmas R1-Q1, R1-Q2 and R1-Q3 will be resolved (end state reached)

of the game are modeled and learners' traces are analyzed accordingly. Next steps will be to assess feedback efficiency depending on learner's needs. Currently we designed a small set of Petri nets patterns and we believe that an interesting research direction is to identify core design patterns of Petri net for a broad games mechanics.

Figure 1 :

 1 Figure 1: Global architecture of the framework

Figure 2 :

 2 Figure 2: An attack/heal system modeled with inhibitor arc (a) and without inhibitor arc (b)

Figure 3 :

 3 Figure 3: Activable behavior pattern (a) and pickable behavior pattern (b) modeled with Petri nets formalism

  The logicRel is distributed of * (AND) over + (OR) to process expressions like (𝑙 𝑥 * 𝑙 𝑦 ) + (𝑙 𝑥 * 𝐿 𝑧 ) rather than 𝑙 𝑥 * (𝑙 𝑦 + 𝐿 𝑧 ).

Fig 8

 8 Fig 8 presents a screenshot of the monitoring module 1 that we integrate into the Unity environment. In this screenshot we implemented previous example and we changed the constraint of the previous example (the frozen door): "opening the door requires the boiler enabled OR the key in the avatar's hand". Formally, first we select the game object to apply the constraint on, the Door (1). Then we select one of its action, the turnOn action, renamed "Open" (2) and we add two constraints, the first is linked with the Key and

1 Figure 4 :Figure 5 :

 145 Figure 4: Monitoring model

State1Figure 6 :

 6 Figure 6: Examples of Petri net adaptations depending on logic relations

Figure 7 :Figure 8 :

 78 Figure7: Petri net of "The frozen door" game level built automatically with a AND constraint "The door can be opened if the boiler is enabled AND the key is in hand" (a) and with an OR constraint "The door can be opened if the boiler is enabled OR the key is in hand" (b)

Figure 10 :

 10 Figure 10: The eight small Petri nets used to model the 17 enigmas of E-LearningScape

Figure 11 :

 11 Figure11: Full Petri net of the pin panel built from "unique action", "activable", "activationCount" and "answerQuestion" patterns (dotted area is detailed in section 5.2)

Figure 13 :

 13 Figure 13: Inhibit choice1 if another was selected before

The monitoring module used in this research is part of the FYFY project: https://github.com/Mocahteam/FYFY, accessedMarch 11,

E-LearningScape: https://github.com/Mocahteam/E-LearningScape, accessed February 03, 2022

[START_REF] Bernard Berthomieu | The tool TINA -construction of abstract state spaces for Petri nets and time Petri nets[END_REF] https://sapiens-uspc.com/projets-innovants/learningscape-2/, accessed February 03,

ACKNOWLEDGMENTS

The authors would like to thank Jessica Lament, Séverine Maillet, Sophie Delamasantière-Boutal and John James Boutal for their contributions to reviewing and improving this paper.