Hiroaki Ono
email: ono.hiroaki.340@m.kyushu-u.ac.jp

Ousseynou Azdine Nait-Ali

Kane Diallo

Guillaume Benoit

Sylvie Castagnet

O Kane Diallo

Influence of pressure cycling on damage evolution in an unfilled EPDM exposed to high-pressure hydrogen

Keywords: Cavitation, Cracking, In-situ observation, Photography, Image analysis, Rubber

This study aims to reveal the internal damage evolution process in a transparent ethylene propylene diene rubber (EPDM) under high-pressure hydrogen cycles (9 and 15 MPa). Damage accumulation of EPDM was tracked from in-situ pictures during cycling. Several dedicated image processing routines allowed the discrimination of mechanisms (separated cavities, clusters and cracks) and sometimes the qualification of their morphology (size distribution, number, ratio of cavities reappearing at any cycle). Numerous small cavities were observed at any cycle, some of them being clustered under the highest pressure. Only part of them systematically appeared again. Some of these cavities inflated and "absorbed" small cavities around them when clustered. Finally, a few cracks were nucleated from some large cavities and grew, following a "stop and grow" process.

Introduction

Rubber materials are generally used for gas sealing. In fuel cell vehicle (FCV) and hydrogen station for fuelling hydrogen gas to FCV, hydrogen gas is stored in high-pressure proof tanks under high-pressure up to 70 and 90 MPa, respectively. And the high-pressure hydrogen gas flows in the pipe lines and fuelling nozzle of FCV and of the station. Fuelling hydrogen from the station to FCV is done by using deferential pressure between FCV and the station. Rubber materials are used as a high-pressure hydrogen gas seal O-ring in these systems. The rubber materials are thus cyclically exposed to high-pressure hydrogen gas.

It has been reported that rubber materials exposed to high-pressure gas, including argon, nitrogen, carbon dioxide and methane, exhibit internal cavitation and cracking [START_REF] Gent | Nucleation and growth of gas bubbles in elastomers[END_REF]Stewart1970;[START_REF] Briscoe | Gas-induced damage in elastomeric composites[END_REF][START_REF] Briscoe | Explosive decompression failure" of rubbers: a review of the origins of pneumatic stress induced rupture in elastomers[END_REF][START_REF] Embury | High-pressure gas testing of elastomer seals and a practical approach to designing for explosive decompression service[END_REF]). These mechanisms are caused by the reduction of solubility of dissolved gas accompanying with rapid removal of ambient gas pressure surrounding the rubber. Also in the case of high-pressure hydrogen gas, occurrence of these types of mechanisms has been reported [START_REF] Jaravel | On key parameters influencing cavitation damage upon fast decompression in a hydrogen saturated elastomer[END_REF][START_REF] Jaravel | Experimental real-time tracking and diffusion/mechanics numerical simulation of cavitation in gas-saturated elastomers[END_REF][START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions-support to a relevant modeling framework[END_REF][START_REF] Yamabe | Fracture analysis of rubber sealing material for high pressure hydrogen vessel[END_REF]Yamabe andNishimura 2009, 2013;Kogaetal.2011). These types of mechanisms can obviously lead to the fatal fracture of highpressure hydrogen gas seal O-ring. This kind of damage could be observed only when cyclic exposure test is finished and the O-ring is taken out from the test machine. Thus, it has been possible to observe the resulting damage, but not the evolution of the damage process. For confidence in the high-pressure hydrogen seal O-ring, understanding how damage evolves is important.

Cavitation inside of rubber materials induced by single and cyclic mechanical loading has deeply been investigated by using not only mechanical tests but also in-situ photography, small angle X-ray scattering and X-ray tomography [START_REF] Lindsey | Triaxial fracture studies[END_REF][START_REF] Bayraktar | Damage mechanisms in natural (NR) and synthetic rubber (SBR): nucleation, growth and instability of the cavitation[END_REF][START_REF] Cristiano | An experimental investigation of fracture by cavitation of model elastomeric networks[END_REF][START_REF] Hocine | Experimental and finite element investigation of void nucleation in rubber-like materials[END_REF][START_REF] Legorju-Jago | Fatigue initiation and propagation in natural and synthetic rubbers[END_REF][START_REF] Zhang | Nanocavitation in carbon black filled styrene-butadiene rubber under tension detected by real time small angle x-ray scattering[END_REF][START_REF] Zhang | Opening and closing of nanocavities under cyclic loading in a soft nanocomposite probed by real-time small-angle x-ray scattering[END_REF]. In general, it has been accepted that the cavitation induced by mechanical loading originates from dilatation of a "precursor" by local hydrostatic pressure.

In the case of cavitation by gas exposure, including pressurization and depressurization process, it is known that a gas dissolved in a "cavity-precursor" of a rubber material during pressurization expands the precursor from the inside during depressurization [START_REF] Gent | Nucleation and growth of gas bubbles in elastomers[END_REF]Stewart1970;[START_REF] Yamabe | Influence of fillers on hydrogen penetration properties and blister fracture of rubber composites for O-ring exposed to high-pressure hydrogen gas[END_REF][START_REF] Jaravel | Experimental real-time tracking and diffusion/mechanics numerical simulation of cavitation in gas-saturated elastomers[END_REF].

As limiting to only hydrogen gas exposure cases, Yamabe and Nishimura were probably first researchers who investigated the cavitation inside rubber materials induced by high-pressure hydrogen gas pressurization and subsequent depressurization [START_REF] Yamabe | Influence of fillers on hydrogen penetration properties and blister fracture of rubber composites for O-ring exposed to high-pressure hydrogen gas[END_REF]. They defined cavity and cracks induced by high-pressure hydrogen gas exposure as "blister fracture". Subsequently, several researchers have been developing this field. [START_REF] Jaravel | On key parameters influencing cavitation damage upon fast decompression in a hydrogen saturated elastomer[END_REF] reported that cavity appearance time after decompression is shortened by elongation, and damage quantities depend on both maximum pressure and depressurization rate in silicone rubber samples under the single high-pressure hydrogen exposure test that was conducted for different maximum pressure (0.1-27 MPa), different depressurization rate (0.2-90 MPa/min) and under elongation. They also attempted to simulate the cavity growth scenario in the above experimental systems using a model with a single initial hole existing at the sample center and took into account internal pressure of cavity, hydrogen content of a cavity, of a rubber matrix and of ambient [START_REF] Jaravel | Experimental real-time tracking and diffusion/mechanics numerical simulation of cavitation in gas-saturated elastomers[END_REF]. Then they indicated that cavity growth inside of rubber can be represented by diffusivity of hydrogen gas and elastic modulus of rubber. [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions-support to a relevant modeling framework[END_REF] also conducted in-situ observation for transparent EPDM under single highpressure hydrogen exposure test with different maxi-mum pressure and different depressurization rate, and reported that the number of cavities and the mean cavity diameter is increased by increment and fastening of maximum pressure and of depressurization, respectively.

Above progress of investigations about internal damage of rubber materials by hydrogen exposure considered only single exposure-the test including only one set of pressurization-saturation-depressurization. In the case of real use of high-pressure hydrogen Oring, as mentioned previously, the O-ring is cyclically exposed to high-pressure hydrogen gas-pressurizationsaturation-depressurization-kept at lower pressure and subjected to the subsequent pressurization cycles. Yamabe et al. and Koga et al. attempted the cyclic exposure test for confirmed condition to evaluate cycling effect on internal damage and revealed an incremental number of cracks with cycle progress (Yamabe and Nishimura 2013;K o g ae ta l .2011). However, it is still unknown how cracks appear and grow through cycling. Thus the investigation of damage evolution under cyclic high-pressure hydrogen gas has not been clarified enough, only the resulting damage has been reported. Hence, in-situ optical tracking of damage morphology during pressurization and depressurization cyclic process is a powerful technique to reveal the damage evolution by cyclic hydrogen exposure.

In this paper, cyclic exposure test up to 9 and 15 MPa were conducted for in-situ observation to chase damage progress in transparent EPDM through the cycling. The test at 9 MPa was employed focusing on cavity growth and the test at 15 MPa was employed for both cavity growth and evolution from cavity to crack.

This paper aims to reveal damage evolution processes under high-pressure hydrogen gas exposure cycling up to 9 and 15 MPa by in-situ photography observation.

Experimental

Sample

For optical damage observation, transparent ethylenepropylene-diene rubber (EPDM) was prepared. The sample was vulcanized under 170 • Cfor7min,froma compound composed of 100 phr (per hundred rubber) of EPDM polymer (ESPREN505, Sumitomo Chemical Co., Ltd.), 0.5 phr of stearic acid and 1.6 phr of dicumyl peroxide as a vulcanized agent. The vulcanizate was obtained as a 150 mm × 150 mm × 2m m sheet from which 20 mm × 20 mm × 2 mm samples were cut out.

High-pressure hydrogen cycles test

The high-pressure hydrogen cycles tests were conducted in a pressure tight vessel with a φ 28 mm × 70 mm thick sapphire glass for optical observation. Figure 1 illustrates the schematics of unit pressure sequences of cyclic hydrogen exposure test for 9 and 15 MPa saturation pressure. The compression rate was 1 MPa/min up to a saturation pressure (P sat ) of 9 or 15 MPa. Samples were kept at P sat for 60 min. The decompression rate was 2.5 MPa/min down to atmospheric pressure.

The authors revealed the effect of the decompression rate on damage morphology of transparent EPDM during a single pressure cycle [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions-support to a relevant modeling framework[END_REF].

In that paper, seven decompression rates (0.75-30 MPa/min) were employed. A middle range value of decompression rate (2.5 MPa/min) was selected here to address cycling.

The time between the end of decompression at cycle (n) and the beginning of compression at cycle (n + 1) was 60 min. Such cycle, as shown in Fig. 1,w a s repeated 5 and 6 times, respectively, for 9 and 15 MPa test. For each cycle, the time elapsed from the beginning of decompression was called T dec .

Time resolved observation

Figure 2 exhibits the schematic illustrations of the insitu observation system, an example of a resulted image and the magnified image showing minimum detectable damage. Images were taken with a CCD camera (Sony XCD SX 90) fitted with an Avenir TV Zoom Lens (12.5-75 mm F18). The resolution was 23 µ m/pixel. The image acquisition period was (1) 2 s over the first Fig. 2 Schematic illustrations of the in-situ observation system, an example of resulted image and the magnified image showing minimum detectable damage 1200 s after the beginning of compression, (2) then 600 s over the range from 1800 to 4200 s, (3) 2 s between 4202 and 6060 s, (4) and finally 60 s from 6120 to 8460 s.

Image processing and analysis

Figure 3 shows a schematic illustration of image processing routines. Fiji plug-in of Image J (Available from: http://fiji.sc/Fiji) was used for all image processing and analysis of the high-pressure tests at 15 MPa. Visilog software was used for the low pressure one (9 MPa). The first "(1) Removing background noise" step consisted of subtracting the image of beginning of 1st cycle to each image to remove artefacts such as dust or shaping defects that did not take part to the final damage. Several damage processes affected the samples during cyclic exposure. The aim of the following procedure was to isolate them from each other. Four different treatments were applied depending on the damage process being addressed. They are detailed in Fig. 3,asthe processes A-C in the "(2) Image processing routines for each evaluation".

(A) Cavity separation The aim of this routine was to analyze the size of separated cavities, so-called by opposition to clustered ones. As shown in the left picture, the eye could not separate some contacted cavities from each other. The first step of the routine was thus to remove such clusters (step 1). Then, the remaining recognizable contacted cavities were divided by 2-pixel thick line (step 2). Obtained images were binarized with a 20 threshold on the grey level scale. The average area of cavities (not necessarily spherical) was computed, and the diameter of equivalent spherical cavities could be calculated from it. It was called D in the following.

In the case of low saturation pressure test (9 MPa), no clusters were observed. A segmentation/erosion/ spherical reconstruction process, equivalent to steps 2 and 3, were applied to separate cavities from each other and extract their diameter and the location of centroids [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions-support to a relevant modeling framework[END_REF].

(B) Extraction of the damage area overlapped between cycle n and cycle n + 1

The aim was to be able to quantify persisting damage between two successive cycles. After noise removal, all binarized pictures taken during cycle n were summed up to integrate damage occurring at any time of the cycle (step 1). The same process was applied to the pictures stack of cycle n + 1. These summed up images were superimposed and colored yellow in areas that appeared to be common damage areas between cycle n and cycle n + 1 (step 2). The superimposed image was binarized again (yellow area was black and the other color was white) (step 3). This allowed the quantifying of the damage fraction of cycle n survival (meaning observed again) at cycle n + 1. This common area was called "surviving area" (F surv ) in the following. ter of each crack. This was used to appreciate the cracks inflation during one given cycle or between two cycles.

Results

Samples were exposed to several cycles of hydrogen pressure up to a maximum pressure (P sat ) of 9 or 15 MPa. In the former case, only separated cavities were detected whereas in the latter case, other forms of damage appeared, like clustering or cracking. General features of the damage morphology are depicted for these more or less severe exposure conditions in the first following section. Each damage mode is later considered more in-depth in the next paragraphs.

In this paper, the terms "cavity" and "crack" were both used to depict the damage morphology observed in 9 and 15 MPa tests. Such terms usually refer to the shape of defects (spherical vs. thin and elongated), associated with variable stress concentrations and usually handled by damage mechanics or fracture mechanics. In the present case, terms were particularized to the observed damage morphology. "Cavity" referred to a smaller defect than a "crack". As detailed in the following sections, cavities could disappear, or re-appear and grow, during cycling. Cracks referred to irreversible large defects, growing from coalescent surrounding cavities up to sizes comparable to the sample thickness and sometimes bursting at the surface of the sample.

Evolution of damage morphology

This first section gives a general description of the observed damage morphology in samples exposed to cycles up to 9 or 15 MPa. Each damage process will be quantified in the following sections. The low-pressure test (9 MPa) is considered first, due to the reduced number of activated damage processes compared to the high-pressure test (15 MPa). Typical damage observed in samples cycled up to 9 MPa is thus illustrated in Fig. 4.

Pictures correspond to the maximum number of cavities for each cycle. Such states were reached after shortening times with cycling. The number of cavities clearly decreased with cycling, meanwhile the average size increased. By carefully considering pictures, it could be detected too that some cavities appeared at each cycle as shown in Fig. 4 whereas some others were not systematically observed. The former ones are called "surviving cavities" in the following.

The thickness of the chamber door and the small size of the window clearly limited the spatial resolution of observations and, subsequently, the ability to track cavity expansion in the thickness direction. Some attempts showed that cavities grow in the thickness direction too but 2D pictures on the edge of the sample were not good enough for detailed analysis.

Under more severe conditions, i.e., up to a higher saturation pressure of 15 MPa, the damage morphology was more complex. As attested by Fig. 5, clusters and large cracks co-existed with separated cavities.

During the first decompression (Fig. 5a-c), separated cavities began to appear (see cycle 1 in Fig. 5a). Then, the number of cavities, as well as their size, increased with time. Unlike in the 9 MPa test, some cavities formed clusters, as shown in Fig. 5b. After maximum damage, a deflation process started during the last part of cycle 1. When the sample was com- pressed again at the beginning of cycle 2, both separated cavities and clusters deflated with the pressure increase, and finally disappeared, as shown in Fig. 5c.

The same scenario was repeated at cycle 2. In cycle 3, cavities and clusters were observed again at 1000 after the beginning of the third decompression same as cycle 2, as shown in Fig. 5d. However, from 1129 s, some of the cavities restarted to inflate and turned to cracks, as shown in Fig. 5e. Such cracks were called "active" as long as their size evolved, either with time during a given cycle, or from one cycle to another.

During the compression process, clusters and active cracks deflated with the pressure increase and finally disappeared, like in cycles 1 or 2. At cycle 4, cavities and active cracks all appeared at the same places as at cycle 3 (Fig. 5f). Active cracks continued growing. Some of the remaining cavities began to inflate again and became active cracks too. The active cracks which had propagated between cycles 3 and 4 turned to "inactive" cracks at cycle 5, meaning that their size did not evolve any more until the end of cycle 6 (as shown in Fig. 5g,h).

Process 1: separated cavities

This elementary damage mechanism was observed in both samples cycled under 9 and 15 MPa. The case of tests at 9 MPa will be considered first; in such moderate decompression conditions, all cavities can be considered as "separated cavities" following the above defined terminology. Pictures were processed to obtain the number and size of cavities. Figure 6 displays the maximal number of cavities for each cycle (left dark grey columns). It significantly decreased, especially between cycles 1 and 2, and was almost divided by two after 5 cycles. The right columns (light grey columns) correspond to the number of "surviving" separated cavities, i.e., the number of separated cavities already observed at cycle 1 in Fig. 6a and at the previ-ous cycle in Fig. 6b. The graphs show that the view of a cumulative damage (i.e. a progressive growth of the same population of cavities cycle after cycle) is erroneous. Indeed, separated cavities observed at cycle 1 did not systematically appear again during the following cycles. However, their relative number increased with cycling. A large majority of the separated cavities observed at cycles 4 and 5 were already visible at the previous cycle and more generally from the first cycle. A more detailed analysis showed that surviving cavities previously reached a diameter larger than 190 µm.

Figure 7a represents the size distribution of separated cavities for each cycle, when the number of cavities was the largest. Histograms of the major population are shifted towards high values and narrowed with cycling, supporting homogenization of cavity size with cycling. A minor population of small cavities also appeared at cycles 2 and 3 but later disappeared. Figure 7b shows the evolution of the average diamet e ro fF i g .7a through cycling. The average diameter increased with cycle until cycle 3 and became constant in remaining subsequent cycles.

Let us consider now more severe decompression conditions. Images taken from samples cycled up to 15 MPa were first processed as detailed in Fig. 3 [process (A)] to isolate separated cavities. The area A of each separated cavity was measured and used to calculate the diameter D of an equivalent spherical cavity with identical area, as shown in Eq. ( 1).

D = 2 A π 1 2
(1)

Figure 8a shows the separated cavity diameter distribution over 0.1 mm wide ranges for cycles 1-6 at T dec = 1000 s for all cycles. This time corresponded to the maximum value of damage occupied area.

As shown in Fig. 8a, the curves show approximately the normal distribution for cycles 1 and 2, with an aver- age value close to the mode value of 0.3 mm. This damage pattern is consistent with the major population observed in Fig. 7a in samples tested up to 9 MPa.

From cycles 3 to 6, curves clearly did not obey normal distribution any more. Indeed, the curves showed approximately the same shape as cycles 1 and 2 in the range from 0.1 to 0.6 mm but a shoulder appeared above 0.6 mm. Although the mode value was unchanged (0.30 mm), the average values increased from cycles 3 to (respectively 0.35, 0.38, 0.39 and 0.37 mm).

Judging from size distribution of separated cavities in a 15 MPa test, two different populations could be distinguished. The small separated cavities (Cav S ) corresponded to the main peak in size distribution as shown in (1) and plotted in Fig. 6b. D S was almost the same for all cycles (0.30 mm). D L slightly increased between cycles 3 and 5 and was almost constant at approximately 0.8 mm between cycles 5 and 6.

To highlight the origin of large cavities observed from cycle 3, a few single separated cavities (called a-d) were tracked along cycling. Their location is depicted in Fig. 9 in pictures taken at T dec = 1000 s. Figure 8b shows an increase in their diameter. Cavities a, b, c and d appeared at the same place thorough all cycles, but it is important to notice that during the first cycles, they pertained to collections of very close small cavities whereas they later appeared as separated cavities in the last cycles. The effect of "surviving" cavities was addressed too in high-pressure conditions, in the same way as for 9 MPa tests. Figure 10 shows the damage fraction of cycle n survived in cycle n + 1(F surv ), as calculated by Eq.

(2).

F surv = A n ∩ A n+1 A n (2)
where A n is the area occupied by damage in cycle n and A n+1 is the area occupied by damage in cycle n + 1. It must be highlighted that both small and large separated cavities were taken into account in such analysis.

Images were summed up to take into account all damage history in cycle n and cycle n + 1, according to process (B) illustrated in Fig. 3. As shown in Fig. 10, F surv significantly increased; it was almost twice as large after 6 cycles as at the first cycle: for cycles 6 and 1, F surv was, respectively, 0.81 and 0.49. This result was fully consistent with that obtained at lower pressure (Fig. 6b).

Process 2: cracking

This damage mechanism was only observed at the higher pressure (15 MPa). To track the evolution of the few actual cracks, images were processed following the routine C illustrated in Fig. 3. The diameter of equivalent circular cracks was then calculated in the same way as previously done for separated cavities, from Eq. ( 1).

Through cycles 3-4, six cracks were nucleated, indexed as crack A to F in Fig. 9. Through cycles 4-5, and then cycles 5-6, respectively, two and one more new cracks were also observed, which were, respectively, indexed as crack G, H and I in Fig. 9.

As shown in Fig. 9, damage morphology was still composed of cavities at T dec = 1000 s in cycle 3. Cracking begun from part of these cavities being the "crack precursor" of crack A-F indexed in Fig. 9.

Figure 11 shows life time of crack A; (a) time evolution of the equivalent diameter [calculated from Eq. ( 1)] of crack A for cycles 1-4 as an example of crack A-I, (b) magnified images of crack A corresponding to each selected time written in the Fig. 11a with arrow. The images of cycles 5 and 6 were added to show "inactive" states. As shown in Fig. 11, crack A had not been "crack" yet, just a cavity in cycles 1 and 2. In other cracks (B-I), the cavities began to appear at cycles 1 or 2. A first growth step was observed at the beginning of cycle 3, followed by a "plateau" which means a tentative stop of the cavity growth as shown in Fig. 11a, b as Cycle 3, T dec = 360 and Cycle 3, T dec = 2417.

The equivalent diameter of cavity during plateau was similar at cycle 3: 0.82, 0.69, 0.77, 0.71, 0.71 and 0.60 mm, respectively, for cracks A to F. Theses cavities are assignable to Cav L judging from their size, and their abundance is equivalent to at least 10% of Cav L in the size range of 0.6-0.9 mm. Thus, remaining 90% of Cav L was still cavity till the end of cycle 3. This constant stage for cracks A to F lasted until T dec = 2417, 1129, 2597, 1129, 1129 and 2837 s, respectively. Following that, the cavity growth restarted until the beginning of compression of the next cycle and becoming the large crack as shown in the images of Cycle 3, T dec = 2472 and Cycle 3, T dec = 3917 in Fig. 11b.

During the decompression process of the following cycle (cycle 4), it was very interesting to note that the crack quickly re-opened to the previous maximum size measured at cycle 3, as shown in image Cycle 4, T dec = 360 in Fig. 11. Then, the following propagation stage was slower, until reaching a maximum equivalent size to that of cycle 3, except crack B, C and E.

Figure 12 exhibits magnified images focusing on crack B, C and E at T dec = 754, 1194 and 1756 in cycle 4. As shown in Fig. 12, crack B, C and E coalesced. This coalesced crack also continued to grow until the maximum size.

After reaching maximum size, the equivalent diameter quickly decreased. This corresponds to the burst of the crack; as shown in images of Cycle 4, T dec = 1400 and T dec = 1402 in Fig. 11, the change of the brightness of the crack can be found. This originates from the shrink of the crack. Taking into account that the shrink was completed at least within 2 s, it could be thought that the crack bursted with the quick release of hydrogen gas from the inside of the crack to the sample surface.

Figure 13 shows the time evolution of the equivalent diameter [calculated from Eq. ( 1)] of some of these cracks: (1) crack A for cycles 3-4 as an example of crack A-F (top), (2) Crack G for cycles 4-5 (middle), and (3) crack I for cycles 5-6 (bottom). As shown in Fig. 13, the same process as crack A to F through cycles 3-4 (stop and grow in a cycle and quick re-opening and burst in subsequent cycle) was observed for propagation of crack G, H and crack I, respectively, through cycles 4-5 and through cycles 5-6, as illustrated in the middle and bottom of Fig. 13, respectively.

As long as this step propagation process occurred, the crack was called "active" crack. Some of them finally burst at the surface of the sample. No more Fig. 11 Tracking of life time of crack A as an example of crack A-I; a time evolution of the equivalent diameter [calculated from Eq. ( 1)] of crack A for cycles 1-4, b magnified images of crack A corresponding to each selected time written in the Fig. 11awitharrow .The images of cycles 5 and 6 was added to show "inactive" state hydrogen could be stored inside the crack during following cycles and the size did not change any more (as shown in image g and h of Fig. 3 and Cycle 4, T dec = 1600 in Fig. 11). The crack was then called "inactive".

Discussion

The two pressure conditions explored (9 and 15 MPa), respectively, focus on the evolution of separated cavities and on the transition from cavitation to cracking. Judging from above results, a damage evolution sce-nario until large cracking could be proposed. It is illustrated in Fig. 14. Each process in Fig. 14 is described below: 4.1 Damage appeared as clusters and separated small cavity: Cav S in first cycles

Cycles 1 and 2 of the 15 MPa test showed only cluster of cavities and small separated cavities. The clustering effect could be brought closer to the onset of satellite cavities growing around primary ones, reported by Gent and Tompkins in CO 2 , Ar and N 2 in several kinds of rubber samples [START_REF] Gent | Nucleation and growth of gas bubbles in elastomers[END_REF]. Jaravel et al. also reported this damage process in hydrogen decompression tests for silicone rubber [START_REF] Jaravel | On key parameters influencing cavitation damage upon fast decompression in a hydrogen saturated elastomer[END_REF]. At lower pressure (9 MPa), this clustering effect was not observed. Small separated cavities exhibited smaller average size at 9 MPa (diameter below 0.4 mm), compared to 15 MPa (diameter below 0.6 mm). [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions-support to a relevant modeling framework[END_REF] reported that the number and the average diameter of cavity increase with the decompression rate increase for 9 MPa hydrogen decompression test. In the case of the test at 15 MPa test and the test at 9 MPa for this paper, the saturation pressure was different but the decompression rate was same at 2.5 MPa/min. Hence, it was indicated that the cavity size difference between the 15 MPa test and the 9 MPa test depends on the difference of dissolved hydrogen content and differential pressure between inside of the cavity and the ambience of the sample originated from saturation pressure.

4.2 Parts of the small cavities (Cav S ) disappear as the cycle progresses

According to Fig. 8a, the abundance of separated large cavities (Cav L ) increased to the detriment of small separated ones (Cav S ). According to Fig. 10, the ratio of "surviving" damage F surv increased. When considering cycles 1 and 2, for which separated cavities are "small" ones Cav S , it means that approximately 50% of Cav S of cycle 1 disappeared in cycle 2. In the same way, test at 9 MPa showed a global trend to nucleate less numerous but larger separated cavities with cycling. At this pressure too, the ratio of surviving cavities increased, suggesting the growth of a non-negligible fraction of small cavities, but the disappearing of another part of them.

Parts of Cav S survive and change to Cav L with cycling

Part of the population of small cavities Cav S dead with cycling, as mentioned above, means the residual Cav S survive through the cycling in the test at 15 MPa. This tendency, i.e., the disappearing of smaller cavities and appearing of larger cavities through the cycling, was the same in the test at 9 MPa, which was also discussed above. According to Fig. 8b, cavity a to d, which were picked up as surviving separated cavities, had a diameter below 0.6 mm (i.e. within Cav S range) at cycles 1 and 2, and later exceeded 0.6 mm (i.e. within Cav L range) after cycle 3. These results indicated that some small cavities inflated as a result of a surviving and growing process with cycling. Moreover, this growing process sometimes interfered with cluster evolution. As observed for instance in cavities c and d in Fig. 9, the collection of small cavities surrounding these cavities at the first cycles disappeared step by step with cycling. Small surrounding cavities seemed to be trapped by the most growing one.

4.4 Parts of the Cav L evolve to an active crack According to Fig. 8a, the size distribution and the average equivalent diameter of large separated cav-Fig. 13 Comparison of the time evolution of the equivalent crack diameter of cracks during successive cycles: Crack A at cycles 3 and 4 (top), Crack G observed at cycle 4 and 5 (middle) and Crack I observed at cycles 5 and 6 (bottom). T dec is the decompression time of each cycle; period 1 refers to the decompression stage, 2 to the atmospheric pressure stage and 3 to the following compression stage. Cycle 6 have no decompression process because cycle 6 was the last cycleforthetest ities increased with cycling. Focusing on the crackprecursor cavity size, they were almost the same at 0.7 mm. According to the "Process 2: cracking" section, the cavity diameter just before the onset of cracks A to F was approximately 0.7 mm, which was within the Cav L diameter range. One could wonder whether this size of 0.7 mm could be regarded as a size criterion of the crack-precursor cavity, considering that no cracks were observed in the 9 MPa test in which no cavities exceeded 0.7 mm. However, according to Fig. 8a, some cavities with diameters larger than 0.7 mm still existed until cycle 6 in the 15 MPa test. Finally, this seemingly "critical" size of 0.7 mm could not be regarded as fracture criterion. It could be argued that not only the crack-precursor cavity causes this, but also the surrounding stress field of that was part of the conditions for crack initiation.

The cracks always exhibited a larger diameter than the sample thickness, namely, the cracks always propagated along in-plane direction. Lindsey reported that the tearing direction is perpendicular to the principal tensile stress direction judged by the result of triaxial Fig. 14 Schematic illustration of damage shape evolution through the hydrogen exposure cycle tensile test by using the poker-chip test for negative hydrostatic pressure effect on the damage propagation [START_REF] Lindsey | Triaxial fracture studies[END_REF]. It was thus assumed that the crack propagation also activated the principal stress direction perpendicular to the in-plane due to the change of the surrounding stress field, which possibly originates from the size change of cavities except the crackprecursor. The norm of this "mode I"-like process also corresponds to the shortest diffusion path from the core of the sample to the nearest free surface. As the damage mechanism is known to arise from diffuso-mechanical coupling, it could also affect the crack propagation direction.

The active cracks finally burst and turn to inactive cracks

This process was evidenced from Figs. 9, 11 and 13.It was assumed that the burst-rapid deflation-of the crack was caused by the rapid hydrogen gas desorption from inside of the active crack to the outside of the rubber matrix through the pass due to the crack reaching the surface of the rubber.

Conclusion

This study focused on the evolution of damage during successive pressure release in a rubber exposed to highpressure hydrogen cycles, with a special interest in the transition from cavitation to cracking. The work was based on in-situ direct optical observations of transparent EPDM samples exposed to 9 or 15 MPa of hydrogen pressure.

Several image processing routines were applied to separately analyse cavities and large crack. Clusters evolution was also considered. These different observations can be summarized as follows. Damages appeared as separated cavities (categorized as "small cavities") and also as clusters for the highest exposure pressure. The average size of these cavities increased with the exposure pressure. Part of these small cavities disappeared during the next cycle but the others still survived and expanded to become relatively large cavities. Regardless, there was always a non-negligible ratio of small cavities at any cycle. Large cavities inflated with cycling and reached a "plateau". A few of these large cavities suddenly restarted to grow and suddenly nucleated "active crack" at one cycle. At the next cycle, the active cracks instantaneously re-opened up to the largest size reached during the previous cycle, propagated slower than decompression stage until reaching a maximum equivalent diameter and then suddenly burst. After burst at the surface of the sample, the driving force of the trapped gas vanished, due to the direct connection to the free surface. Cracks did not propagate anymore and turned to "inactive cracks".

A key observation from this work was that damage evolution cannot be viewed as a cumulative process of cavities which systematically re-appear at any cycle and finally coalesce. More complicated coupled diffuso-mechanical processes seem to govern damage evolution at the local scale.

Fig. 1

 1 Fig. 1 Schematic illustration of a unit of pressure sequence for 9 and 15 MPa cyclic exposure. The units were repeated 5 and 6 times, respectively, for9and15MPatest

Fig. 3

 3 Fig. 3 Overview of the various image processing routines applied for the extraction and the focused analysis of the different damage processes affecting EPDM samples during cyclic exposure to hydrogen at 15 MPa

Fig. 4

 4 Fig. 4 Evolution of cavity fields observed along cycling up to 9 MPa. The cavities circled with red line means cavities appeared at each cycle defined as "surviving cavities" in this paper

Fig. 5 Fig. 6

 56 Fig. 5 Damage morphology observed in samples cycled up to 15 MPa in; a cycle 1 at 360 s after beginning of decompression (T dec = 360), b cycle 1 at T dec = 1000, c cycle 1 at T dec = 4418, d cycle 3 at T dec = 1000, e cycle 3 at T dec = 2500, f cycle 4 at T dec = 1000, g cycle 5 at T dec = 1000 and h cycle 6 at T dec = 1000

Fig. 7 a

 7 Fig. 7 a Evolution of the size distribution of cavities with cycling at 9 MPa and 2.5 MPa/min (at the maximum size of cavities for each cycle). The dark zone in the plot means the region below the lower size limit of camera resolution. b Averaged cavity size evolution through the cycling

  Fig 8a. The large separated cavities (Cav L ) corresponded to the shoulder of the main peak towards high diameter values as shown in Fig 8a, i.e. above 0.6 mm. The average equivalent diameter of each population (respectively D S and D L ) was calculated following Eq.

Fig. 9 a

 9 Fig. 9 a Views of damage at T dec = 1000 s at each cycle, including some separated cavities (a-d) and cracks (A-I) quantitatively analyzed in the work. b The magnified images focusing on cavity a-d

Fig. 10

 10 Fig. 10 The damage fraction of cycle n surviving in cycle n + 1 (F surv )

Fig. 12

 12 Fig. 12 Magnified images focusing on crack B, C and EatT dec = 754, 1194 and 1756 in cycle 4

Acknowledgements Authors are grateful to Pr. S. Nishimura from Kyushu University (Japan) for kindly providing the material of this study. This work was partially Funded by the French Government program "Investissements d'Avenir" (LABEX INTER-ACTIFS, reference ANR-11-LABX-0017-01).


J.-B Le Cam

J R Samaca Martinez

X Balandraud

E Toussaint

J Caillard

Revisiting the mechanisms involved in rubber deformation using experimental thermomechanics

ABSTRACT: Most of phenomena involved in deformation of rubber depend on temperature and have distinguishable thermal and calorimetric signatures. However, since the pioneer investigations being those conducted by Gough and Joule, studies were dedicated more to mechanical response, and the thermal aspects of the deformation of rubber were not really explored experimentally. Revisiting the rubber deformation using experimental thermomechanics should offer therefore new perspectives to better understand damage and deformation mechanisms. In the present study, temperature variations are measured during the mechanical tests by means of infrared thermography. The heat sources produced or absorbed by the material due to deformation processes are deduced from the temperature variations by using the heat diffusion equation. The calorimetric signatures of the most important effects in rubber deformation have been characterized. The results bring information of importance for the understanding and the modeling of physical phenomena involved in the rubber deformation.

Most of these phenomena depend on temperature and have distinguishable thermal and calorimetric signatures. This is the reason why a thermomechanical analysis of rubber deformation should improve our knowledge of the mechanisms involved in rubber deformation, including entropic elasticity, reinforcement by fillers, strain-induced crystallization and stress softening. Among the possibilities available to measure temperature variations during material deformation, infrared thermography appears to be a more and more interesting approach. This technique has been widely applied to metals, polymers and composite materials (see for instance [START_REF] Chrysochoos | Thermal and dissipative effects accompanying luders band propagation[END_REF] and [START_REF] Berthel | Thermoelastic analysis of fatigue dissipation properties of steel sheets[END_REF])), but rarely to elastomeric materials [START_REF] Trabelsi | Crystallization and Melting Processes in Vulcanized Stretched Natural Rubber[END_REF][START_REF] Pottier | Study on the use of motion compensation techniques to determine heat sources. Application to large deformations on cracked rubber specimens[END_REF][START_REF] Toussaint | Combining displacement, strain, temperature and heat source fields to investigate the thermomechanical response of an elastomeric specimen subjected to large deformations[END_REF]Samaca Martinez et al. 2013a, 2013b). This is mainly due to difficulties in extending the measurement to the large deformations undergone by rubber [START_REF] Cam | Thermomechanical Analysis of the Singular Behavior of Rubber: Entropic Elasticity, Reinforcement by Fillers, Strain-Induced Crystallization and the Mullins Effect. Part 2: quantitative calorimetric analysis[END_REF].

The present paper aims first at presenting how infrared thermography and heat source calculation can be carried out in the case of the large deformations undergone by rubbers. Second, the paper deals with the thermomechanical analysis of the physical phenomena involved in rubber deformation. In order to decouple the various phenomena, several chemical compositions of rubber were used.

INTRODUCTION

The first studies dealing with thermomechanical properties of rubber date from the end of the 19th century, the pioneer investigations being those conducted by [START_REF] Gough | A description of a property of Caoutchouc or India Rubber[END_REF] and [START_REF] Joule | On some thermodynamic properties of solids[END_REF]. They showed that rubber behaves mainly as an entropic elastic material, for which the retractive force is purely determined by changes in entropy and the internal energy does not change with deformation at all [START_REF] Holzapfel | Nonlinear Solid Mechanics[END_REF]. Subsequently, studies were dedicated more to the mechanical response, and the thermal aspects of the deformation of rubber were not really explored experimentally. The first studies investigating the mechanical properties of rubber date from the beginning of the 20th century. A major result was the observation by [START_REF] Bouasse | Courbes de traction du caoutchouc vulcanisé[END_REF] of stress softening during the first mechanical cycles. This phenomenon was studied more precisely by [START_REF] Holt | Behavior of rubber under repeated stresses[END_REF] and [START_REF] Mullins | Effect of stretching on the properties of rubber[END_REF] and was then referred to as the "Mullins effect". Since these first studies, numerous other physical phenomena involved in the deformation of rubber have been highlighted, among them viscosity induced by fillers and strain-induced crystallization. Nevertheless, these phenomena are still not clearly understood, and purely mechanical studies have reached their limits in this field.

One of the main issues is therefore to link the physical phenomena, which occur at the microscopic scale, to the corresponding calorimetric variation measured at the macroscopic scale.

HEAT SOURCE CALCULATION FROM THE TEMPERATURE MEASUREMENT

The heat sources produced or absorbed by the material itself were studied within the framework of the thermodynamics of irreversible processes.

In this paper, 'heat source' is used in this paper to designate the heat power density (in W/m 3 ) that is produced or absorbed by the material itself. The temperature fields were measured at the surface of a flat specimen using an Infrared (IR) camera. As the tests performed were assumed to be homogeneous in terms of strain and stress, and as rubbers have a very low thermal diffusivity, the temperature fields were nearly homogeneous. Therefore the 3D heat diffusion equation ( 1) can be reduced to a "0D" formulation as shown in (Samaca Martinez et al. 2013b, 2013c). Assuming that the heat exchanges are proportional to the temperature difference with the outside environment, the heat diffusion equation in the Lagrangian description can be written:

ρ θ τ S ρθ ρ θ EVkh θ S θ V V , ɺ ⎛ ⎝ ⎝ ⎝ ⎞ ⎠ ⎠ ⎠ (1)
with

SdS hT SdS S MC T T + d d d d d (2)
where:

• S h is the heat source produced or absorbed by the material due to stretching; • S TMC is the heat source due to thermomechanical couplings. It contains the thermoelastic coupling term, mainly composed of so-called entropic coupling. It will be demonstrated in the sequel that strain-induced crystallization also contributes to S TMC ; • d 1 is the mechanical dissipation (always positive). It is related to any irreversible mechanical phenomenon, such as viscosity for example; • ρ and C E , Vk are the density and specific heat at constant strain E and internal state variables V k , respectively. • θ is the temperature variation from the initial temperature (here considered in the unstretched state at the beginning of the test). • τ is a characteristic time that accounts for the heat exchange with the outside environment. In practice, it can be experimentally determined by identification from a natural return to ambient temperature. This is done for different values of stretch ratio λ, which is defined as the ratio between the current and initial lengths in the considered direction. For the present experiments, the characteristic time τ was measured as an affine function of the stretch ratio λ.

Lastly, the heat source S h is divided by the product ρC E , Vk , leading to a quantity s in °C/s:

S S C h S EV C k V V == h + ρC C θ τ , θ ⋅ (3)
In the following, the ratio

SC EV C k V V / S S , ρC C
will be named "heat source s" for the sake of simplicity. This equation will be used to calculate the heat source s from the temperature variation θ.

EXPERIMENTAL SETUP

Material and specimen geometry

In order to highlight the calorimetric signature of the numerous phenomena involved in the deformation of rubber, i.e. entropic elasticity, straininduced crystallization and crystallite melting, viscosity and stress softening, non-crystallizable Styrene-Butadiene Rubber (SBR) and crystallizable Natural Rubber (NR) were used, with different filler amounts. SBR was filled with two different amounts of carbon black, 5 and 50 phr (parts per hundred of rubber in weight). They are respectively denoted SBR5 and SBR50 in the following. It should be noted that fillers in SBR5 are used to improve processing; they have no significant effect on material stiffness. NR was unfilled (NR0) and filled with 50 phr of carbon black (NR50). In case of the present NR formulations, the characteristic stretch ratios at which crystallization and crystallite melting occur are denoted by λ c and λ m respectively, and are close to 4 and 3 for NR0 and close to 1.8 and 1.6 for NR50. The specimen geometry was a thin dumbbell-shaped specimen, whose width, height and thickness were equal to 5 mm, 10 mm and 1.4 mm respectively.

Loading conditions

The mechanical loading corresponded to cyclic uniaxial tensile loading. It was applied under prescribed displacement using an INSTRON 5543 testing machine. The signal shape was triangular in order to ensure a constant strain rate during loading and unloading. The loading rate and the nominal strain rate were equal to ±300 mm/min and ±0.5 s -1 , respectively. The test corresponded to series of uniaxial mechanical cycles at four different maximum stretch ratios. The number of cycles was chosen in such a way that the mechanical response was stabilized. It was equal to 3 for SBR5, SBR50, NR0 and 5 for NR50. The following maximum stretch ratios were chosen for each material tested:

For the SBR5, the four maximum stretch ratios were λ 1 = 2, λ 2 = 3, λ 3 = 3.5 andλ 4 = 4; For the SBR50, the four maximum stretch ratios were λ 1 = 2, λ 2 = 3, λ 3 = 4 andλ 4 = 4.5; For the NR0, the four maximum stretch ratios were λ 1 = 2, λ 2 = 5, λ 3 = 6 and λ 4 = 7.5; For the NR50, the four maximum stretch ratios were λ 1 = 1.4, λ 2 = 2, λ 3 = 4 and λ 4 = 6.

The λ i were chosen different from one formulation to another. First, this is due to the fact that the stretches at failure were different (4.2 for SBR5 and 4.8 for SBR50, 7.9 for NR0 and 6.3 for NR50). Second, in NR0 and NR50, one stretch ratio is chosen inferior to that at which crystallization begins (about 4 for NR0 and about 1.6 for NR50).

Full temperature field measurement

Temperature field measurements were performed using a Cedip Jade III-MWIR infrared camera, which features a local plane array of 320 × 240 pixels and detectors with a wavelength range of 3.5-5 µm. Integration time was equal to 1500 µs. The acquisition frequency f a was 147 Hz. The thermal resolution, namely the noise-equivalent temperature difference, was equal to 0.02°C for a temperature range of 5-40 °C. The calibration of the camera detectors was performed with a black body using a Non-Uniformity Correction (NUC) procedure. During the measurements, the external heat sources were reduced by using a black box surrounding the specimen, featuring a small window for the IR camera to be able to observe the gauge zone of the specimen. A suitable movement compensation technique was used to track the small zone at the centre of specimen surface during the test (see for instance [START_REF] Pottier | Study on the use of motion compensation techniques to determine heat sources. Application to large deformations on cracked rubber specimens[END_REF])).

RESULTS AND ANALYSIS

Mechanical and thermal responses

The mechanical response of NR0, NR50, SBR5 and SBR50 is given in Figures 1(a),(b),(c) and (d), respectively. These figures present the nominal stress, defined as the force per initial surface ratio, versus the stretch ratio. Several comments can be made:

1. NR0 and SBR5 did not exhibit significant stress softening, i.e. mechanical cycles had no effect on the mechanical response (see Fig. 1(a) for NR0 and 1(c) for SBR5). This is classically observed in unfilled or barely-filled rubbers; 2. No significant permanent strain was observed in NR0; 3. In SBR5, no significant hysteresis loop was observed, but the residual strain reached 15%. This can be explained by the presence of fillers, even in small amounts; 4. In NR0, no hysteresis loop was observed when the maximum stretch ratio did not significantly exceed that at which crystallization starts (about 4 for this formulation); for higher maximum stretch ratios, a significant hysteresis loop was observed. It is now clearly established that this phenomenon is due to the difference in the kinetics of crystallization and crystallite melting [START_REF] Trabelsi | Crystallization and Melting Processes in Vulcanized Stretched Natural Rubber[END_REF][START_REF] Cam | Volume variation in stretched natural rubber: competition between cavitation and stress-induced crystallization[END_REF]; 5. When fillers were added to the compounds, a hysteresis loop was observed in SBR50 (see Fig. 1(d)). Moreover, the permanent strain increased. The mechanical response of NR50 also exhibited a larger hysteresis loop (occurring for stretch ratios lower than for NR0) and larger residual strains.

The thermal responses obtained during the tests are precisely detailed and discussed in (Le [START_REF] Cam | Thermomechanical Analysis of the Singular Behavior of Rubber: Entropic Elasticity, Reinforcement by Fillers, Strain-Induced Crystallization and the Mullins Effect. Part 2: quantitative calorimetric analysis[END_REF]. The conclusion of this study is that temperature variation is influenced by heat exchanges, which complicates the analysis, and it is not possible to decouple the effects of entropic elasticity, viscosity and stress softening, i.e. to decouple the effects of thermomechanical coupling and mechanical dissipation. This is the reason why heat source is more suitable than temperature to analyze the thermomechanical response of rubber.

Calorimetric responses

Figure 2 presents the calorimetric response obtained for each compound during cyclic tests. These figures, which present the heat sources in °C/s versus time, allow us to present successively the different phenomena that affect the calorimetric responses. They are more precisely investigated in the following sections.

Several results are highlighted by these figures:

1. For all the compounds, an increase in the stretch ratio induces an increase in the heat source. This is a consequence of the entropic elasticity of rubber. This is more precisely investigated with SBR5 in the next section, "Heat sources due to entropic elasticity". It can be noted that, at the lowest stretch ratios, a thermoelastic inversion was observed. This is not addressed in this paper, but the reader can refer to [START_REF] Pottier | Study on the use of motion compensation techniques to determine heat sources. Application to large deformations on cracked rubber specimens[END_REF] for further information on the effect of this phenomenon on the heat sources. In NR, when the stretch ratio at which crystallization begins was exceeded (about 4.2 for NR0 and 1.6 for NR50), a high increase in the heat source was observed. This is investigated in the "Heat sources due to strain-induced crystallization and crystallite melting" section. 2. In filled NR, the stabilized heat source is close to the maximum heat source obtained in unfilled NR at a given maximum stretch ratio, which was not true for SBR. Indeed, at λ = 4, the heat source in SBR50 was approximately 4 times that in SBR5. This is addressed in detail in the section "Heat sources due to filler effect".

In NR0 and SBR5, the calorimetric response did not depend on the number of cycles at a given maximum stretch. When fillers were added to the compounds, the first mechanical cycle at a given maximum stretch ratio led to a higher maximum heat source, but the same minimum heat source as for the other cycles. Moreover, the stabilization of the calorimetric response occurred at the third cycle in SBR50 and at the fifth cycle in NR50. This has a strong similarity with the stress softening phenomenon, and is more precisely investigated in the section "The calorimetric signature of the Mullins effect".

Heat sources due to entropic elasticity

In order to discuss the calorimetric response due to entropic elasticity only, SBR5 was chosen. Indeed, SBR5 did not exhibit any viscosity or stress softening and was not subjected to straininduced crystallization. It can be noted that NR0 could also have been chosen for this discussion, but for maximum stretch ratios inferior to λ c . Figure 3 presents the heat source versus the stretch ratio obtained for the first (stabilized) cycle for each maximum stretch ratio applied: 2, 3, 3.5 and 4 in Figs. 3(a), (b), (c) and (d), respectively. It can be noted that the residual strain induced buckling and that the surface to be observed did not remain flat in that case. Consequently, the temperature measurement is incorrect close to zero strain. This is the reason why in these figures and the following ones there is a grey zone indicating the range of stretch ratios for which the heat source assessments were not valid.

As shown in these figures, heat source variations during loading and unloading were the same in absolute value whatever the maximum stretch ratio applied (in these figures, the light continuous lines correspond to the absolute value of the heat source during unloading). This means that no dissipation occurs during the mechanical cycles, which is in good agreement with the fact that no significant mechanical hysteresis was observed.

Heat sources due to strain-induced crystallization and crystallite melting

The effects of strain-induced crystallization and crystallite melting were studied with NR0. This compound exhibited neither a hysteresis loop due to viscosity nor stress softening. Therefore the singularities in the calorimetric response can only be due to strain-induced crystallization and crystallite melting. As previously explained, four series of three uniaxial mechanical cycles were applied with four increasing maximum stretch ratios, series #1 with λ 1 = 2, series #2 with λ 2 = 5, series #3 with λ 3 = 6 and series #4 with λ 4 = 7.5. When the maximum stretch ratio was inferior to λ c , the heat source evolutions for loading and unloading were symmetrical. This means that no mechanical dissipation occurred. For series with a maximum stretch ratio equal to 5, the heat source evolutions were not symmetrical. During loading, the heat source evolves in a quasi-linear manner until reaching a stretch ratio close to 4. The only change in the microstructure in NR is the strain-induced crystallization. It starts at a stretch ratio of approximately 4. Consequently, the only phenomenon that changes the temperature at stretch ratio inferior to 4 is the thermoelasticity. As the heat source increased quasi-linearly during loading and decreased during unloading, the heat source evolution is mainly due to entropic thermoelasticity. Let us recall that a heat source evolution due to energetic thermoelasticity at constant strain rate does not depend on the stretch level and is negative during loading and positive during unloading. A dissymmetry was observed for higher maximum stretch ratios. This indicates that the heat sources are not caused only by entropic elasticity. Moreover, the area under the curves during loading and unloading was equal, meaning that no heat was produced due to mechanical dissipation. Consequently, the only explanation for the dissymmetry is the occurrence of crystallization during loading, and a difference in the kinetics of crystallization and crystallite melting (the latter during unloading). This is in good agreement with studies reported in the literature [START_REF] Trabelsi | Crystallization and Melting Processes in Vulcanized Stretched Natural Rubber[END_REF]. Concerning the stress-strain curve, a hysteresis loop began to form. It is associated with the crystallization/melting phenomenon, and not with mechanical dissipation. Indeed, if no crystallization occurred, no hysteresis loop was observed for the strain-stress relationship. When the maximum applied stretch ratio increased, loading-unloading dissymmetry increased. From a mechanical point of view, the area of the hysteresis loop also increased. Once again, as the heat produced was equal to the heat absorbed, no mechanical dissipation was detected, while a mechanical hysteresis loop was observed. It should be noted that crystallization induces a strong increase in the heat source, and from stretch ratios equal to 6 during the loading phase, instead of increasing continuously, the heat source decreased. This means that heat continues to be produced (it remains positive), but at a lower rate. This phenomenon could be due to several causes:

1. the fact that this level of stretch ratio would tend to approach crystallinity saturation if any; 2. an increase in the contribution of internal energy;

a less exothermal crystallization process. It should be noted that the hysteresis area in terms of the strain-stress relationship is higher than previously, again with no mechanical dissipation detected.

Heat sources due to filler effect

When fillers are added to the compounds, stress softening is observed. This phenomenon is investigated in the next section. In order to focus only on the effects of fillers on the stationary calorimetric response, the stabilized cycles obtained for SBR50 and NR50 are considered. They are presented in Figures 5 and6, respectively. For NR50, only the last two maximum stretch ratios applied are presented, due to the fact that a significant permanent set (compared to the maximum stretch ratio applied) was observed for the first two maximum stretch ratios applied. This permanent set induced buckling which disturbed the temperature measurement. Consequently, we have not calculated heat sources for these cycles.

As in the previous figure, the global response in terms of the heat sources versus the stretch ratio during the test is indicated, as well as the stabilized calorimetric response at given maximum stretch ratios. Heat source evolutions obtained with SBR50 and NR50 exhibit load-unload dissymmetry. This appears clearly by comparing the heat source obtained during loading with its absolute value during unloading. As SBR is not subjected to strain-induced crystallization and crystallite melting, this shows that mechanical dissipation is produced, even if the calorimetric response is stabilized. This mechanical dissipation is due to viscosity, which is induced (in NR) or amplified (in SBR) by adding fillers. In NR50, the same phenomenon was observed. Moreover, in NR50, adding fillers did not significantly increase the heat source produced. As expected, the increase (decrease) in the heat source produced (absorbed) during loading (unloading) due to the onset of crystallization (melting had finished) occurred at a lower stretch ratio than in NR0.

The calorimetric signature of the Mullins effect

From a mechanical point of view, a major result obtained from the mechanical tests was the observation of the decrease in rubber stiffness during the first mechanical cycles. This phenomenon is referred to as "the Mullins effect" in the literature. Up to now, this effect has mainly been investigated from a mechanical point of view, while its thermal and calorimetric signatures might also contain information of importance about it. Figure 7 presents the heat source versus stretch ratio for series of three first cycles at four increasing maximum stretch ratios in SBR50. First, whatever the maximum stretch ratio, the heat source produced is higher during the first cycle. The following cycles are stabilized. This is the first similarity with the Mullins effect. Second, a decrease in the heat source appears for stretch ratios lower or equal to the maximum stretch previously applied. When the stretch ratio exceeds the maximum stretch ratio previously applied, the calorimetric response greatly increases. The loss in the heat source increases with the increasing maximum stretch ratio. Measurement of the calorimetric response allows us to deduce the mechanical dissipation induced by stress softening. It can be calculated by the difference in mechanical dissipation between the first and the stabilized cycles.

The results are more difficult to analyze in crystallizable filled natural rubber. Even though straininduced crystallization and crystallite melting did not change this tendency, during the first loading the heat source due to crystallization was "hidden" in the total heat source (including mechanical dissipation due to damage). This is precisely detailed in (Samaca Martinez et al. 2014).

CONCLUSION

This study has addressed the calorimetric response associated with deformation processes in different formulations of rubbers: filled and unfilled noncrystallizable SBR and crystallizable NR. These formulations have enabled us to investigate the calorimetric signature of the main physical phenomena involved in rubber deformation: entropic elasticity, strain-induced crystallization, reinforcement by fillers and stress softening. The rubber formulations were chosen in such a way that these phenomena could be studied separately. The main results can be summarized as follows:

-Entropic elasticity leads to heat production (absorption) when the rubber is stretched (relaxed). The heat source-strain relationship is quasi-linear. Load-unload heat source curves are symmetrical.

-The kinetics and therefore the calorimetric signature of strain-induced crystallization differs from that of crystallite melting, so that no symmetry is observed in terms of heat source between loading and unloading. Nevertheless, the areas under the load-unload curves are equal, meaning that crystallization and melting occur without detectable corresponding mechanical dissipation. -Fillers induce viscosity and therefore mechanical dissipation (positive heat source) during both loading and unloading, which increases when the maximum stretch ratio increases. In filled NR, significant heat production is observed at a lower stretch ratio than in unfilled NR, and mechanical dissipation is detected. -Numerous similarities are observed between the mechanical and the calorimetric responses. First, the stabilization of the calorimetric response is reached at the end of the first cycle, similarly to the mechanical response. Second, the higher the maximum stretch ratio applied, the higher the heat source decrease between the first and second cycles. It should be noted that, contrary to the mechanical response, the calorimetric response enables us to detect changes in the characteristic stretch ratios at which strain-induced crystallization starts and crystallite melting is complete.

Figure 1 .

 1 Figure 1. Loading conditions and mechanical responses obtained for the four compounds during cyclic uniaxial tensile tests.

Figure 2 .

 2 Figure 2. Heat source evolution obtained for the four compounds during cyclic uniaxial tensile tests.

Figure 4 (

 4 Figure 4 (left) presents the heat source versus the stretch ratio during the test. The first series of cycles, for stretch ratios superior to λ c , are presented in the diagrams on the right-hand side.

Figure 3 .

 3 Figure 3. Heat source versus stretch ratio obtained for SBR5. The light continuous lines correspond to the absolute value of the heat source during unloading.

Figure 4 .

 4 Figure 4. Heat source versus stretch ratio obtained for NR0. The light continuous lines correspond to the absolute value of the heat source during unloading.

Figure 5 .

 5 Figure 5. Heat source versus stretch ratio obtained for SBR50. The light continuous lines correspond to the absolute value of the heat source during unloading.

Figure 6 .

 6 Figure 6. Heat source versus stretch ratio obtained for NR50. The light continuous lines correspond to the absolute value of the heat source during unloading.

ACKNOWLEDGEMENTS

The authors would like to acknowledge the "Manufacture Française des Pneumatiques Michelin" for supporting this study.


est destinée au dépôt

Recognition of a pathogen avirulence (AVR) protein triggers a set of immune responses grouped under the term Effector-Triggered Immunity (ETI). Pathogens can escape ETI by different mechanisms including complete deletion or down-regulation of the AVR genes, point mutations allowing recognition to be evaded while maintaining the virulence function of the AVR protein, and acquisition of new effectors able to suppress recognition. In some cases, the effector that suppresses ETI can itself be recognized by an R gene. A few examples of such mechanism were recently described in fungi but the underlying mechanisms have remained unexplained.

The Dothideomycete Leptosphaeria maculans, causal agent of oilseed rape stem canker, is one of the fungal pathogens in which suppression of ETI by the presence of an AVR gene has been demonstrated. The AVR gene AvrLm4-7 was reported to hide AvrLm3 from Rlm3-mediated recognition and AvrLm5-9 from Rlm9-mediated recognition (Plissonneau et al., 2016;Ghanbarnia et al., 2018). The way AvrLm4-7 suppresses recognition of AvrLm3 and AvrLm5-9 by their cognate R genes is not known: the presence of AvrLm4-7 does not impede AvrLm3 and AvrLm5-9 expression and yeast twohybrid assays suggested the absence of a physical interaction between AvrLm4-7, AvrLm5-9 and AvrLm3. While AvrLm5-9 and AvrLm3 share a low level of identify, no homology was found with AvrLm4-7. Blondeau et al. (2015) previously determined the crystal structure of AvrLm4-7. We recently determined of the crystal structure of AvrLm5-9. Surprisingly, despite a lack of sequence similarity, AvrLm5-9 shares structural analogies with AvrLm4-7 (a four stranded anti-parallel sheet and two disulfide bridges in common). This suggests that the antagonistic interaction between AvrLm5-9 and AvrLm4-7 could be due to a competition for a same plant target. Structure-informed and pattern-based searches identified a larger number of structural analogues among L. maculans effector candidates (including AvrLm3) and a limited number of structural analogues in other phytopathogenic fungi (including ECP11-1 from Passalora fulva, which shares homologies with AvrLm3; Mesarich et al., 2018). We determined the crystal structure of ECP11-1 and confirmed it shared structural analogies with AvrLm4-7 and AvrLm5-9. Transformants of L. maculans expressing ECP11-1 triggered Rlm3-mediated recognition, and that recognition was suppressed by AvrLm4-7, suggesting either that ECP11-1 has the same function as AvrLm3, or that its structure is sufficiently close to that of AvrLm3 to be recognized by Rlm3.

  Two avirulence genes of Leptosphaeria maculans, displaying suppressive interactions, share a common structural pattern and are part of a larger effector family Noureddine Lazar, Yohann Petit-Houdenot, Nacera Talbi, Inès Li de la Sierra-Gallay, Zelie Emilie, Karine Blondeau, Angela van de Wouw, Bénédicte Ollivier, Carl H. Mesarich, Alexander Idnurm, et al. To cite this version: Noureddine Lazar, Yohann Petit-Houdenot, Nacera Talbi, Inès Li de la Sierra-Gallay, Zelie Emilie, et al.. Two avirulence genes of Leptosphaeria maculans, displaying suppressive interactions, share a common structural pattern and are part of a larger effector family. Effectome meeting, Oct 2019, Lauret, France. hal-04456391 Two avirulence genes of Leptosphaeria maculans, displaying suppressive interactions, share a common structural pattern and are part of a larger effector family N. Lazar 1 , Y. Petit-Houdenot 2 , N. Talbi 2 , I. Li de la Sierra-Gallay 1 , E. Zélie 1 , K. Blondeau 1 , A. van de Wouw 3 , B. Ollivier 2 , C. Mesarich 4 , A. Idnurm 3 , H. van Tilbeurgh 1 and I. Fudal 2 1 Institute for Integrative Biology of the Cell (I2BC), CEA, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91198, Gif-sur-Yvette cedex, France 2 UMR1290 BIOGER, INRA / AgroParisTech/ Université Paris-Saclay, Thiverval-Grignon, France 3 School of BioSciences, University of Melbourne, Melbourne, Australia 4 School of Fundamental Sciences, Massey University, Palmerston North, New Zealand.


domain: phys spi

text:
Levitated magnetic particles as ac magnetic field sensors A. T. M. Anishur Rahman * Electromagnetic waves are widely used in sophisticated defense applications, biomedicine and fundamental science. Their efficient detection determines how we communicate, defend against adversaries, diagnose diseases and perform search and rescue operations. In this article, exploiting the precession of a levitated magnetic particle in ultra-high vacuum, it is shown that weak electromagnetic waves down to femtotesla level can be detected. It is also shown that such a sensor has a large dynamic range over a millitesla, is continuously tunable over many gigahertz and can detect frequencies with a sub-hertz resolution. It is argued that the new magnetometer can be used as a receiver for electromagnetic waves in areas such biomedicine, search and rescue, and defense.

The ability to detect weak electromagnetic (EM) fields has applications in fundamental science [START_REF] Penzias | A Measurement of Excess Antenna Temperature at 4080 Mc/s[END_REF][START_REF] Thornton | A population of fast radio bursts at cosmological distances[END_REF][START_REF] Rahman | Ultrawideband axion search using a faraday haloscope[END_REF], radar [START_REF] Spezio | Electronic warfare systems[END_REF], biomedicine [START_REF] Li | A review on recent advances in doppler radar sensors for noncontact healthcare monitoring[END_REF][START_REF] Zhao | Immune responses to multi-frequencies of 1.5 ghz and 4.3 ghz microwave exposure in rats: Transcriptomic and proteomic analysis[END_REF], search and rescue [START_REF] Thi Phuoc | Review-microwave radar sensing systems for search and rescue purposes[END_REF] and consumer electronics [START_REF] Pedersen | Mobile phone antenna performance 2023[END_REF]. Traditionally, such signals are detected using antennas which once made cannot be changed and have a limited frequency range of operations. In principle, EM fields can be detected using either an electric or a magnetic field sensor. Examples of electric field sensors include Rydberg atoms [START_REF] Fancher | Rydberg atom electric field sensors for communications and sensing[END_REF][START_REF] Jing | Atomic superheterodyne receiver based on microwave-dressed rydberg spectroscopy[END_REF]. Such a sensor has a large frequency range of operation e.g., from MHz to THz [START_REF] Fancher | Rydberg atom electric field sensors for communications and sensing[END_REF][START_REF] Yuan | Quantum sensing of microwave electric fields based on rydberg atoms[END_REF]. Their sensitivities in the free space configuration can be on the order of a µV/m (3 × 10 -14 T) [START_REF] Fancher | Rydberg atom electric field sensors for communications and sensing[END_REF][START_REF] Jing | Atomic superheterodyne receiver based on microwave-dressed rydberg spectroscopy[END_REF][START_REF] Gordon | Weak electric-field detection with sub-1 Hz resolution at radio frequencies using a Rydberg atom-based mixer[END_REF]. Such electric field detectors detect discrete frequencies and require tunable lasers. Existing magnetic field sensors such as atomic vapours [START_REF] Budker | Optical magnetometry[END_REF] and the nitrogen-vacancy centre (NVC) in diamond based sensors [START_REF] Graham | Fibercoupled diamond magnetometry with an unshielded sensitivity of 30 pT/ √ hz[END_REF] are predominantly used as dc field sensors, although some progress has been made towards the detection of EM fields using NVCs in diamond [START_REF] Meinel | Heterodyne sensing of microwaves with a quantum sensor[END_REF]. Such magnetic field sensors have a relatively small dynamic range e.g., ≤ µT . Atomic vapour-based sensors also require magnetic shields. Superconducting quantum interference devices-based magnetometers are excellent magnetic field sensors but require cryogenic temperatures [START_REF] Couëdo | High-Tc superconducting detector for highly-sensitive microwave magnetometry[END_REF].

Levitation in vacuum provides a contactless and a near-frictionless environment. This makes levitated particles very susceptible to external stimuli making them extremely good sensors. For example, using the centerof-mass motions of such particles zeptonewton scale force sensitivity has been achieved [START_REF] Ranjit | Zeptonewton force sensing with nanospheres in an optical lattice[END_REF]. Likewise, exploiting the rotational motion of a levitated particle, torque as small as 4.7 × 0 -28 N m has been measured [START_REF] Ahn | Ultrasensitive torque detection with an optically levitated nanorotor[END_REF]. Among levitated particles, magnetic particles are unique in the sense that they contain an extra degree of freedom e.g., the spin which makes them even more versatile. The coupling between the spin and the other degrees of freedom of a levitated magnetic particle has not been explored yet but is promising for developing new technologies and exploring fundamental physics [START_REF] Rahman | Large spatial schrödinger cat state using a levitated ferrimagnetic nanoparticle[END_REF]. Recently, levitated magnets in cryogenic conditions have been used for detecting dc magnetic fields [START_REF] Ahrens | Levitated ferromagnetic magnetometer with energy resolution well below h[END_REF][START_REF] Jackson Kimball | Precessing ferromagnetic needle magnetometer[END_REF].

In this article, using the precessional motion of a levitated magnetic particle in ultra-high vacuum, it is shown that extremely weak electromagnetic waves of strength femtotesla (3 × 10 -7 V/m) can be detected. Such a magnetometer has a dynamic range over a millitesla, can be continuously tuned for many GHz and detect frequencies with sub-hertz precessions. It is also shown that the direction of arrival of the EM wave can be determined relatively easily. Consider a magnetic sphere of moment µ polarized along its easy magnetization axis is levitated inside a vacuum chamber. A homogeneous dc magnetic field B a is applied along the +z axis which ensures µ aligns with B 0 (Fig. 1) i.e., µ = [0 0 µ s ], where µ s is the saturated magnetic moment of the levitated particle. Consider also that an electromagnetic plane wave whose frequency (ω) and strength we aim to determine is propagating in the +z direction and illuminates the magnetic particle. The interaction between the EM wave and the spins or magnetization initiates a spin precession in the ferromagnetic particle [START_REF] Kittel | Introduction to solid state physics[END_REF][START_REF] Gilbert | A phenomenological theory of damping in ferromagnetic materials[END_REF] giving rise to components of magnetic moment in the xy plane. The dynamics of spin precession can be modeled using the Landau-Lifshitz-Gilbert equation [START_REF] Gilbert | A phenomenological theory of damping in ferromagnetic materials[END_REF]

dµ dt = γµ × B - α µ s (µ × dµ dt ), (1)
where B = b 0 cos ωt x + σb 0 sin ωt ŷ + B 0 ẑ, γ is the gyromagnetic ratio, B 0 = B a + B an with B an being the effective field associated with the magnetocrystalline anisotropy, and α > 0 is the dimensionless Gilbert damping constant. b = b 0 cos ωt x+σb 0 sin ωt ŷ is the magnetic component of the EM field and σ determines its direction of rotation e.g., σ = -1 (+1) corresponds to clockwise (anticlockwise) rotation. b is the field that we aim to detect. Equation ( 1) has no general analytical solution. However, when b ≪ B 0 , one can approximate µ z ≈ µ s and dµz dt ≈ 0. Under these conditions [START_REF] Xi | Precessional dynamics of single-domain magnetic nanoparticles driven by small ac magnetic fields[END_REF][START_REF] Mitsumata | Control of gilbert damping using magnetic metamaterials[END_REF], the steady state solution of ( 1) is given by

µ x = b 0 γm s (γB 0 + σω) 2 + α 2 ω 2 cos (ωt -β), (2)
where β is the phase difference between b and the precession of spins or magnetization and is given by β = tan -1 [ αω γB0+σω ]. A similar expression for µ y exists and can be found in the supplementary material. Importantly, when σ = -1, µ x has a Lorentzian profile and has a full-width half-maximum linewidth of ≈ 12α 2 γ 2 B 2 0 . µ x reaches its maximum when ω = γB 0 . This is known as the ferromagnetic resonance (FMR). On FMR the amplitude of µ x is b x µ s /αB 0 . In contrast, when σ = +1, the rotating EM field opposes the precession of the magnetization [START_REF] Denisov | Magnetization of nanoparticle systems in a rotating magnetic field[END_REF] and hence the induced magnetic moment in the x -y plane is several orders of magnitude smaller compared to the FMR case since, in our case, ω is in gigahertz and for most material [START_REF] Chang | Nanometer-thick yttrium iron garnet films with extremely low damping[END_REF][START_REF] Maier-Flaig | Temperature-dependent magnetic damping of yttrium iron garnet spheres[END_REF][START_REF] Schreiber | Gilbert damping and g factor in Fe˙xCo1˙x alloy films[END_REF][START_REF] Barati | Gilbert damping in binary magnetic multilayers[END_REF] 0 < α ≪ 1. Note that when B a is applied along the -z axis, a counterclockwise rotating (σ = +1) EM field can excite FMR [START_REF] Denisov | Magnetization of nanoparticle systems in a rotating magnetic field[END_REF].

In a ferromagnetic material spins and thus magnetization are connected to the crystal lattice via magnetocrystalline anisotropy [START_REF] Kittel | Introduction to solid state physics[END_REF]. This provides a link between the internal (spin) and the mechanical degrees of freedom of a magnetic object. For a levitated magnet, this connectivity means that when the magnetization starts to precess, the lattice and thus the levitated particle attempt to follow it [START_REF] Jackson Kimball | Precessing ferromagnetic needle magnetometer[END_REF][START_REF] Xi | Precessional dynamics of single-domain magnetic nanoparticles driven by small ac magnetic fields[END_REF][START_REF] Keshtgar | Magnetomechanical coupling and ferromagnetic resonance in magnetic nanoparticles[END_REF][START_REF] Fadeev | Ferromagnetic gyroscopes for tests of fundamental physics[END_REF]. Precessions of levitated magnets can be detected using the optical interferometric scheme used in levitated optomechanics [START_REF] Ahn | Ultrasensitive torque detection with an optically levitated nanorotor[END_REF][START_REF] Rashid | Precession motion in levitated optomechanics[END_REF][START_REF] Rahman | An analytical model for the detection of levitated nanoparticles in optomechanics[END_REF][START_REF] Reimann | Ghz rotation of an optically trapped nanoparticle in vacuum[END_REF]. For a levitated magnet to precess mechanically, however, it must overcome the resistance that it encounters from the residual gas molecules inside the vacuum chamber. The resistive torque [START_REF] Kuhn | Full rotational control of levitated silicon nanorods[END_REF] that an object faces at a precession frequency Ω p is IΩ p Γ g , where I is the moment of inertia, and Γ g is the damping rate due to gas molecules. In contrast, the driving torque due to the incoming EM field is given by the right hand side of (1) normalized by γ [START_REF] Kittel | Introduction to solid state physics[END_REF]. At equilibrium when the driving and the resistive torques are equal, the precession frequency is

Ω p = 1 IΓ g b 0 µ s ω (γB 0 + σω) 2 + α 2 ω 2 , (3)
Equation ( 3) alone cannot uniquely determine the frequency of the unknown EM wave. This is because when the driving torque is less than the resistive torque, the magnet would precess at a lower rate and hence taking the mechanical precession rate as the frequency of the EM wave would be inaccurate. This, however, can be avoided by measuring the precession frequency as a function of the damping rate or the gas pressure P (Γ g ∝ P ). As P decreases, Ω p increases. But, when Ω p reaches the frequency of the unknown EM field, Ω p remains constant as Γ g decreases. This is a consequence of the driven nature of mechanical motions which prefer to synchronize with the frequency of the driving fields [START_REF] Xi | Precessional dynamics of single-domain magnetic nanoparticles driven by small ac magnetic fields[END_REF][START_REF] Keshtgar | Magnetomechanical coupling and ferromagnetic resonance in magnetic nanoparticles[END_REF] (see [START_REF] Thornton | A population of fast radio bursts at cosmological distances[END_REF]). At the critical damping rate Γ c g where Ω p = ω, the mechanical precession frequency directly determines the frequency of the EM wave while the strength of the unknown EM field is

b 0 = IΓ c g µ s (γB 0 + σω) 2 + α 2 ω 2 . (4)
From ( 4) it is clear that our detector is most sensitive at the ferromagnetic resonance. The occurrence of FMR can be ensured by checking that the mechanical precession frequency (can be separately measured) is equal to γB 0 . Otherwise, B a can be adjusted such that it matches the mechanical precession frequency. Moreover, since on FMR the driving torque is maximum, the mechanical precession at γB 0 is expected to happen at a higher Γ c g compared to non-resonant cases. When the FMR condition is satisfied, the minimum detectable field is

b 0 = 2αργB 0 Γ c g r 2 5M , (5)
where we have substituted I = 2ρvr 2 /5 and µ s = vM with r, ρ, v and M being the radius, the mass density, the volume and the saturation magnetization of the levitated magnet. Parameters such as ρ and M in (5) can be found from the relevant bulk material while α, Γ c g and r can be measured experimentally in situ [START_REF] Maier-Flaig | Temperature-dependent magnetic damping of yttrium iron garnet spheres[END_REF][START_REF] Rahman | Burning and graphitization of optically levitated nanodiamonds in vacuum[END_REF][START_REF] Rahman | Measurement of the motional heating of a levitated nanoparticle by thermal light[END_REF].

Let us consider numerical examples. Any magnetic materials can be used as long as they can be levitated. Levitation can be carried out either using a Paul trap [START_REF] Penny | Sympathetic cooling and squeezing of two colevitated nanoparticles[END_REF][START_REF] Delord | Electron spin resonance from NV centers in diamonds levitating in an ion trap[END_REF][START_REF] Bykov | 3d sympathetic cooling and detection of levitated nanoparticles[END_REF] or optical tweezers [START_REF] Ranjit | Zeptonewton force sensing with nanospheres in an optical lattice[END_REF][START_REF] Ahn | Ultrasensitive torque detection with an optically levitated nanorotor[END_REF][START_REF] Rahman | Measurement of the motional heating of a levitated nanoparticle by thermal light[END_REF][START_REF] Vovrosh | Parametric feedback cooling of levitated optomechanics in a parabolic mirror trap[END_REF][START_REF] Rahman | Optical levitation using broadband light[END_REF]. Figure 2a shows the minimum detectable field as a function of the residual gas pressure when a yttrium iron garnet (YIG, a weak magnet) or a cobalt (a strong magnet) sphere (r = 100 nm) is used as a sensor. The frequency of the EM field is assumed to be ω/2π = 5 GHz and is resonant with the FMR of the levitated magnet. We also used Γ g ≈ 4P ρrvg with v g being the velocity of gas molecules [START_REF] Beresnev | Motion of a spherical particle in a rarefied gas. part 2. drag and thermal polarization[END_REF]. At an experimentally viable pressure of P = 10 -9 mbar (Γ g = 2.41 × 10 -6 rad.), the minimum detectable field is ≈ 2 × 10 -15 T with cobalt and ≈ 1 × 10 -15 T with YIG. When converted to electric fields (cb 0 ), these are equivalent to 6 × 10 7 V/m and 3 × 10 7 V/m, respectively, where c is the speed of light. It is evident that despite cobalt being a strong magnet, YIG is preferable. This is because YIG has a significantly lower Gilbert damping constant [START_REF] Chang | Nanometer-thick yttrium iron garnet films with extremely low damping[END_REF][START_REF] Maier-Flaig | Temperature-dependent magnetic damping of yttrium iron garnet spheres[END_REF] compared to cobalt [START_REF] Schreiber | Gilbert damping and g factor in Fe˙xCo1˙x alloy films[END_REF] resulting in a lower α/M ratio which provides better sensitivities. At a higher vacuum, which is within experimental reach, even better sensitivities seem feasible (Fig. 2a). Assuming that the target sensitivity is 10 -15 T, Fig. 2b shows the minimum required residual gas pressure as a function of ω when a YIG sphere is used as a sensor. Evidently, EM fields of lower frequencies can be detected at higher gas damping. This is because, at a lower precession frequency, the resistive torque is lower (∝ Ω p ). It is also evident that smaller levitated magnets are better sensors since such objects have significantly lower moments of inertia (∝ r 5 ) requiring smaller driving torques for initiating precessions.

An essential attribute of an EM sensor is to detect signals of different frequencies. Here, this can be accom-plished in a resonant manner by adjusting B a , see [START_REF] Thornton | A population of fast radio bursts at cosmological distances[END_REF]. As B a changes, the frequency of the ferromagnetic resonance changes along with it. B a can be delivered using an electromagnet allowing the detection of arbitrary frequencies within the range of operation of our magnetometer. The highest frequency that our sensor can detect is limited by the maximum tensile stress (≈ ρΩ 2 p r 2 p ), arising from the mechanical precession, that a levitated magnet can withstand [START_REF] Schuck | Ultrafast rotation of magnetically levitated macroscopic steel spheres[END_REF][START_REF] Ahn | Optically levitated nanodumbbell torsion balance and ghz nanomechanical rotor[END_REF], where r p depends on θ. Mechanical rotations over 5 GHz have been demonstrated [START_REF] Ahn | Ultrasensitive torque detection with an optically levitated nanorotor[END_REF] and tens of GHz have been predicted [START_REF] Ahn | Optically levitated nanodumbbell torsion balance and ghz nanomechanical rotor[END_REF]. The lowest frequency that a precessing magnetometer can sense is determined by the saturation magnetization and/or the magnetocrystalline anisotropy of a levitated magnet. In this context, weak magnets such as YIG are preferable since their saturation magnetizations are low. Using levitated YIG spheres as sensors, frequencies in the MHz range can be detected [START_REF] Lee | Ferromagnetic resonance of a YIG film in the low frequency regime[END_REF][START_REF] Bryant | Subsaturation ferromagnetic resonance in yttrium iron garnet spheres[END_REF].

Magnetometers capable of detecting frequencies within a narrow band are highly sought after in spectroscopy [START_REF] Fancher | Rydberg atom electric field sensors for communications and sensing[END_REF]. At a given B a , our magnetometer is sensitive to frequencies within the ferromagnetic resonance linewidth which is dictated by the Gilber damping constant e.g., ≈ 12α 2 γ 2 B 2 0 . For a high selectivity, materials with low α's are essential. In this regard, YIG is an excellent candidate which is known for its extremely narrow FMR linewidth [START_REF] Chang | Nanometer-thick yttrium iron garnet films with extremely low damping[END_REF][START_REF] Maier-Flaig | Temperature-dependent magnetic damping of yttrium iron garnet spheres[END_REF][START_REF] Serga | YIG magnonics[END_REF]. Using YIG spheres as sensors frequency selectivity of a few MHz can be achieved [START_REF] Fancher | Rydberg atom electric field sensors for communications and sensing[END_REF]. Importantly, within this frequency band, our magnetometer can resolve frequencies with a sub-hertz resolution. This is a result of highly accurate frequency measurements of driven oscillators. For example, rotation frequencies of driven levitated objects have been measured with accuracies better than a mHz [START_REF] Jin | Quantum control and fast rotation of levitated diamonds in high vacuum[END_REF][START_REF] Kuhn | Optically driven ultrastable nanomechanical rotor[END_REF].

The ability to detect fields of different strengths is known as the dynamic range of a sensor. In our case, it is determined by the angle of precession assuming that a levitated magnet is already precessing at ω. As the strength of the EM field increases, a levitated magnet continues to precess at ω. However, its angle of precession (Fig. 1) increases. This remains true as long as the strength of the unknown magnetic field is ≪ B a . If the unknown field becomes comparable to B a , the approximation made in deriving (2) e.g., b 0 ≪ B a breaks down. Interestingly, as the precession angle increases with the increasing b 0 , the detection of precession becomes easier implying an enhanced sensitivity. This is in contrast with other magnetometers which become less sensitive as the strength of the field increases [START_REF] Budker | Optical magnetometry[END_REF][START_REF] Graham | Fibercoupled diamond magnetometry with an unshielded sensitivity of 30 pT/ √ hz[END_REF][START_REF] Couëdo | High-Tc superconducting detector for highly-sensitive microwave magnetometry[END_REF]. Generally, B a is in tens of millitesla implying a large dynamic range e.g., femtotesla to millitesla for a levitated magnet-based magnetometer.

The direction of arrival (DA) of an EM wave is important in many areas of engineering and physics including in defense [START_REF] Robinson | Determining the angle-of-arrival of a radio-frequency source with a Rydberg atom-based sensor[END_REF][START_REF] Huang | Machine-learning-enabled metasurface for direction of arrival estimation[END_REF] and astrophysics [START_REF] Thornton | A population of fast radio bursts at cosmological distances[END_REF]. In our case, the direction of arrival k can be found from the direc-tion of B a . In deriving (4) we assumed that b is in the x -y plane and hence, in this simple case, the direction of arrival is along the z axis. To further differentiate between the arrival along the -z and the +z directions, the sensor, due to its small size (equivalent to a small vacuum chamber), can be temporarily blocked using an EM absorber [START_REF] Elmahaishi | A review on electromagnetic microwave absorption properties: their materials and performance[END_REF] from one of the two sides. In the event the EM arrives from the blocked side, the levitated magnet will stop precessing thus determining the arrival direction. For an arbitrary arrival, the direction of B a can be adjusted such that b becomes perpendicular to B 0 . This can be ensured by checking that the maximum precession frequency (Ω p = ω, see above) is achieved at the highest possible Γ c g . This is because when k is not parallel to B a , the magnetic component of the EM wave is not orthogonal to B a . As a result, the torque exerted by b is not maximal (see Eq. ( 3)) requiring a reduced Γ c g for the magnet to precess at ω. In the extreme case, when k ⊥ B a , there is no torque and the magnet cannot precess.

In conclusion, we have theoretically shown that a levitated magnet in high vacuum is capable of detecting electromagnetic fields with a femtotesla sensitivity. When converted to an electric field, this is better than a 3 × 10 -7 V/m. Benefits of the new magnetometer include a large dynamics range covering from femtotesla to millitesla, highly accurate frequency measurements with a resolution better than a mHz, a large frequency range of operation from MHz to tens of GHz, and a high selectivity. With its high sensitivity, the new magnetometer can potentially be used as a radar receiver. Importantly, due to its high sensitivity and the ability to be configured to sense different frequencies by merely changing the externally applied magnetic field, it can be useful in fields including biomedicine [START_REF] Li | A review on recent advances in doppler radar sensors for noncontact healthcare monitoring[END_REF][START_REF] Zhao | Immune responses to multi-frequencies of 1.5 ghz and 4.3 ghz microwave exposure in rats: Transcriptomic and proteomic analysis[END_REF], search and rescue [START_REF] Thi Phuoc | Review-microwave radar sensing systems for search and rescue purposes[END_REF] and defense [START_REF] Thi Phuoc | Review-microwave radar sensing systems for search and rescue purposes[END_REF] where the ability to sense at different frequencies is crucial.

FIG. 1 .

 1 FIG.1. Precession of a levitated magnet when exposed to electromagnetic waves. The light blue background represents a homogeneous dc magnetic field Ba. A circularly polarized electromagnetic (EM) wave propagating along the z axis initiates spin precession which subsequently induces a mechanical precession. The mechanical angle of precession is denoted by θp. The wavevector of the electromagnetic wave is represented by k. The magnetic field associated with the EM wave is in the x -y plane.

FIG. 2

 2 FIG. 2. a) The minimum detectable magnetic field as a function of residual gas pressure when the precession frequency or the frequency of the unknown EM wave is 5 GHz with a YIG (M = 1.4 × 10 5 A/m and α = 5.58 × 10 -5 ) or a cobalt (M = 1.4 × 10 6 A/m and α = 0.01) sphere (r = 100 nm) as a sensor. b) The minimum required gas pressure for sensing EM fields of different frequencies with b0 = 10 -15 T when a YIG sphere (100 nm) is used.