Manos Mavrakis

M Angeles Juanes
email: majuanes@cipf.es

The compass to follow: focal adhesion turnover

How cells move is a fundamental biological question. The directionality of adherent migrating cells depends on the assembly and disassembly (turnover) of focal adhesions (FAs). FAs are micron-sized actin-based structures that link cells to the extracellular matrix. Traditionally, microtubules have been considered key to triggering FA turnover. Through the years, advancements in biochemistry, biophysics, and bioimaging tools have been invaluable for many research groups to unravel a variety of mechanisms and molecular players that contribute to FA turnover, beyond microtubules. Here we discuss recent discoveries of key molecular players that affect the dynamics and organization of the actin cytoskeleton to enable timely FA turnover and consequently proper directed cell migration.

INTRODUCTION

Directionality of adherent cells mostly depends on tight coordination of membrane protrusions at the leading edge, assembly of adhesive points known as focal adhesions (FAs), contraction of the cell body, and disassembly of mature FAs [START_REF] Gardel | Mechanical integration of actin and adhesion dynamics in cell migration[END_REF][START_REF] Ridley | Cell migration: integrating signals from front to back[END_REF][START_REF] Sengupta | The principles of directed cell migration[END_REF].

Decades of investigations have revealed that FAs are composed of a large repertoire of proteins that organize at the nanoscale level in three interconnected layers running parallel to the cell membrane, rather than in three independent layers [START_REF] Orré | Molecular motion and tridimensional nanoscale localization of kindlin control integrin activation in focal adhesions[END_REF][START_REF] Legerstee | A layered view on focal adhesions[END_REF][START_REF] Orré | The inner life of integrin adhesion sites: From single molecules to functional macromolecular complexes[END_REF][START_REF] Kanchanawong | Nanoscale architecture of integrin-based cell adhesions[END_REF]. Such FA organization starts with the association of integrin transmembrane receptors connecting the extracellular matrix (ECM) to the plasma membrane [START_REF] Vicente-Manzanares | Adhesion dynamics at a glance[END_REF][START_REF] Vicente-Manzanares | Integrins in cell migration--the actin connection[END_REF] (Figure1).

Integrins bind to adaptor proteins, including the focal adhesion kinase (FAK) and paxillin which altogether compose the bottom or signaling layer. On top of those, forming the core or force transduction layer, lie vinculin and talin which link the actin cytoskeleton to the ECM [START_REF] Orré | Molecular motion and tridimensional nanoscale localization of kindlin control integrin activation in focal adhesions[END_REF][START_REF] Legerstee | A layered view on focal adhesions[END_REF][START_REF] Orré | The inner life of integrin adhesion sites: From single molecules to functional macromolecular complexes[END_REF][START_REF] Case | Molecular mechanism of vinculin activation and nanoscale spatial organization in focal adhesions[END_REF][START_REF] Case | Integration of actin dynamics and cell adhesion by a three-dimensional, mechanosensitive molecular clutch[END_REF]. Zyxin, vasodilator-stimulated phosphoprotein (VASP), and α-actinin, among other proteins are enriched at the top together with actin filaments (F-actin), forming the so-called actin regulatory layer [START_REF] Legerstee | A layered view on focal adhesions[END_REF][START_REF] Orré | The inner life of integrin adhesion sites: From single molecules to functional macromolecular complexes[END_REF][START_REF] Vicente-Manzanares | Integrins in cell migration--the actin connection[END_REF][START_REF] Nordenfelt | Direction of actin flow dictates integrin LFA-1 orientation during leukocyte migration[END_REF][START_REF] Swaminathan | Actin retrograde flow actively aligns and orients ligand-engaged integrins in focal adhesions[END_REF][START_REF] Kumar | Local Tension on Talin in Focal Adhesions Correlates with F-Actin Alignment at the Nanometer Scale[END_REF]] (Figure1). This complex nanoarchitecture of the FAs is assembled at the leading edge of migrating cells within minutes. Nascent adhesions grow in size and develop into focal complexes that will fully mature, then will disassemble, and eventually will reassemble (Figure1). Those association and dissociation cycles are key to permitting efficient and persistent motility [START_REF] Sengupta | The principles of directed cell migration[END_REF][START_REF] Vicente-Manzanares | Adhesion dynamics at a glance[END_REF][START_REF] Seetharaman | Cytoskeletal crosstalk in cell migration[END_REF]. The precise spatial and temporal control of FA turnover is achieved by multiple mechanisms, including changes in the cytoskeletal dynamics, the coordination of signaling pathways and trafficking of integrins [START_REF] Gardel | Mechanical integration of actin and adhesion dynamics in cell migration[END_REF][START_REF] Sengupta | The principles of directed cell migration[END_REF][START_REF] Case | Molecular mechanism of vinculin activation and nanoscale spatial organization in focal adhesions[END_REF][START_REF] Seetharaman | Cytoskeletal crosstalk in cell migration[END_REF][START_REF] Paul | Endocytic trafficking of integrins in cell migration[END_REF].

In this review we highlight what we have learnt and what the puzzling pieces are related to two main questions:

(i) Which actin nucleation and elongation factors assemble actin filaments at FAs? How do they control FA turnover?

(ii) How does actin at FAs interplay with other components of the cytoskeleton such as microtubules and septins to control FA turnover?

Actin nucleation and elongation factors at focal adhesions

Several actin nucleation promoting factors have been found in close proximity to or at

FAs, including the actin-related protein 2/3 (Arp2/3), formins, Ena/VASP, and the tumor suppressor Adenomatous polyposis coli (APC) (Figure 1).

The Arp2/3 complex nucleates branched actin filaments upon stimulation of the Wiskott-Aldrich syndrome protein (WASP)/Scar and WAVE (WASP family veroprolinhomologue), and this actin is coupled to integrin-mediated adhesion [START_REF] Papalazarou | The cell pushes back: The Arp2/3 complex is a key orchestrator of cellular responses to environmental forces[END_REF][START_REF] Gautreau | Nucleation, stabilization, and disassembly of branched actin networks[END_REF][START_REF] Case | Adhesive F-actin waves: a novel integrin-mediated adhesion complex coupled to ventral actin polymerization[END_REF]. Depletion of the Scar/WAVE regulatory complex leads to FAK activation that results in fewer but more stable FAs [START_REF] Tang | Loss of Scar/WAVE complex promotes N-WASP-and FAKdependent invasion[END_REF]. Similarly, fewer FAs with reduced disassembly rates and longer lifetimes were found in a mouse embryonic fibroblast (MEF) model depleted of Nckap1, a conserved subunit of the WAVE regulatory complex [START_REF] Whitelaw | The WAVE regulatory complex is required to balance protrusion and adhesion in migration[END_REF]. The Arp2/3 complex can interact transiently with vinculin and/or FAK, and the 'hybrid' Arp2/3 complex is recruited to FAs [START_REF] Demali | Recruitment of the Arp2/3 complex to vinculin: coupling membrane protrusion to matrix adhesion[END_REF][START_REF] Serrels | Focal adhesion kinase controls actin assembly via a FERM-mediated interaction with the Arp2/3 complex[END_REF][START_REF] Chorev | Regulation of focal adhesion formation by a vinculin-Arp2/3 hybrid complex[END_REF][START_REF] Swaminathan | The FAK-Arp2/3 interaction promotes leading edge advance and haptosensing by coupling nascent adhesions to lamellipodia actin[END_REF]. For instance, Arp2/3-vinculin increases the levels of the Arp2/3 at FAs where it promotes FA formation [START_REF] Demali | Recruitment of the Arp2/3 complex to vinculin: coupling membrane protrusion to matrix adhesion[END_REF][START_REF] Chorev | Regulation of focal adhesion formation by a vinculin-Arp2/3 hybrid complex[END_REF]. The Arp2/3-FAK interaction [START_REF] Serrels | Focal adhesion kinase controls actin assembly via a FERM-mediated interaction with the Arp2/3 complex[END_REF] recruits Arp2/3 to the nascent adhesions where it 'tempers' the phosphorylation on Y397 of FAK, thereby reducing the formation of FAK-dependent nascent adhesions [START_REF] Serrels | Focal adhesion kinase controls actin assembly via a FERM-mediated interaction with the Arp2/3 complex[END_REF][START_REF] Swaminathan | The FAK-Arp2/3 interaction promotes leading edge advance and haptosensing by coupling nascent adhesions to lamellipodia actin[END_REF]. In addition, the formation of nascent adhesions at the leading edge is facilitated by the actin-binding protein lamellipodin (Lpd). Ldp has not been localized at FAs. Then, its effects on adhesions may be indirect, perhaps in part due to its interaction with the small GTPase Rac1 or WAVE regulatory complex, both of which associate with adhesions [START_REF] Dimchev | Lamellipodin tunes cell migration by stabilizing protrusions and promoting adhesion formation[END_REF][START_REF] Rottner | Interplay between Rac and Rho in the control of substrate contact dynamics[END_REF].

The Diaphanous-related formin actin nucleator, Dia1 (also known as mDia, DIAPH1

or DRF1) was reported to localize to FAs in mouse embryonic fibroblasts (MEFs) using fluorescent protein fusions [START_REF] Valencia | Force-dependent activation of actin elongation factor mDia1 protects the cytoskeleton from mechanical damage and promotes stress fiber repair[END_REF]. Dia1 mediates actin polymerization at FAs to facilitate the formation and elongation of dorsal stress fibers (SF) [START_REF] Hotulainen | Stress fibers are generated by two distinct actin assembly mechanisms in motile cells[END_REF][START_REF] Tojkander | Generation of contractile actomyosin bundles depends on mechanosensitive actin filament assembly and disassembly[END_REF][START_REF] Lehtimäki | Generation of stress fibers through myosin-driven reorganization of the actin cortex[END_REF]. Dorsal SFs are noncontractile actin bundles located at the cell front which connect to the substrate via a FA at one end, but can reorganize and generate ventral SFs, which are thick bundles connected to FAs at both ends [START_REF] Hotulainen | Stress fibers are generated by two distinct actin assembly mechanisms in motile cells[END_REF][START_REF] Tojkander | Generation of contractile actomyosin bundles depends on mechanosensitive actin filament assembly and disassembly[END_REF][START_REF] Lehtimäki | Generation of stress fibers through myosin-driven reorganization of the actin cortex[END_REF] (Figure 1). Dia1-driven actin polymerization activity promotes the formation and maturation of FAs via Rho-associated protein kinase (ROCK) activation [START_REF] Valencia | Force-dependent activation of actin elongation factor mDia1 protects the cytoskeleton from mechanical damage and promotes stress fiber repair[END_REF][START_REF] Hotulainen | Stress fibers are generated by two distinct actin assembly mechanisms in motile cells[END_REF][START_REF] Zaoui | Memo-RhoA-mDia1 signaling controls microtubules, the actin network, and adhesion site formation in migrating cells[END_REF]. Dia2 formin has not been specifically found at FAs [START_REF] Valencia | Force-dependent activation of actin elongation factor mDia1 protects the cytoskeleton from mechanical damage and promotes stress fiber repair[END_REF][START_REF] Gupton | mDia2 regulates actin and focal adhesion dynamics and organization in the lamella for efficient epithelial cell migration[END_REF]. However, reports showed that Dia2 nucleates actin to facilitate FA assembly in migrating PtK1 epithelial cells and in MEFs [START_REF] Valencia | Force-dependent activation of actin elongation factor mDia1 protects the cytoskeleton from mechanical damage and promotes stress fiber repair[END_REF][START_REF] Gupton | mDia2 regulates actin and focal adhesion dynamics and organization in the lamella for efficient epithelial cell migration[END_REF].

In vitro work showed that the human FH1/FH2 domain-containing protein 1 (FHOD1) formin does not de novo polymerize actin filaments, but it binds dynamically to and bundles existing actin filaments [START_REF] Schönichen | FHOD1 is a combined actin filament capping and bundling factor that selectively associates with actin arcs and stress fibers[END_REF]. In contrast, cell biology data suggested that FHOD1 nucleates actin filaments and bundles them in vivo [START_REF] Schönichen | FHOD1 is a combined actin filament capping and bundling factor that selectively associates with actin arcs and stress fibers[END_REF][START_REF] Shi | Active FHOD1 promotes the formation of functional actin stress fibers[END_REF][START_REF] Iskratsch | FHOD1 is needed for directed forces and adhesion maturation during cell spreading and migration[END_REF]. The controversy between in vitro and in vivo data was resolved by a later study which showed that purified human FHOD1 does not nucleate actin filaments from rabbit skeletal muscle actin, in agreement with [START_REF] Schönichen | FHOD1 is a combined actin filament capping and bundling factor that selectively associates with actin arcs and stress fibers[END_REF]; but it is capable to nucleate actin from non-muscle actin, (e.g. from Acanthamoeba actin) [START_REF] Patel | Drosophila and human FHOD family formins nucleate actin filaments[END_REF]. FHOD1 is expressed in both muscle and nonmuscle cells although its localization differs and that might be important for effectively carrying out its cellular functions [START_REF] Patel | Drosophila and human FHOD family formins nucleate actin filaments[END_REF]. In many cell lines, including fibroblasts or osteosarcoma cells, FHOD1 has been found localized as clusters at integrin sites [START_REF] Schönichen | FHOD1 is a combined actin filament capping and bundling factor that selectively associates with actin arcs and stress fibers[END_REF][START_REF] Shi | Active FHOD1 promotes the formation of functional actin stress fibers[END_REF][START_REF] Iskratsch | FHOD1 is needed for directed forces and adhesion maturation during cell spreading and migration[END_REF]. Targeting of FHOD1 to adhesions depends on direct interaction with Src family kinases. Src-dependent phosphorylation at Y99 of FHOD1 seems to bring FHOD1 in close proximity to ROCK1 at the plasma membrane to enable downstream FHOD1 activation by ROCK1 phosphorylation [START_REF] Schönichen | FHOD1 is a combined actin filament capping and bundling factor that selectively associates with actin arcs and stress fibers[END_REF][START_REF] Shi | Active FHOD1 promotes the formation of functional actin stress fibers[END_REF][START_REF] Iskratsch | FHOD1 is needed for directed forces and adhesion maturation during cell spreading and migration[END_REF]. Activation of FHOD1 might induce actin SFs that generates force to sustain integrin cluster growth/maturation of FAs [START_REF] Schönichen | FHOD1 is a combined actin filament capping and bundling factor that selectively associates with actin arcs and stress fibers[END_REF][START_REF] Shi | Active FHOD1 promotes the formation of functional actin stress fibers[END_REF][START_REF] Iskratsch | FHOD1 is needed for directed forces and adhesion maturation during cell spreading and migration[END_REF].

In short, FHOD1 seems to share similar actin nucleation activity than other formins [START_REF] Patel | Drosophila and human FHOD family formins nucleate actin filaments[END_REF]. Discrepancy between studies in number of actin filament number and/or organization using FHOD1 depleted MEF cells could be due to differences in knockdown efficiency, endogenous FHOD1 expression levels or experimental growth conditions that influence mechanical response (e.g. soft/stiff substrate) [START_REF] Valencia | Force-dependent activation of actin elongation factor mDia1 protects the cytoskeleton from mechanical damage and promotes stress fiber repair[END_REF][START_REF] Iskratsch | FHOD1 is needed for directed forces and adhesion maturation during cell spreading and migration[END_REF].

The major driver that elongates SFs at FAs seems to be Dia1 [START_REF] Valencia | Force-dependent activation of actin elongation factor mDia1 protects the cytoskeleton from mechanical damage and promotes stress fiber repair[END_REF]. Supporting this is the fact that actin polymerization at FAs occurs in pulses which are coincident with Dia1-driven actin activity, which is stimulated by SF tension and was proposed to help prevent SF damage [START_REF] Valencia | Force-dependent activation of actin elongation factor mDia1 protects the cytoskeleton from mechanical damage and promotes stress fiber repair[END_REF]. Moreover, both Dia1 knockdown MEF cells and MEFs treated with the pan-formin inhibitor SMIFH2 resulted in 75% suppression of the rate of SF elongation in photobleaching experiments [START_REF] Valencia | Force-dependent activation of actin elongation factor mDia1 protects the cytoskeleton from mechanical damage and promotes stress fiber repair[END_REF]. Pharmacological treatment with the Arp2/3 inhibitor, CK666, did not show any effect on the rate of SF elongation in MEF cells [START_REF] Valencia | Force-dependent activation of actin elongation factor mDia1 protects the cytoskeleton from mechanical damage and promotes stress fiber repair[END_REF]. However, it could be that Dia1 may not be the sole actin nucleation factor contributing to that safeguard SF mechanism at FAs [START_REF] Valencia | Force-dependent activation of actin elongation factor mDia1 protects the cytoskeleton from mechanical damage and promotes stress fiber repair[END_REF].

Proteins from the Ena/VASP family -Mena, VASP or Ens/VASP-like (Evl) -are linked to the actin cytoskeleton through their association with vinculin and are known as drivers of actin filament elongation [START_REF] Legerstee | A layered view on focal adhesions[END_REF][START_REF] Faix | Ena/VASP proteins in cell edge protrusion, migration and adhesion[END_REF]. It is unknown if any member of the family contributes to the Dia1-mediated safety mechanism against mechanical damage.

Something perhaps worth to investigate since Puleo and coworkers reported that Evl polymerizes actin at FAs to promote mechanosensing in order to reinforce maturation of FAs, and consequently promote directed motility [START_REF] Puleo | Mechanosensing during directed cell migration requires dynamic actin polymerization at focal adhesions[END_REF]. Although this study showed that SMIFH2 had no effect on the motility of MCF7 cells, the authors acknowledged that some level of formin could remain active, masking any potential collaborative role between formins and Evl [START_REF] Puleo | Mechanosensing during directed cell migration requires dynamic actin polymerization at focal adhesions[END_REF]. Systematic re-expression of the individual Ena/VASP proteins in a B16-F1 triple knockdown (Mena, VASP, Evl) showed a higher rescue in FA size and vinculin levels after re-introduction of Evl than Mena or VASP [START_REF] Puleo | Mechanosensing during directed cell migration requires dynamic actin polymerization at focal adhesions[END_REF]. This could seem unexpected since cellular levels of Mena and VASP are higher than Evl levels in B16-F1 [START_REF] Puleo | Mechanosensing during directed cell migration requires dynamic actin polymerization at focal adhesions[END_REF]. However, FRAP assays showed that Evl is more stably associated at FAs, which could explain its important function at FAs [START_REF] Damiano-Guercio | Loss of Ena/VASP interferes with lamellipodium architecture, motility and integrin-dependent adhesion[END_REF]. Since Mena and VASP still rescued to some extent the defects found in the triple knockdown [START_REF] Damiano-Guercio | Loss of Ena/VASP interferes with lamellipodium architecture, motility and integrin-dependent adhesion[END_REF],

future experiments will reveal how they cooperate with Evl to capture actin filament barbed ends to regulate force at FAs.

Regarding FA disassembly, evidence shows that APC-driven actin nucleation activity promotes FA disassembly to facilitate directed cell migration, and this activity is inhibited by the microtubule end-binding protein EB1 [START_REF] Juanes | Adenomatous polyposis coli nucleates actin assembly to drive cell migration and microtubule-induced focal adhesion turnover[END_REF][START_REF] Juanes | The role of APC-mediated actin assembly in microtubule capture and focal adhesion turnover[END_REF][START_REF] Juanes | EB1 Directly Regulates APC-Mediated Actin Nucleation[END_REF]. Using as tool an APC mutant impaired in nucleating actin filaments (named APC-m4), it was shown that APC-driven actin nucleation is critical for sustaining levels, organization and dynamics of F-actin near/at FAs [START_REF] Juanes | Adenomatous polyposis coli nucleates actin assembly to drive cell migration and microtubule-induced focal adhesion turnover[END_REF][START_REF] Juanes | The role of APC-mediated actin assembly in microtubule capture and focal adhesion turnover[END_REF]. In addition, the APC-m4 mutant presented reduced density of FA components such as Zyxin and (phosphorylated)-Paxillin, -Src and -FAK [START_REF] Juanes | The role of APC-mediated actin assembly in microtubule capture and focal adhesion turnover[END_REF]. That phenotype could be a consequence of the actin disorganization found in the APC-m4. Supporting this hypothesis is the fact that organized F-actin provides force that sculpts the molecular organization of integrins at focal adhesions which is critical for activating integrins to sense directional extracellular cues in fibroblasts and in leukocytes [START_REF] Nordenfelt | Direction of actin flow dictates integrin LFA-1 orientation during leukocyte migration[END_REF][START_REF] Swaminathan | Actin retrograde flow actively aligns and orients ligand-engaged integrins in focal adhesions[END_REF]. In addition, the FA phenotype found in APC-m4 cells agrees with an independent work linking actin-microtubules-Src at FAs [START_REF] Su | MACF1 promotes preosteoblast migration by mediating focal adhesion turnover through EB1[END_REF]. Overexpression of MACF1/ACF7, a microtubule-actin crosslinker from the spectraplakin family, increases the levels of phosphorylated Src at FAs [START_REF] Su | MACF1 promotes preosteoblast migration by mediating focal adhesion turnover through EB1[END_REF]. Phosphorylated Src binds to EB1 at FAs, resulting in less EB1 available to bind APC and promote FA turnover [START_REF] Su | MACF1 promotes preosteoblast migration by mediating focal adhesion turnover through EB1[END_REF]. Together, it is tempting to propose that the arrival of MACF1 at FAs activates Src, dislodges EB1-APC interaction and this relieves APC-mediated actin nucleation activity at FAs. The pool of actin filaments nucleated by APC may be key to organize/remodel F-actin at/near FAs which in turn may impact on the organization of FA components to facilitate timely disassembly. These scenarios are speculative, and others could be considered.

Actin interplay with other cytoskeletal components to control focal adhesion turnover

The speculative reduction in forces at/from FAs in the APC-m4 mutant could explain the FA's inability to sense and/or respond to captured microtubules and autophagosome delivery to promote timely FA disassembly [START_REF] Juanes | The role of APC-mediated actin assembly in microtubule capture and focal adhesion turnover[END_REF]. Supporting this idea is the fact that forces within FAs may exert feedback on microtubules to complete FA turnover [START_REF] Gupta | Single cell rigidity sensing: A complex relationship between focal adhesion dynamics and large-scale actin cytoskeleton remodeling[END_REF][START_REF] Burridge | Focal adhesions, stress fibers and mechanical tension[END_REF][START_REF] Akhmanova | Touch, grasp, deliver and control: functional cross-talk between microtubules and cell adhesions[END_REF], and vicecersa: acetylated microtubules tune the mechanosensitivity of FAs to promote cell adhesion and migration of astrocytes [START_REF] Seetharaman | Microtubules tune mechanosensitive cell responses[END_REF].

Whilst there is some controversy on whether microtubules get captured at the vicinity or at the FAs, it is clear that microtubules play an important role in FA disassembly [START_REF] Seetharaman | Cytoskeletal crosstalk in cell migration[END_REF]. Pioneering live imaging experiments showed that microtubules use actin SFs as tracks to direct them to the vicinity of FAs [START_REF] Kaverina | Targeting, capture, and stabilization of microtubules at early focal adhesions[END_REF]. Later works showed that microtubule guidance along SFs requires the crosslink protein MACF1/ACF7 to reach FAs [START_REF] Leung | Microtubule actin crosslinking factor (MACF): a hybrid of dystonin and dystrophin that can interact with the actin and microtubule cytoskeletons[END_REF][START_REF] Wu | ACF7 regulates cytoskeletal-focal adhesion dynamics and migration and has ATPase activity[END_REF].

Microtubules get captured and stabilized around FAs through their interaction with KANK1 and talin which in turn binds to actin [START_REF] Bouchet | Talin-KANK1 interaction controls the recruitment of cortical microtubule stabilizing complexes to focal adhesions[END_REF][START_REF] Rafiq | A mechano-signalling network linking microtubules, myosin IIA filaments and integrin-based adhesions[END_REF]. Microtubules retract and regrow to contact several times FAs until FAs reach full maturation [START_REF] Seetharaman | Cytoskeletal crosstalk in cell migration[END_REF][START_REF] Kaverina | Targeting, capture, and stabilization of microtubules at early focal adhesions[END_REF]. Fully mature FAs may be 'sensed somehow' by the microtubules which will deliver autophagosomes to the FA, and will concomitantly retract [START_REF] Kenific | NBR1-dependent selective autophagy is required for efficient cell-matrix adhesion site disassembly[END_REF][START_REF] Sharifi | Autophagy Promotes Focal Adhesion Disassembly and Cell Motility of Metastatic Tumor Cells through the Direct Interaction of Paxillin with LC3[END_REF]. Autophagosomes contain LC3 and NBR1 that interact with Src-dependent phosphorylated Paxillin and Zyxin [START_REF] Sharifi | Autophagy Promotes Focal Adhesion Disassembly and Cell Motility of Metastatic Tumor Cells through the Direct Interaction of Paxillin with LC3[END_REF][START_REF] Sandilands | Autophagic targeting of Src promotes cancer cell survival following reduced FAK signalling[END_REF].

Those interactions activate signaling events that may be involved in the removal of phosphorylated FA components which may lead to their disintegration [START_REF] Sandilands | Autophagic targeting of Src promotes cancer cell survival following reduced FAK signalling[END_REF].

Another cytoskeletal element that interplays with actin to control FA maturation, is septin proteins. Mammalian septins are found as heterohexamers and heterooctamers that can self-assemble into septin filaments which are thought to be the functional form of septins in cells [START_REF] Mostowy | Septins: the fourth component of the cytoskeleton[END_REF][START_REF] Cavini | The structural biology of septins and their filaments: an update[END_REF]. Knockdown of SEPT2 or SEPT9 in migrating MDCK cells led to a higher number of smaller FAs closer to the cell edge [START_REF] Dolat | Septins promote stress fiber-mediated maturation of focal adhesions and renal epithelial motility[END_REF]. SEPT2 knockdown cells showed that assembly and disassembly rates of nascent adhesions were the same as in control cells, but that the stabilization phase was abolished resulting in short-lived FAs that fail to mature [START_REF] Dolat | Septins promote stress fiber-mediated maturation of focal adhesions and renal epithelial motility[END_REF]. SEPT2 knockdown in Cancer-Associated Fibroblasts (CAFs) was further reported to lead to fewer FAs [START_REF] Calvo | Cdc42EP3/BORG2 and Septin Network Enables Mechanotransduction and the Emergence of Cancer-Associated Fibroblasts[END_REF], whereas SEPT2 or SEPT9 knockdown in murine and human melanoma cells led to a decrease in FA size [START_REF] Farrugia | CDC42EP5/BORG3 modulates SEPT9 to promote actomyosin function, migration, and invasion[END_REF]. How septins affect FA maturation and thus the lifetime of FAs is not clear, in particular since septins are largely excluded from FAs, as shown by immunostainings in NRK [START_REF] Schmidt | Functional interdependence between septin and actin cytoskeleton[END_REF], MDCK [START_REF] Dolat | Septins promote stress fiber-mediated maturation of focal adhesions and renal epithelial motility[END_REF] and U2OS cells [START_REF] Martins | Human septins organize as octamerbased filaments and mediate actin-membrane anchoring in cells[END_REF]. The role of septins in FA stabilization is most likely due to their association with FA-anchored actin SFs. Dolat et al showed that septins at the leading lamella of migrating MDCK cells localize to the interface of transverse arc SFs and dorsal SFs, getting recruited to the distal ends of dorsal SFs after the assembly of nascent FAs [START_REF] Dolat | Septins promote stress fiber-mediated maturation of focal adhesions and renal epithelial motility[END_REF]. SEPT2 knockdown in these cells resulted in a more disorganized lamellar actin meshwork, with less, shorter and more short-lived dorsal SFs [START_REF] Dolat | Septins promote stress fiber-mediated maturation of focal adhesions and renal epithelial motility[END_REF]. SEPT2 knockdown in CAFs led to a depletion of actomyosin ventral SFs, which are otherwise heavily decorated by septins in wild-type CAFs [START_REF] Calvo | Cdc42EP3/BORG2 and Septin Network Enables Mechanotransduction and the Emergence of Cancer-Associated Fibroblasts[END_REF], whereas SEPT7 and SEPT9 knockdown in melanoma cells also affected the generation of ventral actin SFs, which are also decorated by septins in wild-type cells [START_REF] Farrugia | CDC42EP5/BORG3 modulates SEPT9 to promote actomyosin function, migration, and invasion[END_REF]. SEPT9 knockdown in U2OS cells, or impairing septins from polymerizing altogether, compromised both septin-SF association and actomyosin ventral SF integrity, emphasizing an essential role for septin heterooctamers, but not heterohexamers, in generating or/and maintaining SFs [START_REF] Martins | Human septins organize as octamerbased filaments and mediate actin-membrane anchoring in cells[END_REF].

Given the importance of septins in the generation and/or maintenance of FA-anchored SF and the fact that purified mammalian septin hexamers and octamers cross-link actin filaments into bundles [START_REF] Iv | Insights into animal septins using recombinant human septin octamers with distinct SEPT9 isoforms[END_REF][START_REF] Mavrakis | Septins promote F-actin ring formation by crosslinking actin filaments into curved bundles[END_REF], the current hypothesis is that septins, through their actin filament cross-linking activity, maintain the integrity and organization of SFs, which in turn are required for the stabilization and maturation of FAs [START_REF] Livne | The inner workings of stress fibers - from contractile machinery to focal adhesions and back[END_REF]. Septins could also impact FA maturation indirectly by affecting the accumulation of mechanical tension on SFs; whether septins directly affect SF actomyosin contractility is an ongoing matter of investigation. In addition to, or independently from their actin filament cross-linking activity, septins could stabilize SFs and thus contribute to FA maturation by anchoring SFs at the plasma membrane. Purified mammalian septin hexamers bind lipid membranes [START_REF] Szuba | Membrane binding controls ordered self-assembly of animal septins[END_REF][START_REF] Bridges | Micron-scale plasma membrane curvature is recognized by the septin cytoskeleton[END_REF][START_REF] Tanaka-Takiguchi | Septin-mediated uniform bracing of phospholipid membranes[END_REF], purified human septin octamers can simultaneously bind lipid membranes and actin filaments [START_REF] Martins | Human septins organize as octamerbased filaments and mediate actin-membrane anchoring in cells[END_REF], thus providing actin-membrane anchoring, whereas ventral SF-associated septins are closely opposed to the plasma membrane [START_REF] Martins | Human septins organize as octamerbased filaments and mediate actin-membrane anchoring in cells[END_REF], supporting altogether such a scenario. The identification of the actin-and membrane-binding domains of septins will be key for dissecting their role in FA maturation. A third way in which septins could affect FA turnover is through their direct association with microtubules leading to the stabilization of the latter [START_REF] Kuzmić | Septin-microtubule association via a motif unique to isoform 1 of septin 9 tunes stress fibers[END_REF].

There is no experimental evidence supporting this specific scenario, but the availability of mutants for disrupting or enhancing septin-microtubule association now enables this hypothesis to be tested [START_REF] Kuzmić | Septin-microtubule association via a motif unique to isoform 1 of septin 9 tunes stress fibers[END_REF]. Of note, specific SEPT9 isoforms can have different functions, as reported for cancer cell migration [START_REF] Verdier-Pinard | Septin 9_i2 is downregulated in tumors, impairs cancer cell migration and alters subnuclear actin filaments[END_REF]. How the long SEPT9 isoform i1, which binds both microtubules and actin SFs, and the long SEPT9 isoform i3, which binds only actin SFs, compare with respect to their impact on FA dynamics, has not been tested given that the SEPT9 knockdowns in the mentioned studies did not differentiate among the two isoforms. The identification of the actin-binding domain of septins will further enable to test whether potential septin-mediated actin-microtubule cross-talk is relevant for their role in FA turnover. Finally, a recent study identified a complex between SEPT2, SEPT9 and the Rho-GAP, ARHGAP4, which impacts FA assembly and disassembly by regulating integrin-b1 levels [START_REF] Kang | ARHGAP4-SEPT2-SEPT9 complex enables both up-and down-modulation of integrin-mediated focal adhesions, cell migration, and invasion[END_REF], proposing a yet distinct mechanism on how septins affect FA turnover.

Conclusion and future directions

Our understanding on how FAs turn over has rapidly expanded using recent technologies. For example, advanced biochemistry and bioimaging, biophysical modeling, and quantitative analysis have been used to unravel novel players and/or decipher mechanisms underlying the FA dynamics.

However, many observations remain puzzling, in particular how septins contribute to FA maturation. Given that septins can bind both SFs and microtubules in cells, it is tempting to speculate that septin-mediated SF-microtubule cross-talk interplays with FA dynamics. Comparing how the long SEPT9 isoforms i1 and i3 differentially impact FA turnover is a promising avenue of research for future studies aiming to test this hypothesis. Identifying actin-and membrane-binding domains on septins, and thus being able to decouple actin-specific contributions, will be also key for testing the relevance of such a hypothesis. Simultaneous live imaging of septins, actin SFs,

  Figure 1

ACKNOWLEDGEMENTS

This work was supported by grants from the Academy of Medical Science/the Wellcome Trust/ the Government Department of Business, Energy and Industrial Strategy/the British Heart Foundation/Diabetes UK Springboard Award [SBF006\1070], and the CIDEGENT Excellent Research Program of the Valencian regional government CIDEGENT/2021/026 to M.A.J.

CONFLICT OF INTEREST STATEMENT

The authors declare no competing financial interests.


Thomas Grippi
email: thomas.grippi@thalesaleniaspace.com

Stéphanie Béhar-Lafenetre

Holger Friedrich

Daniel Haas

Uwe Schenderlein

Sylvain Marinel

Charles Manière

Grain growth modeling for gas pressure sintering of silicon nitride based ceramics

Keywords: Silicon nitride, Si 3 N 4, Gas pressure sintering, grain growth, model, 1

The complexity of liquid phase sintering of silicon nitride implies the use of advanced gas pressure sintering (GPS) industrial process. Sintering additives have been jointly developed to allow the sintering of this challenging material to reach high mechanical properties. The latter properties and GPS process of silicon nitride are strongly influenced by the microstructure development during the thermal treatment under temperature close to 1800°C. In particular, grain growth is a key parameter that needs to be identified. In this work, an experimental design is used to explore microstructural development during the final stage of sintering. The grain size, and porosity data are extracted for different holding times and temperatures to allow the identification of the experimental grain growth. The study is extended to the impact of the heating kinetics regime and the influence of the gas pressure furnace environment. 𝑘 0 Grain growth pre-exponential factor (m 1+p •s -1 ) 𝑄 𝐺 Grain growth activation energy (J•mol -1 ) t Time (s) t 0 Beginning of holding time (s)

Graphical abstract

Introduction

Si 3 N 4 ceramics are widely used in components for optical space applications due to their outstanding mechanical properties and stability in harsh environment. With its high strength, its stiffness and its low coefficient of thermal expansion, silicon nitride is well suited for stable space structures. However, it is extremely difficult to densify Si 3 N 4 by pressure-less sintering process, due to the high degree of covalent bonding between nitrogen and silicon, with very low diffusion coefficient of nitrogen or silicon in Si (at 1400°C) in alumina [START_REF] Oishi | Self-Diffusion of Oxygen in Single Crystal and Polycrystalline Aluminum Oxide[END_REF][START_REF] Doremus | Diffusion in alumina[END_REF][START_REF] Matsubara | A set of constitutive functions for dried body to predict entire deformation process of ceramic products during firing[END_REF]. It is well established that conventional pressure-less sintering methods are hardly applicable to produce pure, dense Si 3 N 4 -based ceramics. Most of the sintering processes adapted to silicon nitride to reach dense parts involve the application of pressure: hot pressing [START_REF] Bowen | Hot-pressing and the phase transformation in silicon nitride[END_REF][START_REF] Bowen | Hot-Pressing of Si 3 N 4 with Y 2 O 3 and Li 2 O as Additives[END_REF], gas pressure sintering (GPS) [START_REF] Mitomo | Influence of powder characteristics on gas pressure sintering of Si 3 N 4[END_REF][START_REF] Mitomo | Pressure sintering of Si 3 N 4[END_REF], SPS [START_REF] Lukianova | Microstructure of Spark Plasma-Sintered Silicon Nitride Ceramics[END_REF], etc. Among all those sintering methods, the space industry favored the GPS process as it allows sintering of large size parts with complex shapes, since the applied pressure is ensured by a gas. To reach a high density (>98%), liquid phase sintering (LPS) has been developed by using sintering aids.

Additives for silicon nitride sintering have been widely explored to date (Y 2 O 3 , Al 2 O 3 , MgO, SiO 2 , etc.) [START_REF] Riley | Silicon Nitride and Related Materials[END_REF][START_REF] Huang | Phase Equilibria Diagrams of High Temperature Non-oxide Ceramics[END_REF][START_REF] Lange | Phase Relations and Stability Studies in the Si 3 N 4 -SiO 2 -Y 2 O 3 Pseudoternary System[END_REF][START_REF] Krasil'nikov | Investigation of the Structure of Si3N4-Based Ceramic with Al2O3 and Y2O3 Additives[END_REF]. Fully dense silicon nitride composite based ceramics can now be produced by combining GPS and LPS processes. However, the sintering process applied to large and complex parts for space industry may lead to macroscopic defects such as deformations, cracks, heterogeneous shrinkage, etc. Therefore, it is of primary importance to anticipate and minimize the risk of occurrence of these defects. In this context, the modeling of the sintering behavior and FEM simulation of the parts using the continuum mechanic approach are needed. Consequently, it is mandatory to collect all the physical parameters required to implement a comprehensive model of Si 3 N 4 GPS-LPS sintering [START_REF] Park | Master sintering curve formulated from constitutive models[END_REF][START_REF] Bollina | Master sintering curve concepts applied to fulldensity supersolidus liquid phase sintering of 316L stainless steel powder[END_REF][START_REF] Wang | Estimate of the Activation Energies for Boundary Diffusion from Rate-Controlled Sintering of Pure Alumina, and Alumina Doped with Zirconia or Titania[END_REF][START_REF] Raether | Investigation of sintering mechanisms of alumina using kinetic field and master sintering diagrams[END_REF]. Among all those input data, there are heat transfer parameters of the whole furnace and sample configuration, the grain growth behavior and the densification kinetics that can be extracted from dilatometric data. This paper is dedicated to the grain growth behavior of Si 3 N 4 during the GPS-LPS sintering process. The latter is a key aspect in the understanding of the final stage sintering microstructural development. In addition grain growth may influence the diffusion path motion in the liquid phase and the final part mechanical resistance. The grain growth is a consequence of the dissolution-precipitation mechanism (Ostwald ripening) [START_REF] Sun | A research on the grain growth of WC-Co cemented carbide[END_REF] and can be influenced by the nature of the liquid phase composition [START_REF] Hotta | Inhibition of grain growth in liquid-phase sintered SiC ceramics by AlN additive and spark plasma sintering[END_REF]. In this article, the grain growth behavior is identified based on isothermal tests [START_REF] Rahaman | Sintering of Ceramics[END_REF] and analytic modeling.

Experimental

Thermal cycle

The furnace used is located at the Fraunhofer ISC, Center for High Temperature Materials and Design HTL, Bayreuth [START_REF] Raether | Optical dilatometry for the control of microstructure development during sintering[END_REF]. This device is a vertically oriented cylindrical furnace. The heating is provided by graphite tubes located radially inside the furnace. The furnace's wall, its top and base sections are both water-cooled. The studied material is a grade of silicon nitride provided by QSIL. Two debinded cylinders ø10x10mm samples are sintered for each cycle. Each sample is in a different configuration:

 Configuration 1 refers to the "window configuration", which is used for optical dilatometry. In this configuration the sample is located inside two crucibles with two holes each in opposite positions for sample observation (see Figure 1).

 Configuration 2 refers to the "closed crucible configuration", a smaller closed crucible, in which a sample is located, as a witness sample for comparison. This sample configuration is more stable and less sensitive to the atmosphere and gas flow/convection that could differ from one cycle to another (see Figure 1).

In both configurations, the sample is surrounded by scrap material (Si 3 N 4 crushed elements made of the same composition as the samples) that reproduces the conditions of sintering in an industrial furnace.

Figure 1 Cross section of the sintering configuration

The objective of this study is to conduct the microstructure analysis during the intermediate and final stage of densification (grain growth, pore size and coalescence, density, etc.). To conduct this study, 8 isothermal interrupted tests are planned: 4 at Tsinter -100 K and 4 at Tsinter, with a 5 K/min high temperature heating rate, Tsinter being a common temperature reached during the sintering process, close to 1800°C (Figure 2). A last test is planned at Tsinter -200 K without dwelling in order to have 3 different temperatures during the heating ramp (anisothermal behavior study case). For all experiments, the thermal cycles were run and the heating was switched off when the different temperature-time points were achieved. The aim is to reveal the interrupted cycle microstructure. This study is made by taking a representative industrial sintering cycle as a reference. Interruptions of the cycles at different times are conducted (full holding time, 0.5*holding time, 0.25*holding time and 0 min holding time) by maintaining the same sintering conditions: gas pressure, heating ramp, samples configuration (see Figure 2).

Data collecting and grain size determination

The microscope used was JEOL JSM 7200F equipped with a Robinson-type back scattered electron diffusion detector (BED-C). Each sample was cut at half height and mirror polished.

A micrograph at magnification x500 and at least 6 pictures between the center and the border at magnification x10.000 were taken as shown in Figure 3. The first step of the grain growth measurement consists in a linear intercept. This step is done manually, the user of this method counts the grain borders and calculates the mean grain size with a stereological factor [START_REF] Underwood | Quantitative Stereology for Microstructural Analysis[END_REF][START_REF]Ceramic Microstructures[END_REF].

𝐿 = 𝑋 𝑀𝑁 𝐿 (1)
By knowing the length X of the line on the micrograph (for instance the border of the microstructure picture on the left Figure 4), M the magnification and NL the number of intersections, the mean intercept length L can be determined. To take into account the multiphase microstructure (amorphous phase, porosity, …), the ratios of these phases were estimated for each image (see right picture in Figure 4) As it is recommended by the work of Wurst and Nelson [START_REF] Wurst | Lineal Intercept Technique for Measuring Grain Size in Two-Phase Polycrystalline Ceramics[END_REF], the expression of N L is replaced by N L = N ss + N sv /2, where N ss refers to a solid-solid boundary and N sv refers to a solid-vapor boundaries (the porosity and in our case, also the amorphous phase is counted as thick grain boundary). Concerning the value of X in presence of porosity and liquid phase, the effective value of X is corrected by multiplying this term by the Si 3 N 4 granular phase ratio [START_REF] Wurst | Lineal Intercept Technique for Measuring Grain Size in Two-Phase Polycrystalline Ceramics[END_REF].

Based on multiple-linear intercepts and the different SEM images, a mean intercept (L) value can be calculated. The mean grain size is calculated via a stereological factor from a 1dimension value to 3-dimensions [START_REF] Mendelson | Average Grain Size in Polycrystalline Ceramics[END_REF]: The Figure 4 shows the SEM microstructures of samples previously mirror-polished. The typical Si3N4-based ceramic microstructure is observed: Si3N4 grains and amorphous secondary phase (coming from the liquid phase at high-temperature).

𝐺 = 1.56𝐿 ̅ (2)

Grain growth model identification method

In order to determine the grain growth behavior during all the industrial thermal cycle, the linear intercept method (Figure 2) will be employed to determine the average grain size of each test. The collected data will be fitted mathematically for the two temperatures to determine the experimental grain growth kinetics. The isothermal grain growth behavior will be identified and modeled with GNU Octave, for verification. As the grain growth shows a slightly different regime in the ramping stage, a second anisothermal identification will be conducted to correct the first behavior to be identified (isothermal behavior). Finally, the grain size prediction in the whole thermal cycle will be compared to the experimental data for a final check.

Results and discussion

SEM characterizations

The grain growth can be seen as a function of temporal interruptions on SEM micrographs at x10,000 in Figure 5. The density evolution during the temperature holding has also been compared as a function of holding at "Tsinter" in red and "Tsinter-100K" in black, as can be observed on micrographs at x500. The apparent relative density stagnates in the case of the isotherm at "Tsinter-100K", at 89% of relative density. For the isotherm at "Tsinter", 99.7% of relative density is reached by the end of the cycle. Each picture gives a mean grain size (with linear intercept method [START_REF] Underwood | Quantitative Stereology for Microstructural Analysis[END_REF][START_REF]Ceramic Microstructures[END_REF][START_REF] Wurst | Lineal Intercept Technique for Measuring Grain Size in Two-Phase Polycrystalline Ceramics[END_REF]). For each cycle, the analysis has resulted into a cloud of data reported in Figure 6. As a first check, it is interesting to compare both configurations (configurations 1 and 2) with the values of isotherms at Tsinter. As can be seen, for both configurations the grain size varies from a mean value of 0.75 µm to 1.00 µm at the end of the cycle (Figure 6). The difference is negligible and shows that there is no drastic microstructure difference between both configurations. As the 2 configurations are very similar, the presented study focuses on data from the configuration 1 (crucible with the window). Indeed, this resulting grain growth model will be implemented with the final sintering model that will be based on the sample in this same configuration i.e. in the "window" crucible, to be studied in-situ (TOM dilatometry configuration).

From the method of analysis, it has been observed that data extracted from micrographs captured in zones closer to samples' borders, lead to larger grain values (see in Figure 6).

These larger mean diameters, near the surface, may be explained by pollution elements, vapor transports [START_REF] Rahaman | Sintering of Ceramics[END_REF][START_REF] Hotta | Inhibition of grain growth in liquid-phase sintered SiC ceramics by AlN additive and spark plasma sintering[END_REF] but not temperature gradients as it is distributed like a thin skin. As the granular growth study is focusing on the core behavior and not precisely on surface aspects, these values have been taken out from the data processing.

Figure 6 Grain growth as a function of holding time for 2 configurations at Tsinter

Isothermal grain growth identification

As described in the previous part, each sample has been analyzed and a grain size has been determined for each cycle, based on SEM pictures. The grain size versus time and temperature can now be computed. From grain size values, the evolution of G has been fitted mathematically. The grain growth fitting function uses the following formula:

𝐺 𝑓𝑖𝑡 = 𝐴 × 𝑡 𝐵 + 𝐶 ( 3
)
where t is a time variable and A, B and C are fitting constants.

The fitted expressions and experimental data points are reported in Figure 7. From this 𝐺 𝑓𝑖𝑡 function, the grain growth rate (𝐺 ̇𝑓𝑖𝑡 ) has been determined.

In liquid phase sintering, the grain growth may come from a dissolution-precipitation mechanism (Ostwald ripening) and obeys similar grain growth kinetics than in the solid-state case [START_REF] Rahaman | Sintering of Ceramics[END_REF].

𝐺 ̇= 𝐾(𝑇)

𝐺 𝑝 𝑤𝑖𝑡ℎ 𝐾(𝑇) = 𝑘 0 exp (-

𝑄 𝐺 𝑅𝑇 ) (4)
If the rate controlling grain growth mechanism is the diffusion through the liquid phase, the exponent (p) equals 2. This exponent's value equals 1 for interface reaction mechanisms.

In order to determine experimentally the grain growth mechanism from the experimental grain growth kinetic data, the previous rate equation is linearized.

ln(𝐺 ̇) = ln(𝐾(𝑇)) -𝑝 × ln(𝐺) (5)
The linear regression is reported in Figure 8 a., for each temperature. From these curves, a mean value of p=1.93 is obtained indicating a dominant grain growth mechanism of diffusion through the liquid phase. From each temperature intercept in Figure 8 This first couple of parameters has been determined using the data from the isothermal conditions. As can be inferred from the Figure 9 below, the parameters are reliable in the case of the isothermal conditions, on the model computed analytically.

Figure 9 Modeling of grain growth for 2 different temperatures holding on Octave

In a second investigation, the same parameters have been used to compute the full cycle shown below on Figure 10. In this particular case, the parameters identified should be able to fit the heating part experimental data and then the holding part data. However, as shown in Figure 10, these parameters have failed to reproduce the whole cycle starting from a 0.27 µm mean grain size (measured after debinding process). It can be deduced that different grain growth kinetics lead to this apparent gap between the grain growth rate in anisothermal and isothermal cases. In Figure 5, a high proportion of small grains are present in the beginning.

These grains have a high solubility (Ostwald ripening) and may be a possible explanation for these two grain growth regimes. Faster grain growth rate may be required in the heating part.

A phase transition from initial α grains to β phase may also explain such regime change of grain growth during the heating.

Tsinter

Tsinter-100K 0,25 0,5 1

Figure 10 Isothermal grain growth behavior applied to the whole sintering cycle.

Anisothermal (ramping) stage grain growth identification

In order to model the whole cycle grain growth, the behavior is split into two behaviors :

-Anisothermal grain growth behavior during the heating part of the cycle.

-Isothermal grain growth behavior during the holding temperature.

To identify the anisothermal grain growth regime, the experimental data collected between "Tsinter-200K", "Tsinter-100K" and "Tsinter" will be employed. The analysis is processed in a similar way to the first part described above. The grain growth mechanism parameter (p) identified in isothermal regime is assumed to be the same as in the anisothermal case. The parameter required to be adapted to this case is the activation energy Q G and the preexponential factor k 0 . To do so, grain size data (0min holding times points) have been fitted for each temperature with the following function that corresponds to the grain growth behavior in ramping stages:

𝐺 𝑓𝑖𝑡 = 𝐴 × 𝑒 𝐵𝑡 (7)
where t is a time variable in second, A and B are constants. The anisotherm fitting is reported in Figure 11.

Figure 11 Mathematical fit for grain growth during the heating part (5K/min heating rate)

From this fitting, the grain growth rate 𝐺 ̇ has been expressed. Assuming the same mechanism (p parameter identified before), it is possible to calculate the anisotherm 𝐾(𝑡) formula:

𝐾(𝑡) = 𝐺 𝑝 𝐺 ̇ (8)
Using the following anisotherm linearization, k 0 and Q G can be determined.

ln (𝐺 𝑝 𝐺 ̇) = 𝑄 𝐺 𝑅𝑇 + ln (𝑘 0 ) (9)
From the linearization in Figure 12, we obtain for the anisotherm regime: Q G =262kJ/mol and k 0 =1,009E-15 m 1+p •s -1 . y = 6,15E-09e

General grain growth model verification

In this part, as the two models are fitting for their own part, the combination of both approaches must be carried out to have the most comprehensive model for each temperature rate, anisothermal and isothermal stages. The two models have been combined with a sigmoidal function (f 0 ) that switches parameters identified in isothermal (case 1) to parameters identified in ramping (case 2). In the present case, this sigmoid has done a switch for the parameters Q G and k 0 : from 𝑄 𝐺1 and k 01 for case 1 to 𝑄 𝐺2 and k 02 for case 2 at the right time.

{ 𝑓 0 = 1 1+𝑒 -(𝑡-𝑡 0 ) 𝑄 𝐺 = 𝑄 𝐺1 (1 -𝑓 0 ) + 𝑄 𝐺2 (𝑓 0 ) 𝑘 0 = 𝑘 01 (1 -𝑓 0 ) + 𝑘 02 (𝑓 0 ) (10)
The experimental grain growth is well fitted by these combined grain growth models (see Figure 14). This whole model is now fully comprehensive of the cycle variations.

Conclusion

Based on micrographs of quenches on an industrial sintering cycle, it was possible to extract the microstructural development behavior of Si 3 N 4 during a gas pressure sintering. This liquid phase sintering process exhibits a grain growth behavior by diffusion through the liquid phase with a grain size exponent of p=2. The grain growth model points out a complex behavior with higher grain growth kinetics in the beginning of the cycle during the heating. This dissimilar grain growth behavior is possibly explained by the high proportion of small grains that easily dissolves in the initial microstructure or to phase transition. With a step like function, a comprehensive model allows reproducing the grain growth experimental data for the whole industrial cycle. This grain growth model will be coupled to a densification model for a comprehensive simulation of silicon nitride sintering

Acknowledgement

The author thanks Christelle Bilot for her help in the SEM images of this study. We would like to thank Thales Alenia Space for the PhD subvention of the author (CIFRE Thesis Contract CIFRE N° 2019/1366). A collaboration between four entities has been needed, FCTI

(industry) has provided the debinded specimens, the advanced gas sintering equipment of the Fraunhofer HTL has been used for the experimental sintering test, the modeling/simulation study is processed at the CRISMAT laboratory, and the assessment of the results for space applications is done by Thales Alenia Space. Figure 11 Mathematical fit for grain growth during the heating part (5K/min heating rate) Figure 12 Linear regression to identify Grain growth activation energy during the heating Figure 13 Grain growth model with a new range of parameters adapted to heating part. Figure 14 Full grain growth fitting anisothermal case and isothermal case

Nomenclature𝐿

  Linear intercept grain size (m) X length X of the line on the micrograph (m) 𝑁 𝐿 the number of intersections M Magnification 𝑟 Mean particle radius (m) R Gas constant 8.314 (J•mol -1 •K -1 ) T Temperature (K) 𝐺 ̇ Grain growth rate (m•s -1 ) 𝐺 Grain size diameter (m) 𝐺 0 Initial grain size diameter (m) 𝑝 Grain growth rate exponent 𝐾 Grain growth factor (m 1+p •s -1 )

3 N 4 . 1 (

 341 As an example the grainboundary diffusion of nitrogen is about D at 1400°C)[START_REF] Petzow | [END_REF][START_REF] German | Sintering Theory and Practice[END_REF][START_REF] Rahaman | Sintering of Ceramics[END_REF][START_REF] Bernache-Assollant | Frittage : aspects physico-chimiques Partie 2 : frittage en phase liquide[END_REF] in Si 3 N 4 , this is very low compared to the typical value of D

Figure 2

 2 Figure 2 Schematic of interrupted cycles study, Tsinter~1800°C

Figure 3

 3 Figure 3 Schematic view of the range of SEM micrographs taken for each sample.

Figure 4 (

 4 Figure 4 (Left) linear intercept on a Si 3 N 4 polished surface sample micrograph at x10.000. (Right) colored micrograph for the calculation of phase proportions (Si 3 N 4 grains in blue, the amorphous secondary phase in green and pores in red)

Figure 5

 5 Figure 5 SEM micrographs for each quench, at magnification x500 (top), at magnifying

Figure 7

 7 Figure 7 Mathematical fit for 2 different temperatures and grain size as a function of holding

Figure 8

 8 Figure 8 (a.) Linearization and identification of the parameter p, (b.) Linearization and

Figure 12

 12 Figure 12 Linear regression to identify Grain growth activation energy during the heating

Figure 13

 13 Figure 13 Grain growth model with a new range of parameters adapted to heating part.

Figure 14

 14 Figure 14 Full grain growth fitting anisothermal case and isothermal case

7 .Figure captions Figure 1

 7captions1 Figure captions

Figure 4 (

 4 Figure1Cross section of the sintering configuration Figure2Schematic of interrupted cycles study, Tsinter~1800°C Figure3Schematic view of the range of SEM micrographs taken for each sample. Figure4(Left) linear intercept on a Si 3 N 4 polished surface sample micrograph at x10.000. (Right) colored micrograph for the calculation of phase proportions (Si 3 N 4 grains in blue, the amorphous secondary phase in green and pores in red) Figure5SEM micrographs for each quench, at magnification x500 (top), at magnifying x10.000 (middle) and corresponding relative density(bottom graph). Figure6Grain growth as a function of holding time for 2 configurations at Tsinter Figure7Mathematical fit for 2 different temperatures and grain size as a function of holding time Figure8(a.) Linearization and identification of the parameter p, (b.) Linearization and identification of Q G and k 0 Figure9Modeling of grain growth for 2 different temperatures holding on Octave Figure10Isothermal grain growth behavior applied to the whole sintering cycle. Figure11Mathematical fit for grain growth during the heating part (5K/min heating rate) Figure12Linear regression to identify Grain growth activation energy during the heating Figure13Grain growth model with a new range of parameters adapted to heating part. Figure14Full grain growth fitting anisothermal case and isothermal case

Figure 6 Figure 9

 69 Figure1Cross section of the sintering configuration Figure2Schematic of interrupted cycles study, Tsinter~1800°C Figure3Schematic view of the range of SEM micrographs taken for each sample. Figure4(Left) linear intercept on a Si 3 N 4 polished surface sample micrograph at x10.000. (Right) colored micrograph for the calculation of phase proportions (Si 3 N 4 grains in blue, the amorphous secondary phase in green and pores in red) Figure5SEM micrographs for each quench, at magnification x500 (top), at magnifying x10.000 (middle) and corresponding relative density(bottom graph). Figure6Grain growth as a function of holding time for 2 configurations at Tsinter Figure7Mathematical fit for 2 different temperatures and grain size as a function of holding time Figure8(a.) Linearization and identification of the parameter p, (b.) Linearization and identification of Q G and k 0 Figure9Modeling of grain growth for 2 different temperatures holding on Octave Figure10Isothermal grain growth behavior applied to the whole sintering cycle. Figure11Mathematical fit for grain growth during the heating part (5K/min heating rate) Figure12Linear regression to identify Grain growth activation energy during the heating Figure13Grain growth model with a new range of parameters adapted to heating part. Figure14Full grain growth fitting anisothermal case and isothermal case

2,25E-04x