F Gr Éban

G Monteil

X Roizard
email: xavier.roizard@ens2m.fr

Influence of the structure of blanked materials upon the blanking quality of copper alloys

Keywords: Blanking profile, Burr, Work hardening, Precipitates

Improving of the productivity of the blanking industry of leadframes (supports for electronic components) involves several research areas. The blanking ability of copper alloys is the topic of this study. In this paper, we propose to determine the influential parameters of blanked materials with respect to the facies of blanked profiles.

For this purpose, the blanking tests on usual copper alloys as well as original alloys, in particular, in terms of microstructure, enabled us to show that the blanking profile was strongly influenced by the mechanical characteristics and the microstructure of the alloys. As a matter of fact, the more work hardened those are (depending upon the making process of the sheet), the larger the fracture zone, is for its own leading to a weaker sheared zone, detrimental as for to the overall blanking quality.

The burr resulting from blanking, it is only conditioned by the microstructure and, in particular, by the precipitates present in the copper matrix. Its formation is not influenced by the rate of work hardening. As for the roll-over zone, it depends mainly on the blanking geometry (punch/die clearance).

Introduction

The industrial making of leadframes (metal supports of electronic components) is obtained through high speed blanking. Metal blanking is performed on thin laminated sheets (0.1-2 mm) at a frequency in the range of 300-800 strokes min -1 . The sheets are made of low alloy copper to guarantee the high electrical conductivity (>4.6 × 10 7 S/m) necessary for the functioning of leadframes. The objective of this study is to determine the influence of the blanked material on the quality of blanking, and more exactly the influence of the composition, work hardening and microstructure of the copper alloys on the blanked material part edge and on the quantity of initiated burr.

Principle of the tests

Blanking process of different alloys

In order to compare the influence of the blanked material upon the stamping result, all of the tested alloys were blanked with an industrial progressive precision tool at high speed. The characteristics of the blanking operation are given in Table 1.

The couple blanking punch/die was run in beforehand, by blanking several thousand pieces, because the behaviour of the tools in relation to the blanked material is more selective after wear.

All of the studied copper alloys are listed in Table 2.

Measurement of the blanking quality

The quality of blanking is expressed by means of the ratio of the different zones of the blanked part edge expressed as a percentage of the whole blanked part edge length: roll-over zone, sheared zone, fracture zone and burr. In this study, the burr is differentiated from the other zones of the blanked part edge, which, together, form the edge profile, including the roll-over, the sheared and fracture zones. The final expressed value for each zone is the average of five measurements resulting from five different samples. The uncertainty of this measurement (here set at 2 standard deviations) has been estimated once per type of measurement after 20 tests repeated under the same operating conditions.

Measurement of the blanked part edge profile

Traditionally [START_REF] Tekiner | An experimental study for the effect of different clearances on burr, smooth-sheared and blanking force on aluminium sheet metal[END_REF][START_REF] Hambli | Blanks oft: a code for sheet metal blanking processes optimization[END_REF], the blanked part edge profile is measured from a perpendicular cut on the blanked plane, thus expressing the different zones of the profile according to their length along the profile under inspection (Fig. 1).

However, to reduce scatter in results, due to the irregularity of the blanked profile, particularly for the sheared and fracture zones and often noticed in the case of copper alloys (Fig. 2), the mean position of the boundary of the different zones of the profile has been measured on a 0.5 mm length by photograph and further image analysis.

In all the tests performed in the present study, the roll-over zone is constant and represents 10% of the thickness of the sheet. Considering the variety of studied alloys and their characteristics (cf. Table 2), what can be deduced from this is that the roll-over zone is, in most cases, conditioned by the punch/die clearance [2] and not significantly by the properties of the copper alloy sheets. Some authors, however, have noticed that the roll-over zone increases slightly when the coefficient of work hardening increases [START_REF] Hatanaka | Simulation of sheared edge formation process in blanking of sheet metals[END_REF]. Below, the sheared and fracture zones are regarded as representing 100% of the profile of the blanked part edge.

Measurement of the burr

Blanking is carried out on a progressive tool, consequently, the complex geometry of the leadframe is obtained in several press strokes, which has the effect of crushing the burr (Fig. 3). The burr cannot therefore be measured by its height and its width, as is traditionally done [START_REF]Données matériaux en Découpage-Emboutissage[END_REF]. The quantitative characterization of the burr was therefore performed after undergoing deformation in several stages during the forming process, by means of the image analysis sequence on the scanning electronic microscope (SEM) image (Fig. 4).

However, the volume of the distorted burr can be regarded as being representative of the burr size, since the distortions are mainly plastic (the volume remains therefore constant).

Thus, burr is characterized by its crushed area, the sample lying on a flat plane and the burr being oriented towards the upper side (Fig. 4).

From the SEM photograph, the outline of the area of the flattened burr is drawn "manually" with the aid of dedicated image analysis software (Sigmascan ® ) which allows the area of this previously defined zone to be calculated.

Results and discussion

The results presented in this paper are related to the systematic study of the influence of the composition, microstructure and level of work hardening on the blanking quality of several copper alloys classically used in the manufacturing industry of leadframes.

Influence of the composition of the sheets

Five copper alloys of different compositions, exhibiting the same state of work hardening H12 (termed half hardened state) were blanked (wt%):

-CuFe2.3P0.03, -CuFe0.1P0.03, -CuCo0.28P0.08, -CuNi0.27P0.01, -Cua1 (no alloyed copper with 99.9% of copper).

Profile of the blanked part edge

The composition of the copper alloys has a weak influence on the profile of the blanked part edge as shown in Fig. 5. The sheared zone represents, for all the samples studied, 75-82% of the profile. The magnitude of the measuring uncertainty is of 4%.

For all the copper alloys, the material's work hardening is noticed along the sheared zone of the blanked part edge profile (the first 50-60 m according to the nature of the alloy), a sign that the strain level is increasing (Fig. 6). At a distance greater than 60 m from the top surface, hardness becomes constant in the fracture zone since a maximum has been reached, corresponding to the cracking limit. The microhardness values given in the figures are the average of five different tests. The uncertainty is set at 2 standard deviations and results from a set of 20 measurements for each test type.

In contrast, pure copper Cua1 becomes softer during this first stage of plastic deformation, which is characteristic of the material's dynamic recrystallization phenomenon. This phenomenon is confirmed by the refinement of the copper grains being reduced from 22 to 5 m in the sheared zone (Fig. 7). It can be assumed that the refinement of the grains brings about an increase in the sheared zone by increasing the ductility of the sheet. Cua1 is not an alloyed copper, its temperature of recrystallization (250 • C for an annealing duration of 30 ′ ) is therefore lower than that of the other alloys studied in the present paper (400-600 • C).

Blanking burr

The composition of the alloys has a strong influence on the quantity of burr built up during blanking as shown in Fig. 8. The CuFe2.3P0.03 and CuNi0.27P0.01 alloys produce 35% less burr than the CuFe0.1P0.03 alloy.

Two hypotheses are proposed to explain the differences of burr quantity among the tested alloys:

-All of the alloys are in a state of strain H12, however, the alloying elements have a more or less important hardening effect during laminating and blanking. Then the difference of burr would be explained by the different mechanical behaviours. -The blanked alloys have roughly identical grain sizes (around 10 m, except for Cua1 with 22 m grain size). On the contrary, the alloying elements precipitate mostly into copper. The difference of the nature and morphology of the precipitates can bring about differences in behaviour during the manufacturing process, notably during the burr formation.

Discussion

The choice of the composition of copper alloy has a strong influence on the quality of blanking, even in the case of a small quantity of allowing elements. The influence of the alloying elements is particularly strong upon the quantity of burr formed during blanking which can vary up to 35%. The variations in the blanked part edge profile are smaller than for the burr. As a result, the influence of the state of work hardening and the microstructure of the copper alloys on the blanking quality remains to be qualified.

Influence of the degree of initial work hardening of blanked material

In this part, three alloys CuFe0.1P0.03 in work hardened states H14, H12 and annealed, were blanked. These alloys are of an absolutely similar composition and observation by means of transmission electron (TEM) microscopy has demonstrated the similarity of their microstructure (grain size, precipitation distribution, and nature).

Blanked part edge profile

The profile of the blanked part edge of the CuFe0.1P0.03 is hardly influenced by its state of work hardening (Fig. 9). Taupin et al. [START_REF] Taupin | Material fracture and burr formation in blanking results of FEM simulations and comparison with experiments[END_REF] noticed the same behaviour for steels of different hardness. The harder the steel was, the more extended the fracture zone was, which must have been linked to reduced toughness. Wong and Das [START_REF] Wong | Analysis of stresses in bar cropping[END_REF] consider that there are two main modes of fracture of the sheets according to the nature of the material:

-If the material is ductile, the blanked part edge profile exhibits an aspect which is almost completely sheared. -If the material is brittle, cracks will spread suddenly from the beginning of plastic flow and the part edge profile of the blanked sample exhibits an almost completely fractured aspect. The annealed alloy CuFe0.1P0.03 is regarded as being very ductile, while an increasing work hardening level lowers its resilience and extends of the fractured area in the blanked part edge profile.

Blanking burr

The quantity of burr formed during the blanking of the three alloys CuFe0.1P0.03 is identical (Fig. 10). This result demonstrates the independence of the burr with respect to the work hardening. Otherwise, burr growth seems to be independent of the profile, since the latter shows strong variations (fractured zone from 71 to 87%) without affecting the quantity of burr.

Influence of the microstructure

To study the influence of the microstructure, the results of the blanking of two CuNiP alloys are compared, CuNi0.27P0.01 H12 and CuNi0.27P0.01 H12*, of similar level of work hardening, but for which the annealing temperature before rolling is higher than 200 • C limiting the precipitation of hard nickel-phosphorus composites, which can be under the forms Ni2P, Ni3P or Ni5P2 according to Crampton [START_REF] Crampton | The copper rich alloys of the copper-nickel-phosphorus system[END_REF] and Voloshko and Rozenberg [START_REF] Voloshko | The copper corner of the Cu-Ni-P equilibrium diagram[END_REF].

Blanked part edge profile

The CuNi0.27P0.01 H12 alloy gives a rather more fractured zone than the CuNi0.27P0.01 H12* alloy (Fig. 11), while the latter has a lower hardness (cf. Table 2), which contradicts against the previous results of Section 3.2.1. It seems therefore that nickel-phosphorus precipitates favour the ductility of the sheets during blanking.

It can be systematically seen in Fig. 12 that the facies of the fracture zone of the blanked profile contain dimples. However, the number of dimples is smaller in the case of the alloy H12* (Fig. 12), linked to its smaller amount of precipitates. Brokken [START_REF] Brokken | Numerical modelling of ductile fracture in blanking[END_REF] also compares the precipitation of a second carbide phase in steel with the number of dimples of the fracture zone examined from the blanked part edge profile.

Blanking burr

The quantity of burr is significantly more important in the case of the CuNi0.27P0.01 H12* alloy (Fig. 13), emphasizing the influence of the nickel-phosphorus precipitates.

But can be considered that the nickel-phosphorus precipitates present in the CuNi0.27P0.01 alloy are like so many zones of crack initiation during the growth of the fracture zone (cf. number of dimples in Fig. 11) and of the burr. The larger the number of precipitates is the weaker the burr is.

Discussion

It is shown that the level of work hardening of the sheetsduring lamination -changes the shape of the blanked part edge profile significantly. In contrast, the level of work hardening does not have any influence on the quantity of burr formed during the blanking process. It appears that precipitates influence the quality of blanking both for the faces of the blanked zone and for the quantity of burr.

Conclusion

The quality of sheet metal blanking can be checked by quantifying the relative importance of the roll-over, sheared and fracture zones and burr height. Quality is all the better as the sheared zone is important in the blanked part edge profile, because it respects the desired geometry in the blanking metal process. One can notice that there are noticeable differences in the blanking quality according to the nature of the copper alloy sheet.

Main conclusions are the following:

-The roll-over is hardly influenced by the geometry of the blanking process (blanking punch/die clearance). For the copper alloys under study, the roll-over zone represents, in all cases, 10% of the thickness of the sheet. -The sheared and fracture zones are mainly influenced by the mechanical properties of the sheet. The more work hardened it is (depending on the technique of sheet making), the greater the fracture zone is. It can be easily understood that the fracture occurs all the earlier as resilience is weak. Therefore, it leads to a weaker sheared zone, more detrimental to the overall good quality of the blanking process. -Burr growth is independent of the work hardening state of the sheet, it is hardly influenced, on the contrary, by the precipitates. The CuNi0.27P0.01 alloy for which the precipitation is enhanced sees its quantity of burr diminish by 25%.

N.B.: The uncertainties of measurements correspond to two root mean squares.

Fig. 1 .

 1 Fig. 1. Traditional blanked profile.

Fig. 2 .

 2 Fig. 2. CuFe0.1P0.03 H14 blanking profile (optical microscope ×500).

Fig. 3 .

 3 Fig. 3. Burr crushing: (a) blanking burr after one strike and (b) blanking burr in a progressive tool.

Fig. 4 .

 4 Fig. 4. Illustration of blanked piece.

Fig. 5 .

 5 Fig. 5. Blanking profile of copper alloy H12.

Fig. 6 .

 6 Fig. 6. (a) Principle of hardness measure along the shear zone and (b) hardness of the strip along the shear zone.

Fig. 7 .

 7 Fig. 7. Microstructure of pure copper Cual H12 (SEM × 1300): (a) before blanking and (b) after blanking along the shear zone.

Fig. 8 .

 8 Fig. 8. Crushed burr surface of copper alloys H12.

Fig. 9 .

 9 Fig. 9. Blanking profile of CuFe0.1P0.03.

Fig. 10 .

 10 Fig. 10. Crushed burr surface of CuFe0.1P0.03.

Fig. 13 .

 13 Fig. 13. Crushed burr surface of the alloy CuNi0.27P0.01 H12 and H12*.

Fig. 12 .

 12 Fig. 12. Variation of the number of dimples on the fracture zone of the alloy CuNi0.27P0.01 H12 and H12*.

Table 1

 1

	Blanking parameters
	Punch material		Tungsten carbide
	Clearance		10 m
	Blanking speed		550 strokes min -1
	Work-piece thickness		0.254 mm
	Punch dimension (rectangular)	0.5 mm × 10.5 mm
	Table 2
	Copper alloys studied (mass%)
	Copper alloys	State	Vickers hardness (H v 200 g)
	CuFe2.3P0.03	H12	128 ± 2
	CuFe0.1P0.03	H14	146 ± 2
	CuFe0.1P0.03	H12	124 ± 2
	CuFe0.1P0.03	Annealed	76 ± 2
	CuCo0.28P0.08	H12	126 ± 2
	CuNi0.27P0.01	H12	107 ± 2
	CuNi0.27P0.01	H12*	98 ± 2
	Cua1	H12	99 ± 2


Adel Benarbia
email: adel.benarbia@univ-nantes.fr

Vincent Sobotka
email: bvincent.sobotka@univ-nantes.fr

Nicolas Boyard
email: cnicolas.boyard@univ-nantes.fr

Christophe Roua
email: dchristophe.roua@cogit-composites.com

Fused filament fabrication: Numerical adhesion modeling suitable for semicrystalline polymers

Keywords: Fused Filament Fabrication (FFF), Heat Transfer, Adhesion, Interfacial bonding, Crystallization, Phase transformation, Semicrystalline polymer, Numerical simulation

come

Introduction

In recent years, additive manufacturing has been the subject of many studies, regardless of the type of technology. The Fused Filament Fabrication (FFF) process being the most widespread, a great deal of attention has been paid to it. It is one of the affordable additive manufacturing technologies with a simple technology and a fast manufacturing speed. Nevertheless, one of the notable limitations of the FFF process is the decrease of the mechanical properties, compared to conventional processes like injection molding. These mechanical properties are low in the direction normal to the interfaces, resulting in high anisotropic properties [START_REF] Popescu | FDM process parameters influence over the mechanical properties of polymer specimens: A review[END_REF]. The decrease in mechanical properties is related to the presence of porosities and the low interfacial strength between filaments. A lot of researches have been performed to improve the quality of the parts manufactured with this process [START_REF] Popescu | FDM process parameters influence over the mechanical properties of polymer specimens: A review[END_REF][START_REF] Lepoivre | Heat Transfer and Adhesion Study for the FFF Additive Manufacturing Process[END_REF].

Adhesion in the FFF process takes place in several steps. First, the material melted by the nozzle is deposited on the previous filament, intimate contact occurs and interfaces develop with the other filaments. This phenomenon called coalescence is strongly controlled by the rheological behavior of the material and the surface tension but also external pressures such as nozzle crushing. When coalescence begins, the healing process starts with the diffusion of polymer chains through the interfaces in order to create a real molecular bond with the others filaments [START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF].

Coalescence and healing are only possible if the chains present at the interfaces have sufficient molecular mobility [START_REF] Barocio | Interlayer fusion bonding of semi-crystalline polymer composites in extrusion deposition additive manufacturing[END_REF]. Molecular mobility is governed by the thermodynamic state of the polymers, for the amorphous phase, mobility is restricted below the glass transition [START_REF] Gennes | Reptation of a Polymer Chain in the Presence of Fixed Obstacles[END_REF]. Finally, in the case of semicrystalline polymers, crystallization may limit the adhesion phenomenon. These mechanisms are therefore strongly dependent on the rheological behavior evolving with the temperature and the crystalline degree of the polymer [START_REF] Lo | Interfacial adhesion mechanisms in incompatible semicrystalline polymer systems[END_REF].

Studies were conducted to predict the thermal history of the FFF process and its influence on interfacial adhesion. A large majority of the studies consider the heat transfer at the interface assuming a homogeneous temperature within the thickness of the filament [START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF][START_REF] Costa | Estimation of filament temperature and adhesion development in fused deposition techniques[END_REF]. But some studies have proposed thermal modeling at the filament scale in order to estimate more precisely the interface temperature [START_REF] Lepoivre | Heat Transfer and Adhesion Study for the FFF Additive Manufacturing Process[END_REF]. From these works, numerical Finite Element Method (FEM) models were developed and showed good agreement with experimental tests in the case of amorphous polymers.

However, for semicrystalline polymers, even if some works take into account the influence of crystallization and melting enthalpies o heat transfer, none of them realize a real link with the adhesion physics [START_REF] Greco | Statistical and kinetic approaches for linear low-density polyethylene melting modeling[END_REF]. The only studies that propose to predict the healing degree as a function of crystalline degree assume that adhesion stops abruptly at the beginning of crystallization [START_REF] Barocio | Prediction of the Degree of Bonding in the Extrusion Deposition Additive Manufacturing Process of Semi-Crystalline Polymer Composites[END_REF]. In spite of this approximation the authors obtained consistent results with the experimental. Nevertheless, several studies showed that despite the presence of crystals, for low crystallinity degree, the adhesion mechanism can continue because the amorphous phase still has a certain level of mobility [START_REF] Carotenuto | Non-Isothermal Crystallization Kinetics of an Ethylene-Vinyl-Acetate. II. Time-Temperature-Crystallinity-Superposition[END_REF][START_REF] Minale | Nonisothermal Crystallization Kinetics of an Ethylene-Vinyl-Acetate: I Calorimetry Versus Rheology[END_REF]. In recent years, a promising time-temperature-crystallization-superposition (TTCS) model has been studied in order to predict the evolution of molecular mobility as a function of temperature and crystallinity and could allow a better understanding of adhesion with semicrystalline polymer [START_REF] Carotenuto | Non-Isothermal Crystallization Kinetics of an Ethylene-Vinyl-Acetate. II. Time-Temperature-Crystallinity-Superposition[END_REF][START_REF] Pantani | Evolution of iPP Relaxation Spectrum during Crystallization: Evolution of iPP Relaxation Spectrum during Crystallization …[END_REF].

Multiphysics modeling of FFF process

In this work, heat transfers, phase transformations and adhesion physics are implemented in a coupled multiphysics model. The model does not represent the rheology of the deposited filament. Assuming an instantaneous flow, a non-deformable oblong section will be taken to build the model. The simulation is performed in a single computational step where the use of boolean conditions allows the model to evolve in order to represent the successive filaments deposition. This allows a strong coupling between heat transfers and phase transformations.

The numerical resolution was performed on a 2D model simulated by FEM on COMSOL ® software. In agreement with the work of [START_REF] Lepoivre | Heat Transfer and Adhesion Study for the FFF Additive Manufacturing Process[END_REF], we develop a dimensionless modelling with the simplifying hypothesis that the thermal exchanges occur mainly in the plane orthogonal to the deposition direction. A comparison between 2D and 3D models validated this assumption. The deposition is simulated by the domain activation method and the boundary conditions change according to the deposition steps.

Heat transfer and phase transformation are computed on the whole domain. At each calculation step, the healing degree is evaluated on filaments interfaces in order to predict their evolution over time as a function of temperature and crystalline degree.

The driving mechanisms of coalescence are more difficult to quantify and can make the simulation complex. Indeed, some authors agree that the phenomenon is driven only by viscous effects and surface tension [START_REF] Lepoivre | Heat Transfer and Adhesion Study for the FFF Additive Manufacturing Process[END_REF][START_REF] Balani | Influence of printing parameters on the stability of deposited beads in fused filament fabrication of poly(lactic) acid[END_REF]. While others describe a strong influence of the printing nozzle pressure on the filament geometry [START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF].

Based on these studies, our model will consider an oblong geometry with a constant and homogeneous section without defects. The oblong geometry allows to take into account the porosities present between the filaments. The fillet size will be based on micrographic observations of a printed part. An 8 filament high wall was simulated in one step. The model being adaptive, it is possible to simulate a wall of several filaments wide as shown Fig. 1.

Heat transfer

The thermal model is represented in Fig. 1. In the domain, we solve the energy equation coupled with crystallization and melting. The source term present in the energy equation term depends on ∆H the crystallization and melting enthalpies and α the crystalline degree. In order to represent the effects of the heating nozzle, the initial temperature of each new filament is considered uniform and set to TNOZZLE to represent the extrusion set point. The bottom surface of the first filament is fixed at the temperature TBED throughout the simulation process in order to represent the printing plate setting. Heat transfer between the environment and the deposited filaments are considered only convective. It is considered a constant chamber temperature TChamber throughout the deposit. The boundary conditions are re-evaluated at each deposit thanks to boolean variables in order to make the model evolve with the deposit. Thermal contact resistances (TCR) were applied between the filaments to represent contact imperfection [START_REF] Lepoivre | Heat Transfer and Adhesion Study for the FFF Additive Manufacturing Process[END_REF].

The characterizations of thermal conductivity 𝑘𝑘, heat capacity Cp and density 𝜌𝜌 were performed on PAEK (Polyaryletherketone). The thermal property measurements were performed above the glass transition in the semi-crystalline solid state taking into account the influence of temperature. We will consider a linear variation of the thermal properties with temperature and will realize the simplifying assumption that the values are identical whatever the crystalline degree.

Model parameters such as layer thickness, printing speed, nozzle temperature are chosen as representative of real process [START_REF] Zanjanijam | Fused Filament Fabrication of PEEK: A Review of Process-Structure-Property Relationships[END_REF].

Phase transformation: Crystallization/melting

In order to predict the evolution of the crystalline degree as a function of the thermal history, a phase change model including crystallization and melting was added to the model.

Given the high heating rates of the interface during deposition, we will consider that the material undergoes a simple melting without simultaneous melting and recrystallization [START_REF] Baumard | Caractérisation des cinétiques de cristallisation du poly(éther éther cétone) (PEEK) par mesure de la chaleur spécifique en calorimétrie différentielle à balayage rapide[END_REF]. It is therefore possible to model the two mechanisms separately by considering a transition temperature TTransition at 300°C.

 Crystallization kinetics

In the case of the FFF process, crystallization generally takes place in non-isothermal conditions. In order to model the formation and growth of crystals we chose the Nakamura model eq [START_REF] Popescu | FDM process parameters influence over the mechanical properties of polymer specimens: A review[END_REF] which is the most widely used in the literature [START_REF] Nakamura | Some aspects of nonisothermal crystallization of polymers. I. Relationship between crystallization temperature, crystallinity, and cooling conditions[END_REF].

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = 𝐾𝐾 𝑁𝑁𝑁𝑁𝑁𝑁 (𝑇𝑇). 𝐺𝐺(𝑑𝑑) , 𝑤𝑤𝑤𝑤𝑑𝑑ℎ 𝐺𝐺(𝑑𝑑) = 𝑛𝑛 (1 -𝑑𝑑)[-ln(1 -𝑑𝑑)] 1-1 𝑛𝑛 ( 1
)
where 𝑑𝑑 is the relative crystalline degree, 𝐾𝐾 𝑁𝑁𝑁𝑁𝑁𝑁 (𝑇𝑇) the Nakamura kinetics coefficient calculated by 𝐾𝐾 𝑁𝑁𝑁𝑁𝑁𝑁 (𝑇𝑇) = 𝐾𝐾 𝑁𝑁𝐴𝐴𝐴𝐴𝑁𝑁𝐴𝐴𝐴𝐴 (𝑇𝑇) 1/𝑛𝑛 obtained in isothermal conditions and n the Avrami index [START_REF] Avrami | Phase Change, and Microstructure Kinetics of Phase Change[END_REF].

Furthermore, in the FFF process, when the previously crystallized filament is heated due to the new deposit, it is possible that at low interfacial temperatures the polymer is partially melted. The kinetics of crystallization from a partially melted state is not well explored and no model is available in the literature. We will therefore assume that during a recrystallization from a partial melt, the kinetics is not impacted and therefore the polymer behaves as if it had simply stopped the crystallization. As our PAEK has too high crystallization kinetics to be characterized by differential scanning calorimetry (DSC), a test campaign was performed on fast scanning calorimeter (FSC) to estimate the thermal dependency of the isothermal Avrami coefficients 𝐾𝐾 𝑁𝑁𝐴𝐴𝐴𝐴𝑁𝑁𝐴𝐴𝐴𝐴 (𝑇𝑇) over the large temperature range [160°C -300°C].

 Melting kinetics

The melting kinetics has been further explored in the case of thermoplastic polymers. It has been shown that the thermo-kinetic models available in the literature taking into account the heating rate only allow to represent a restricted range of rates [START_REF] Greco | Statistical and kinetic approaches for linear low-density polyethylene melting modeling[END_REF]. Moreover, these models cannot predict the interruption of melting during isotherm or cooling. In several works, authors made the assumption that the melting behavior depends only on the temperature by considering an instantaneous mechanism independent of the heating rate over a certain range [START_REF] Nicodeau | Continuous welding modeling of thermoplastic matrix composites n[END_REF]. In this study, we also have chosen this approach by taking a statistical distribution model of the crystalline lamellae thickness which is temperature dependent eq [START_REF] Lepoivre | Heat Transfer and Adhesion Study for the FFF Additive Manufacturing Process[END_REF].

𝑑𝑑 = 𝑑𝑑 𝑚𝑚𝑚𝑚𝑚𝑚 . 𝑀𝑀𝑀𝑀𝑀𝑀𝑑𝑑𝑤𝑤𝑛𝑛𝑀𝑀(𝑇𝑇) (2)
Melting function was estimate by temperature integration of the melting peak measured on DSC and weighting 𝑑𝑑 𝑚𝑚𝑚𝑚𝑚𝑚 in order to represent the melting under conditions of incomplete crystallization, case where 𝑑𝑑 𝑚𝑚𝑚𝑚𝑚𝑚 < 1.

Implementation of the melting and crystallization models in the same simulation model allows to predict the melting zone and time available for the interfacial healing. For example, it can be seen in Fig. 2 that the interface melted during the deposition because of the heat provided by the hot filament.

Healing coupled with crystallization degree

Prediction of healing at interfaces was also added in the modelling taking into account the coupling of an anisothermal healing model and a time-temperature-crystallization-superposition model

 Healing model

In order to represent the adhesion kinetics in the anisothermal case, we proposed to use the empirical model largely defined in eq [START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF].

𝐷𝐷 𝐻𝐻𝐻𝐻𝑚𝑚𝐻𝐻𝐻𝐻𝑛𝑛𝐻𝐻 (𝑑𝑑) = �� 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑤𝑤𝐻𝐻𝐻𝐻𝑤𝑤𝐻𝐻𝑛𝑛𝐻𝐻 𝑡𝑡 0 4 (3)
were 𝐷𝐷 𝐻𝐻𝐻𝐻𝑚𝑚𝐻𝐻𝐻𝐻𝑛𝑛𝐻𝐻 is the healing degree and 𝑑𝑑 𝑤𝑤𝐻𝐻𝐻𝐻𝑤𝑤𝐻𝐻𝑛𝑛𝐻𝐻 the experimental measured welding times to achieve complete welding of an interface under isothermal conditions.

Several authors showed that in the case of several thermoplastics, the welding times can be considered as equal to the molecule reptation times 𝑑𝑑 𝑟𝑟𝐻𝐻𝑟𝑟𝑡𝑡𝑚𝑚𝑡𝑡𝐻𝐻𝑟𝑟𝑛𝑛 [START_REF] Wool | Polymer interfaces : Structure and strength[END_REF] with a good agreement with the experiment. In order to simplify the numerical solution, an integrative form was proposed by [START_REF] Lepoivre | Heat Transfer and Adhesion Study for the FFF Additive Manufacturing Process[END_REF]. Taking into account the assumption that the relaxation times are equal to welding time, we obtain eq [START_REF] Barocio | Interlayer fusion bonding of semi-crystalline polymer composites in extrusion deposition additive manufacturing[END_REF] with ∆𝑑𝑑 the calculation step, and t the simulation time step..

[𝐷𝐷 𝐻𝐻𝐻𝐻𝑚𝑚𝐻𝐻𝐻𝐻𝑛𝑛𝐻𝐻 ] 𝑡𝑡+1 = � ∆𝑑𝑑 𝑑𝑑 𝑟𝑟𝐻𝐻𝑟𝑟𝑡𝑡𝑚𝑚𝑡𝑡𝐻𝐻𝑟𝑟𝑛𝑛 + [𝐷𝐷 𝐻𝐻𝐻𝐻𝑚𝑚𝐻𝐻𝐻𝐻𝑛𝑛𝐻𝐻 ] 𝑡𝑡 4 � 1 4 (4)
 Time / Temperature / Crystallization Superposition (TTCS)

In order to represent the evolution of the molecular mobility during the process, it is necessary to know the relaxation time of the polymer according to its local environment. To our knowledge, no study takes into account the crystalline degree in its adhesion model. Indeed, while the evolution of molecular mobility as a function of temperature is well known with the time-temperaturesuperposition TTS, few studies focus on the influence of the presence of crystals on polymer diffusion.

But recently, some studies have converged to present a time-temperature-crystallizationsuperposition TTCS principle eq (5) based on experimental approaches [START_REF] Pantani | Evolution of iPP Relaxation Spectrum during Crystallization: Evolution of iPP Relaxation Spectrum during Crystallization …[END_REF].

𝜏𝜏(𝑇𝑇, 𝑑𝑑) = 𝑎𝑎 𝑡𝑡 (𝑇𝑇 𝐴𝐴𝑅𝑅𝑅𝑅 , 𝑑𝑑 0 , 𝑇𝑇) . 𝑎𝑎 𝜉𝜉 (𝑇𝑇 𝐴𝐴𝑅𝑅𝑅𝑅 , 𝑑𝑑 0 , 𝑑𝑑) . 𝜏𝜏(𝑇𝑇 𝐴𝐴𝑅𝑅𝑅𝑅 , 𝑑𝑑 0 ) ( 5
)
were 𝑇𝑇 𝐴𝐴𝑅𝑅𝑅𝑅 and 𝑑𝑑 0 are respectively the temperature and crystalline degree reference and 𝜏𝜏 the reptation time.

The model proposes a shift of rheological properties similar to thermal shift but with the crystalline degree. The postulated TTCS is very promising and has shown its consistency on several polymers. Moreover, several previous studies have shown that the evolution of rheological properties is proportional to the volume of crystallites and not to the shape or size of crystals [START_REF] Lamberti | Crystallinity and Linear Rheological Properties of Polymers[END_REF]. These observations suggest that they can be applied to all types of crystallization. Moreover, the principle of TTCS allows to correctly represent the state of molecular mobility in cases of isothermal and anisothermal crystallization. Its general validity must be evaluated in order to apply it widely to all polymer families. Nevertheless, being the only model available in the literature, we will use it to predict the evolution of the mobility depending on the temperature and the crystalline degree. The thermal shift factors were fitted using the Williams-Landel-Ferry equation (WLF) [START_REF] Williams | The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids[END_REF] and the crystalline shift factors using a power law as proposed by defined [START_REF] Carotenuto | Non-Isothermal Crystallization Kinetics of an Ethylene-Vinyl-Acetate. II. Time-Temperature-Crystallinity-Superposition[END_REF] in eq [START_REF] Lo | Interfacial adhesion mechanisms in incompatible semicrystalline polymer systems[END_REF].

ln[𝑎𝑎 𝑡𝑡 (𝑇𝑇 𝐴𝐴𝑅𝑅𝑅𝑅 , 𝑑𝑑 0 , 𝑇𝑇)] = -𝐶𝐶1(𝑇𝑇 -𝑇𝑇 𝐴𝐴𝑅𝑅𝑅𝑅 ) 𝐶𝐶2 + (𝑇𝑇 -𝑇𝑇 𝐴𝐴𝑅𝑅𝑅𝑅 ) 𝑎𝑎𝑛𝑛𝑑𝑑 𝑎𝑎 𝜉𝜉 (𝑇𝑇 𝐴𝐴𝑅𝑅𝑅𝑅 , 𝑑𝑑 0 , 𝑑𝑑) = 10 𝐵𝐵1.𝛼𝛼 𝐵𝐵2 (6)
where C1, C2, B1 and B2 are fitting coefficients. Rheological measurements were carried out in our laboratory between 360°C and 400°C under nitrogen in the amorphous state in order to estimate the thermal shift factor. The model coefficients of the crystallinity shift factor are taken from the literature [START_REF] Carotenuto | Non-Isothermal Crystallization Kinetics of an Ethylene-Vinyl-Acetate. II. Time-Temperature-Crystallinity-Superposition[END_REF].

As previously explained, by integrating all these models in the same study, the multiphysics model is able to predict the evolution of reptation time as a function of temperature and crystalline degree in anisothermal condition. In order to explain the model, we have calculated the evolution of the relaxation time for several constant cooling rates. The results are shown in Fig. 3. Fig. 3.b represents the evolution of the creep time as a function of temperature and crystalline degree Fig. 3.a. We notice that at high speed, the polymer remains amorphous during the cooling and the material follows the TTS law. But when the speed decreases and the material crystallizes, the reptation time increases quickly from the first moment of crystallization.

Results and discussion

As said previously, a wall of 8 filaments high and 1 filament wide is simulated. All results are evaluated in the middle of the interface between the 5 th and 6 th layer so the results will begin at 32s. The simulations carried out from the multiphysics model allow to estimate the evolution of the crystallinity rate and the degree of healing according to the thermal history at the interface.

Simulations were performed with the TTCS model and with the TTC model in order to show the importance of taking into account the crystallization on the adhesion. Both cases were simulated with the same parameters except that the crystalline shift factor was numerically fixed at 1 to model the TTS.

With the conditions used in the simulation, it can be seen in Fig. 4, that the 5 th filament crystallizes rapidly during cooling. During the next deposition, the heat generated by the upper filament partially melts the interface. The difference in the evolution of the degree of healing between the TTCS and TTS models describes the low molecular mobility of the chains due to the presence of crystals at the interface. Indeed, the final degree of healing is divided by 10.

To show the influence of the thermal environment on the adhesion degree, we varied the chamber and bed temperatures between 150°C and 270°C as shown Fig. 5. It is first noticed for temperatures lower than 200°C, i.e. where the cooling is fast, that the filament crystallizes completely before the next deposition. In these cases, only melting can allow a noticeable decrease of crystalline degree and thus a better adhesion. Because during deposition the interface temperature is not high enough to allow a total fusion, the healing degree are very low (<0.09). At high temperatures, the crystals are not yet formed at the time of deposition (T≥250°C) therefore a better healing degree is obtained. This numerical study allows to estimate the process range in order to obtain a correct interface with a complete crystallization at the end of printing. Other parameters such as nozzle temperature, printing speed, number of beads per layer, deposit order can be tested to evaluate their influence on crystallization and adhesion.

Conclusion

A multiphysics model was developed to represent the thermal history, crystalline and adhesion degree evolution at the filament scale in the FFF process. Modeling development allowed to calculate the evolution of the crystalline degree on the whole domain and in particular at the interfaces. Indeed, the model describes crystalline degree remaining after the melting generated by the deposit and the time before recrystallization. A brief parametric analysis demonstrates the nonnegligible influence on the crystalline evolution on interfacial adhesion. In fact, it is shown that complete healing is only possible if the crystalline degree is close to zero. This model allows us to understand the influence of many printing parameters on adhesion and therefore to propose groups of parameters in order to improve the printing quality.

In order to confirm that the model correctly predicts the evolution of the crystalline degree as well as the effective adhesion degree of the printed part, an experimental study will have to be carried out and the adhesion values confronted with mechanical tests. Moreover, a next study will aim to model more finely the melting kinetics as well as the recrystallization kinetics through an experimental campaign.

Fig. 1 -

 1 Fig. 1 -Schematic representation of the case study

Fig. 2 -

 2 Fig. 2 -Remelting of the filament surface after a second hot filament deposition

Fig. 3 -

 3 Fig. 3 -Relative crystalline degree (A) and the reptation time evolution under different cooling rate (B)

Fig. 4 -Fig. 5 -

 45 Fig. 4 -Evolution of computed temperature, crystalline and healing degree with TTS and TTCS model