Marie-Ingrid Richard
email: mrichard@esrf.fr

Stéphane Labat

Maxime Dupraz

Jérôme Carnis

Lu Gao

Michaël Texier

Ni Li

Longfei Wu

Jan P Hofmann

Mor Levi

Steven J Leake

Sergey Lazarev

Michael Sprung

@ Emiel

J M Hensen

Eugen Rabkin

Olivier Thomas

Anomalous Glide Plane in Platinum Nano-and Microcrystals

Keywords: glide planes, dislocations, Platinum, Bragg coherent diffraction imaging, defects

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In face centred cubic (fcc) metals, plasticity is generally thought to occur by the motion of dislocations in well defined planes and directions. These groupings of planes and directions are called slip systems, and are the conventional way of describing plastic deformation caused by dislocation slip. In fcc metals, slip generally occurs on {111} planes in <110> directions. [START_REF] Barrett | Structure of Metals, Third Edition: Crystallographic Methods, Principles and Data[END_REF][START_REF] Hirth | Theory of dislocations[END_REF] The perfect Burgers vector, b, is a 2 <110> (a being the lattice unit), which is a close packed direction, and represents the shortest repeat length in the crystal. Other uncommon slip planes are sometimes encountered in fcc crystals, [START_REF] Hirth | Theory of dislocations[END_REF] for example: {100}, {110}, {121} or {212}. They are either observed at elevated temperatures ({100}, 3 {110} 4 or {212} [START_REF] Bhatr | Observations of {212} h100i slip system in face centered cubic austenitic stainless steel[END_REF] ) or on deformed specimens ({100}, 6,7 {110} 8 or {121} [START_REF] Lee | Anomalous slip in an FCC system[END_REF] ). These non-octahedral slip may result through cross-slip in the <110> direction. [START_REF] Barrett | Structure of Metals, Third Edition: Crystallographic Methods, Principles and Data[END_REF] As defects have a critical influence on the properties and performances of materials, it is critical to establish the fundamental slip planes for crystal plasticity models. Most of the experimental observations on slip planes have been performed on bulk crystals. The energy and mobility of defects in single nano-objects may be different compared to their bulk counterparts due to surface effects. [START_REF] Kinney | Computer simulation of surface effects on defect production in tungsten[END_REF] At the nanoscale, the number of defects (e.g., dislocations) in a crystal is reduced and their behavior cannot be determined by a statistical approach. Crystal defects are not always detrimental for performances as they can instead give rise to specific functionalities, such as improving adsorption affinity or enhancing the catalytic activity. [START_REF] Behrens | The Active Site of Methanol Synthesis over Cu/ZnO/Al2O3 Industrial Catalysts[END_REF][START_REF] Ulvestad | Topological defect dynamics in operando battery nanoparticles[END_REF][START_REF] Nilsson Pingel | Influence of atomic site-specific strain on catalytic activity of supported nanoparticles[END_REF] But quantifying the lattice distortions associated with defects as well as imaging the structure of defects in three-dimensions (3D) are challenging at the nanoscale. Accurate identification of the dislocation line, slip planes as well as the knowledge of the strain at the scale of defects are essential to the understanding of fundamental defect behaviors and the underlying physics.

To investigate defects and associated slip planes in small objects, we employed a noninvasive technique called Bragg Coherent Diffraction Imaging (BCDI). [START_REF] Robinson | Coherent X-ray diffraction imaging of strain at the nanoscale[END_REF] BCDI is a lens-less imaging technique based on inverse microscopy, which employs highly accurate digital methods to replace the x-ray imaging lenses. This technique combines 3D coherent diffraction measurements of Bragg peaks and phase retrieval algorithms to obtain a spatial reconstruction of individual nanoscale objects. The technique is sensitive to atomic displacement and strain. The retrieved object in direct space is a complex field, whose modulus encodes the electron density ⇢(r) of the object, and its phase, the displacement field u(r) projected onto the diffraction vector. The displacement field information is crucial in identifying the character of single dislocations. Thanks to the high brilliance, coherence and flux of synchrotron x-ray beams, BCDI has shown its capability to image single defects in nano-crystals [START_REF] Ulvestad | Topological defect dynamics in operando battery nanoparticles[END_REF][START_REF] Ulvestad | Single Particle Nanomechanics in Operando Batterie via Lensless Strain Mapping[END_REF][START_REF] Labat | Inversion Domain Boundaries in GaN Wires Revealed by Coherent Bragg Imaging[END_REF][START_REF] Ulvestad | Avalanching strain dynamics during the hydriding phase transformation in individual palladium nanoparticles[END_REF][START_REF] Ulvestad | 3D Imaging of Twin Domain Defects in Gold Nanoparticles[END_REF][START_REF] Yau | Bragg coherent diffractive imaging of single-grain defect dynamics in polycrystalline films[END_REF] together with an unmatched sensitivity (picometer resolution) to displacement fields. [START_REF] Labat | Inversion Domain Boundaries in GaN Wires Revealed by Coherent Bragg Imaging[END_REF] BCDI can be used to understand the impact of buried single defects on properties under in situ conditions. [START_REF] Ulvestad | Topological defect dynamics in operando battery nanoparticles[END_REF][START_REF] Yau | Bragg coherent diffractive imaging of single-grain defect dynamics in polycrystalline films[END_REF] Here, we apply BCDI to image and characterise in 3D the structure of defects in single faceted platinum (Pt) crystals. A {110}-type glide plane has been observed in two Pt crystals prepared via a chemical and physical process. This paper presents evidence for this anomalous non-octahedral glide plane in fcc sub-micrometer crystals. The {110} slip plane is associated with preferential nucleation of defects at the surface of the crystals.

Non-invasive structural characterisation of defects allows understanding defect behaviors in confined crystals and will facilitate defect engineering to enhance material performances.

Results

Two types of Pt particles have been prepared via a chemical (electrochemical deposition and faceting) and a physical (dewetting) process. For both samples, a high proportion of perfect (defect free) particles are observed. In this paper, we will concentrate on defective particles. The two crystals measured here are a tetrahexahedral (THH) Pt particle grown on a glassy carbon substrate 20 by electrodeposition and faceting and a dewetted Pt particle epitaxially grown on a sapphire substrate (see Experimental Section). The three-dimensional BCDI measurements were performed by measuring the scattered intensity in the vicinity of the specular 111 Pt Bragg reflection at the ID01 beamline of the European Synchrotron (ESRF) and at the P10 beamline of PETRA III synchrotron. Experimental details are given in the Experimental Section. The reconstructed density and phase are displayed with a voxel size of 3 nm 3 (3.5 nm 3 ) for the THH Pt particle (dewetted Pt particle). The resolution of the reconstruction is estimated to 18.3 nm (THH Pt particle) and 27 nm (dewetted Pt particle) using the Phase-Retrieval Transfer Function (PRTF), [START_REF] Chapman | High-resolution ab initio three-dimensional x-ray diffraction microscopy[END_REF][START_REF] Cherukara | Anisotropic nano-scale resolution in 3D Bragg coherent diffraction imaging[END_REF] which measures how well the retrieved Fourier amplitude matches the square root of the measured diffraction intensity. Details about the phase retrieval process are described in the Experimental Section.

The phase map obtained from the inversion of the BCDI measurement at the 111 Bragg reflection encodes the displacement field, u 111 , along the [111] direction, which is normal to the substrate surface (either glassy carbon for the THH particle or sapphire for the dewetted particle).

Figure 1 shows the reconstructed phase (proportional to u 111 ) and strain (✏ 111 ) fields of the THH crystal. The isosurface of the reconstructed Pt particle is drawn at 35% of the maximum electron density for the (a) top and (b) bottom views. The particle is strongly faceted with {201} planes, which correspond to high-index facets of the THH shape. [START_REF] Tian | Synthesis of tetrahexahedral platinum nanocrystals with high-index facets and high electro-oxidation activity[END_REF] Discontinuities appear in the reconstructed phase field as indicated by black arrows in Figures 1(a) and(c).

They are the signature of defects within the crystallite. They are observed as a line in the phase plot, which can be described as a sharp phase boundary ended by 2⇡ phase vortices (region where the phase varies from -⇡ to ⇡). The two phase vortices are the signature of two dislocation cores. The phase jump in the reconstruction links the two dislocations (see Figures 1(a)-(c)), suggesting that they consist of two ends of the same dislocation line.

The strain has been derived from the displacement field: ✏ zz = @u 111 @z and using the procedure described in Ref. [START_REF] Hofmann | 3D-resolved lattice strain tensor measurements of specific crystallographic defects extracted from a bulk sample[END_REF] (see Figure 1(d)). Note that the out-of-plane ✏ zz strain is different around the two segments (labelled as A and B) of the dislocation, implying that they are of different types. Figure 2 shows the reconstructed phase and modulus (i.e. electron density) of the dewetted Pt crystal. The isosurface is drawn at 35% of the maximum electron density. As well, a discontinuity appears in the reconstructed phase field (see Figure 2(a)), which is the signature of a dislocation. Note that for the dewetted Pt particle, the dislocations appear pinned at the free surface as well as at the Pt/substrate interface (see Figure 2(b)).

Different methods have been applied to determine the dislocation lines after phase retrieval, like derivative-based method [START_REF] Ulvestad | rapid identification of dislocation lines in coherent diffractive imaging via a min-max optimization formulation[END_REF] for dislocation core and/or guided algorithms [START_REF] Clark | Three-dimensional imaging of dislocation propagation during crystal growth and dissolution[END_REF][START_REF] Ulvestad | Identifying Defects with Guided Algorithms in Bragg Coherent Diffractive Imaging[END_REF] (min-max optimisation formulation). Here, the 3D dislocation line orientation is determined using the spatial amplitude variation of the retrieved density/modulus (see Figures 2(b) and 3). Simulations of defects in BCDI have shown low amplitude along the dislocation line. [START_REF] Clark | Three-dimensional imaging of dislocation propagation during crystal growth and dissolution[END_REF] Large lattice strains near the dislocation core lead to scattered intensity beyond the numerical aperture of the detector, resulting in a loss of amplitude at the dislocation core. In Figures 2(b) and 3, line contours are observed, which appear as regions or pipes of lower density. The spatial modulus variation is the signature of dislocations, which is made of two segments (labelled A and B) for the Pt THH crystal (see Figures 1 and3) and also of two segments (labelled A' and B') for the Pt dewetted crystal (see Figure 2 For the Pt dewetted particle, the two ([101] and [2 12]) segments lie in a (10 1) plane (see Figure 2(c)). Note that the spatial resolution of the BCDI is not sufficiently high to observe a possible dissociation of the dislocations.

Discussion

The {111}-type plane is the major operative slip system in fcc structures, as the {111} planes are the most densely packed planes in fcc metals, like Pt. [START_REF] Hirth | Theory of dislocations[END_REF] For the Pt THH particle, the A segment is along the [111] direction and, thus, cannot lie in a {111} plane. Also, for the Pt dewetted particle, the A' segment is along the [2 12] direction and, thus, cannot lie in a {111} plane. The configuration of these two defects is unexpected in fcc metals.

Note that here, we concentrated on a single crystal reflection, g=111. This provides only one component of the lattice displacement field. A dislocation gives rise to diffraction contrast if g•b is non zero. [START_REF] Steeds | Dislocation arrangement in copper single crystals as a function of strain[END_REF][START_REF] Williams | Transmission Electron Microscopy: A Textbook for Materials Science[END_REF] This means that x-rays are sensitive to a dislocation if the displacement field generated by the dislocation has a non-zero projection onto the particular Bragg peak. Possible Burgers vectors in face centred cubic materials (like, Pt) are a 2 h110i or h100i. In our case, as g=111, dislocation lines with a Burgers vector of type [1 10], [ 10 1] or [01 1], for instance, can not be measured as they are invisible for this condition (g•b = 0). To get the full picture of the defects as well as the full lattice displacement and strain fields, measurements of at least three (independent) non-colinear reflections are needed. [START_REF] Labat | Inversion Domain Boundaries in GaN Wires Revealed by Coherent Bragg Imaging[END_REF][START_REF] Hofmann | 3D lattice distortions and defect structures in ion-implanted nano-crystals[END_REF] Nevertheless, as in the two particles, one segment of the measured dislocations is along a direction (either [111] or [ 2 12]) that cannot lie in a {111} plane; with only one reflection, we can assert that the configuration of the defects in the two particles is unusual in fcc.

For the Pt dewetted particle, the glide plane of the two (A' and B') segments can be determined straightforwardly as the segments lie in a (10 1) plane (see Figure 2). This implies that the glide plane of the defects corresponds to an anomalous {110}-type plane. For the THH particle, a single plane can not contain the two segments as well as its junction, it is then necessary to determine the Burgers vector and the type (screw, edge or mixed) of the two main segments of the dislocation to estimate their possible glide planes. As we know the direction of the dislocation line of the A segment ( t = [111]) and that in fcc crystals, the Burgers vector is either a 2 h110i or h100i, the only possible glide plane for the A segment, that fulfills b ⌦ t, is of {110}-type. Interestingly, for both the dewetted and THH Pt particles, an anomalous {110}-type glide plane is observed.

To determine the type (screw, edge or mixed) of the two (A and B) main segments of the dislocation in the Pt THH particle, we investigated the reconstructed phase and displacement field, u 111 , around the two segments. It gives a continuum description of how the atoms are displaced from their equilibrium positions and gives insight of the nature of the dislocation loop. Figure 4(a) shows the reconstructed phase along the [111] direction as a function of azimuthal angle, ✓, for segments A and B. For segment A, the phase, proportional to u 111 , varies quasi-linearly around the segment (shift of 2⇡ over 360 ). For segment B, periodic modulations of the phase are observed as a function of ✓. If we consider a screw dislocation along the z-direction in a cubic crystal and in an isotropic infinite media (the Zener or elastic ratio for Pt being equal to 1.6, this implies that Pt is close to an isotropic elastic media), then the displacement field is given by: 2

u z (r, ✓) = b ✓ 2⇡ , ( 1
)
where b is the modulus of the Burgers vector and ✓ measures the angle around the dislocation core. As the phase corresponds to g 111 • u(r) (where g 111 is the scattering vector at the 111 Pt Bragg reflection), a linear variation of the displacement field leads to a linear variation of the phase. The phase around the A segment varies quasi-linearly. This demonstrates that the dislocation should have a screw component along the [111] direction. An edge component of the A segment may exist but can not be detected. As only the g=111 reflection has been measured, the edge component is invisible for this condition. The A segment is then the screw component of a mixed dislocation with a Burgers vector of type b = a 2 h110i or ah001i (the two possible Burgers vectors in fcc). For these two Burgers vectors, the phase, (✓), will vary linearly, as experimentally observed (see Figure 4 ), respectively. Interestingly, this defect configuration reproduces quite well the observed variations of the phase around the two segments (Figure 4(a)). Thus, the A segment is lying in an anomalous (10 1) glide plane, while it is highly probable that the B segment is lying in a (111)-type glide plane. Here, a single reflection allows us to unambiguously characterise the two segments (type and Burgers vector) by measuring and simulating the displacement field around them.

The unusual glide plane observed for the A segment in the THH Pt particle can be (1) a pure h110i {110} slip system, i.e. induced by emission of dislocations from sources lying in {110} planes, (2) the consequence of cross-slip from h110i {111} to h110i {110} or (3) a zig-zag configuration, i.e. that the observed line lies in {111} planes at a small scale (at the limit of our resolution) with a mean direction parallel to the trace of {110} planes. The latter case may be ruled out, as it has been observed that zig-zag dislocations exhibit crossslip events separated by distance generally larger than our spatial resolution (10 nm). [START_REF] Pichaud | Cross-slip at very low stresses in highly perfect Cu crystals[END_REF][START_REF] Pichaud | Further topographic observations of zigzag stress-induced dislocations in copper crystals[END_REF][START_REF] Pichaud | Slip in copper slightly deformed at room temperature[END_REF] The second hypothesis can also be ruled out, as it is energetically unfavorable to cross-slip from {111} planes into {110} planes. Note that it has been observed that high temperature and high stacking-fault energy like in Pt may reduce the energy necessary for the crossing from {111} to {110}. [START_REF] Pichaud | Slip in copper slightly deformed at room temperature[END_REF] In our case, the hypothesis of a pure <110> {110} glide is the most probable. The non-octahedral {110} slip planes are unusual in fcc metals, but they have been observed in particular conditions: (1) compression tests at high temperature, [START_REF] Hazif | Cross-Slip on {110} Planes in Aluminum Single Crystals Compressed along h100i Axis[END_REF] (2) deformation experiments on Al single crystals, [START_REF] Martin | The importance of noncompact slip in some close-packed metallic structures[END_REF] (3) high speed deformation experiment on Al 8 and (4) in slightly deformed (±1.2 MPa) Cu single crystals at room temperature. [START_REF] Pichaud | Slip in copper slightly deformed at room temperature[END_REF] In the latter case, the high perfection of the crystal and the activation of surface sources may explain the possibility of pure {110} slip plane. Indeed, in this case, the pure {110} slip plane was associated with preferential surface nucleation as influenced by image forces.

Image forces become significant when a dislocation is close to a surface. A pure {110} slip plane can thus be encountered in nanostructures in contrast to bulk fcc materials as free surfaces can favor surface nucleation in {110} planes as observed in Ref. [START_REF] Pichaud | Slip in copper slightly deformed at room temperature[END_REF] For the Pt dewetted particle, the two segments lie in a (10 1) plane. As the particle has been formed after dewetting at high temperature (1000 C), it is possible that the dislocations form at high temperature during the cooling process (thermal stress) as slip in {110} planes is known to be thermally activated. [START_REF] Caillard | Some aspects of cross-slip mechanisms in metals and alloys[END_REF] Several groups have observed a non-octahedral slip in Pt in the past and during deformation (see for examples, Refs. [START_REF] Mclean | Non-octahedral slip in platinum polycrystals[END_REF][START_REF] Kalsar | Texture evolution in platinum: the role of nonoctahedral slip[END_REF] ). It coincides with the {100}<011> slip system. Here, a {110} slip system is observed. The thermal stress is mainly a shear parallel to the substrate, and this may act differently on the particle as compared to a normal compression/deformation. The surface nucleation barriers probably play a decisive role here. The Pt dewetted particle has been annealed at 450 C and under different gas conditions. As shown in Figure 5, we observed no movement of the two segments upon heating of the particle to 450 C and under different gases. This stability can be explained by the fact that the dislocations have been formed at a higher temperature (dewetting at 1000 C). Note that unseen defects, verifying the invisibility condition at the 111 Bragg reflection, may also explain the stability of the two segments in the particle. As discussed above, for the Pt THH particle (growth at room temperature), it is thus highly possible that the A segment (along [111]) lies in a pure {110} slip plane. Then, the junction and B segment (along [101]) may result from cross-slipping from {110} into {111} and heterogeneous crossslipping from a {111} plane into another {111} plane, respectively, as it is energetically favorable to cross-slip into {111} planes. The junction being located at the center of the particle, this region has a different atomic environment/stress state that could facilitate the cross-slip.

Conclusions

In summary, we measure the 3D displacement field in single Pt particles using BCDI. The technique enables a characterisation of the 3D morphology of defects. Our results shed light on anomalous glide {110}-type planes in platinum crystals. Two Pt particles prepared either via a chemical or a physical process and having very different morphologies show dislocations in {110} glide planes. The {110} slip plane is associated with preferential nucleation of defects at the surface of the nanostructures. More studies will be needed to generalise this phenomenon for other fcc nanostructures. BCDI will allow 3D structural visualisation of dislocations under external conditions like heterogeneous catalytic gas phase, temperature or mechanical force changes and will facilitate strain and defect engineering.

Methods/Experimental

Sample preparation.

The tetrahexahedral (THH) Pt particles were electrochemically synthesized on glassy carbon (GC) electrodes by a square-wave-potential method with a saturated calomel reference electrode (SCE) and a Pt foil counter electrode in 2 mM H 2 PtCl 6 and 0.1 M H 2 SO 4 electrolyte following the procedure in Ref. [START_REF] Tian | Synthesis of tetrahexahedral platinum nanocrystals with high-index facets and high electro-oxidation activity[END_REF] The GC electrode is first subjected to a potential of +1.20 V for 2 s to clean the surface and then -0.35 V for 60 ms to create Pt nuclei. The Pt nuclei grows to THH particles by applying a square-wave-potential between +0.04 V and +1.09 V at 100 Hz for 10 min. In the case of the dewetted Pt particles, a 30-nm thick Pt film was deposited on sapphire with an electron beam evaporator. The Pt nanocrystals have their c-axis oriented along the [111] direction normal to the (0001) Al 2 O 3 substrate. A standard photolithography method was employed to prepare a patterned layer of photoresist on sapphire prior to the electron beam evaporation of Pt. The lithographic processing route ensured that a number of dewetted Pt particles are well-separated from their neighbors and that only one crystallite is irradiated by the incoming x-ray beam. The diameter of the particles ranges between 200 nm and 600 nm.

Experimental set-up.

The x-ray experiments were carried out at the upgraded ID01 beamline of the ESRF synchrotron and at the P10 beamline of PETRA synchrotron. At the ID01 beamline, the beam size was focused down using a Fresnel zone-plate (FZP). The resulted beam size was about 310 nm (horizontally) x 170 nm (vertically). A coherent portion of the monochromatic xray beam was selected with high precision slits by matching their horizontal and vertical gaps with the transverse coherence lengths of the beamline 60 µm (horizontally) and 235 µm (vertically) before the FZP). The intensity distribution around the 111 Pt reflection was measured in a coplanar diffraction geometry with the sample surface mounted horizontally.

The scattered x-rays were detected using a two-dimensional (2D) Maxipix pixel detector [START_REF] Ponchut | a fast readout photon-counting X-ray area detector for synchrotron applications[END_REF] (516 x 516 pixels of 55 µm). The sample was mounted on a fast xyz piezoelectric stage with a lateral stroke of 100 µm and a resolution of 2 nm, sitting on a hexapod that was mounted on a (3+2 circles) goniometer. At the P10 beamline of PETRA III synchrotron at DESY, the BCDI experiment was performed at a beam energy of 8 keV (wavelength of 1.55 Å). The beam size was focused down to 0.9 µm (vertically) ⇥ 1.1 µm (horizontally) using Be compound refractive lenses. The sample was mounted in a dedicated water-cooled gasflow chamber with the substrate surface oriented in the horizontal plane on a high-precision (1 µm) XYZ-stage that was mounted on a 6-circle Huber diffractometer. The diffracted beam was recorded with a 2D EIGER 4M photon-counting detector (2070x2167 pixels with pixel size of 75 µm x 75 µm) positioned on the detector arm at a distance of 1.839 m. We measure the 111 Pt Bragg reflection in three-dimensions by rotating the particle around the Bragg angle (20 ) through 3 , in steps of 0.012 . The detector was positioned at an out-of-plane of 39.26 .

Phase retrieval.

Phase retrieval was carried out on the raw diffracted intensity data using PyNX package, [START_REF] Mandula | Ptycho : a computing library for X-ray coherent diffraction imaging of nanostructures[END_REF] imposing at each iteration that the calculated Fourier intensity of the guessed object agrees with the measured data. Defective pixels for experimental data and gaps in the detector were masked and were not used for imposing the reciprocal space constraint mentioned above. The initial support, which is the constraint in real space, was estimated from the autocorrelation of the diffraction intensity. A series of 1400 Relaxed Averaged Alternating Reflections (RAAR [START_REF] Luke | Relaxed averaged alternating reflections for diffraction imaging[END_REF] ) plus 200 Error-Reduction (ER 39 ) steps including shrink wrap algorithm [START_REF] Marchesini | X-ray image reconstruction from a diffraction pattern alone[END_REF] were used. The phasing process included a partial coherence algorithm to account for the partially incoherent incoming wave front. [START_REF] Clark | High-resolution three-dimensional partially coherent diffraction imaging[END_REF] The reconstruction was then corrected for refraction and absorption, the small size of the particles ensuring that dynamical diffraction effects could be neglected. [START_REF] Hu | Dynamic diffraction artefacts in Bragg coherent diffractive imaging[END_REF] After removing the phase ramp and phase offset, [START_REF] Carnis | BCDI Python package -version 0.0.5[END_REF] the data was

6 THH

 6 (b)). It was possible to index the direction of the two segments courtesy of the scattering vector g (here, 111 Bragg reflection) and the different types of facet of the crystal: {201}-type facets for the Pt particle and low-index {001}, {110} and {111} facets for the dewetted Pt particle. For the Pt THH particle, the segment A is along [111] and segment B is along either [110], [011] or [101]; we arbitrarily fix its direction to [101]. For the Pt dewetted crystal, the segment A' is along [101] and the segment B' is along a h2 12i-type direction (here, arbitrarily fixed to [2 12]). Note that the [111]-segment of the dislocation loop of the Pt THH particle is not perfectly straight. As observed in Figure 3(a), it is slightly curved close to the crystal surface. The plane containing the [101]-segment and the curved junction towards the [111]-segment is a ( 131) plane, meaning that the dislocation loop rotates in a ( 131) plane.

3 a ⇥ b 100 ✓

 3100 (a)) since g 111 • u(r) equals to ✓ (g 111 ⇥u 110 ⇥ cos( ) = 2⇡ p 3 a ⇥ b 110 ✓ 2⇡ ⇥ cos( ) = ✓, with b 110 = a p 2 and , being the angle between the [111] and [110] directions; or g 111 ⇥u 100 ⇥ cos( ) = 2⇡ p 2⇡ ⇥ cos( ) = ✓, with b 100 = a and , being the angle between the [111] and [100] directions). The B segment shows a 2⇡-modulation. It cannot be an edge dislocation as a ⇡-modulation of the phase around the dislocation core should be observed. As the two A and B segments are close together inside the crystal, we simulated the superposition of the out-of-plane displacement (u 111 ) and phase ( 111 ) of the two segments supposing a common Burgers vector ( a 2 [101]), which is along the dislocation line of the B segment ( t = [101], see Figure 4(b)). As the two lines emerge at the sample surface, they are connected and must have the same Burgers vector. In this case, A and B segments are a mixed dislocation ( t = [111] and b = a 2 [101]) and a screw dislocation ( t = [101] and b = a 2 [101]

Figures

  Figures

Figure 1 :

 1 Figure 1: Phase and out-of-plane strain of the reconstructed THH Pt crystal. (a) Top and (b) bottom views of the isosurface of the reconstructed phase (in radians) of the Pt particle drawn at 35% of the maximum density. The amplitude is used to draw the isosurface, while the phase is projected onto the isosurface as the colormap. The black and red arrows indicate a phase defect and directions, respectively. The 3D grid has a spacing of 50 nm in Figs. (a) and (b). (c)-(d) Two-dimensional cut of the phase and out-of-plane strain, ✏ zz , at a particle height of 240 nm, respectively.

Figure 2 :

 2 Figure 2: Phase and modulus of the reconstructed dewetted Pt crystal. Reconstructed (a) phase and (b) modulus. Slice of the reconstructed modulus (c) and phase (d).The isosurface is drawn at 35% of the maximum electron density.

Figure 3 :

 3 Figure 3: Dislocation loop and possible slip planes of the Pt THH particle. (a) Wireframe plots of the reconstructed electron density of the Pt particle drawn at 35% of the maximum density. The direction of the two segments of the dislocation loop is evidenced. (b) Visualisation of the plane containing the [101]-segment and the curved junction towards the [111]-segment. The 3D grid has a spacing of either 20 nm (along Z in Fig. a) or 50 nm.

Figure 4 :Figure 5 :

 45 Figure 4: Phase evolution around segments. (a) Reconstructed and simulated (red dashed lines) phase as a function of azimuthal coordinate ✓ for segments A and B of the THH Pt particle. The error bars correspond to the observed variation of the phase over a distance of 25 nm from the center of each defect. (b) Superposition of the 111 phase of the two segments supposing a common Burgers vector ( a 2 [101]), which is along the dislocation line of the B segment ( t = [101]).

Acknowledgement

The authors are grateful to ESRF (proposal HC4760) and PETRA (proposal 20180962EC) synchrotrons for allocating beamtime. The measurement was performed at the ID01 beamline of the European synchrotron (ESRF) and at the P10 beamline of the PETRA III synchrotron at DESY, a member of the Helmholtz Association (HGF). We thank the ID01 and P10 beamline staff for excellent support during the experiment. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 818823) and from the DI-NACS ANR project (ANR-21-CE08-0033-01). The research leading to this result has been supported by the project CALIPSOplus under the Grant Agreement 730872 from the EU Framework Programme for Research and Innovation HORIZON 2020. We also wish to thank the support by a grant from the Ministry of Science and Technology, Israel and from the Centre National de la Recherche Scientifique (CNRS), France.

finally interpolated onto an orthogonal grid for easier visualization.


Mallorie Tourbin
email: mallorie.tourbin@ensiacet.fr

Sylvie Schetrite

Béatrice Biscans

Christine Frances

Aggregation of concentrated colloidal silica dispersions in flow

Keywords: nanoparticles, dense suspensions, stability, aggregation, correlation photon spectroscopy, acoustic spectroscopy

come

INTRODUCTION

The success of nanoparticles is due to the industrial interest for many applications of this type of particles. But the usual properties of the nanoparticles (optical properties, texture, rheology, bioavailability …), in the form of powders or suspensions, are closely associated to the particle size and to their dispersion state. Processing nanoparticles induces a series of problems during the generation, handling, conveying or storage steps. An inherent problem of the different methods of generation or treatment of nanoparticles in dense suspensions is the difficulty of controlling the process of aggregation and then of preserving the quality and the stability of the products. The physicochemical properties of the medium and the hydrodynamic conditions (shear, transport flow,…) are decisive for the aggregation phenomenon. So, it is essential to know the influence of these parameters in order to control and model this process during the generation or treatment steps. The aim of this study is to follow the aggregation process of concentrated colloidal silica suspensions placed under conditions of flow (shearing, transport) identical to those met in real processes. One major problem deals with the characterization of the properties of dense suspensions. Indeed, optical methods usually used to characterize size often require doing measurements in diluted conditions. However dilution can involve a change of the dispersion state of the system, modifying the physico-chemical properties of the suspension, thus inducing an aggregation or de-agglomeration process of the dispersed phase. So, the main objective of the present paper is to define an appropriate method for the characterization of dense suspensions.

MATERIALS AND METHODS

Materials

A colloidal silica suspension, Klebosol 30R50 (Clariant, France), having an initial solid content of 30% w/w was used. The suspension is stabilised by Na + ions and has a pH of about 8-9. Its density is about 1.9 g/ml (silica density is 2.2g/ml) and its specific surface around 50m 2 /g. Figure 1 shows a photography of dried powder obtained using a scanning electronic microscope (LEO 435VP). Silica particles are spherical and have a diameter of about 80 nm. The particles are organised themselves into agglomerates looking like grapes when the colloidal suspension is dried on a desiccation balance. Deionised water (pH~6 and Λ~5-7 µS) was used to prepare the diluted suspensions. The solid content of these suspensions was determined with a precision of 10 -2 %, by the measure of the humidity rate by thermogravimetry using a Halogen Moisture Analyser HB43 desiccation balance (Mettler Toledo).

Characterization measurements

The stability of the different silica suspensions was measured with a Turbiscan MA 2000 apparatus (Formulaction, France). The studied dispersion was poured into a flat-bottomed glass cylindrical cell. A LED that emits in the near infrared (λ=850 nm) gives light to the glass tube and two photodetectors collect the transmitted and retrodiffused light by the dispersion. The detection head scans the entire height of the sample (about 65 mm), acquiring transmission and backscattering data. The apparatus allows detecting the phenomenon of particles migration (creaming, sedimentation) and particle size variations. [START_REF] Mengual | multiple light scattering measurement for concentrated emulsion and suspension instability analysis[END_REF] [1]

Figure 2: Principle of Turbiscan measurements

Zeta potential measurements of a diluted suspension were made by electrophoresis with a Zetasizer 3000 apparatus (Malvern Instruments). Furthermore, pH measurements were made with a standard MeterLab Radiometer analytical PHM210 pH-meter and a combined pH electrode PHC3001-8 which is especially designed for aqueous suspensions.

The size of the suspended particles was measured by two different ways: correlation photon spectroscopy and acoustic spectroscopy. The correlation photon spectroscopy allows measuring size below the limit of the static light diffusion and diffraction methods (≈ 50 nm). For this study, a Zêtasizer Nano S (Malvern Instruments) was used. The principle of this apparatus is based on the Stokes-Einstein law which gives the Brownian diffusion coefficient of the particles according to their size and the physical properties of the medium. The retrodiffused light (angle of 173°) is detected and permits to determine the particle size distribution of the suspensions.

The size distribution of nanoparticles suspended in liquids is a critical parameter in a multitude of industries and applications. Particle size analysis of concentrated systems using conventional light scattering methods often requires severe dilution. Uncertainties arise due to the unknown behaviour of such dense dispersions during dilution. Another current question is how representative the sample is. So a better understanding of aggregation phenomena would result if measurements were done in true processing conditions with respect to solid content and hydrodynamic conditions. Sound waves interact in a similar manner to light waves but have the advantage that they can travel through concentrated suspensions and emulsions. Ultrasonic attenuation spectroscopy is capable for examining the concentrated or optically opaque systems without any dilution. Such in-situ characterization of concentrated systems makes the acoustic method very useful because it does not need to modify the dispersion state of the suspending particles.

In this work, we have access to the commercial acoustic spectrometer Ultrasizer (Malvern Instruments, United Kingdom). Particle size analysis using acoustic spectroscopy consists of:

 Predicting attenuation and velocity spectra by mathematical modelling for any particle size distribution (PSD) and volume concentration,  Measuring attenuation and velocity spectra,  Inverting the mathematical model to extract both the PSD and the volume concentration from the measured spectra. The Ultrasizer measures the acoustic attenuation spectrum of the sample over the frequency range of 1-160 MHz. The range used for a given measurement depends on the attenuation characteristics of the samples (particles size, solid content). The interaction of ultrasound waves with particles has been comprehensively described through a complete mathematical framework which accurately predicts the attenuation spectrum of any suspension. A rigorous but complex model has been published by [START_REF] Epstein | The absorption of sound in suspensions and emulsions. I. Water fog in air[END_REF] [2] and [START_REF] Allegra | Attenuation of Sound in Suspensions and Emulsions: Theory and Experiments[END_REF] [3], which predicts the scattering of an acoustic plane wave by a sphere of a given size. When a sound wave passes through a suspension, changes occur to the wave as well as to the two phases of the particulate medium. A particle presents a discontinuity to sound propagation and the wave scatters with a redistribution of the acoustic energy throughout the volume before being detected at the receiver (scattering and diffraction losses). In addition, absorption phenomena occur when particles move relative to the suspending medium and the mechanical energy degrades into heat (viscous losses) and temperature differences develop between phases with mechanical energy going again into heat (thermal losses). [START_REF] Epstein | The absorption of sound in suspensions and emulsions. I. Water fog in air[END_REF] and [START_REF] Allegra | Attenuation of Sound in Suspensions and Emulsions: Theory and Experiments[END_REF] have shown that any of these attenuation phenomena can be accurately characterized through fundamental equations based on the laws of conservation of mass, energy, and momentum, the thermodynamic equations of state, and stress-strain relations for isotropic elastic solids or viscous, by formulating the Wave Equations which describe the interaction between the sound wave and the particles.

Aggregation experiments

Aggregation experiments were carried out in the Ultrasizer batch cell. The tank used has a specific geometry (Figure 3). It is a water jacketed agitated vessel of 470 mL capacity. The stirrer, a three-bladed impeller, was located very near to the bottom of the tank. The stirring speed was fixed at 600 rpm. The experimental set-up shown in Figure 4 also includes a thermostatic bath. 440 mL of a 0.8 M sodium chloride solution was firstly poured into the tank and stirred. When the desired operating temperature (25°C) was reached, 11 mL of Klebosol 30R50 was added instantaneously to obtain the desired solid concentration. The attenuation spectrum versus frequency was measured after different periods of time. Thermal conductivity (J.m -1 .s -1 .°C -1 )

1.37 5.9525 10 -1

Heat capacity (J.kg -1 .°C -1 ) 7.42 10 2 4.1785 10 3

Shear rigidity (N.m -2 )

3.13 10 10 Viscosity (Po)

9.0300 10 -3 Attenuation coefficient -factor (dB.inch -1 ) 1.46 10 - 3 2.00 10 -1

Attenuation coefficient -exponent 2.00 1.976

RESULTS AND DISCUSSION

Characterization of initial suspensions

Physicochemical stability

The pH of the solution has a strong influence on the stability of colloidal silica suspensions. Indeed, the gelation rate is very high for pH 4-5. However, below this critical zone of pH, and around the isoelectric point (pH ~ 2), the suspension is stabilized by hydrogen bonds that are formed between the silica particles and the dispersed phase. Above this pH, and more particularly for pH>8, the suspension is stabilized by electrostatic repulsions. The initial silica suspension has a solid content of 30 % wt. and a pH of 8.97. Additionally, the stability of this suspension is strengthened by sodium ions that adsorb on the silanol groups (Si-OH), which increase the charge of the particle surfaces and increase the range of the electrostatic repulsion forces. Figure 5 shows the evolution of pH versus the solid content of the suspension. The zeta potential of a very diluted Klebosol suspension is ξ = -56.1 mV. That corroborates the stabilization of the silica suspension by electrostatic repulsions.

The stability of two Klebosol 30R50 suspensions having different solid contents was measured analyzing the transmitted and backscattered light fluxes by the dispersions. Figures 6a and6b report the transmission profiles versus the cell height of the two suspensions measured each 40 min during approximately 5 days and a half. Those profiles are regular on the whole cell height and remain invariant during the measure. Therefore, the silica suspension remains stable and homogeneous in this range of concentration. Thus, the homogeneity of the suspension is not affected by major destabilization phenomena such as particle migration (creaming, sedimentation), particle size variation or aggregation (coalescence, flocculation).

Particle sizing

Photons correlation spectroscopy

For very diluted suspension (wt.<1%), the mean size measured by photon correlation spectroscopy with the Zetasizer NanoS is about d=85.3 nm and is almost constant for the different concentrations (Figure 7) (reproducibility ~1%). This is in good agreement with the observations made using scanning electronic microscopy. There is an almost linear decrease of the mean size measured when the mass concentration is higher than 1wt %. Two phenomena could explain this trend : the multiple scattering of the wave due to the excessive concentration of particles and/or a modification of the structural factor due to the interaction between particles [START_REF] Xu | Particle characterisation: light scattering methods[END_REF] [4]. Thus, 1 wt % appears to be the limit concentration to obtain a correct analysis with this apparatus concerning Klebosol 30R50. Figure 8 shows the particle size distributions measured for the concentration lower than 1% wt. It represents the size distributions calculated by the software with the partial list square method.

Acoustic Spectroscopy

The Ultrasizer measures the attenuation of the ultrasonic wave through the suspension for different frequencies. However, it is more pertinent to consider the excess of attenuation due only to the silica particles in the medium and not to the intrinsic attenuation of each phase. The excess of attenuation, α e , is:

pc pd t e c c     ) 1 (    
(2) were α t is the total attenuation, α pd the attenuation of the dispersed phase, α pc the attenuation of the continuous phase and c the volume concentration of the dispersed phase. The equation of the excess of attenuation function versus the frequency, f, is determined from the attenuation spectrum of water and doing the hypothesis that the attenuation of the solid phase is negligible compared with the attenuation of the liquid phase:

) 0083 , 0 0047 , 0 )( 1 ( ) ( 2 f f c f t e       (3)
On Figure 9, it can be seen that for different concentrations of Klebosol 30R50, the attenuation spectra are parallel (reproducibility ~ 1.5%). The trend of the curves is characteristic of the visco-inertial behaviour of the suspensions.

In theory, the attenuation is proportional to the concentration of the suspended particles and moreover, when the visco-inertial effects are predominant, the attenuation is also proportional 19% v/v 12,60% 9,50% 6,30% 3,20% 0,60%

to the wave frequency. Then, normalizing the spectra by the frequency and the concentration, the curves would be superimposed. But the normalized spectra (Figure 10) do not coincide (the distance between two curves goes from 3 to 20 %). The trend of these curves highlights the existence of the multiple scattering of the acoustic wave in the suspension that modifies the behaviour of the wave. Multiple scattering is increased at higher concentrations. That is why for higher concentrations the curves are almost linear while for the more diluted suspensions, the curves are more curved.

To take into account the multiple scattering of the wave through the suspension, a correction factor was determined in such a way the curves be superimposed. Then, the modified data were used to determine the particle size distributions corresponding to each spectrum. The results are reported in Figure 11. It can be seen that the distributions obtained are in good agreement with the results gave by the microscopy and the photons correlation spectroscopy.

0 0,5 1 1,5 2 2,5 3 3,

Aggregation kinetics

The silica aggregation process was analyzed for given operational parameters (electrolyte concentration, silica concentration, temperature and stirring speed). The Ultrasizer is used to quantify the change in size of the suspending particles as a function of time. For times between 0 and 150 min, the spectra are quite similar, indicating that the particle dispersion is not undergoing important destabilization by aggregation. At further times, distinct changes in the spectra are observed: there is an increase in attenuation at low frequencies and a decrease at high frequencies (the gradient of the curves decreases). It can also be concluded that the aggregation process is almost completed within 8 hours (480 min) because the spectra are not still evolving.

Taking into account the multiple scattering of the ultrasonic wave through the suspension, the particle size distributions of the suspension during the destabilization is shown in Figure 14a.

The aggregation of particles is illustrated by the shift of the distribution towards the largest sizes. At the beginning (t = 0 + ), a few minutes after the addition of the silica suspension in the sodium chloride solution, the monomodal log-normal curve of the particle size distribution has a mean size of 80 nm. This is in very good agreement with the results obtained for the characterization of the initial suspension. Until t = 240 min, the mean size does not change a lot (it increases from 80 nm to 100 nm) but the geometric deviation increases, as shown on Figure 14b where it can be seen that the width of the distribution is increasing. An important change in size as aggregation proceeds is seen once the process time exceeded 270 min: the distribution becomes bimodal. There are two populations: one of about 90 nm (primary particles) which becomes narrower (the particles of this size are less and less numerous) and another one of about 500-700 nm which becomes larger (the aggregates of this size are more and more numerous) at the same time. Later, for t > 360 min, the particle population seems to have three modes: the later two and another one of about 10 µm but the Ultrasizer software is not able to draw a trimodal distribution.

From these results, it can be supposed that the silica aggregation process is the succession of two aggregation stages probably corresponding to two distinct mechanisms: the perikinetic aggregation and the orthokinetic aggregation. First, the collisions between primary particles of 80 nm are brought about by a slow perikinetic mechanism for particle sizes smaller than 100nm leading to a slow increase of the width of the distribution from the primary particle size towards the larger ones. This stage is followed by a rapid orthokinetic mechanism for the coarse particles resulting in the apparition of large aggregates that form a second and probably a third population until about 10 µm. The perikinetic aggregation is relatively slow (4 hours in the conditions of this experiment) because the motions of the small particles are brought about by diffusion and the repulsive colloidal interactions increase the stability of the suspension, which reduces the collision frequency. But during the orthokinetic aggregation (about 1h30), the collisions of coarse aggregates result from the fluid motion and then, the collision frequency is higher because the particles overcome easily the potential barrier as a result of their relative motion [START_REF] Schaer | Study of silica particle aggregation in a batch agitated vessel[END_REF] [START_REF] Schaer | Study of silica particle aggregation in a batch agitated vessel[END_REF].

CONCLUSIONS

In this study, the aggregation of colloidal silica is observed in absence of any other phenomena such as nucleation or particle growth. Various techniques (multiple light scattering, photons correlation spectroscopy, acoustic spectroscopy) were used to characterize the properties of concentrated suspensions of nanoparticles. Based on the acoustic spectrometry, the evolution of the attenuation spectra and of the calculated particle size distributions during a destabilization process showed the existence of two stages of aggregation having different kinetics: a slow perikinetic aggregation of small primary particles (until about 100 nm) and a fast orthokinetic aggregation of larger aggregates (until more than 10 µm).

Figure 1 :

 1 Figure 1: SEM photography of Klebosol 30R50 (x 50000)

Figure 3 :

 3 Figure 3: batch cell of the Ultrasizer Figure 4: Experimental set-up

  Diluting the suspension until 1 wt. %, the pH of the suspension does not fluctuate a lot and remains between 7.73 and 9.25.

Figure 5 :

 5 Figure 5 : Evolution of pH versus solid content of Klebosol 30R50

Figure 6b :

 6b Figure 6b: Diffuse transmitance of Klebosol 30R50 3 % wt.

Figure 6a :

 6a Figure 6a: Diffuse transmitance of Klebosol 30R50 30 % wt.

Figure 7 :Figure 8 :

 78 Figure 7: Mean diameter of the suspending particles of the Klebosol 30R50 versus concentration

Figure 9 :

 9 Figure 9: Excess attenuation spectra of Klebosol 30R50 versus frequency for different concentrations 0,1 1

Figure 11 :

 11 Figure 11: Particles size distributions of different Klebosol 30R50 suspensions measured with the Ultrasizer

Figure 12 :

 12 Figure 12: Attenuation spectra of Klebosol 30R50 during the aggregation process

Figure 14b :

 14b Figure 14b: Differential particle size distributions during the aggregation of Klebosol 30R50 (wt. 1%) by sodium chloride solution (0,8M) for 0+< t <240 min.

Figure 14a :

 14a Figure 14a: Differential particle size distributions during the aggregation of Klebosol 30R50 (wt. 1%) by sodium chloride solution (0,8M) for 0+< t <300 min.0

Table 1 .

 1

	Density (kg.m -3 ) Sound speed (m.s -1 ) Thermal dilatation (°C -1 )	Silica 2.20 10 3 5.96 10 3 1.56 10 -6	aqueous solution of NaCl (0.8 M) 9.9524 10 2 1.4970 10 3 2.5677 10 -4

Table 1 :

 1 Physical properties of silica and 0,8M aqueous sodium chloride solution

ACKNOWLEDGEMENTS

The authors gratefully acknowledge R. Tweedie and M. Terray from Malvern Instruments for their scientific support on acoustic spectroscopy analysis. The authors also acknowledge the French research organization FERMaT for its support.


Victor V Braguta
email: vvbraguta@theor.jinr.ru

Maxim N Chernodub
email: maxim.chernodub@univ-tours.fr

Artem A Roenko
email: roenko@theor.jinr.ru

Dmitrii A Sychev
email: sychev.da@phystech.edu

Negative moment of inertia and rotational instability of gluon plasma

Using first-principle numerical simulations of the lattice SU(3) gauge theory, we calculate the isothermal moment of inertia of the rigidly rotating gluon plasma. We find that the moment of inertia unexpectedly takes a negative value below the "supervortical temperature" Ts = 1.50(10)Tc, vanishes at T = Ts, and becomes a positive quantity at higher temperatures. The negative moment of inertia indicates a thermodynamic instability of rigid rotation. We derive the condition of thermodynamic stability of the vortical plasma and show how it relates to the scale anomaly and the magnetic gluon condensate. The rotational instability of gluon plasma shares a striking similarity with the rotational instabilities of spinning Kerr and Myers-Perry black holes.

Introduction. The moment of inertia I is a quantity that expresses the resistance of a physical body to angular acceleration around a certain axis. In thermodynamic equilibrium, all physical objects have positive moments of inertia implying that in order to achieve an angular acceleration, one needs to apply a torque [START_REF] Landau | Mechanics[END_REF].

A negative moment of inertia of a body would imply that its acceleration generates a torque itself as if the physical body has a negative mass. Impossible in thermal equilibrium, this effect can be achieved in nonequilibrium, open systems. In mechanics, the realization of a I < 0 system requires presence of an active component such as a motor [START_REF] Lončar | Negative-inertia converters: Devices manifesting negative mass and negative moment of inertia[END_REF]. In electronics, the relevant example is played by electrical negative-impedance converters identified as an active electric circuit with negative resistivity [START_REF] Chen | The circuits and filters handbook[END_REF]. A negative moment of inertia can also be realized in rotating Casimir systems associated with negative vacuum energy [START_REF] Chernodub | Permanently rotating devices: extracting rotation from quantum vacuum fluctuations?[END_REF][START_REF] Chernodub | Rotating Casimir systems: magneticfield-enhanced perpetual motion, possible realization in doped nanotubes, and laws of thermodynamics[END_REF][START_REF] Flachi | Quantum vacuum, rotation, and nonlinear fields[END_REF]. In addition, the negativity of isothermal moment of inertia can be achieved in thermodynamically unstable systems such as rotating black holes [START_REF] Bernard | Action principle and partition function for the gravitational field in black-hole topologies[END_REF][START_REF] Prestidge | Dynamic and thermodynamic stability and negative modes in schwarzschild-anti-de sitter black holes[END_REF][START_REF] Reall | Classical and thermodynamic stability of black branes[END_REF][START_REF] Monteiro | Thermodynamic instability of rotating black holes[END_REF][START_REF] Altamirano | Thermodynamics of rotating black holes and black rings: Phase transitions and thermodynamic volume[END_REF].

In our paper, we show that the rigidly rotating gluon plasma possesses, in thermal equilibrium, a negative moment of inertia (I < 0) below the temperature T s = 1.50 [START_REF] Monteiro | Thermodynamic instability of rotating black holes[END_REF] T c .

(

) 1
where T c is the deconfining transition temperature in a non-rotating plasma. We call T s the "supervortical temperature" since at T = T s , the rigidly rotating plasma looses its moment of inertia, I(T s ) = 0, in a distant similarity with a superconductor which looses its resistivity at a certain critical temperature. Rotating quark-gluon plasma (QGP) with temperatures around the supervortical temperature (1) is routinely produced in relativistic heavy-ion collisions. Such plasma can have exceptionally high vorticity of the order of ω ≈ (9 ± 1) × 10 21 s -1 ∼ 0.03 fm -1 c ∼ 7 MeV [START_REF] Adamczyk | Global Λ hyperon polarization in nuclear collisions: evidence for the most vortical fluid[END_REF]. The properties of vortical QGP can be probed via spin polarization of produced hadrons which provide us with an opportunity to confront theoretical methods with experimental results [START_REF] Becattini | Polarization and Vorticity in the Quark-Gluon Plasma[END_REF][START_REF] Huang | Vorticity and Spin Polarization in Heavy Ion Collisions: Transport Models[END_REF]. Theoretical approaches to the thermodynamics of rotating QGP always assume a rigid rotation of the system, drastically simplifying analytical treatment [START_REF] Victor | Rotating quantum states[END_REF][START_REF] Victor | Rotating fermions inside a cylindrical boundary[END_REF].

The global consensus on the phase diagram of rotating quark-gluon plasma is still lacking. The thermal transition from hadronic to the QGP phase is accompanied by the restoration of the chiral symmetry and the deconfinement of color. There is a general agreement in the community that the rigid rotation, according to all model estimates, should reduce the critical temperature of the chiral transition in the fermionic sector [START_REF] Hao-Lei Chen | Analogy between rotation and density for Dirac fermions in a magnetic field[END_REF][START_REF] Jiang | Pairing Phase Transitions of Matter under Rotation[END_REF][START_REF] Chernodub | Interacting fermions in rotation: chiral symmetry restoration, moment of inertia and thermodynamics[END_REF][START_REF] Chernodub | Effects of rotation and boundaries on chiral symmetry breaking of relativistic fermions[END_REF][START_REF] Wang | Quark matter under rotation in the NJL model with vector interaction[END_REF][START_REF] Zhang | Chiral phase transition inside a rotating cylinder within the Nambu-Jona-Lasinio model[END_REF][START_REF] Sadooghi | Inverse magnetorotational catalysis and the phase diagram of a rotating hot and magnetized quark matter[END_REF].

However, the situation with the deconfining transition is not clear: the rigid rotation should either drive plasma to the deconfinement phase [START_REF] Chen | Gluodynamics and deconfinement phase transition under rotation from holography[END_REF][START_REF] Fujimoto | Deconfining Phase Boundary of Rapidly Rotating Hot and Dense Matter and Analysis of Moment of Inertia[END_REF][START_REF] Golubtsova | Heavy quarks in rotating plasma via holography[END_REF][START_REF] Shi Chen | Perturbative Confinement in Thermal Yang-Mills Theories Induced by Imaginary Angular Velocity[END_REF][START_REF] Golubtsova | Probing the holographic model of N = 4 SYM rotating quarkgluon plasma[END_REF] or, with another scenario, should not affect the system at the rotational axis, forming, at high vorticity, an inhomogeneous confining-deconfining phase (the inverse hadronization effect) [START_REF] Chernodub | Inhomogeneous confiningdeconfining phases in rotating plasmas[END_REF]. While signatures of the inhomogeneity are seen in kinematic variables in numerical simulations of pure gluon plasma [START_REF] Chernodub | Inhomogeneity of rotating gluon plasma and Tolman-Ehrenfest law in imaginary time: lattice results for fast imaginary rotation[END_REF], the numerical first-principle simulations have also revealed that the bulk critical temperature of the deconfining phase transition grows with the increase of the angular frequency [START_REF] Braguta | Study of the Confinement/Deconfinement Phase Transition in Rotating Lattice SU(3) Gluodynamics[END_REF][START_REF] Braguta | Influence of relativistic rotation on the confinement-deconfinement transition in gluodynamics[END_REF]. Moreover, it turned out that gluons and fermions have opposite effects on the critical temperature in rotating systems. It seems that the gluon sector wins in this contest and the deconfinement as well as the chiral critical temperatures increase with the rotation [START_REF] Braguta | Thermal phase transitions in rotating QCD with dynamical quarks[END_REF].

Thus, the model-based analytical approaches and the first-principle numerical simulations of rigidly rotating gluon plasma do not match. To explore this puzzle deeper, we look, in our work, at the mechanical properties of the rotating gluon plasma.

Angular momentum and moment of inertia. A mechanical response of a thermodynamic ensemble to a rigid rotation with the angular velocity Ω can be quantified in terms of the conjugated variable, the total arXiv:2303.03147v1 [hep-lat] 6 Mar 2023 angular momentum J which includes orbital and spin parts. These quantities determine the relation between the energy E (lab) in the inertial laboratory frame and the energy in the co-rotating, non-inertial reference frame, E = E (lab) -J Ω [START_REF] Landau | Mechanics[END_REF]. The angular momentum J can be expressed via either the energy E or the free energy F = E -T S in the co-rotating frame:

J = - ∂E ∂Ω S = - ∂F ∂Ω T , (2)
where we used dE = T dS -J dΩ and dF = -SdT -J dΩ.

The moment of inertia is a scalar quantity,

I(T, Ω) = J(T, Ω) Ω = - 1 Ω ∂F ∂Ω T . (3)
which fixes a relation between the angular momentum J (T, Ω) = I(T, Ω)Ω and the angular velocity Ω = Ωe of rotation around a fixed axis e. Thus, the basic thermodynamic quantities (2) and the moment of inertia (3) can be determined with the help of the free energy F in the co-rotating reference frame.

For simplicity, we start our discussion with a cylindershaped gluon plasma with a radius R, rigidly rotating with the angular frequency Ω around the symmetry axis. We consider slowly rotating gluon plasma implying the velocity

v R = ΩR , (4)
at the boundary of the system to be non-relativistic, v 2 R 1. As we work with a large system size, R ∼ (a few) fm, the condition of the slowness of rotation also requires that the angular velocity should be much smaller than the intrinsic QCD energy scale, Ω Λ QCD . Moreover, in the whole range of temperatures in our work, T

(1.0 ∼ 2.0)T c , the boundary effects can also be neglected because the spatial thermal correlation lengths in the strongly interacting gluonic plasma at T T c are of the order of Λ -1 QCD or shorter, while below T c the correlations are governed by the glueball masses M 0 ++ = 1.653 [START_REF] Golubtsova | Heavy quarks in rotating plasma via holography[END_REF] GeV [START_REF] Athenodorou | The glueball spectrum of su(3) gauge theory in 3 + 1 dimensions[END_REF] which correspond to even shorter correlation lengths. These physical conditions (system size R, temperature range T , and rotational frequency range Ω) correspond to physical conditions of vortical plasma created at RHIC in noncentral relativistic heavy-ion collisions [START_REF] Adamczyk | Global Λ hyperon polarization in nuclear collisions: evidence for the most vortical fluid[END_REF].

Thermodynamics and velocity at the boundary. Several theoretical approaches to rigidly rotating gluon plasma [START_REF] Huang | Vorticity and Spin Polarization in Heavy Ion Collisions: Transport Models[END_REF][START_REF] Fujimoto | Deconfining Phase Boundary of Rapidly Rotating Hot and Dense Matter and Analysis of Moment of Inertia[END_REF] express its thermodynamic properties as a function of the angular frequency Ω, suggesting independence (or a mild dependence) of the thermodynamics on the size of the system perpendicular to the axis of rotation [START_REF] Fujimoto | Deconfining Phase Boundary of Rapidly Rotating Hot and Dense Matter and Analysis of Moment of Inertia[END_REF]. However, first-principle numerical simulations indicate that the size dependence is very pronounced [START_REF] Braguta | Study of the Confinement/Deconfinement Phase Transition in Rotating Lattice SU(3) Gluodynamics[END_REF][START_REF] Braguta | Influence of relativistic rotation on the confinement-deconfinement transition in gluodynamics[END_REF] (see also discussion in Ref. [START_REF] Chernodub | Interacting fermions in rotation: chiral symmetry restoration, moment of inertia and thermodynamics[END_REF]). Moreover, it appears that the thermodynamics of the slowly rotating system can be expressed as a function of the velocity of the system at the boundary (4). This statement, valid at least in the O(Ω 2 ) order, implies that the thermodynamic variables incorporate angular frequency only via the common product ΩR [START_REF]pressure does not depend on the combination Ω/ΛQCD[END_REF].

Neglecting a shape change for the slowly rotating plasma, the moment of inertia can be taken as an Ωindependent quantity, I = I(T, R). [START_REF] Hereafter | [END_REF] Then, the thermodynamic relation [START_REF] Chen | The circuits and filters handbook[END_REF] implies that

F (T, R, Ω) = F 0 (T, R) - 1 2 I(T, R)Ω 2 , (5)
where F 0 ≡ F (lab) (Ω = 0) is the free energy of the nonrotating gas. The quadratic term Ω 2 has a minus sign in the co-rotating free energy (5) because this term represents a centrifugal energy responsible for particle runaway forces directed outwards of the axis of rotation. For a classical system, the moment of inertia is determined by the mass (energy) distribution ρ(T, x ⊥ , Ω), where x ⊥ is the radial coordinate normal to the axis of rotation. The effect of non-relativistic rotation on the uniform spatial mass distribution can be neglected, ρ(T, x ⊥ , Ω) = ρ 0 (T ), implying that the moment of inertia takes the following familiar form:

I(T, R, Ω) = V d 3 x x 2 ⊥ ρ(T, x ⊥ , Ω) = π 2 L z R 4 ρ 0 (T ) , (6)
where the integral is taken over the volume V = πL z R 2 of the rotating cylinder. While the derivation of Eq. ( 6) is valid for a nonrelativistic rotation of a classical system, the system itself can be described by relativistic thermodynamics similarly to the gluon plasma where the relation (6) holds as well [START_REF] Landau | Statistical Physics[END_REF]. In this case, the quantity ρ 0 , which determines the number of degrees of freedom that couple to rigid rotation, can have another origin. For example, ρ 0 in a Casimir system is negative so that the moment of inertia can take negative values as well [START_REF] Chernodub | Permanently rotating devices: extracting rotation from quantum vacuum fluctuations?[END_REF][START_REF] Chernodub | Rotating Casimir systems: magneticfield-enhanced perpetual motion, possible realization in doped nanotubes, and laws of thermodynamics[END_REF].

Combining Eqs. ( 5) and ( 6) we get that the co-rotating free energy can be expressed via the velocity v R at the border of the cylinder x ⊥ = R, given in Eq. ( 4):

F (T, v R ) = F 0 (T, R) 1 + 1 2 K 2 v 2 R + O v 4 R , (7)
where F 0 is the free energy in the absence of rotation.

The response of the system with respect to the rigid rotation is represented in Eq. ( 7) by the dimensionless coefficient K 2 = V ρ 0 /(2F 0 ), where V = πLR 2 is the volume of the cylinder. The dimensionless moment of inertia K 2 relates the moment of inertia I to the free energy at a vanishing angular frequency, F 0 (notice that F 0 < 0):

I(T ) ≡ lim Ω→0 I(T, Ω) = -K 2 (T )F 0 (T )R 2 . (8)
Thus, the angular frequency Ω enters the free energy [START_REF] Bernard | Action principle and partition function for the gravitational field in black-hole topologies[END_REF] only as the velocity v R at the boundary of the system (4).

Scale anomaly and equation of state. The free energy density, f , is determined via the scale (trace) anomaly [START_REF] Boyd | Thermodynamics of SU(3) lattice gauge theory[END_REF]:

T µ µ = -T 5 d dT f T 4 . (9)
Integrating ( 9), we get the free energy density:

f (T ) = -T 4 T 0 dT T T µ µ (T ) T 4 , (10)
highlighting the role of the anomaly T µ µ = 0. When defining the integral in Eq. ( 10), we used the fact that the anomalous trace vanishes rapidly at low temperatures, T µ µ ∼ exp(-M/T ), due to the mass gap M = 0. The trace of the energy-momentum tensor T µ µ is computable in lattice simulations thus making it possible to calculate the free energy density [START_REF] Monteiro | Thermodynamic instability of rotating black holes[END_REF] numerically from the first principles. However, this thermodynamically transparent expression can be rewritten in the lattice form which is more suitable for numerical computations [START_REF] Boyd | Thermodynamics of SU(3) lattice gauge theory[END_REF]:

f (T ) T 4 = -N 4 t β β0 dβ ∆s(β ) , T = 1 N t a(β) , (11a) ∆s
(β) = s(β) T =0 -s(β) T , s = S N t N z N 2 s , (11b)
where the SU (N c ) lattice coupling β = 2N c /g 2 0 is expressed in terms of the bare continuum coupling g 0 , while the lattice action S will be specified below. In Eq. (11a), the lower integration limit β 0 is chosen in a deep confinement phase where the integrand, represented by the difference (11b) in the expectation values of the action at vanishing and finite temperature, is negligibly small.

The lattice formula (11a) has the same meaning as the continuum relation [START_REF] Monteiro | Thermodynamic instability of rotating black holes[END_REF], with the right-hand-side of Eq. (11a) expressed via the lattice form of the scale anomaly. The latter includes the running coupling which is computable via the scale dependence of the lattice spacing a = a(β).

The lattice form (11a) is also suitable for direct calculation of the free energy density f in the non-inertial co-rotating reference frame. To this end, the action S should be formulated in the curved background with the Euclidean metric corresponding to the co-rotating frame [START_REF] Yamamoto | Lattice QCD in rotating frames[END_REF]:

g E µν =     1 0 0 x 2 Ω I 0 1 0 -x 1 Ω I 0 0 1 0 x 2 Ω I -x 1 Ω I 0 1 + x 2 ⊥ Ω 2 I     , (12)
written in the Euclidean coordinates x µ = (x 1 , . . . , x 4 ), where x 4 = -it is the imaginary time coordinate and

x 2 ⊥ = x 2 1 + x 2 2 .
The system rotates around the x 3 axis.

The angular velocity in Eq. ( 12) is put in the purely imaginary form Ω I = -iΩ to avoid the sign problem [START_REF] Yamamoto | Lattice QCD in rotating frames[END_REF]. The expressions for the Minkowski spacetime can be obtained by the analytical continuation. In particular, the velocity v R at the boundary (4) becomes imaginary v I = -iv R , with the following relation:

v 2 I = -v 2 R . (13)
The action in the co-rotating frame in the continuum Euclidean spacetime has the following form:

S = 1 4g 2 d 4 x √ g E g µν E g αβ E F a µα F a νβ , (14)
where g E µν = (g µν E ) -1 is the curved Euclidean metric ( 12) with the determinant g E = det (g µν ) = 1 and F a µν is the field strength of SU(3) gauge field.

Numerical first-principle results. We discretize rotating terms in the action [START_REF] Huang | Vorticity and Spin Polarization in Heavy Ion Collisions: Transport Models[END_REF] following Ref. [START_REF] Braguta | Influence of relativistic rotation on the confinement-deconfinement transition in gluodynamics[END_REF][START_REF] Yamamoto | Lattice QCD in rotating frames[END_REF] and use the tree-level improved Symanzik gauge action for the terms without rotation [START_REF] Curci | Symanzik's Improved Lagrangian for Lattice Gauge Theory[END_REF][START_REF] Luscher | Computation of the Action for On-Shell Improved Lattice Gauge Theories at Weak Coupling[END_REF]. To set the temperature scale we use the results from Ref. [START_REF] Beinlich | String tension and thermodynamics with tree level and tadpole improved actions[END_REF].

Our calculations are performed on the lattices of size 2 with N t = 5, 6, 7, 8. We used periodic boundary conditions in all directions. For the zero temperature subtraction (11b) we use data from the lattice with the same spatial sizes and N t = 40. The imaginary velocity at the boundary is identified with the velocity at the middle of the boundary side, v I = Ω I L s /2, where L s = a(N s -1) is the length of the lattice in the x and y directions. More details on numerical simulations can be found in Appendix.

N t × N z × N 2 s = N t × 40 × 41
In the inset of Fig. 1, we show the normalized difference of lattice action densities ∆s (11b) which enters the free energy density (11a). At vanishing velocity of the rotation, v I = 0, we recover the known result [START_REF] Boyd | Thermodynamics of SU(3) lattice gauge theory[END_REF][START_REF] Sz | Precision SU(3) lattice thermodynamics for a large temperature range[END_REF]. The steep rise of ∆s, which happens close to the critical coupling β β c , points to the first order nature of the phase transition in the non-rotating plasma.

As the imaginary velocity v I increases, the transition shifts towards smaller lattice couplings β signaling that the critical temperature T c = T c (v I ) decreases as the imaginary angular frequency Ω I (the velocity v I of the rotation) raises. This result is in agreement with previous numerical calculations [START_REF] Braguta | Study of the Confinement/Deconfinement Phase Transition in Rotating Lattice SU(3) Gluodynamics[END_REF][START_REF] Braguta | Influence of relativistic rotation on the confinement-deconfinement transition in gluodynamics[END_REF][START_REF] Braguta | Thermal phase transitions in rotating QCD with dynamical quarks[END_REF].

The normalized free energy density in the co-rotating frame, -f /T 4 , calculated via Eq. (11a), is shown in Fig. 1. This quantity is a monotonically raising function of the temperature T at all imaginary velocities v I , indicating the presence of a plateau at T → ∞ for each fixed v I .

The free energy density f , shown in Fig. 1, is a rising (diminishing) function of v 2 I at fixed temperature T < T s (T > T s ). This property can be quantified by fitting of the free energy density at a fixed temperature T by a 13) and ( 4), for the same lattice. Periodic boundary conditions are employed. parabolic function

f (T, v I ) = f 0 (T ) 1 - 1 2 K 2 (T )v 2 I , (15)
where f 0 and K 2 serve as fit parameters (we remind that f 0 < 0). The expression in the Euclidean spacetime [START_REF] Victor | Rotating quantum states[END_REF] represents the first two terms of the free energy [START_REF] Bernard | Action principle and partition function for the gravitational field in black-hole topologies[END_REF] in the co-rotating frame in the Minkowski spacetime after the Wick transformation for the boundary velocities [START_REF] Becattini | Polarization and Vorticity in the Quark-Gluon Plasma[END_REF].

The dimensionless moment of inertia K 2 for several lattices and the corresponding continuum limit (a → 0, or, equivalently, 1/N t → 0 at a fixed temperature T ) are shown in Fig. 2. A striking feature of the free energy, Fig. 1, is that the curves corresponding to different v I intersect at the same "supervortical" temperature T s , signaling that at this temperature, the free energy (15) looses, at least for slow rotation v 2 I 1, the dependence on the rotational frequency. Therefore, the rigidly rotating gluon plasma looses its moment of inertia at T = T s . We use this property as a definition of the supervortical temperature, K 2 (T s ) = 0, and show its continuum limit in Fig. 2.

The continuum limit of the dimensionless moment of inertia can be well reproduced by a rational function

K (fit) 2 (T ) = K (∞) 2 - c T /T c -1 , (16)
where the best-fit parameters are the high-temperature asymptotics K (∞) 2 = 2.23 [START_REF] Yamamoto | Lattice QCD in rotating frames[END_REF] and the slope coefficient c = 1.11 [START_REF] Chernodub | Effects of rotation and boundaries on chiral symmetry breaking of relativistic fermions[END_REF]. The high-temperature limit of the moment + C/N 2 t at Nt → ∞ or, equivalently, a → 0. The best fit ( 16) of the continuum curve is shown by the dashed black line. The position of the supervortical temperature Ts, extrapolated to continuum limit, is marked by the vertical line which separates the unstable (T < Ts) and stable (T > Ts) regimes of rigid plasma rotation. The error estimations (the shaded regions for Ts and K (cont.) 2

, and the bars of the data) include both statistical and systematic uncertainties.

of inertia, K (∞) 2
, is a non-universal quantity that may depend on the geometry of the rotating system and the boundary conditions.

In order to estimate systematic errors related to our determination of T s , we repeat the whole procedure with several methods of numerical integration in Eq. (11a) and several upper limits v (max) I for the fit of the free energy density by the function [START_REF] Victor | Rotating quantum states[END_REF]. The final result for the lattices with periodic boundary conditions is given in Eq. ( 1), where the estimated uncertainty incorporates both statistical and systematic contributions.

As mentioned above, the absence of the massless excitations in the deconfinement phase implies the independence of our results on the type of boundary conditions in the transverse spatial directions, provided the spatial volume is large enough. To verify this property, we calculated the K 2 coefficient for the system with open boundary conditions and found an agreement with the periodic lattices, albeit more significant uncertainties. The corresponding supervortical temperature for the open system, T s /T c = 1.53 [START_REF] Victor | Rotating quantum states[END_REF] agrees with the estimate for periodic boundary conditions [START_REF] Landau | Mechanics[END_REF].

(In)stability and sign of the moment of inertia. Formally, the negative moment of inertia, observed in the region of temperatures T < T s , might imply that the rotation causes the quark-gluon plasma to cool down (the faster rotation, the lower thermal energy). However, physically, this counter-intuitive effect, contradicting the kinematic Tolman-Ehrenfest picture [START_REF] Tolman | Temperature Equilibrium in a Static Gravitational Field[END_REF][START_REF] Tolman | On the Weight of Heat and Thermal Equilibrium in General Relativity[END_REF], indicates that the rigid rotation is impossible thermodynamically. The plasma experiences an instability which makes the rotation non-rigid. Similar instabilities occur in curved gravitational backgrounds of rotating Kerr and Myers-Perry black holes [START_REF] Bernard | Action principle and partition function for the gravitational field in black-hole topologies[END_REF][START_REF] Prestidge | Dynamic and thermodynamic stability and negative modes in schwarzschild-anti-de sitter black holes[END_REF][START_REF] Reall | Classical and thermodynamic stability of black branes[END_REF].

For a system in stable equilibrium at a given temperature T and angular velocity Ω, any deviation from the equilibrium should obey the following condition [START_REF] Landau | Statistical Physics[END_REF]:

δE -T δS -ΩδJ > 0 , (17)
which implies that all eigenvalues of the inverse Weinhold metric, defined in the thermodynamic space [START_REF] Weinhold | Metric geometry of equilibrium thermodynamics[END_REF],

g (W ),µν = - ∂ 2 f (T, Ω) ∂X µ ∂X ν , X µ = (T, Ω i ) , (18)
must be positively defined (see a discussion in [START_REF] Monteiro | Thermodynamic instability of rotating black holes[END_REF]).

The positivity of the matrix ( 18) is achieved provided the specific heat at constant angular momenta C J and the eigenvalues (spectrum) of the tensor of isothermal differential moment of inertia I ij ≡ I ji are positive quantities:

C J > 0 , C J = T ∂S ∂T J , (19) spec
(I ij ) > 0 , I ij = ∂J i ∂Ω j T . (20)
The inequality ( 19) is a standard requirement for the thermodynamic stability [START_REF] Landau | Statistical Physics[END_REF]. Given the cylindrical geometry of our system, the condition (20) reduces to the requirement I > 0 for the principal moment of inertia (3) at infinitesimally slow rotations, Ω → 0. In terms of the coefficient K 2 , Eqs. ( 7) and ( 8), the thermodynamic stability requires:

K 2 (T ) > 0 (thermodynamic stability), (21)
which is violated below the supervortical temperature, T < T s . This instability has a thermodynamic origin. It has no obvious relation to the hydrodynamic instabilities that might be generated by the viscous flow of hot gluons.

Moment of inertia and scale anomaly. The moment of inertia I is directly related to the trace anomaly as one can see from Eqs. ( 8), [START_REF] Monteiro | Thermodynamic instability of rotating black holes[END_REF], and (15):

I(T ) = -T 4 T 0 dT T T µ µ (2) (T ) T 4 , (22)
where T µ µ (2k) are the moments of the anomalous trace

T µ µ (2k) (T ) = 1 (2k)! ∂ 2k ∂v 2k I T µ µ (T, v I ) v I =0 , (23)
for k ∈ Z. The equation ( 23) can readily be written for the real angular velocity using the correspondence [START_REF] Becattini | Polarization and Vorticity in the Quark-Gluon Plasma[END_REF].

Role of the magnetic gluon condensate. Using Eq. ( 3) or Eq. ( 20), we can derive, via a path-integral representation of Yang-Mills theory in a curved spacetime [START_REF] Huang | Vorticity and Spin Polarization in Heavy Ion Collisions: Transport Models[END_REF], the moment of inertia of gluons, I ≡ I 33 :

I = I fluct + I cond = V d 3 x V d 3 x M 12 0 (x)M 12 0 (x ) T + V d 3 x ( ij F a i3 x j ) 2 + (F a 12 ) 2 (x 2 1 + x 2 2 ) T . (24)
We used Eq. ( 5) and the relation Ω 2 = -Ω 2 I . In Eq. ( 24),

M ij (x) = x i T j0 (r) -x j T i0 (x) , i, j = 1, 2, 3 , (25)
is the local angular momentum, which is related to the Ω → 0 limit of the stress-energy tensor of gluons:

T µν = F a,µα F a,ν α - 1 4 η µν F a,αβ F a αβ . (26)
We also used the notation

O T = O T -O T =0
to denote the thermal part of the expectation value of an operator O. The normalization of the moment of inertia ( 24) is chosen such that the cold vacuum has no inertia.

The first term in the moment of inertia ( 24) describes fluctuations of the angular momentum and has a standard linear-response form

I fluct = (J 3 ) 2 T , J i = 1 2 V d 3 x ijk M jk (x) , (27)
were we used the fact that J 3 = 0 at Ω = 0 at any temperature. One might expect that I fluct > 0 because thermal fluctuations always increase the susceptibility in addition to quantum fluctuations. The second term in (24) involves a nonperturbative magnetic gluon condensate in the static, Ω → 0, limit. Using the SO(3) rotational symmetry and the translational invariance of the plasma in spatial dimensions, we get the relation

F a i3 F a j3 T = δ ij (F a 12 ) 2
T , which can be expressed via the magnetic gluon condensate at a finite temperature (F a ij ) 2 T ≡ 6 (F a 12 ) 2 T . We get:

I cond = 1 3 V d 3 x x 2 ⊥ (F 2 ij ) 2 = π 6 L z R 4 (F a ij ) 2 T . (28)
Surprisingly, this relation has the same form as the classical formula for the moment of inertia [START_REF] Flachi | Quantum vacuum, rotation, and nonlinear fields[END_REF], where the mass density ρ corresponds to the thermal part of the magnetic gluon condensate ρ(T ) = (F a ij ) 2 T /6. The full gluon condensate G 2 (a sum of its electric and magnetic parts) is a phenomenologically important quantity which takes a positive value at T = 0 [START_REF] Shifman | QCD and Resonance Physics. Theoretical Foundations[END_REF][START_REF] Shifman | QCD and Resonance Physics: Applications[END_REF]. It decreases monotonically with the increase of the temperature, implying that the thermal part of the condensate, G 2 T , always takes a negative value [START_REF] Miller | The Gluon condensate in QCD at finite temperature[END_REF][START_REF] Gubler | Recent Progress in QCD Condensate Evaluations and Sum Rules[END_REF]. This property, which is interpreted as melting of the gluon condensate, is in agreement with the relation of the thermal condensate to the scale anomaly: G 2 T = -T µ µ T < 0 (see a discussion in Ref. [START_REF] Boyd | Thermodynamics of SU(3) lattice gauge theory[END_REF]).

However, the magnetic contribution to the scale anomaly (11b) reverses its sign at T 2T c [START_REF] Boyd | Thermodynamics of SU(3) lattice gauge theory[END_REF] indicating that the magnetic part of the thermal gluon condensate becomes positive and implying that I > 0 above 2T c . This effect is associated with the evaporation of the magnetic component of the gluon plasma [START_REF] Chernodub | Magnetic component of Yang-Mills plasma[END_REF][START_REF] Liao | Magnetic Component of Quark-Gluon Plasma is also a Liquid![END_REF] and the associated string dynamics [START_REF] Chernodub | Magnetic component of Yang-Mills plasma[END_REF][START_REF] Ya | Chiral spin symmetry and hot/dense QCD[END_REF]. Thus, the negativevalued condensate in I cond should nullify the positive contribution of the correlator term I fluct in a region below 2T c in agreement with our estimate of the supervortical temperature (1), T s < 2T c .

In short, the thermal magnetic condensate takes a negative value right above T c suggesting that the negative contribution of the condensate I cond overwhelms the positive standard contribution I fluct thus leading to a negative total moment of inertia [START_REF] Chen | Gluodynamics and deconfinement phase transition under rotation from holography[END_REF], I = I fluct + I cond < 0, above T c . At T > T s , the perturbative hot gluons prevail the non-perturbative scale counterpart, and the moment of inertia becomes positive.

The suggested mechanism is also in a qualitative agreement with previous numerical observations indicating that rigid rotation increases the critical transition temperature T c [START_REF] Braguta | Study of the Confinement/Deconfinement Phase Transition in Rotating Lattice SU(3) Gluodynamics[END_REF][START_REF] Braguta | Influence of relativistic rotation on the confinement-deconfinement transition in gluodynamics[END_REF]. Indeed, if the rigid rotation makes the plasma colder, then stronger thermal fluctuations (and, consequently, higher temperature) are needed to destroy the confinement phase in the rotating plasma as compared to the non-rotating plasma. This simple observation explains the effect of raising critical temperature T c with increasing angular frequency Ω. Moreover, the crucial role of the magnetic condensate in our mechanism suggests that this effect should be absent for nongluonic degrees of freedom. The latter hypothesis is perfectly consistent with the recent first-principle observation made, separately, for quarks and gluons in Ref. [START_REF] Braguta | Thermal phase transitions in rotating QCD with dynamical quarks[END_REF] Our paper focuses on the investigation of pure gluon plasma, which exhibits the fundamental nonperturbative properties of its realistic quark-gluon counterpart. To clarify the contribution of quarks to the moment of inertia, we notice that Eq. ( 24) remains also valid in QCD. Namely, the total angular momentum M 12 now includes not only the gluon part [START_REF] Fujimoto | Deconfining Phase Boundary of Rapidly Rotating Hot and Dense Matter and Analysis of Moment of Inertia[END_REF], but also the orbital, ψγ 4 (xD y -yD x )ψ, and spin, i/2 ψγ 4 σ 12 ψ, angular momenta of quarks. While the quark fields make a positive thermal contribution to the fluctuation term I fluct , the gluomagnetic contribution I cond stays negative in QCD [START_REF] Ishii | Hadronic screening masses and the magnetic gluon condensate at high temperature[END_REF]. Following our discussions of gluon plasma, we conclude that I ≡ I fluct + I cond < 0 also for the quarkgluon plasma in a certain range of temperatures. Thus, we believe that the rigid rotation of quark-gluon plasma is also unstable in a certain temperature range around the transition temperature.

Conclusions. All field-theoretical analytical and first-principle numerical approaches dedicated to the investigation of the thermodynamics of rotating quarkgluon plasma consider a rigidly rotating plasma, meaning the angular velocity Ω at all points in the system, regardless of the distance to the rotational axis, takes the same value [55]. In other words, the plasma rotates like a solid body. While this assumption sounds unnatural from, at least, a hydrodynamic point of view (after all, the plasma is not a solid), the rigid rotation is very convenient in treating the system analytically. Moreover, the rigid approximation could sound reasonable for a sufficiently small, slowly rotating system as a first-order approximation to a more complicated, spatially inhomogeneous rotation.

In our work, we show that below the supervortical temperature (1), the rigid rotation of the gluon plasma is thermodynamically unstable even at slow rotation velocities. This effect exhibits a striking similarity with spinning black holes [START_REF] Bernard | Action principle and partition function for the gravitational field in black-hole topologies[END_REF][START_REF] Prestidge | Dynamic and thermodynamic stability and negative modes in schwarzschild-anti-de sitter black holes[END_REF][START_REF] Reall | Classical and thermodynamic stability of black branes[END_REF][START_REF] Monteiro | Thermodynamic instability of rotating black holes[END_REF][START_REF] Altamirano | Thermodynamics of rotating black holes and black rings: Phase transitions and thermodynamic volume[END_REF]. While the back-hole rotational instability is promoted by the curved gravitational background, the instability in the gluon plasma originates from the scale anomaly via the thermal part of the magnetic gluon condensate. Therefore, we conclude that the model of rigid rotation cannot be used, for thermodynamic reasons, for a realistic study of the rotation of the gluon plasma.

Our results also suggest that the puzzling discrepancy between numerical [START_REF] Braguta | Study of the Confinement/Deconfinement Phase Transition in Rotating Lattice SU(3) Gluodynamics[END_REF][START_REF] Braguta | Influence of relativistic rotation on the confinement-deconfinement transition in gluodynamics[END_REF][START_REF] Braguta | Thermal phase transitions in rotating QCD with dynamical quarks[END_REF] and analytical (including holography [START_REF] Chen | Gluodynamics and deconfinement phase transition under rotation from holography[END_REF], Tolman-Ehrenfest kinematic estimations [START_REF] Chernodub | Inhomogeneity of rotating gluon plasma and Tolman-Ehrenfest law in imaginary time: lattice results for fast imaginary rotation[END_REF], hadron-gas approach [START_REF] Fujimoto | Deconfining Phase Boundary of Rapidly Rotating Hot and Dense Matter and Analysis of Moment of Inertia[END_REF], and perturbative arguments [START_REF] Shi Chen | Perturbative Confinement in Thermal Yang-Mills Theories Induced by Imaginary Angular Velocity[END_REF] based on imaginary rotation [START_REF] Chernodub | Inhomogeneous confiningdeconfining phases in rotating plasmas[END_REF]) models' predictions about the critical temperature in the center of rotating QCD (gluon) plasma might originate from the scale anomaly which should be taken into account appropriately. In particular, our work shows that the magnetic gluon condensate -which has a nonperturbative component at any temperature -plays a crucial role in rotating quark-gluon plasma. where Ūµν denotes the clover-type average of four plaquettes, W 1×2 µν is the rectangular loop, Vµνρ is the asymmetric chair-type average of eight chairs [START_REF] Braguta | Influence of relativistic rotation on the confinement-deconfinement transition in gluodynamics[END_REF], and c 0 = 1-8c 1 , c 1 = -1/12. The action [START_REF] Chernodub | Inhomogeneous confiningdeconfining phases in rotating plasmas[END_REF] in the case c 1 = 0 coincides with the lattice gauge action used in Refs. [START_REF] Braguta | Study of the Confinement/Deconfinement Phase Transition in Rotating Lattice SU(3) Gluodynamics[END_REF][START_REF] Braguta | Influence of relativistic rotation on the confinement-deconfinement transition in gluodynamics[END_REF].

For each lattice size we keep the (imaginary) angular velocity in lattice units unchanged with the variation of β, therefore, the linear velocity v I at the boundary of the system remains constant with the changes in temperature. In our simulations the linear velocity takes the following values: v 2 I = 0.000, 0.015, 0.030, 0.045, 0.060, 0.075, 0.090.

To set the temperature scale, we use the results for the string tension from Ref. [START_REF] Beinlich | String tension and thermodynamics with tree level and tadpole improved actions[END_REF]. For the non-rotating lattices with periodic boundary conditions, we use the values of the inverse critical coupling β c taken from Ref. [START_REF] Beinlich | String tension and thermodynamics with tree level and tadpole improved actions[END_REF]. For the case of open boundary conditions, we determine β c from the peak of the Polyakov loop susceptibility.

Simulations are performed using Monte Carlo algorithm, each sweep consists of one heatbath update and two steps of the overrelaxation updates. In finite (zero) temperature simulations typical statistics are about 5000-40000 (2000-10000) sweeps after thermalization for each set of parameters, depending on N t of the finite temperature lattice. The statistical errors are estimated via the jackknife method.

FIG. 1 .

 1 FIG.1. The free energy density f in the co-rotating frame as a function of the temperature T for the Nt = 6 lattice. The vertical line shows the supervortical temperature Ts for this lattice. The inset shows the expectation value of the lattice action density ∆s corresponding to the scale anomaly (11b) as a function of the lattice gauge coupling β. Both plots are given for several imaginary velocities squared v 2 I at the boundary R = Ls/2 ≡ 20a of the system, Eqs. (13) and (4), for the same lattice. Periodic boundary conditions are employed.

5 × 8 × 40 × 41 2 FIG. 2 .

 584022 FIG. 2. The dimensionless moment of inertia K2 of the gluon plasma as a function of the temperature T for several temporal lattice extensions Nt. The red shaded region, with the central values marked by the red solid line, denotes the tinuum extrapolation, K2 = K (cont.) 2

  Re Tr yΩ I ( Vxyτ + Vxzτ ) -xΩ I ( Vyxτ + Vyzτ ) + xyΩ 2 I Vxzy , (29)

Acknowledgments. The authors are grateful to Oleg Teryaev for useful discussions. This work has been carried out using computing resources of the Federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC "Kurchatov Institute", http://ckp.nrcki.ru/ and the Supercomputer "Govorun" of Joint Institute for Nuclear Research.

Appendix: Simulation details

We use the following lattice gluon action: