L Farge

R Spiegel

S André

C Noûs

R Lainé

S Hoppe

Development of Plasticity in Vitrimers Synthesized from a Semicrystalline Polymer using Injection Molding

Keywords: Vitrimers, Cross-linking, PBT, Mechanical properties

The main objective of this work is to analyze finely the influence of cross-linking on the mechanical properties of PBT (Polybutylene Terephtalate)-based vitrimers. Tensile tests were carried out at four temperatures (80°C, 100°C, 130°C and 160°C) on a series of vitrimer specimens made by injection molding for which the cross-linker concentrations were changed by small increments of 0.25%, from 0% to 2%. The displacement and strain fields were measured on the specimen surfaces through 3 D DIC (Digital Image Correlation) in order to analyse the strain localization/delocalization phenomena that occur during the successive stages of the deformation process. In particular, we measured the yield strain Y  (onset of strain localization), and the strain at neck stabilization NS  (beginning of the strain delocalization phase). When the degree of cross-linking increases, we observed two complementary effects leading to the decrease of the strain range during which plastic instability develops: Firstly Y  increases. Secondly, and this is the main cause of the plastic instability strain range reduction, NS  decreases. This latter effect results from the limitation in extensibility of the macromolecular network in the solid state caused by cross-linking. More specifically, cross-linking leads to two distinct modifications of the macromolecular network that both contribute to the reduction of its extensibility.

The first is the decrease of the chain length between the network nodes. The second is the preorientation of the macromolecular network that occurs during injection molding in the case of vitrimers with high viscosities due to cross-linking. Eventually, when Y NS  → , the suppression of the yield point on the tensile curve was observed for the most cross-linked vitrimers. Furthermore, in the temperature range of investigation,

NS

 was found to be independent of temperature.

Introduction

Semi-crystalline polymers deformed in tension at test temperatures between the glass transition and melting temperatures exhibit ductile behavior and can be deformed to large strains without breaking, typically greater than 300% (or true strain>1). [START_REF] Ward | Mechanical properties of solid polymers[END_REF][START_REF] Vincent | [END_REF][3] At the beginning of the deformation process, the strain is uniform along the specimen gauge length. Plastic instability leading to strain localization (necking) begins when the first Considère condition is met-i.e., at the yield point when the nominal stress reaches a maximum. Next, the nominal stress decreases before reaching a minimum as the strain in the neck increases much faster than in the other specimen parts. Neck is stabilized when the material in this region has hardened sufficiently to cause other regions of the specimen to deform instead. This "strain delocalization" phase is called "neck stabilization" or "neck propagation". [4][5][6][7] Without deformation field measurements but through a qualitative analysis of the characteristic tensile curves of different semi-crystalline polymers, Vincent has inferred indirectly that neck stabilization begins at the nominal stress minimum. This is the second Considère condition. [START_REF] Vincent | [END_REF] At the microstructure level, strain hardening leading to neck stabilization is considered to be caused by the molecular network limitation in extensibility. [8][9] The nodes of this network consist initially of chain entanglements and connection points between adjacent crystals linked by tie molecules. 10 At large strain levels, when neck stabilization occurs, the lamellae are disintegrated into small fragments and the deformation process depends chiefly on the entangled amorphous network characteristics. 11- 12 The higher the network node density, the lower the strain level at which neck stabilization occurs.

Furthermore, if a polymer has a certain degree of orientation before deformation, the remaining extensibility of the macromolecular network in the solid state will be reduced and the strain at neck stabilization will be lowered. [13][14][15] In addition to the just described scheme, some semi-crystalline polymers like polyethylene may also exhibit a second yield point, 7,[16][17] but this was never observed for PBT (Polybutylene Terephtalate), the semi-crystalline polymer studied in this work.

Vitrimers are a new family of polymers that was invented in the beginning of the 2010s by Leibler and collaborators. [18][19][20][21] They are synthesized by "introducing" exchangeable chemical cross-links between the chains of a precursor polymer. The mechanism of cross-link migration is thermally activated. For the vitrimers with thermoplastic precursors, when the exchange reaction speed is low in the solid state, "vitrimerization" leads to an enhancement of the dimensional stability thanks to cross-linking. By contrast, in the molten state, the speed of exchange reactions increases and in spite of the presence of cross-links between the chains, the material may remain readily formable.

In 2017, Demongeot et al. developed an experimental protocol to "vitrimerize" PBT. 22 In addition to this semi-crystalline polymer, the reactive mixture contains a diepoxid, the cross-linker, and Zn(II), the catalyst of the transesterification reaction that causes cross-linking and bond exchange. DSC (Differential Scanning Calorimetry) measurements have shown that the vitrimer glass transition and melting temperatures are nearly the same as those of their PBT precursor. [22][23] Only a slight decrease of the crystallinity degree was observed for the vitrimers relatively to the pristine PBT. The cross-linking is assumed to take place in the inter-crystalline amorphous phase. A very schematic representation of the specific microstructure of PBT-based vitrimers is shown in Figure S1.

In a recent study, we reproduced the Demongeot et al. experimental protocol and we showed that the deformation process differs greatly between the pristine PBT and vitrimers obtained with cross-linker concentrations of 1% and 2%. 23 We found that the vitrimer nominal stress curves increase monotonically, which means that the yield point was suppressed for these materials. However, the physical mechanisms leading to the suppression of the necking behavior in vitrimers were not analysed in details. This is the purpose of the present paper. To this end, we carried out tensile experiments on the pristine PBT and on a series of PBT-based vitrimers for which the diepoxide cross-linker concentrations have been varied in small increments (≈0.25 wt%) from 0% to 2%. This makes it possible to study in a quasi-continuum way the evolution of the mechanical properties between two extreme cases, namely:

1) that of the pristine PBT, which is in line with the semi-crystalline polymers classical scheme recalled above, 2) that of vitrimers with high cross-linker concentrations that exhibit no yield point on the nominal stress curve.

In order to analyse specifically the strain localization/delocalization phenomena that occur during the deformation process, 3D Digital Image Correlation (3D DIC) measurements were performed to obtain the strain and the displacement fields at the specimen surfaces both in the Eulerian and Lagrangian descriptions. From a more general perspective, our work brings new insight of how the cross-linking of the amorphous phase of semi-crystalline polymers can fundamentally modify the development of plasticity in these materials, in particular when they are made by injection molding.

1 Experimental section

Vitrimer production

Along with the PBT precursor, the chemical reagents that were introduced in the µcompounder (DSM Xplore) at 270°C are the diepoxide-the cross-linker-the Zn(II) catalyst and an anti-oxidant. The PBT precursor was provided by DuPont in pellet form (Material reference: CRASTIN FGS600F40). The diepoxide was the bisphenol-A epichlorhydrin resin DER332 produced by Olin epoxy. It is further referenced by the common abbreviation "DGEBA". The Zn(II) acetylacetonate hydrate was provided by Sigma Aldrich, as well as the anti-oxidant (Irganox 1010). In short, the reaction between PBT and the DGEBA resin first leads to the extension of the PBT chains and to the appearance of lateral hydride groups along these chains. Next, in presence of the Zn(II) catalyst, the transesterification reaction occurs leading to cross-linking and bond exchanges.

The rotational speed of the two µcompounder screws was 100 rpm. The force exercised on the barrel initially increased and then became constant after about 5 minutes, which suggests that the crosslinking reaction was over. The melt remained for 8 minutes in the µcompounder and was then transferred to a shooting pot and injected into a mold in shape of a tensile specimen (Figure S2). During injection, the flow direction corresponds to the specimen tensile axis. More details about the synthesis protocol can be found elsewhere. [22][23] We synthesized seven different polymers denoted Vi where the subscript designates the cross-linker (DGEBA) percentage by weight (wt%). V0 is simply the pristine PBT. The detail of the prepared reactive mixtures is indicated in table 1. Between V0.5 and V2 the DGEBA wt% was increased by small increment of about 0.25%. This made it possible to analyze precisely the dependence of the vitrimer mechanical properties on the degree of cross-linking. The presence of DGEBA slightly modifies the specimen color that goes from a very light brown (V0) to a more pronounced brown (V2) (Figure S3).

Mechanical testing

The measurements were performed on a Bose 3000 tensile machine equipped with a thermal chamber (Figure S4). The dog bone specimens (see Figure 1a and1b) includes a 4×4×4 mm 3 cubic central part and were specifically designed so that necking is always initiated in the specimen center. The relative speed of the tensile machine crossheads was 0.02 mms -1 during the tests. The tensile machine upper limit for the grip relative displacement is 20 mm and the tests were stopped when this limit was reached. The tensile experiments were performed for the 7 Vi materials (see Table 1) at 80°C, 100°C, 130°C and 160°C. These temperatures are above the PBT glass transition temperature (Tg≈60°C) and below the melting temperature (Tm≈225°C). In these conditions, these materials show no brittle behavior and can be deformed to very large strains.

As it is customary in continuum mechanics, a material point is identified by its initial position X (undeformed or Lagrangian coordinate) on the undeformed specimen. The current position of the X material point at time t is denoted ( ) , x X t (deformed or Eulerian coordinate). The origin was chosen so that 0 xX == corresponds to the specimen center where necking is initiated. The material points that are initially situated in the central cube of the undeformed specimen are identified during the tensile test as belonging to the

  22 X mm  -+ interval. Obviously, during tensile testing, if 0 X  and 0 t  , then ( ) , x X t X 
. In order to perform 3D DIC displacement and strain measurements, a random speckle pattern was applied on the specimen surface by projecting small black paint droplets with an airbrush. The images were recorded by each of the two cameras every 1s through a glass window placed on the oven front face (Figure S4).  is the local true strain defined by ( )

, ln dx xt dX   =  
. dX is the initial size of a very small length element taken along the longitudinal axis and dx is its current size during the test. The maximum strain in the neck center is denoted M  . The nominal and true stresses are defined by

0 N FS  =
and FS  = respectively. F is the tensile force.

0 S and S are the initial and current areas of the central cross-section. Assuming an isochoric deformation process, it can readily be checked that the true stress in the neck center is given by ( )

exp M N    = .
A strain map example is shown in Figure 1c. It corresponds to a measurement obtained during a test performed on the pristine PBT (V0) at 80°C. In Figure 1d, we have also represented the strain profile () x  that was directly measured along the deformed specimen. With the displacement measurement, ( )

, u X t x X
=-, it is easy to obtain the strain ( , ) Xt  as a function of the undeformed coordinate ( X )-i.e., in the Lagrangian description. In Figure 1d, we have added the () X  profile, which is logically narrower than () x  . X  is the full width at half maximum of () X  (see Figure 1d). By definition X  is relative to the undeformed state and can therefore be considered as a measure of the material mass that is included in the most deformed part of the neck; the smaller X  , the more localized the strain.

The strain rate associated with a given material point-identified by X -was calculated by centered differences: ( )

( ) ( ) ,, , 2
X t t X t t Xt t   +  - - = 

Dynamic Mechanical Analysis (DMA) experiments

DMA experiments were performed in tension mode using a Metravib+300 analyzer at a heating rate of 1°C/min from the room temperature to 250°C. The specimen dimensions were 20×10×4 mm 3 . The amplitude and frequency of the applied tensile strain were 0.25×10 -3 and 10Hz respectively.

Small Angle X-ray Scattering (SAXS) experiments

The SAXS measurements were performed on the SWING beamline at the Soleil synchrotron. The wavelength (  ) was 0.077489 nm and the q range ( E abruptly drops below our apparatus measurement range. At temperatures smaller than 200°C, the V1 and V2 vitrimer behaviors are roughly the same as that of the pristine PBT. The melting temperatures (Tm ) of the two vitrimers are slightly smaller compared to V0.

Above Tm, the vitrimer behaviors drastically differ from that of the pristine PBT: a rubber plateau appears on the vitrimer ' E curves (see the inset in the central part of Figure 2). The level of this plateau is rised when the vitrimer DGEBA wt% is increased. This proves that the addition of the DGEBA resin in the µcompounder has actually led to the vitrimer cross-linking, and that the degree of cross-linking depends directly of the DGEBA wt% as it has already been demonstrated in previous works. 22

Figure 2 '

E (storage modulus) versus temperature for V0, V1 and V2.

Characteristic tensile curves and definition of the strain range of plastic instability

The main objective of this part is to highlight and analyse through full-field strain (  ) and strain rate (  ) measurements the successive strain localization and delocalization stages that can occur during the plastic deformation process. Based on these observations, we were able to objectively identify the high and low bounds of the strain range of plastic instability-strain range during which strain localization occurs and neck develops.

The N  versus M  curve measured for V0 at 80°C is shown in Figure 3. This curve exhibits a maximum at the yield strain

0.19 M Y  ==
, and a minimum at

0.71 M NS  ==
. In Figure 4, we have plotted the  curves measured at the beginning of the test for different material points situated in the specimen central part (X=0, 0.4 mm, 0.6 mm, 0.8 mm and 1.0 mm). Initially, the strain rates are the same for all these material points. The curves begin to separate when strain localizes, then  becomes maximum in the specimen center ((X=0), which is observed approximately at the yield point (

0.19 Y  =
). This confirms that the lower bond of the strain range of plastic instability is the yield strain In Figure 5, we show the strain rate (  ) profiles that were measured along the specimen length, which we plot here as function of the Lagrangian coordinate X. The first profile was obtained before the yield point, at point A (

0.1 M  =
) of the ( )

M N



curve (Figure 3). This strain rate profile is nearly constant in the specimen central part, which confirms that strain localization has not taken place. The slight decrease observed at either side of the curve is due to the variation of the cross-section along our dog bone specimen. For the yield point B (

0.19 M y  ==
), strain localization has started and the  profile clearly becomes sharper. For point C, (

0.5 M  =
), the narrowing of the peak width has become more pronounced and the strain rate in the center has more than doubled. This trend has been reversed at the nominal stress minimum (point D,

0.71) M NS  ==
): the strain rate curve has broadened and the strain maximum in the specimen center has significantly decreased. At strains slightly larger than 0.71

NS  =
, neck stabilization can be observed simply and directly thanks to our 3D DIC measurements:

The strain rate maximum is no longer situated in the neck center but in two symmetrical points around the center. This is barely observable for the strain rate profile measured at

0.85 M  = , but it is clear at 1 0.87 M NS  ==
(see the inset in the right upper part of Figure 5). In the following, we note 1 NS  , the minimum value of M  for which we detected that the strain rate maximum is no longer in the specimen center, but in two symmetrical points on each side of the center. The Lagrangian quantity X  is the distance on the undeformed specimen between the two material points where the stain rate is maximum. Note that X  defined in Figure 5 is different from X  defined in Figure 1d. During neck stabilization, X  increases:

0.9 X mm  = at 0.87 M  = , 3 X mm  = at 1 M  = and 5 X mm  = at 1.1 M  =
(see Figure 5). It can be checked that the two symmetrical maxima that appear on the strain rate profiles when

1 M NS  
correspond to material points that are situated in the neck shoulders at the time of measurement. 6,7 Therefore, the X  increase means that the neck gradually extends to all the material points of the specimen. At the end of the test,

4 X mm  
: the strain rate maxima-and therefore the neck shoulders-correspond then to material points situated outside the central cube that can be seen on the undeformed specimen. (see Figure 1a).

By analyzing the evolution of the X  variable during the test (see the definition of X  in Figure 1d), it is possible to confirm that neck stabilization has begun at 1 NS  . Initially, X  decreases, which is the combined effect of stress inhomogeneity due to specimen geometry, initial material softening, and strain localization after the yield point. For

1 0.85 M NS  =
, X  reaches a minimum and then rises.

Logically, when the strain rate is no longer maximum at the material point located in the specimen center, the bell curve of strain in the Lagrangian description ( ( )

X 
, Figure 1d) begins to broaden. The quantity of substance (i.e., the mass) included in the specimen most deformed part increases: the neck extends toward the less deformed regions of the specimen; neck stabilization is in progress.

With our 3D DIC measurements-i.e., through direct observation of strain delocalization-we have defined a threshold strain, 1 NS  , characterizing the neck stabilization beginning. 1

NS

 was found to be slightly larger than NS  , which was also checked for all the tests performed during this study (see section 2.4.3). We have therefore decided to set the upper bound of the strain range of plastic instability to NS  -i.e, at the nominal stress minimum, which is in accordance with the second Considère condition. In the following, we will consider that the strain range of plastic instability extends from Y  to NS  . However, the conclusions of the study would have been the same if we had chosen

1 NS
 for the upper bound of this interval. Let us note that NS  corresponds to the definition of the natural draw ratio as given by Ward and Sweeney 1 or Séguéla [START_REF] Vincent | [END_REF] .This quantity is closely related to strain hardening and its measurement makes it possible to compare the limitation in extensibility of polymers. These points will be illustrated in section 2.4.3.

The definition of the strain range of plastic instability was introduced by analyzing the tensile behavior of the pristine PBT (V0), which exhibits significant strain localization. We will now examine the opposite case of the V2 vitrimer. The evolution of the nominal stress measured at 160°C for V2 shows nor maximum neither minimum (Figure 6), which suggests that deformation occurs in a quasi-uniform way along the specimen. This is confirmed by the analysis of the strain rate profiles (inset in Figure 6) that are nearly constant throughout the test in the specimen central flat part. Compared to Figure 5, the curves in the Figure 6 inset appear to be noisy. This is due to the reduced contrast between the black paint and the V2 specimen brown surfaces (see Figure S3), moreover accentuated by the fact that the maximum of the strain rate scale is five times smaller in the inset of Figure 6 than in Figure 5.

We

Influence of cross-linking on the vitrimer mechanical behavior

In Figure 7, we show the evolution of the nominal stress observe on the curves shown in Figure 7 and 8: At the four test temperatures, the higher the DGEBA content-and thus the degree of cross-linking-the larger the stress. It should be noted that the strain rates M  are not the same for the different materials during the tests. The strain rate becomes smaller when the degree of cross-linking increases (see Figure S5 for the tests performed at 80°C). It is well known that the stress increases when the strain rate is increased. If the tests would had been carried out at a same constant strain rate-for example: 2.5 10 -3 s -1 , roughly the average values of the curves of Figure S5-the curves shown in Figure 8 would have been slightly modified. Nevertheless, the effect of cross-linking would have been even more noticeable since the gaps between the curves would have increased. We have also calculated the evolution of the strain hardening coefficient 4,24 ( )

ln M M d    =
as a function of M  . This coefficient is independent of strain rate if the true strain-true stress relationship can be written in the general form: ( ) ( ) ( )

, fg      =
as it is the case for the phenomenological G'sell and Jonas model that is widely used for describing the tensile behavior of semi-crystalline polymers. [24][25] The ( ) M  curves presented in Figure S6 confirm that an increase in DGEBA wt% leads to an increase in strain hardening, an expected consequence of cross-linking. [26][27] The Figures 7 and8 curves show that the tensile properties of vitrimers can be precisely tuned by controlling the cross-linker concentration. The V05 and V0 nominal stress curves are very close but the effect of cross-linking can still be demonstrated through the  strain hardening coefficient that is higher for V05 than for V0 (see Figure S6). Furthermore, at the end of the tensile test, an abrupt increase of the stress can be observed for V0 and V05 at the four temperatures (see the arrows at the end of the Figure 7 curves). This may correspond to the high strain range when the polymer chain are in fullextension. 28 The final stress increase always occurs at smaller strains for V05 than for V0. This suggests that the molecular network was indeed modified, and that the node density is larger for V05 than for V0. Recrystallization that was already observed for PBT at large strains could also contribute to the final stress increase visible in the Figure 8 curves. 29 The transition between the YPTC behavior and the NYPTC behavior with the increases in DGEBA wt% can be seen at the four temperatures on the curve sets shown in Figure 7. At 80°C for example, the V0, V0.5 and V0.75 curves clearly exhibit a yield point. By enlarging the corresponding curves, it can be checked that this remains true for V1 but not for V1.27. At 100°C, the transition is also observed between the V1 and V1.27 curves. The transition is between the V0.75 and V1 curves at 130°C and 160°C. The evolution between the YPTC and NYPTC behaviors takes place in a very gradual or continuous manner. This is well illustrated for the measurements performed at 160° where the V0.75 (YPTC behaviour) and V1 (NYPTC behavior) curves are very close and look the same if they are not enlarged. ) values are reported in table 2 (line 2 and 3) for all the test temperatures. As expected, when the test temperature of a given Vi material increases, the yield strain increases and the yield stress decreases. More interestingly, it can be checked that the higher the DEGBA concentration, the higher the yield strain-i.e., the higher the strain localization onset (see line 3 of Tables 2a, 2b, 2c and 2d). Instead, we would have expected the cross-linking to hinder the polymer deformation and thus result in lower yield strains ( Y  ). A literature review was conducted to clarify this apparent contradiction. We found that the yield strain (and stress) of a semi-crystalline polymers may strongly depend on the initial orientation degree of the material. [30][31] To assess whether our materials exhibited some degree of pre-orientation, we carried out SAXS experiments on our samples.

Effect of Pre-orientation

The SAXS patterns measured for the undeformed V0, V05, V1, V1.5 (DGEBA wt%1.5%≈1.58%) and V2 polymers are shown in Figure 9. The X-ray beam passed through the specimens along the thickness direction. The diffuse ring associated with the spherulitic morphology is clearly visible on the V0 and V05 patterns. In Figure 10, we show the intensity profiles measured along the rings corresponding to max h q q cte == , where max h q is the value at maximum intensity along the horizontal axis. For V0, V0.5, V1, V1.5, and V2, the max h q values are 0.52, 0.50, 0.44 ,0.43 and 0.42 nm -1 respectively. The profiles of Figure 10 reveals that V0 and V05 are weakly anisotropic, and that the anisotropy is slightly more pronounced for V05 than for V0. On the other hand, the V1, V1.5 and V2 patterns are strongly anisotropic (note the log scale for the y axis in Figure 10), and are characteristic of the presence of a "shish-kebab" type morphology. [32][33] The "shish-kebab" morphology is induced by shear at the mold walls during polymer injection and is concentrated in the specimen skin regions. The increase in shear stress at the mold walls is due to the increase in polymer viscosity caused by cross-linking. In the case of polymers with high viscosities, the occurring of oriented morphologies is inherent to the injection process and must be taken in consideration when implementing this type of shaping process. Moreover, it was shown in previous studies that the shish kebab morphology is preserved when the polymer is annealed below, near or above the melting temperature. 34 Accordingly, after annealing the polymer remains strongly oriented.

In short, when the degree of cross-linking is increased, the material degree of pre-orientation increases as well (see Figure 10), which contribute to increase the yield strain as already reported in published works. 30,31 The material pre-orientation degree seems to have a significant influence on the tensile properties. This can be illustrated by analyzing the behaviors of the V0, V0.5 and V1 materials. The V0 and V0.5 SAXS patterns are very similar (see Figure 9 and 10) and the tensile properties of these two materials are also nearly the same (see Figure 7 and8). On the other hand, the V1 SAXS pattern differs significantly from those of V0 and V0.5, so is its tensile behavior. Figure 10 Intensity profiles versus  (azimuthal angle, defined on the V0 pattern in Figure 9) at

NS

 that correspond to characteristic steps of the neck stabilization phase beginning (see section 2.2) are reported in lines 5 and 6 of table 2. For V1 at 130°C and 160°C, the measurement of

1 NS
 may remain possible even if the nominal stress curve exhibit no maximum or minimum values, which suggests that moderate localization/delocalization effects may still be present. This is to be confirmed by the analysis of the X  curves (see section 2.4.5). . This confirms the conclusion of section 2.2: strain delocalization is objectively observed shortly after NS  . Neck stabilization occurs approximately at this strain value. It is well reasonable to consider that NS  is the upper bound of the strain range of plastic instability.

V 0

V 0.5 V 1 V 1.5 V 2
Flow and tensile direction



At the four test temperatures, the same trend is observed for NS  and 1 NS  (Figure 11): the higher the DGEBA concentration-and therefore the degree of cross-linking-the smaller the NS  and 1

NS

 values, which shows that neck stabilization (strain delocalization) occurs at smaller strains for the strongly cross-linked polymers. As previously mentioned,

NS

 can be seen as an indicator of the polymer limitation in extensibility. This is illustrated and commented in Figure S7 where the evolution of M  is represented for V0, V0.5, V0.75 and V1 at 80°C. In this figure, it can be checked that the lower NS  , the lower the level of the plateau where "strain saturation" (limitation in extensibility) is observed.

Furthermore, by analyzing the ( ) M  (strain hardening coefficient) curves it can also be understood that an increase in strain hardening results in a decrease of (see V0 at 80°, Figure S6). Therefore, when the ( )

M 
curve is shifted upward (increase in strain hardening) the NS  value becomes smaller.

Two complementary effects occurring at the macromolecular network level can explain the increase in strain hardening leading to the decrease of the strain at neck stabilization as observed when the degree of cross-linking is rised: Firstly, due to cross-linking, "chemical" nodes are added to the "physical" entanglements nodes (see Figure S1). The network node density increases, which results in a reduction of the intrinsic molecular network extensibility and therefore in a reduction of the strain level at which neck stabilization occurs. Secondly, the polymer pre-orientation can be seen as a prior extension in the melt. Before applying, the tensile stress (

0 M  = )
, the macromolecular network is already stretched, which limits the strain increment that is necessary to reach the network limitation in extensibility. The influence of the pre-orientation degree on the limitation in extensibility of semicrystalline polymers is reported in several published works. [13][14][15]

Effects of temperature and strain rate

In the explanation concerning the decrease of the strain at neck stabilization with cross-linking that we provide at the end of the previous section (section 2.4.3), we have only considered the influence of the network topological characteristics. We also wanted to assess the possible effect of temperature or strain rate. In Figures 12a and12b, NS  and 1

NS

 are plotted against the temperature for V0, V0.5 and V0.75, the three materials for which these quantities are available at the four temperatures. The NS  and 1

NS

 evolutions show only slight variations with temperature, without it being possible to identify a clear trend. We also carried out tensile tests at different strain rates for V0.

NS

 was found to be unchanged, 0.71 0.01 NS  =, for crosshead speeds of 0.060 mms -1 , 0.020 mms -1 and 0.0067 mms -1 (See Figure S8). We can note that this result is contained in the G'sell and Jonas model. [24][25] As previously mentioned, this model involves that ( ) M  is independent of strain rate, the same applies to Clearly, for all the curves corresponding to DGEBA concentrations lower than 1.56%: the smaller the DGEBA, the smaller X  . This confirms that strain localization is more pronounced when the polymer is weakly cross-linked. At 80°C (Figure 13a) the X  curve measured for V1.27 is clearly below that corresponding to V1.56 though no maximum can be observed on the corresponding nominal stress curves. This shows that moderate strain localization (necking) solely due to plastic instability-i.e., independent of specimen geometry-can occur even if no yield point appears on the nominal stress curve. This is also true at the other temperatures, for example at 130°C: the V1 and V1.27 tensile curves show no yield point (Figure 7c) but the V1 X  curve is significantly below that of V1.27 (Figure 13b). As for V1.56 and V2, the X  curves are very close. It can be assessed that the weak variations of these two curves are then fully dependent of the specimen geometry and the associated stress concentrations phenomena, as for any "plastically homogeneous" materials (materials that show no plastic a b instability). In other words, the plastic instability leading to strain localization is completely suppressed for the vitrimers with the highest cross-linking degrees. All the X  curves increase at the end tests. As previously mentioned, this indicates the occurring of strain delocalization (see section 2.2). We placed symbols (filled circles) to indicate the X  curve minima in Figure 13. It can be checked that the strain at these minima are logically close to the 1 NS  values reported in Table 2 (line 6). The strains corresponding to these minima decrease when the DGEBA wt% increases. confirms again that neck stabilization occurs at lower strains when the polymer degrees of cross-linking are increased.

Conclusion

The chief goal of this study was to investigate finely the influence of cross-linking on the mechanical properties of PBT-based vitrimers produced by injection molding. We carried out tensile tests at NYPTC Behavior YPTC Behavior 80°C various temperatures coupled with full-field strain measurements on vitrimers whose cross-linker concentrations have been changed in very small increments. We found that the strain range of plastic instability-phase during which strain localization occurs-decreases when the degree of cross-linking is increased. This gradually leads to the suppression of necking. The reduction of the strain range of plastic instability is mainly due to the decrease of the strain threshold ( NS  ) at which neck stabilization begins.

At the microstructure level, the decrease of NS  is due to the limitation in extensibility of the macromolecular network caused by cross-linking. When the degree of cross-linking is increased, two complementary effects contribute to the network limitation in extensibility: firstly the decrease of chain length between the nodes of the macromolecular network, and secondly the pre-orientation of this network. Pre-orientation of polymers with high cross-linking degrees and therefore with high viscosities like PBT-based vitrimers cannot be avoided in the very common case when injection molding is used to produce them. An interesting extension of this study would be to obtain our tensile specimens through a process like compression molding that preserves material isotropy. It would then be possible to study the specific tensile behavior of non-oriented vitrimers.

We also found that NS  does not depend on temperature or strain rate, which confirms that the properties of the macromolecular network is the primary factor to take into account to describe the neck stabilization characteristics.

Figure 1

 1 Figure 1 Strain measurement at the surface of a tensile specimen: central specimen cubic part (a), drawing of the specimens (b), strain map measured when

,In Figure 2 ,

 2  : Bragg's angle) corresponding to the patterns presented in this study is [0.12-1.8]nm -1 . we show the dependence of the storage modulus ( ' E ) measured by DMA versus temperature for V0 (pristine PBT), V1 and V2. As for V0, the curve evolution is in line with what is expected for a semi-crystalline polymer. The glass transition occurs at about 50-60°C. Next, ' E slightly decreases. Due to the presence of the crystalline network, the material is still in the solid state. Melting begins at about 220°C and '

Figure 3 N

 3 Figure 3 N  and X  versus M  for V0 at 80°C. Inset: zoom around point D on the

Figure 4

 4 Figure 4 Strain rate curves  obtained at the beginning of the tensile test for different material points situated in the specimen central flat part (V0, 80°C).

Figure 5

 5 Figure 5 Strain rate profiles  against the Lagrangian coordinate X for different selected M  strains corresponding to the states indexed by points A to G in Figure 3 (V0 at 80°C).

  have briefly presented the two extreme deformation processes possible for our materials: The first behavior type is called "YPTC" (Yield Point on the Tensile Curve, see V0 at 80°C) in the following and is associated with pronounced successive stages of strain localization and delocalization. The second behavior type is hereinafter referred as to "NYPTC" (No Yield Point on the Tensile Curve, see V2at 160°C) and corresponds to quasi-uniform deformation during the test. In the following, we performed several tests at various test temperatures and for vitrimers synthesized with various DGEBA wt% in order to study carefully the transition between these two characteristic behaviors.

Figure 6 N

 6 Figure 6 N  versus M for V2 at 160°C. The strain rate profiles ( ) X 

N

   against M  for all the Vi materials and test temperatures (80°C: Figure 7a, 100°C: Figure 7b, 130°C: Figure 7c and 160°C: Figure 7d). Similarly, the true stress-true strain curves ( ( ) M  ) are plotted in Figure 8. The effect of cross-linking is easy to

  Figure 7

Figure 8 2 . 4

 824 Figure 8 ( ) M  curves, a: 80°C, b: 100°C, c : 130°C and d: 160°C.

Figure 9 SAXS

 9 Figure 9 SAXS Patterns measured before deformation for V0, V0.5, V1, V1.5 (DGEBA wt%=1.5≈1.58) and V2.

q

  maximum intensity along the horizontal axis). The intensity is normalized by the minimum value ( min I ) along the profile and averaged on a ±0.05 nm -1 interval around

NS  , 1 NS  and Y 1 0

 1and1 are plotted in Figure11as a function of the DGEBA wt%. As in the case of the test analyzed in the 2.2 section (V0 at 80°C),1 NS is always slightly larger than NS  , and the difference is very roughly constant:

1 

 1 = curve twice. The first time during its decreasing part at the yield point ( N  maximum) for M Y  = , and the second time during the increasing part that follows, at neck stabilization ( N  minimum) for M NS  =

Figure 11 DGEBA, 1 NS

 111 Figure 11 DGEBA wt% dependence of the yield strain ( Y  , onset of strain localization), and of the strain delocalization thresholds ( NS  , 1 NS  ), a: 80°C, b: 100°C, c : 130°C and d: 160°C.

1 NS

 1 In short, the temperature and strain rate are only secondary effects on NS  and thus on the polymer limitation in extensibility compared to the large primary effects linked to the molecular network characteristics, namely the network node density and its pre-orientation.It is worth noting that in the temperature range of investigation, we have not detected any rise of NS  or that could indicate that a possible thermal of covalent bond migrations could have changed the network extensibility properties.

Figure 12

 12 Figure 12 Temperature dependence of NS  (a) and 1 NS  (b).

  the four test temperatures for the pristine PBT (V0) and the Vi vitrimers are shown in Figure 13 (a: 80°C, b: 130°C) and Figure S9 (a: 100°C and b: 160°C). As illustrated in the inset of Figure 13b for strain profiles measured at 130°C when X  value implies strong strain localization. Therefore, the curves of Figure 13 allow quick and simple comparison of the different materials cross-linking degree effect on strain localization.

Figure 13 2 . 4 . 6 NS  and 1 NS 1 

 1324611 Figure 13 Evolutions of X  against M  at 80°C (a) and 130°C (b).

Figure 14

 14 Figure 14 Strain range of plastic instability ( NS Y  -) versus the DGEBA wt%. The inset illustrates the mechanism leading to the transition between the YPTC and NYPTC behaviors.

Figure

  Figure S3 Colour of specimens. Figure S4 Photography of the experimental set up.

Extensibility

  Figure S7 M  versus l  (crosshead relative displacement) at 80°C for V0, V0.5, V0.75 and V1 (YPTC behavior). This illustration is intended to show that it is possible to compare the limitation in extensibility of different polymers through the measurement of NS  .

Table 2d Table 2

 2d2

	Mechanical properties of PBT (V0) and PBT-based vitrimers (Vi, i≠0), a: 80°C, b: 100°C, d: 130°C
	and d: 160°C. NM: Not Measurable.

Acknowledgements

The authors would like to thank the ICEEL Carnot Institute, the EMPP research department of Lorraine University, the Grand Est Region and ADEME (french agency for ecological transition) for their financial support. We are also very grateful to Jérémy Bianchin who made the injection specimen mold.

Cameras

Oven Shaft Oven window


Jos Käfer
email: jos.kafer@univ-lyon1.fr

Marcos Méndez

Sylvain Mousset

Labile sex expression in angiosperm species with sex chromosomes

Here we review the literature on sexual lability in dioecious angiosperm species with well-studied sex chromosomes. We distinguish three types of departures from strict dioecy, concerning either a minority of flowers in some individuals (leakiness) or the entire individual, which can constantly be bisexual or change sex. We found that for only four of the 22 species studied, reports of lability are lacking. The occurrence of lability is only weakly related to sex chromosome characteristics (number of sex-linked genes, age of the non-recombining region). These results contradict the naive idea that lability is an indication of the absence or the recent evolution of sex chromosomes, and thereby contribute to a growing consensus that sex chromosomes do not necessarily fix sex determination once and for all. We discuss some implications of these findings for the evolution of sex chromosomes, and suggest that more species with well-characterized lability should be studied with genomic data and tools.

Introduction

About 5 to 6 % (more than 15000 species) of flowering plants are dioecious, meaning they have separate female and male individuals [START_REF] Renner | The relative and absolute frequencies of angiosperm sexual systems: dioecy, monoecy, gynodioecy, and an updated online database[END_REF]. Currently, sex-determining systems are known for less than 100 species, mostly through the discovery of sex chromosomes [cf 2 -5]. Dioecy, and therefore sex chromosomes, evolved many times independently in the angiosperms.

The knowledge about sex chromosomes in plants is still scarce: fewer than 30 species have been studied in detail with genomic data [see reviews [START_REF] Charlesworth | Plant sex chromosomes[END_REF][START_REF] Renner | Plant sex chromosomes defy evolutionary models of expanding recombination suppression and genetic degeneration[END_REF][START_REF] Muyle | The evolution of sex chromosomes and dosage compensation in plants[END_REF]. Most described chromosome systems are of the XY type (male heterogamety), but some species have ZW type chromosomes (with female heterogamety). It is thought that the evolution of sex chromosomes starts with a small sex-determining region on a chromosome pair, from which recombination suppression spreads to a larger part of the chromosomes. Initially, the YY or WW genotypes are viable (although they might be infertile). The non-recombining region of the Y or W chromosome is expected to degenerate through the reduced efficacy of selection, which leads to an accumulation of deleterious mutations and transposable elements. Transposable elements might initially cause these chromosomes to grow in size (which seems to be specific to plants), but at later stages, non-functional genes can be lost, leading to a decrease in size and maybe even to the loss of the entire Y or W chromosome.

Recent genomic studies have shown that not all plant sex chromosomes follow similar evolutionary trajectories [cf 5 , 7]. Among the most striking observations is that some relatively recently evolved sex chromosomes have large non-recombining regions and strong chromosome heteromorphy (e.g. Silene latifolia, Coccinia grandis) whereas some older systems have homomorphic sex chromosomes with small non-recombining regions (e.g.

Diospyros lotus, Vitis vinifera).

The process that causes recombination suppression to spread (e.g. interference of the sex-determining genes creating infertile individuals, sexually antagonistic selection) is not yet clear [START_REF] Charlesworth | When and how do sex-linked regions become sex chromosomes?[END_REF][START_REF] Ponnikas | Why do sex chromosomes stop recombining?[END_REF].

Reports of genetic sex determination remain rare compared to the total number of dioecious species, but this is mostly attributed to the difficulty of detecting sex chromosomes, especially if they are of recent origin [START_REF] Ming | Sex chromosomes in land plants[END_REF][START_REF] Muyle | The evolution of sex chromosomes and dosage compensation in plants[END_REF]. Thus, it is suspected that many more species with genetic sex determination will be described as sequencing techniques continue to improve.

Environmental sex determination is less well known in plants. Note that this term is used differently in plants and in animals. In vertebrates, by far the best-studied group, sex is often fixed during their lifetime, thus the influence of the environment occurs at a specific period early in development (although sex changes within an individual's lifetime are documented in fish [START_REF] Bachtrog | Sex determination: why so many ways of doing it?[END_REF]). In plants, the organs for sexual reproduction are produced anew in each flower (or similar structure), thus allowing more flexibility [START_REF] Cronk | The distribution of sexual function in the flowering plant: from monoecy to dioecy[END_REF]. Some striking examples of sex change in plants, also called "sex choice" or "gender diphasy", have been documented [START_REF] Lloyd | Modification of the gender of seed plants in varying conditions[END_REF][START_REF] Schlessman | Gender diphasy ("sex choice[END_REF], for example in Arisaema and Gurania, which change sex according to their size. As their size depends on the growing conditions, this can be termed environmental sex determination [START_REF] Cronk | The distribution of sexual function in the flowering plant: from monoecy to dioecy[END_REF]. It is considered evidence for the absence of genetic sex determination; indeed, in species in which all individuals change sex, no genetically based dimorphism is supposed. Lloyd and Bawa [START_REF] Lloyd | Modification of the gender of seed plants in varying conditions[END_REF] considered the possibility that sex choice might occur in sexually dimorphic species, but concluded that the evidence is weak.

As can be inferred from previous reviews [START_REF] Lloyd | Modification of the gender of seed plants in varying conditions[END_REF][START_REF] Schlessman | Gender diphasy ("sex choice[END_REF][START_REF] Freeman | Sex change in plants: Old and new observations and new hypotheses[END_REF][START_REF] Korpelainen | Labile sex expression in plants[END_REF], variation in sex expression most often concerns a few individuals in a population or a few flowers on a plant. Such variation might be fixed (i.e. some individuals have different patterns of sex expression than the majority) or variable (different patterns of sex expression occur every now and then). In both cases, the plants are able to produce flowers of the sex they normally don't produce. This lability has sometimes been considered a sign for weak genetic differentiation between the sexes [e.g. 5]. Indeed, the main hypothesis for the expansion of sex-linked non-recombining regions has been the accumulation of genes with sex-antagonistic effects [START_REF] Charlesworth | Steps in the evolution of heteromorphic sex chromosomes[END_REF] [but see [START_REF] Jeffries | A neutral model for the loss of recombination on sex chromosomes[END_REF]. Such genes are thought to increase specialization in one of the sexual functions (female or male) and to confer less fitness, or even sterility, to intermediate phenotypes.

Thus, well-differentiated sex chromosomes have sometimes been regarded as an example of irreversible evolution [START_REF] Bull | On irreversible evolution[END_REF] or "evolutionary traps" [START_REF] Bachtrog | Sex determination: why so many ways of doing it?[END_REF]. The existence of deviations from the purely female and purely male individuals is sometimes classed as "subdioecy", a transitory state between hermaphroditism (monocliny) and "full dioecy" [e.g. [START_REF] Charlesworth | Model for evolution of dioecy and gynodioecy[END_REF][START_REF] Ehlers | Inconstant males' and the maintenance of labile sex expression in subdioecious plants[END_REF][START_REF] Spigler | Gynodioecy to dioecy: are we there yet?[END_REF].

While reviews on sex changes or other forms of labile sex expression exist and have noted that some species with sex chromosomes display lability [START_REF] Freeman | Sex change in plants: Old and new observations and new hypotheses[END_REF][START_REF] Korpelainen | Labile sex expression in plants[END_REF], more sex chromosomes have been discovered and have been studied with genomic data since their publication. Here we compile data on lability in species with sex chromosomes for which there are estimates of the number of sex-linked genes and the age of these chromosomes. We show that lability occurs in almost all species investigated, and test whether the occurrence of lability is related to the characteristics of their sex chromosomes.

Methods

Terminology and classification of sexual lability

In the strict sense, sexual lability refers to the variation of sex expression in time. It has been known at least since the early 20 th century, under many names: sex change [START_REF] Pritchard | Change of sex in hemp -Mutilation makes female plants of Cannabis sativa produce male flowers -Change in nutrition probably responsible for the result[END_REF], sex reversal [START_REF] Schaffner | Complete reversal of sex in hemp[END_REF], sex choice [START_REF] Lloyd | Modification of the gender of seed plants in varying conditions[END_REF], sequential hermaphroditism [START_REF] Korpelainen | Labile sex expression in plants[END_REF], among others. The term "leaky dioecy" was coined by Baker and Cox [START_REF] Baker | Further thoughts on dioecism and islands[END_REF] to indicate "the condition that prevails where hermaphroditism or bisexuality occurs at low levels in populations of otherwise dioecious species". According to this definition, leaky dioecy includes trioecious species [START_REF] Darwin | The different forms of flowers on plants of the same species[END_REF], in which male, female and cosexual (usually monoclinous) individuals coexist in the same population. This is however not the sense that most authors nowadays tend to give to the term leakiness [e.g. 25], as it is considered similar to "inconstancy" [START_REF] Schlessman | Incidence, correlates, and origins of dioecy in the island flora of New Caledonia[END_REF]; Cronk [START_REF] Cronk | The distribution of sexual function in the flowering plant: from monoecy to dioecy[END_REF], in this issue, prefers to use the almost forgotten term "pleogamy".

While sexual lability (in the strict sense), leaky dioecy, and trioecy refer to different aspects of sexual expression in plants, they are difficult to separate in practice for two reasons. First, in many cases only a static picture of the sex ratio is available for a given population. Second, sex change often involves bisexual stages, and not simply switches from male to female or vice versa [START_REF] Lovett Doust | Sex and gender dynamics in jack-in-the-pulpit, Arisaema triphyllum (Araceae)[END_REF]. This means that at a given time, more than two sexual forms may be present in the population of sex-changing species. Therefore, some claims of leaky dioecy, subdioecy or trioecy based on a snapshot of sex expression in a population could be cases of sex change.

In addition, as we will show below, plant species can exhibit both sex change and leaky dioecy. In this review, we have adopted a broad definition of sexual lability that includes leaky dioecy and trioecy as special cases, for which sex expression does not necessarily vary in time.

Table 1 is an attempt to classify the different forms of lability that are observed in dioecious plants. Note that this classification is purely based on the observed phenotypic variation, and does not use any knowledge about the possible mechanisms of lability, which are unknown in the vast majority of cases. We chose to separate leakiness and trioecy by reserving the latter term to cases in which individuals are fully monoclinous or monoecious. Leakiness indicates the presence of a minority of aberrant flowers in an individual, which can thus still be considered mostly female or male. To subdivide the major types of lability, we use the classical terms of sexual systems to indicate the phenotype of the aberrant individuals [cf 1], although some prefer to reserve these terms for use at the population-level only [START_REF] Cronk | The distribution of sexual function in the flowering plant: from monoecy to dioecy[END_REF].

Sex chromosome data

We gathered information about sexual lability for species in which the sex chromosomes have been studied with genomic data. Information about many of these species was recently compiled by Renner and Müller [START_REF] Renner | Plant sex chromosomes defy evolutionary models of expanding recombination suppression and genetic degeneration[END_REF], and we added a few corrections and species, notably those for which no chromosome-level assembly was available.

We included species studied with transcriptomic data, but results from low-density genetic maps were not used, nor approaches using RADseq or similar data, because in the absence of a well-assembled genome, they do not permit an estimation of the total number of sex-linked genes.

The original studies of sex chromosomes use different methods to calculate the number of genes in the non-recombining region. All methods have performance limits, but almost no study seeks to quantify the false discovery rate and the miss rate. Furthermore, some rely on well-assembled genomes, others on de novo assemblies with more uncertainties; coverage-based methods more easily recover Y-or W-specific genes, while methods using segregating alleles perform better on XY or ZW gametologs. We chose to use point estimates of the numbers of sex-linked genes and the ages of the sex chromosomes, as the variation among species (approximately two orders of magnitude) is much larger than the confidence intervals per species.

Data on lability

Evidence of departure from strict dioecy was obtained from a review of the literature on reproductive biology of the species selected according to the criteria mentioned above. In some cases, lability has been explicitly reported. In other cases, this information was retrieved from sources documenting taxonomy, sexual system, sex determination mechanisms, teratology in flowers, population sex ratio, or reproductive success within and among reproductive events. We documented all the types of lability reported for each species but not all the references of lability per species. We also documented whether cases of lability were present in other species of the same genus. We have mainly restricted our review to cases of naturally occurring lability, or observations in cultivated plants that were exposed to nearly natural conditions (notably excluding the application of hormones).

Whenever available, the sex of the aberrant flowers was scored, as well as the direction of sex change, the resulting sex of the individuals (monoclinous, monoecious, gynomonoecious or andromonoecious), the frequency of the aberrant individuals, and the viability of the aberrant flowers (whether pollen was viable or viable seeds were produced).

Statistical analyses

Prior to the analysis, we merged closely related species to avoid phylogenetic biases: Silene otites and S.

pseudotites, the Populus species, and the Salix species. These species did not differ in observed lability; for the sex chromosome characteristics, the average value was taken into account in the analysis. We used two statistical tests to assess the effect of sex chromosome characteristics on the presence or absence of the different types of lability: Wilcoxon's rank sum test and binomial regression using a Generalized Linear Model (GLM).

For the binomial regression, we used the presence or absence of the three types of lability, and the presence of absence of any kind of lability. There is only a relatively small number of species with well-described sex chromosomes, so we performed a power analysis to test the robustness of our results. Full descriptions of the GLM and the power analysis are provided in Supplementary File 1.

Results

Information about the presence or absence of the types of lability is shown in Table 2. No lability has been reported for Actinidia chinensis, Diospyros lotus, Salix viminalis and S. purpurea. but there are reports of lability for other species occurring in the same genera: Actinidia deliciosa [START_REF] Testolin | Sex segregation ratio and gender expression in the genus Actinidia[END_REF], Diospyros kaki [START_REF] Akagi | Plant genetics. A Y-chromosome-encoded small RNA acts as a sex determinant in persimmons[END_REF] and D.

egrettarum [START_REF] Venkatasamy | Leaky dioecy in Diospyros (Ebenaceae) endemic to the Island of Mauritius[END_REF], and many Salix species ( [START_REF] Mirski | Exceptions from dioecy and sex lability in genus Salix[END_REF]; see Supplementary Table 1). Fourteen species show more than one type of lability, and four show all kinds of lability.

In the species studied, the non-recombining regions of the sex chromosomes have between seven (Fragaria chiloensis) and 1600 genes (Cannabis sativa, Rumex hastatulus; Supplementary Table 2), but it should be noted that the highest estimates come from studies using transcriptome data, which likely represent only a proportion q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Actinidia chinensis Amborella trichopoda age of non-recombining region (My) number of sex-linked genes lability absent present type q q q q q q q q q Trioecy Leakiness Sex change As shown in Figure 1, there is no clear correlation between the occurrence of lability and characteristics of the sex chromosomes. Table 3 shows the tests for sex chromosome differences for all types of lability. Only one of these tests was significant at the 5% level: species with younger sex chromosomes more often produce stable bisexual (monoclinous or monoecious) individuals. These results were confirmed by the binomial regression analysis (Supplementary File 1). The additional power analysis shows that we can only expect to find significant results when the strongest change in the probability of occurrence of lability (the so-called "switch") lies within the range of values (i.e., ages or number of sex-linked genes) that are effectively observed in the dataset. Thus, if this switch occurs before or immediately at the onset of sex chromosome evolution, or after several tens of millions of years of existence of the sex chromosomes, it cannot be discovered using these data. This also indicates that the significant negative correlation between the occurrence of trioecy and sex chromosome age is a robust result, even with the low sample size.

It is difficult to draw conclusions about the frequency of sex change, leakiness, and the occurrence of monoclinous or monoecious individuals within each of the species, as many studies report observations in a qualitative manner, and the quantitative data presented are often not directly comparable. From the few measurements we gathered, one can infer that the frequencies of aberrant individuals and flowers are often low, except in some cases where the plants were grown in extreme conditions. Species with younger sex chromosomes or fewer genes do not consistently have higher frequencies of sex change or leakiness; some of the highest frequencies of lability have been reported for Cannabis sativa [START_REF] Schaffner | Complete reversal of sex in hemp[END_REF][START_REF] Schaffner | Sex determination and sex differentiation -Sex determination and sex differentiation in the higher plants[END_REF], which has rather large and old sex chromosomes.

Sex change was documented from male to female and vice versa in four species. Leakiness was more frequently reported for males (eight species) than for females (three species) or for both sexes (four species; Table 2).

We illustrate these results by describing two plant species for which detailed studies are available. Silene latifolia stands out among angiosperms as it has strikingly dimorphic sex chromosomes. The Y chromosome is larger than the X, and it is considerably degenerated [START_REF] Papadopulos | Rapid Y degeneration and dosage compensation in plant sex chromosomes[END_REF][START_REF] Muyle | Rapid de novo evolution of X chromosome dosage compensation in Silene latifolia, a plant with young sex chromosomes[END_REF], although these chromosomes evolved only approximately 10 million years ago [START_REF] Krasovec | The mutation rate and the age of the sex chromosomes in Silene latifolia[END_REF]. Sex determination is considered quite stable in S. latifolia [START_REF] Ainsworth | Boys and girls come out to play: The molecular biology of dioecious plants[END_REF]. Completely monoclinous individuals (trioecy) have received some attention as they arise through deletions of parts of the Y chromosome and allow one to investigate the genetic architecture of dioecy in this species [START_REF] Westergaard | The mechanism of sex determination in dioecious flowering plants[END_REF][START_REF] Miller | A sex-chromosome mutation in Silene latifolia[END_REF][START_REF] Lardon | Sexual dimorphism in white campion: complex control of carpel number is revealed by y chromosome deletions[END_REF][START_REF] Fujita | Narrowing down the mapping of plant sex-determination regions using new Y-chromosome-specific markers and heavy-ion beam irradiation-induced Y-deletion mutants in Silene latifolia[END_REF].

Leakiness does not seem to be caused by a mutation, at least in one report where it was found not to be heritable [START_REF] Veltsos | The X chromosome is necessary for ovule production in Silene latifolia[END_REF]. An extensive study of the occurrence of sexual lability has been performed by Frick and Cavers [START_REF] Frick | Aberrations in sex expression in Silene latifolia[END_REF], who found both leakiness and sex changes by monitoring plants in common garden experiments and the greenhouse for several flowering seasons.

In Amborella trichopoda, sex chromosomes of the ZW type have a small non-recombining region that evolved quite recently (less than 16.5 My ago, most probably around 8 My ago) [START_REF] Käfer | A derived ZW chromosome system in Amborella trichopoda, representing the sister lineage to all other extant flowering plants[END_REF]. A detailed study using cultivated individuals during two flowering seasons showed variable proportions of leakiness in males (thus, inconstancy in the literal sense) and sex change in about 1% of individuals [START_REF] Anger | Dioecy in Amborella trichopoda: evidence for genetically based sex determination and its consequences for inferences of the breeding system in early angiosperms[END_REF]. The individual grown in the Santa Cruz arboretum (California), from which the reference genome is derived [START_REF]The Amborella genome and the evolution of flowering plants[END_REF], has been shown to be genetically female (i.e. has both Z and W chromosomes [START_REF] Käfer | A derived ZW chromosome system in Amborella trichopoda, representing the sister lineage to all other extant flowering plants[END_REF]). It is striking that this individual has changed sex, having flowered as mostly male in the past but recently as female. The botanical garden in Bonn also observed sex changes in an A. trichopoda plant [START_REF] Grosse-Veldmann | Amborella trichopodacultivation of the most ancestral angiosperm in botanic gardens[END_REF], and although its history is not completely clear, based on the relatedness φ statistic [START_REF] Manichaikul | Robust relationship inference in genome-wide association studies[END_REF] of 0.361 calculated on resequencing data [START_REF]The Amborella genome and the evolution of flowering plants[END_REF], this individual should be considered a clone of the Santa Cruz individual. Furthermore, sex changes have been reported in cuttings of the Santa Cruz plant [START_REF] Buzgo | Floral developmental morphology of Amborella trichopoda (Amborellaceae)[END_REF].

Discussion

We have shows, using a set of species whose sex chromosomes have been studied with genomic data, that sexual lability is only weakly influenced by the age of the sex chromosomes and the number of sex-linked genes. More precisely, sex change and leakiness did not seem to be influenced by the characteristics of the sex chromosomes, while the occurrence of trioecy diminished with the age of the sex chromosomes but not with the number of sex-linked genes. Importantly, in the vast majority of the species included in this analysis, at least some form of lability has been reported.

These results should be considered preliminary as this review is limited by the availability of data. First, we were able to include only a limited number of species, i.e. those for which the sex chromosomes were sufficiently well-characterized. We cannot rule out that the size of the non-recombining region or the sex chromosome age affect the occurrence of lability. However, it seems plausible that strong correlations, if they existed, would have been visible in this small sample as well, as shown by the power analysis (Supplementary File 1). We do not know if the species included in this review are a representative sample of all species with sex chromosomes, because such chromosomes have simply not been characterized in the vast majority of dioecious species. In addition, differences in the probability of lability between dioecious species with and without sex chromosomes could not be explored because sex-determining mechanisms are only understood in a small subset of dioecious plants.

Second, the estimates of the number of sex-linked genes or the age of the sex chromosomes depend on the methods used in the primary studies. The estimates that are used in this review will most likely become more precise when more genomic resources will be available. Note however that there is large variation in the current sample, and the rank sum tests presented in Table 3 are, to a certain degree, robust to inaccuracies.

Third, reports on sexual lability are scarce and difficult to compare to one another. As lability typically concerns only a few individuals in a population, observations of many plants, preferably including several flowering seasons, are needed to discover spontaneously occurring sex changes, leakiness, or bisexual individuals, as we have described for S. latifolia and A. trichopoda [START_REF] Anger | Dioecy in Amborella trichopoda: evidence for genetically based sex determination and its consequences for inferences of the breeding system in early angiosperms[END_REF][START_REF] Frick | Aberrations in sex expression in Silene latifolia[END_REF]. Such studies seem to have been more frequent before 1990 than today, with some notable exceptions. Not all species have received the same amount of attention from researchers, such that the absence of sexual lability in some species is likely to be the effect of the absence of studies. For the species completely lacking reports of sexual lability, we found evidence of lability in closely related species; it seems thus plausible that all of the studied species are capable of some degree of lability. Finally, we could only compare the presence or absence of lability as quantitative information (e.g. fraction of labile individuals) was available for only a subset of species (Supplementary Table 1).

Despite these limitations, our results show that lability can occur in dioecious plants regardless of the characteristics of their sex chromosomes. These findings are contrary to the naive idea that sex chromosomes represent a stable and irreversible way to fix dioecy [cf 17]. This idea has been largely questioned in recent years, and data in animals show that exceptions to this rule are frequent [e.g. 9]. Whether more subtle effects exist, and whether lability changes in dioecious species before sex chromosomes evolve, remains to be explored but currently the data to test these hypotheses are not available.

We have only limited understanding of the causes of lability. Some aberrant sexual phenotypes have been linked to mutations on the sex chromosomes and can be transmitted to descendants, as is often the case with trioecy (e.g. mutations on the Y chromosome in Carica papaya [START_REF] Wang | Sequencing papaya X and Yh chromosomes reveals molecular basis of incipient sex chromosome evolution[END_REF], Silene latifolia [START_REF] Miller | A sex-chromosome mutation in Silene latifolia[END_REF][START_REF] Lardon | Sexual dimorphism in white campion: complex control of carpel number is revealed by y chromosome deletions[END_REF] and Vitis vinifera [START_REF] Badouin | The wild grape genome sequence provides insights into the transition from dioecy to hermaphroditism during grape domestication[END_REF], or modified X chromosomes in Cannabis sativa [START_REF] Razumova | Molecular cytogenetic analysis of monoecious hemp (Cannabis sativa L.) cultivars reveals its karyotype variations and sex chromosomes constitution[END_REF] and Mercurialis annua [START_REF] Russell | Sex determination in dioecious Mercurialis annua and its close diploid and polyploid relatives[END_REF]). One could hypothesize that leakiness, which involves only some flowers of a plant, is more often the result of "developmental noise" [START_REF] Cronk | The distribution of sexual function in the flowering plant: from monoecy to dioecy[END_REF] or of some local action of hormones than the other types of lability that involve whole plants and that might have genetic causes. However, the picture that emerges from this review is that different types of lability often co-occur, and the distinction might thus be partially artificial. In the case of trioecy with a genetic basis, monoclinous or monoecious individuals can easily be selected and maintained artificially, as is the case for some crop species (Cannabis sativa, Vitis vinifera, Carica papaya).

Leakiness in Mercurialis annua has been the subject of intensive studies, that show that it can be selected for when mates (pollen) are limited [START_REF] Cossard | The rapid dissolution of dioecy by experimental evolution[END_REF], but is also influenced by the environment [START_REF] Cossard | A functional decomposition of sex inconstancy in the dioecious, colonizing plant Mercurialis annua[END_REF][START_REF] Sánchez | Plasticity in sex allocation in the plant Mercurialis annua is greater for hermaphrodites sampled from dimorphic than from monomorphic populations[END_REF]. Such experiments combining the effects of population density, mate availability, environmental conditions and inheritance could be performed for other species with sex chromosomes, and will probably show that sex determination is not entirely genetic nor environmental, as several authors have already suggested [e.g. [START_REF] Renner | Plant sex chromosomes defy evolutionary models of expanding recombination suppression and genetic degeneration[END_REF][START_REF] Golenberg | Hormonal interactions and gene regulation can link monoecy and environmental plasticity to the evolution of dioecy in plants[END_REF].

The wide occurrence of lability in dioecious species suggests that this sexual system is maintained through selection and not because of intrinsic (developmental or genetic) constraints. Indeed, if dioecy was not advantageous, it would be counter-selected and lost, together with the sex chromosomes [cf 101, 104]. For example, if a species went through a short period of monocliny or monoecy, the sex chromosomes could be lost, and a new pair might evolve when the species transitions back to dioecy. If and when such transitions happen is not known, but such events might be an additional mechanism to consider in the study of sex chromosome turnover [cf 6, 105, 106].

Sex chromosome research is currently benefiting from the "genomic revolution": technologies for wholegenome sequencing are improving, their costs are diminishing and many tools are available to infer sex-linkage [START_REF] Muyle | The evolution of sex chromosomes and dosage compensation in plants[END_REF][START_REF] Palmer | How to identify sex chromosomes and their turnover[END_REF][START_REF] Tomaszkiewicz | Y and W chromosome assemblies: Approaches and discoveries[END_REF]. On the contrary, studies including large-scale screening and long-term monitoring of sexual lability seem to have lost popularity (with some exceptions, notably economically important species), as illustrated by the age of the references we used to extract data from. We suppose this is due to the current research context in which long-term studies with uncertain outcomes are disfavoured. So we invite researchers to consider the reports of sexual lability in a species as an incentive to study it; the most time-consuming part of the work is already done. Of course, it is unlikely that sex chromosomes will be found in species in which sex change is common (e.g. Arisaema, Catasetinae, Gurania, Panax [START_REF] Lloyd | Modification of the gender of seed plants in varying conditions[END_REF]) as no genetic dimorphism is expected, but other species seem clearly dimorphic with a substantial, albeit minor, proportion of sex change, such as some Acer, Ilex, Solanum and Atriplex species [START_REF] Freeman | Sex change in plants: Old and new observations and new hypotheses[END_REF].

5 Acknowledgements 6 Authors' contributions JK conceived the study, gathered data on sex chromosomes, carried out data analysis, and drafted the manuscript. MM gathered and compiled data on sexual lability, assisted in interpreting the results, and critically revised the manuscript. SM conceived and performed additional statistical tests. All authors gave final approval for publication and agree to be held accountable for the work performed therein.

Competing Interests

We have no competing interests.

Figure 1 :

 1 Figure1: Presence or absence of the different types of sexual lability in species with well-characterized sex chromosomes. For each species, the circles show whether the type of lability has been observed (red) or not (white): small (inner circles) for trioecy, the intermediate circles for leakiness, and the large (outer) circles for sex change.

  of the genes. The ages of recombination suppression range from 0.25 My (Silene pseudotites) to 33 My (Humulus lupulus).

Table 1 :

 1 Classification of types of lability in dioecious species. The terms "aberrant flowers" and "aberrant individuals" refer to the deviations from strict dioecy.

	Lability type	Sex of aberrant individuals	Sex of aberrant flowers	Frequency of aberrant flowers in aberrant individuals	Temporal variation within an individual's lifetime
		Monoecious	Female or male
	Leakiness	Gynomonoecious Hermaphrodite Andromonoecious Hermaphrodite	Low ( 50%, usually < 10%)	Possibly
			Female or male
		Trimonoecious	plus
			hermaphrodite
	Trioecy	Monoclinous Monoecious	Hermaphroditic Female or male	(almost) 100% Similar proportions	No (or very limited)
	Sex change	Female or male (sequentially)	Female or male (sequentially)	(almost) 100%	Yes (by definition)

Table 2 :

 2 Sexual lability in 22 species with well-described sex chromosomes. The lability types correspond to those summarized in Table1. Am = andromonoecious, Gm = gynomonoecious, M = monoecious, Tm = trimonoecious. For species with reported leakiness, the leaky sex is indicated in parenthesis: f = female, m = male, b = both sexes. For clarity, species without reported lability are indicated with a dash in all columns. For details, see Supplementary Table1. This table also includes references to the literature on sex chromosomes.

			Type of lability
	Species	Sex change Leakiness	Trioecy References
	Actinidia chinensis	-	-	-	[28, 33, 34]
	Amborella trichopoda Yes	Am * (m)		[35, 36]
	Asparagus officinalis		Am * (m)	H	[5, 37-39]
	Cannabis sativa	Yes	M, Gm (f)	M	[21, 22, 32, 40-42]
	Carica papaya	Yes	Am, Tm (m)	H	[5, 43-46]
	Coccinia grandis	Yes	Gm (f)		[47, 48]
	Diospyros lotus	-	-	-	[5, 30]
	Fragaria chiloensis		Am (m)	H	[49-51]
	Humulus lupulus	Yes	M (b)		[52-55]
	Mercurialis annua		M, Tm (b)	M	[5, 56-58]
	Phoenix dactylifera		Am, Gm * (b)		[5, 59]
	Populus alba		M (m)		[5, 60]
	Populus euphratica		M (f)		[5, 61]
	Populus tremula		Am, M (m)		[5, 62, 63]
	Rumex hastatulus		Am, Gm (b)	H	[64-66]
	Salix purpurea	-	-	-	[31, 67, 68]
	Salix viminalis	-	-	-	[31, 67, 69]
	Silene latifolia	Yes	Am, M (m)	H	[70-75]
	Silene otites			H	[76-78]
	Silene pseudotites			H?	[77-79]
	Spinacia oleracea	Yes		M	[5, 80-83]
	Vitis vinifera	Yes	Am * (m)	H	[84-86]

*: evidence that leaky individuals actually change sexual expression between flowering events, i.e. are inconstant in sex expression.

Table 3 :

 3 Summary and tests of the effect of sex chromosome characteristics and sexual lability. Two-sided p-values were calculated using Wilcoxon's rank sum test.

		Median number of sex-linked genes	Median age of non-recombining region (My)
		without lability with lability p-value without lability with lability	p-value
	Sex change	81.5	707.5	0.27	8.08	13.3	0.48
	Leakiness	216	211	0.78	10.0	10.0	0.58
	Trioecy	159	234	1	20.8	5.00	0.021
	Overall	57	216	0.58	19.8	10.0	0.26

  The classical cases of sex change seem to involve plant size or resource status, or maybe external environmental conditions [cf[START_REF] Schlessman | Gender diphasy ("sex choice[END_REF][START_REF] Freeman | Sex change in plants: Old and new observations and new hypotheses[END_REF]. But the example of A. trichopoda, in which cuttings and clones grown in different locations change sex, seems to fit neither of the proposed mechanisms. Furthermore, there could be a role of somatic mutations, leading to an occasional branch of a different sex (leakiness), or even sex change in plants that regenerate frequently, but this hypothesis has only rarely been considered. The study of the causes of lability in species with sex chromosomes will only advance when sex-determining genes start to be described, but this currently is only the case for a handful of species [cf 5]. Sexually labile individuals might help in this enterprise by allowing to identify mutations, differences in gene expression, or epigenetic mechanisms[START_REF] Muyle | Epigenetics drive the evolution of sex chromosomes in animals and plants[END_REF].

We thank Aline Muyle, Helene Badouin and Djivan Prentout for bringing several interesting studies to our attention. Cornelia Löhne (Botanische Gärten der Universität Bonn) and Holly Forbes (University of California Botanical Garden at Berkeley) provided information about cultivated Amborella plants. We also thank the participants of the workshop on plant sex chromosome evolution (June 2021) for stimulating discussions, and in particular Susanne Renner and Niels MÃOEller for the organisation and comments on the manuscript. This work benefited from discussions in the "DiveRS" group funded by the synthesis centre CESAB of the French Foundation for Research on Biodiversity (FRB; www.fondationbiodiversite.fr), and we thank Andrew Helmstetter (CESAB) for critically reading the manuscript.