Ang Qiao

Theany To

Malwina Stepniewska

Haizheng Tao

Laurent Calvez

Xianghua Zhang

Morten M Smedskjaer

Yuanzheng Yue

Deformation mechanism of a metal-organic framework glass under indentation

Revealing the deformation mechanism of brittle materials under sharp contact loading (indentation) is important for their applications since this knowledge is crucial for identifying the origin of flaw and scratch formation on their surfaces. As a newly emerged glass family, metal-organic framework (MOF) glasses have not been studied concerning the mechanism of their indentation-induced deformation. Here, we explore this mechanism for ZIF-62 glass (a typical MOF glass system). The fractions of densification and shear flow during indentation were determined by atomic force microscopy, while the elastic deformation was analyzed via nanoindentation. The results show that ZIF-62 glass deforms primarily through densification and elastic deformation under the sharp contact loading. Pile-ups around indents were not observed, indicating that no or limited shear flow occurs in the glass during indentation. This behavior could be attributed to three structural factors, namely, high free volume, easily densified glass structure, and limited translational mobility of structural units.

Introduction

Inorganic glasses, except some metallic glasses, are well known for their inherent brittleness and low fracture toughness, leading to easy breakage under tensile stresses. [START_REF] Sehgal | Brittleness of glass[END_REF] The ability of these glasses to resist the formation of surface indents, cracks, and scratches is crucial as these defects are the strength-limiting factors. [START_REF] Rouxel | The fracture toughness of inorganic glasses[END_REF][START_REF] Yoshida | Indentation deformation and cracking in oxide glass -toward understanding of crack nucleation[END_REF] As such, hardness is an important glass property, which measures the glass' ability to resist sharp penetration under indentation loading. During the indentation process, three main deformation modes determine the indentation response of the glass, i.e., elastic deformation, shear flow, and densification. [START_REF] Yoshida | Quantitative evaluation of indentation-induced densification in glass[END_REF] Elastic deformation is a reversible deformation that can fully recover upon unloading. [START_REF] Yoshida | Quantitative evaluation of indentation-induced densification in glass[END_REF] Shear flow is a shear-induced volume conservative displacement, usually resulting in pile-ups around the indent. [START_REF] Hermansen | Densification and plastic deformation under microindentation in silicate glasses and the relation to hardness and crack resistance[END_REF] Densification is a non-volume conservative compression that increases the density around the indent. [START_REF] Hermansen | Densification and plastic deformation under microindentation in silicate glasses and the relation to hardness and crack resistance[END_REF] A comprehensive understanding of the indentation deformation mechanism is critical for improving the hardness and crack initiation resistance of glasses. The indentation deformation mechanism of oxide glasses has been well studied, [START_REF] Yoshida | Quantitative evaluation of indentation-induced densification in glass[END_REF][START_REF] Hermansen | Densification and plastic deformation under microindentation in silicate glasses and the relation to hardness and crack resistance[END_REF][START_REF] Rosales-Sosa | Crackresistant Al 2 O 3 -SiO 2 glasses[END_REF][START_REF] Gross | Crack-free high load Vickers indentation of silica glass[END_REF][START_REF] Houérou | Surface damage of soda-lime-silica glasses: indentation scratch behavior[END_REF] whereas such study has not been performed on metal-organic frameworks (MOF) glasses, which is a new family of melt-quenched (MQ) glasses. [START_REF] Bennett | Hybrid glasses from strong and fragile metal-organic framework liquids[END_REF] The majority of currently known MOF glasses are derived from zeolitic imidazolate frameworks (ZIFs). ZIFs feature a zeolite-like threedimensional framework structure, consisting of the metal nodeligand tetrahedral network, which are constructed by coordination bonds. [START_REF] Hayashi | Zeolite A imidazolate frameworks[END_REF][START_REF] Park | Exceptional chemical and thermal stability of zeolitic imidazolate frameworks[END_REF] ZIF glasses possess a disordered, but fully polymerized network, similar to that of silica glass composed by [SiO 4 ] tetrahedral units, [START_REF] Madsen | Ultrahigh-field 67 Zn NMR reveals short-range disorder in zeolitic imidazolate framework glasses[END_REF][START_REF] Greaves | Inorganic glasses, glass-forming liquids and amorphizing solids[END_REF][START_REF] Stepniewska | Observation of indentation-induced shear bands in a metal-organic framework glass[END_REF] but the former ones feature a higher degree of short-range disorder. [START_REF] Madsen | Ultrahigh-field 67 Zn NMR reveals short-range disorder in zeolitic imidazolate framework glasses[END_REF] Since the discovery of the MQ ZIF glasses, some studies have been carried out concerning their mechanical properties. [START_REF] Stepniewska | Observation of indentation-induced shear bands in a metal-organic framework glass[END_REF][START_REF] Li | Mechanical properties and processing techniques of bulk metal-organic framework glasses[END_REF][START_REF] To | Fracture toughness of a metal-organic framework glass[END_REF] Originally, due to the lack of suitable methods for preparing bulk bubble-free ZIF glasses, only nanoindentation studies were performed on ZIF glasses with the nano-or micron-size to measure their elastic modulus and nano-hardness. [START_REF] Li | Mechanical properties and processing techniques of bulk metal-organic framework glasses[END_REF] Recently, new synthesis strategies have emerged (vacuum hot-pressing methods), enabling the preparation of large size (~10 mm) bubble-free ZIF glasses. This makes it possible to measure the macro-scale mechanical properties of ZIF glasses, as done in three recent studies. First, Li et al. studied the creep resistance, strain-rate sensitivity, and scratch properties of a series of MQ ZIF glasses. [START_REF] Li | Mechanical properties and processing techniques of bulk metal-organic framework glasses[END_REF] where Im is imidazolate and bIm is benzimidazolate. [START_REF] Stepniewska | Observation of indentation-induced shear bands in a metal-organic framework glass[END_REF] The shear bands found in the ZIF glass are in strong contrast to the deformation behavior of other types of fully polymerized glasses. Third, To et al. measured the fracture toughness of the bulk ZIF-62 glass by using a self-consistent single-edge precracked beam method. [START_REF] To | Fracture toughness of a metal-organic framework glass[END_REF] The fracture toughness of the glass was found to be ~0.1 MPa m 0.5 , much lower than that of all known brittle oxide and metallic glasses. By combining reactive molecular dynamics simulations and ab initio calculations, the unique fracture behavior was attributed to the weak coordinative Zn-N bonds. Despite these findings, the following questions remain unanswered: what is the indentation deformation mechanism of ZIF glass? What structural factors affect indentation deformation?

To answer these questions, we investigated the indentation deformation of ZIF-62 glasses. Nanoindentation was performed to determine the extent of elastic deformation of the glasses. Sub-T g (where T g is the glass transition temperature) annealing on indented samples were performed in dry argon gas and subsequently, the indents were imaged using laser scanning microscopy and atomic force microscopy (AFM) to detect the morphology and volume changes of the indents upon annealing. In this way, the ratio of the plastic deformation and densification during indentation can be obtained. By comparing the indentation behaviors of the compressed and the standard ZIF-62 glasses, we studied the effect of structural densification on the crack initiation resistance and hardness of ZIF-62 glasses. Finally, the structural origin of the indentation deformation of ZIF glasses was discussed.

Experimental

Preparation of ZIF-62 glasses

The ZIF-62 crystal was first synthesized by solvothermal method (see Supporting Information). The crystalline nature of the as-prepared ZIF-62 sample was confirmed by X-ray diffraction (XRD), as shown in Fig. S1. The melting temperature (T m ) of ZIF-62 crystal determined by calorimetric measurement is 693 K (Fig. S2).

Subsequently, the ZIF-62 crystal powder was heated to 738 K at 10 K/min and held for 5 min, and then cooled to room temperature in a tube furnace (in inert argon atmosphere) and thereby ZIF-62 glass was obtained (See XRD pattern in Fig. S1 and the glass transition peak in Fig. S2).

This glass was crushed to powder using agate mortar. The powder was used to produce bulk ZIF-62 glass by spark plasma sintering (SPS) under vacuum (HP-D 10 SPS, FCT System GmbH, Germany). Specifically, 0.15 g glass powder was filled into a graphite die (10 mm in diameter) protected by papyex foil and a K-type thermocouple was inserted into a hole in the wall of the die and be touched with the sample. The process chamber was evacuated and the temperature was raised to 593 K at 50 K/min by the resistive heat generated by passing a pulsed direct current (max. 0.35 kA) at 4.65 V through the graphite parts. After the dwelling time of 5 min, the samples were cooled at about 50 K•min -1 .

Since ZIF-62 glass was subjected to a pressure of 60 MPa during SPS synthesis, we denominated the as-prepared glass as 'compressed glass'. Note that the T g of the glass obtained at the ambient pressure was measured to be 599 K by a differential scanning calorimeter (DSC). By reheating the compressed glass to 633 K in DSC, and holding it at this temperature for 5 min, we obtained a relaxed glass, which was designated as 'standard glass'. Subsequent to indentation, the standard glass was annealed at 0.9T g in argon for 2 h in order to recover the volume densified by indentation. [START_REF] Yoshida | Indentationinduced densification of soda-lime silicate glass[END_REF] We refer to this sample as 'annealed ZIF-62 glass'.

Characterizations

The density of the glasses was measured at room temperature according to the Archimedes' principle with ethanol as the soaking solution. For the indentation measurements, the standard and compressed ZIF-62 glass samples were ground using SiC paper of grit size 2400, and 4000, and finally polished using a suspension of diamond particles with 1 μm. The hardness was measured using Vickers indentation (CB5000, Nanovea) with 0.49 N loading. The loading and unloading rates were set to ascertain 6 sec loading time (9.8 N•min -1 ), and 15 sec creeps at the highest load was applied. The indent images were subsequently obtained by an optical microscope and the hardness value (H V ) was calculated as:

𝑯 𝐕 = 𝟏. 𝟖𝟓𝟒𝟒 𝑷 ( 𝒅 𝟏 +𝒅 𝟐 𝟐 ) 𝟐 , (1)
where P is the load, and d 1 and d 2 are the lengths of the indent diagonals. To evaluate the crack resistance (CR) of the glasses to sharp contact loading, the indentations were performed under various loads (0.16 N). Dividing the total number of cracks by the total number of corners (four in each) of the Vickers indents leads to the crack probability at each specific load. Cracking resistance curves were obtained by plotting the crack probability against the load. CR is defined as the load, at which the crack probability is equal to 50%.

To clarify the elastic deformation behavior, we measured the load-displacement curves using nanoindentation experiments. This was done using a Triboindenter instrument (Hysitron, Inc., Minneapolis, MN), which has a high data acquisition rate of up to 104 s -1 . The experiments were performed under loads of 1000-4000 with a Berkovich tip.

The laser scanning microscopy (Lext OLS 4100, Olympus) was performed on series Vickers indents to observe topographical images of the indent sites. About 80 different depth of focus images were recorded to generate the 3-dimensional image of an indent. The indentation impressions were also observed using a Ntegra (NT-MDT) AFM. The volume change caused by annealing was measured in tapping mode at room atmosphere (RH ~50%, ~20 o C). The AFM cantilever had a silicon tip with a scanning frequency of 1.4 Hz.

Results

Fig. 1a shows the load-displacement (P-h) curves for the two types of ZIF-62 glasses, i.e., standard and compressed ones. The P-h curves provide a mechanical fingerprint of the deformation response to indentation load, and mechanical properties such as Young's modulus (E) and hardness (H) can be obtained. [START_REF] Vanlandingham | Nanoindentation of polymers: an overview[END_REF][START_REF] Bhushan | Nanoindentation hardness measurements using atomic force microscopy[END_REF][START_REF] Hainsworth | Analysis of nanoindentation load-displacement loading curves[END_REF] Young's modulus can be calculated based on the Oliver-Pharr method. [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] The reduced modulus (E r ) and nano-hardness derived from the nanoindentation measurements, as well as the density and Vicker micro-hardness, are listed in Table Ⅰ. Compared to the standard ZIF-62 glass, the compressed glass possesses a 2% higher density (Table Ⅰ), implying that the pressure used during the SPS process densifies the glass structure. Upon reheating to above T g , the structure of the compressed glass can partially relax. Vickers hardness of the compressed ZIF-62 glass is also higher than that of the standard one, indicating that the pressure-induced densification makes the ZIF-62 glass harder, as found in various oxide glasses. [START_REF] Kapoor | Pressureinduced densification of oxide glasses at the glass transition[END_REF][START_REF] Zhang | Indentation deformation and fracture of densified silicate glass[END_REF] However, this enhancement of hardness is not observed in the nanoindentation experiments on the same glasses, likely because the slight difference in hardness values is hidden within the error range of hardness measurements via the nanoindentation due to the indentation size effect. By calculating the displacement difference, i.e., (h max -h f )/h max , we obtained the elastic deformation of both the compressed and the standard ZIF-62 glasses, which are 58.7% and 50.4%, respectively. This implies that the elastic deformation contributes more than half to the total deformations during indentation. Next, we consider in more detail the unloading curves that can be used to determine the elastic displacements, contact areas, Young's modulus, and nanohardness. When the unloading curve can be fitted by a power-law relationship [Eq. ( 2)], [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF][START_REF] Doerner | A method for interpreting the data from depth-sensing indentation instruments[END_REF][START_REF] Loubet | Microindentation Techniques in Materials Science and Engineering[END_REF] then these mechanical properties can be determined:

𝑷 = 𝜶(𝒉 -𝒉 𝐟 ) 𝒏 , (2)
where  and n are material constants. By using this equation, we fit the unloading curves of the glasses, finding good agreement (Rsquare is 0.99) between the measured and fitted curves (Fig. 1b), implying that the mechanical properties (especially reduced modulus) derived from the nanoindentation measurement are accurate. [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] The fitting parameters are listed in Table Ⅱ.

Then, based on the Oliver-Pharr method, [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF][START_REF] Fischer-Cripps | Nanoindentation[END_REF][START_REF] Wang | A numerical study of factors affecting the characterization of nanoindentation on silicon[END_REF] the reduced modulus (E r ) of the sample can be derived from unloading data using the following equation:

𝑬 𝒓 = √𝝅 𝟐 𝑺 𝜷√𝑨 , ( 3
)
where S is the measured stiffness (S = dP/dh) of the upper portion of the unloading data, β is the shape constant for the nanoindentation

ARTICLE Dalton Transactions

Please do not adjust margins tip (1.034 for the used Berkovich tip), A is the projected area of the elastic contact. Further, Young's modulus (E) of the ZIF-62 glasses could be calculated from the equation [START_REF] Wang | A numerical study of factors affecting the characterization of nanoindentation on silicon[END_REF] :

𝑬 = (𝟏- 𝟐 ) 𝟏 𝑬 𝒓 - 𝟏- 𝒊 𝟐 𝑬 𝒊 , ( 4
)
where v is Poisson's ratio for the ZIF-62 glasses, E i and v i are the parameters for the diamond indenter, which are 1141 GPa and 0.07, respectively [START_REF] Wang | A numerical study of factors affecting the characterization of nanoindentation on silicon[END_REF] . E r values have been obtained by using Eq. (3) (See Tab. Ⅰ). Poisson's ratio of the ZIF-62 glass was reported to be 0.34. [START_REF] Stepniewska | Observation of indentation-induced shear bands in a metal-organic framework glass[END_REF] Hence, Young's modulus data of ZIF-62 glasses are calculated and listed in Table Ⅱ.

Fig. 2 shows laser microscopy images of Vickers indents in the three ZIF-62 glasses under the load of 0.98 N. Generally, the shear flow during indentation results in pile-ups around the indent. [START_REF] Yoshida | Quantitative evaluation of indentation-induced densification in glass[END_REF][START_REF] Hermansen | Densification and plastic deformation under microindentation in silicate glasses and the relation to hardness and crack resistance[END_REF] However, from the images of the indents of the ZIF-62 glasses with the depth scale (Fig. 2 and3a), no clear signs of pile-ups can be observed around the indents. This suggests that a low degree of plastic deformation occurs in ZIF-62 glasses during indentation. The AFM and laser microscopy cross-section profiles (Figs. 3b andc, and Fig. S3) also confirm that there are no pile-ups.

The indent volume of the compressed ZIF-62 is ~40% smaller than that of the standard one (Figs. 3a and3c), implying that significant pre-densification occurs during the SPS process, which Please do not adjust margins could be associated with partial collapse of the framework cavities. This is likely due to the flexibility of the ZIF glass structure with a large size linker and free volume, [START_REF] Bennett | Hybrid glasses from strong and fragile metal-organic framework liquids[END_REF][START_REF] Stepniewska | Observation of indentation-induced shear bands in a metal-organic framework glass[END_REF] in addition to weak Zn-N coordinative bonds. By performing indentation on the standard glass and subsequently sub-T g (0.9T g ) annealing, we can qualify the densification and plastic shear flow contributions to the total indentation deformation by investigating the volume changes (Fig. 3b). [START_REF] Yoshida | Quantitative evaluation of indentation-induced densification in glass[END_REF][START_REF] Hermansen | Densification and plastic deformation under microindentation in silicate glasses and the relation to hardness and crack resistance[END_REF] In addition, since the densification can induce both reversible and irreversible volume changes, the former can recover upon the sub-T g annealing. Therefore, by calculating the volume changes upon annealing (Fig. 3b), we can obtain the ratio of densified volume to total deformation volume, and this ratio is so called "volume recovery ratio". The volume recovery ratio of ZIF-62 glass is found to be 48.4 %. Fig. 4 shows the relationship between Poisson's ratio and the volume ratio of recovery for various glasses reported by Satoshi et al.. [START_REF] Yoshida | Quantitative evaluation of indentation-induced densification in glass[END_REF] Owning to both high Poisson's ratio (~0.34) of ZIF-62 glass [START_REF] Stepniewska | Observation of indentation-induced shear bands in a metal-organic framework glass[END_REF] and a volume recovery ratio as high as (most) soda-lime-silicate glasses, the ZIF-62 glass does not obey the general trend of volume recovery ratio vs. Poisson's ratio reported for many glass formers, e.g., silica and metallic glasses. Since the ZIF glass structure differs from that of oxide and metallic glasses, this might explain the deviation, although its exact origin remains unknown. However, we note that such deviation from the general trend has previously been observed in some oxide glasses, [START_REF] Kapoor | Pressureinduced densification of oxide glasses at the glass transition[END_REF] indicating that Poisson's ratio is not the only factor affecting the volume recovery ratio. A more quantitative study should be conducted to verify the correlation between the volume recovery ratio and Poisson's ratio as well as the glass structural factors.

Fig. 5 shows the indentation cracking resistance curves for both the compressed and the standard ZIF-62 glasses. As mentioned in the experimental section, CR is defined as the load required for generating two radial/median cracks on average, i.e., a 50% crack probability for Vickers indentation. [START_REF] To | On the relation between fracture toughness and crack resistance in oxide glasses[END_REF] Interestingly, the CR values for both compressed and standard samples are almost the same (~3.8 N), as shown in Fig. 5. On the contrary, in most oxide glasses, predensification leads to reduced CR due to the pressure-induced decrease in deformation through densification, as the latter in turn is believed to reduce the residual stress that drives indentation cracking. [START_REF] Rouxel | Driving force for indentation cracking in glass: composition, pressure and temperature dependence[END_REF]

Discussion

We have observed pronounced densification in ZIF-62 glasses during the indentation, with the ratio of densification to the total indentation volume of about 50%. There should be two types of structural changes induced by densification during indentation. [START_REF] Arora | Indentation deformation/fracture of normal and anomalous glasses[END_REF][START_REF] Ramamurty | Hardness and plastic deformation in a bulk metallic glass[END_REF] One involves the breakage of bonds, leading to permanent volume changes, while the other is related to the distortion of structural units and a decrease of the free volume in glass. The latter part of the structural change caused by the densification can recover upon annealing at 0.9T g . Compared to the oxide glasses, the easier densification in the ZIF-62 glasses could be ascribed to its three different structural features, i.e., (1) the softer Zn[Im/bIm] 4 tetrahedral network constructed by the weaker coordination bonds, (2) larger cavity and (3) the presence of larger ligand. [START_REF] Bennett | Hybrid glasses from strong and fragile metal-organic framework liquids[END_REF][START_REF] Stepniewska | Observation of indentation-induced shear bands in a metal-organic framework glass[END_REF] This unique structural unit leads to the high flexibility of the structure due to the compressible and rotatable organic units. The pressure could also distorts the medium-range (< 20 Å) structural units involving a group of Zn[ligand] 4 tetrahedra. Moreover, the penetration of the sharp indenter could induce the breakage of the weak Zn-N coordinative bonds. Thus, in the area subjected to the load, the glass network is compressed, leading to both the reversible changes (e.g., the rotation of organic ligands and distortion of the tetrahedral network), and the irreversible ones (e.g. the breakage of the local bond induced by high stress). We also note that although the cavities in crystalline ZIF-62 partially collapse during melting, some of these cavities survive and remain in the glass structure, but with smaller sizes (4-8 Å). [START_REF] Zhou | Metal-organic framework glasses with permanent accessible porosity[END_REF][START_REF] Frentzel-Beyme | Porous purple glass-a cobalt imidazolate glass with accessible porosity from a meltable cobalt imidazolate framework[END_REF][START_REF] Thornton | Porosity in metal-organic framework glasses[END_REF] These remaining cavities can be compressed upon indentation, thus contributing to the densification. Furthermore, the existence of numerous cavities also gives rise to a high degree of free C g is the ratio between the volume occupied by the ions and the corresponding effective volume of glass, reflecting the free volume in the glass. The calculation of C g is provided in the Supporting Information. As shown in Tab. S1, C g of ZIF-62 glass is found to be 0.08, much lower than that of silica glass (0.45), as well as other glasses. It proves that the free volume of ZIF glass is significantly higher than other glasses. A high degree of free volume in ZIF glass means a low ability to resistant the compression. The easy densification of the ZIF glass structure is also confirmed by the higher ratio of the density increase () to the applied pressure (P) for the ZIF-62 glass, compared to oxide glasses (Fig. 6). [START_REF] Sharma | Relationship between density, viscosity and structure of GeO 2 melts at low and high pressures[END_REF][START_REF] Smedskjaer | Irreversibility of pressure induced boron speciation change in glass[END_REF][START_REF] Wondraczek | Structureenergy map of alkali borosilicate glasses: Effects of pressure and temperature[END_REF][START_REF] Wakabayashi | Compression behavior of densified SiO 2 glass[END_REF][START_REF] Allwardt | Aluminum coordination and the densification of high-pressure aluminosilicate glasses[END_REF][START_REF] Bista | Aluminosilicate melts and glasses at 1 to 3 GPa: Temperature and pressure effects on recovered structural and density changes[END_REF][START_REF] Mashimo | Hugoniot-compression curve of Zrbased bulk metallic glass[END_REF][START_REF] Liu | Density and elasticity of Zr 46 Cu 37.6 Ag 8.4 Al 8 bulk metallic glass at high pressure[END_REF][START_REF] Hong | Pressure and density dependence of the boson peak in polymers[END_REF] In other words, this is most likely due to the fact that the coordination bonds in the former are much weaker than the ioniccovalent bonds in the latter. [START_REF] Stepniewska | Observation of indentation-induced shear bands in a metal-organic framework glass[END_REF] The high flexibility of Zn[ligand] 4 tetrahedra [START_REF] Wharmby | Extreme flexibility in a zeolitic imidazolate framework: porous to dense phase transition in desolvated ZIF-4[END_REF] gives rise to glass structure with a high degree of elastic deformation, i.e., the ZIF glass network undergoes a certain extent of distortion at the early stage of the loading. Moreover, large organic ligands can also be subjected to a certain degree of distortion, leading to a change in their morphology upon pressure. In other words, the high flexibility of the ZIF glass structure contributes to the elastic deformation during indentation. Hence, we observe the pronounced elastic deformation for ZIF-62 glass during indentation.

Upon indentation, the plastic deformation can induce an upwards slippage in oxide and metallic glasses, leading to pile-ups around the indents. [START_REF] Bhowmick | Plastic flow softening in a bulk metallic glass[END_REF][START_REF] Taljat | Development of pile-up during spherical indentation of elastic-plastic solids[END_REF][START_REF] Das | Materials characterization and classification on the basis of materials pile-up surrounding the indentation[END_REF] However, almost no pile-ups are observed in ZIF-62 glasses upon indentation, implying that the shear flow gives little contribution to the indentation deformation. This behavior is evidently similar to that in silica glass. [START_REF] Howell | Correlations between free volume and pile-up behavior in nanoindentation reference glasses[END_REF] The extent of pile-up during indentation is affected by multiple structural factors. Specially, for the studied ZIF glass, three structural factors should be related to the lack of pile-ups. (1) The remaining cavities with size of 4-8 Å in the ZIF glass structure after melt-quenching, [START_REF] Zhou | Metal-organic framework glasses with permanent accessible porosity[END_REF][START_REF] Frentzel-Beyme | Porous purple glass-a cobalt imidazolate glass with accessible porosity from a meltable cobalt imidazolate framework[END_REF][START_REF] Thornton | Porosity in metal-organic framework glasses[END_REF] and the remarkably low C g value indicate that ZIF glass has higher free volume compared to other glasses. It is well known that glasses with high free volume possess a lower tendency to pile-up. [START_REF] Howell | Correlations between free volume and pile-up behavior in nanoindentation reference glasses[END_REF] (2) It has been reported that glasses with high propensity to densify during indentation do not show pile-ups, [START_REF] Kurkjian | Indentation behavior of soda-lime silica glass, fused silica, and single-crystal quartz at liquid nitrogen temperature[END_REF] since such glass structure exhibits densification rather than shear flow. The ZIF glass network can be easily densified due to the weak Zn-N coordination bonds (compared to the ioniccovalent bonds in oxide glasses), which fracture easily upon mechanical loading. [START_REF] To | Fracture toughness of a metal-organic framework glass[END_REF] Moreover, the high free volume also contributes to the easy densification, leading to the significant local deformation rather shear flow. (3) It has been suggested that the indentation induced shear flow in glass is correlated with the viscous behavior of the corresponding melt. [START_REF] Kjeldsen | Role of elastic deformation in determining the mixed alkaline earth effect of hardness in silicate glasses[END_REF] Specifically, for a glass that undergoes no shear flow during indentation, its liquid state often possesses relatively high viscosity that hinders the translational motion of the involved structural units. Silica glass and ZIF-62 glass both possess very high viscosity at T m (10 5.5 and 10 5 Pa•s, respectively). [START_REF] Qiao | A metal-organic framework with ultrahigh glass-forming ability[END_REF] According to previous studies, [START_REF] Sellappan | Composition dependence of indentation deformation and indentation cracking in glass[END_REF][START_REF] Rouxel | Indentation deformation mechanism in glass: Densification versus shear flow[END_REF] densification dominates the indentation deformation of oxide glasses with low Poisson's ratio (< 0.25) or relatively high network polymerization degree, i.e., those with low modified contents, whereas the shear flow is more predominant for glasses with high Poisson's ratio. However, ZIF-62 glass exhibits a pronounced densification and only a minor extent of shear flow upon indentation, although it possesses a relative high Poisson's ratio (~0.34) and a fully polymerized network. This anomalous phenomenon could be attributed to the following scenarios. In contrast to the tetrahedral [SiO 4 ] unit in oxide glasses, the tetrahedral structural unit in ZIF glasses is much bigger, because they are constituted by a metallic node (Zn) and four Im/bIm ligands that are significantly bigger than oxygen. Thus, the structure of ZIF-62 glass is easier to be distorted and densified under mechanical load, rather than shear deformation despite its higher Poisson's ratio (0.34) compared to the polymerized oxide glasses. Moreover, the weaker Zn-N bonds are also the reason for the high degree of densification and only limited shear flow since such bonds readily break and thereby easily induce a deformation of medium-range structure. 9

Conclusions

Based on the indentation study of ZIF-62 glass, we found that more than 50% of the total deformation during load-unloading arises from the elastic deformation, while the densification dominates the permanent volume change during indentation. The easier densification of ZIF-62 glass (compared with other types of network glasses such as silicate glasses) is attributed to both the higher flexibility of the glass network and the higher porosity. Interestingly, the shear flow does not occur in ZIF-62 during indentation, and this distinct feature can be ascribed to three structural factors: high free volume, easily densified Fig. 6. Pressure dependence of density difference (ρ1-ρ0) in glasses. ρ1 and ρ0 are the density values of compressed and initial glasses, respectively. ZIF-62 is from this study. Oxide glasses: GeO2, [START_REF] Sharma | Relationship between density, viscosity and structure of GeO 2 melts at low and high pressures[END_REF] Na2O-CaO-B2O3 (NCB), [START_REF] Smedskjaer | Irreversibility of pressure induced boron speciation change in glass[END_REF] Na2O-B2O3-SiO2 (NBS), [START_REF] Wondraczek | Structureenergy map of alkali borosilicate glasses: Effects of pressure and temperature[END_REF] SiO2, [START_REF] Wakabayashi | Compression behavior of densified SiO 2 glass[END_REF] CaO-Al2O3-SiO2 (CAS), 42 K2O-Al2O3-SiO2 (KAS), [START_REF] Allwardt | Aluminum coordination and the densification of high-pressure aluminosilicate glasses[END_REF] Na2O-Al2O3-SiO2 (NAS), [START_REF] Allwardt | Aluminum coordination and the densification of high-pressure aluminosilicate glasses[END_REF] Na2O-3SiO2 (NS3) and Jadeite [START_REF] Bista | Aluminosilicate melts and glasses at 1 to 3 GPa: Temperature and pressure effects on recovered structural and density changes[END_REF] ; Metallic glasses: Zr-Al-Ni-Cu [START_REF] Mashimo | Hugoniot-compression curve of Zrbased bulk metallic glass[END_REF] and Zr-Cu-Ag-Al [START_REF] Liu | Density and elasticity of Zr 46 Cu 37.6 Ag 8.4 Al 8 bulk metallic glass at high pressure[END_REF] ; Polymer glasses: PIB [START_REF] Hong | Pressure and density dependence of the boson peak in polymers[END_REF] , PS [START_REF] Hong | Pressure and density dependence of the boson peak in polymers[END_REF] and PMPS [START_REF] Hong | Pressure and density dependence of the boson peak in polymers[END_REF]

  Second, Stepniewska et al. reported the cracking patterns and shear bands induced by indentation in the representative ZIF-62 glass with the composition Zn[Im 1.75 bIm 0.25 ],

Fig. 1 .

 1 Fig. 1. Load-displacement curves for ZIF-62 glasses. (a) Loading-unloading curves obtained under 2000 N maximum load for compressed and standard ZIF-62 glass. hmax is the maximum displacement at the maximum loading and hf is the final depth after unloading; (b). Unloading curves for the compressed and standard ZIF-62 glasses. Solid curves: the power-law fits [Eq. (2)].[START_REF] Loubet | Microindentation Techniques in Materials Science and Engineering[END_REF]

Fig. 2 .

 2 Fig. 2. Laser microscopy images of Vickers indents. (a) Standard, (b) annealed, and (c) compressed ZIF-62 glasses.

Fig. 3 .

 3 Fig. 3. Three dimensional images and cross-section profiles for indents upon 0.98 N load on the ZIF-62 glasses. (a). 3-D images captured by the laser microscopy for (left to right) the standard, annealed and compressed ZIF-62 glasses; (b). Cross-section profiles obtained by AFM measurements for the standard and annealed ZIF-62 glasses; (c). Cross-section profiles obtained by AFM measurements for the compressed and standard ZIF-62 glasses.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Relationship between Poisson's ratio and the volume recovery ratio from indentation. Comparison of the data for ZIF-62 glass with that of other reference glasses from refs. 4, 28, 29, 30. Poisson's ratio of ZIF-62 glass is obtained from ref. 14.

Table 1 .

 1 Comparison of the mechanical properties for the ZIF-62 glasses

	Samples	Density	Microhardness	Nanohardness	Reduced modulus	Crack	Elastic
		(g/cm 3 )	(GPa)	(GPa)	(GPa)	resistance (N)	recovery (%)
	Compressed ZIF-62 glass	1.5060.005	0.590.02	0.700.03	7.540.03	3.8	58.7
	Standard ZIF-62 glass	1.4810.003	0.520.02	0.740.03	6.770.02	3.9	50.4
	Annealed ZIF-62 glass	N/A	0.530.02	0.670.02	6.670.02	N/A	50.5

Table 2 .

 2 Fitting parameters for the loading and unloading curves. The unloading curves were fitted by the power-law proposed by Loubet et al. [Eq. (2)].[START_REF] Loubet | Microindentation Techniques in Materials Science and Engineering[END_REF]

	Samples	Constant 	Constant n	R-Square	Young's modulus (GPa)
	Compressed ZIF-62 glass	0.880.01	1.4040.002	0.998	6.660.03
	Standard ZIF-62 glass	0.920.01	1.4240.002	0.999	5.970.02

  are taken from publications. and strong steric hindrance of the fully polymerized network structure. The lack of shear flow during indentation is indirectly connected with the significantly higher viscosity of ZIF-62 glass compared with other not-fully polymerized network glasses.

	1 - 0 )/P (gcm -3 GPa -1 )	0.1 0.2 0.3 0.4	Jadeite CAS ZIF-62 glass Oxide glasses: GeO2 NCB SiO2 KAS NAS N3S	Metallic glasses: Zr-Al-Ni-Cu Zr-Cu-Ag-Al Polymer glasses: PIB PS PMPS
	(
		0.0
		0	4	8	12
				P (GPa)

structure

Acknowledgements

This work was supported by VILLUM Fonden (grant no. 13253) and National Natural Science Foundation of China (No. 51802236).

Conflicts of interest

There are no conflicts to declare.


Prosenjit Haldar

Sen Mu
email: senmu@u.nus.edu

Bertrand Georgeot

Jiangbin Gong

Christian Miniatura

Gabriel Lemarié
email: lemarie@irsamc.ups-tlse.fr

Prethermalization and wave condensation in a nonlinear disordered Floquet system

Periodically-driven quantum systems make it possible to reach stationary states with new emerging properties. However, this process is notoriously dicult in the presence of interactions because continuous energy exchanges generally boil the system to an innite temperature featureless state.

Here, we describe how to reach nontrivial states in a periodically-kicked Gross-Pitaevskii disordered system. One ingredient is crucial: both disorder and kick strengths should be weak enough to induce suciently narrow and well-separated Floquet bands. In this case, inter-band heating processes are strongly suppressed and the system can reach an exponentially long-lived prethermal plateau described by the Rayleigh-Jeans distribution. Saliently, the system can even undergo a wave condensation process when its initial state has a suciently low total quasi-energy. These predictions could be tested in nonlinear optical experiments or with ultracold atoms.

Introduction. The nonequilibrium dynamics of periodically driven quantum systems has been under intensive scrutiny over the past few years [START_REF] Mikael C Rechtsman | Photonic oquet topological insulators[END_REF]. It has been shown that interesting stationary states could emerge after appropriate Floquet engineering [619]. However, the interplay between interactions and temporal driving generally induces heating processes which force the system into a featureless state of innite temperature [2022].

Dierent strategies have been discussed to prevent this detrimental heating, e.g. many-body localization [23 28], or coupling to a bath [START_REF] Karthik | Controlled population of oquet-bloch states via coupling to bose and fermi baths[END_REF][START_REF] Iadecola | Occupation of topological oquet bands in open systems[END_REF]. An alternate route is to use high-frequency driving to maintain the system in a long-lived metastable state before heating to innite temperature takes over. Known as prethermalization, this strategy works for both quantum [8,[START_REF] Bukov | Prethermal oquet steady states and instabilities in the periodically driven, weakly interacting bose-hubbard model[END_REF]3338] and classical systems [3942].

In this Letter, we consider a variant of the quantum kicked rotor (QKR), a paradigmatic model of quantum chaos [START_REF] Haake | Quantum Signatures of Chaos[END_REF][START_REF] Felix | Simple models of quantum chaos: Spectrum and eigenfunctions[END_REF], in the presence of a weak nonlinear Gross-Pitaevskii (GP) interaction term. In the absence of interactions, the QKR shows dynamical localization [START_REF] Casati | Stochastic behavior of a quantum pendulum under a periodic perturbation[END_REF], a phenomenon analogous to Anderson localization [START_REF]50 Years of Anderson Localization[END_REF] but in momentum (p) space [START_REF] Grempel | Quantum dynamics of a nonintegrable system[END_REF] which has been observed in a number of cold atom experiments [4852]. The presence of GP interaction terms challenges dynamical localization: transport is no longer frozen and anomalous diusion rules the spreading of wavepackets [5,[START_REF] Dl Shepelyansky | Delocalization of quantum chaos by weak nonlinearity[END_REF][START_REF] Gligori¢ | Interactions destroy dynamical localization with strong and weak chaos[END_REF][START_REF] Lellouch | Dynamics of the mean-eld-interacting quantum kicked rotor[END_REF].

Analogous observations have been made in disordered nonlinear Schrödinger chains, see e.g. [5762].

On the other hand, it is known that the propagation of a random initial wave in a nonlinear medium can give rise to Rayleigh-Jeans (RJ) thermalization and wave condensation phenomena as observed in nonlinear optics [7,[START_REF] Sun | Observation of the kinetic condensation of classical waves[END_REF][START_REF] Aschieri | Condensation and thermalization of classical optical waves in a waveguide[END_REF] and explained by wave turbulence theory [1,[START_REF] Nazarenko | Wave turbulence[END_REF][START_REF] Nazarenko | Wave turbulence[END_REF]. A similar situation happens when an initial plane wave propagates in a disordered nonlinear medium, disorder inducing the required wave randomization [4]. This conguration is particularly interesting because (i) it is the natural setting to observe the coherent backscattering (CBS) [7073] and coherent forward scattering (CFS) eects [7480] when interactions are absent and (ii) it allows to control the initial energy of the wave and thus the route to wave condensation in nonlinear systems.

In this Letter, we study RJ thermalization and wave condensation phenomena for the nonlinear QKR and establish when they happen in this Floquet system: the quasi-energies of the system should form suciently narrow bands as compared to their separation. This happens when the kick and disorder strengths are smaller than a characteristic threshold. In this case, heating processes associated to inter-band transitions are strongly suppressed and the system can reach an exponentially long-lived prethermal plateau where RJ thermalization takes place. Moreover, wave condensation at the bottom of the fundamental Floquet band is observed. Ultimately, at exponentially large times, the system heats up to the innite-temperature state.

Model. Our model features the nonlinear QKR with a GP interaction term in p-space [START_REF] Dl Shepelyansky | Delocalization of quantum chaos by weak nonlinearity[END_REF]:

i∂ t ψ(p, t) = H(t)ψ(p, t) + gN a |ψ(p, t)| 2 ψ(p, t) (1)
where H(t) = p2 /2 -K cos θ j δ(t -j) ( = 1, moment of inertia I = 1). Here, θ ∈ [-π, π) is the rotor angle, p = -i∂ θ the momentum, K the kick strength, g the nonlinear strength, and N a the number of atoms. Because ψ(θ + 2π, t) = ψ(θ, t), we have Nonlinearity quickly erases these CBS-CFS interference effects. For suciently small W and K, W + K < ∼ π/2, the system reaches a prethermal equilibrium when t > τeq, with a quasi-stationnary neq(θ). Ultimately, after a time τ boil which can be exponentially large, Floquet heating brings the system to an innite temperature state with uniform n∞(θ).

ψ - n+1 (p) = e -iα(p) e -igN |ψ + n (p)| 2 ψ + n (p), (2)
ψ + n+1 (θ) = e iK cos θ ψ - n+1 (θ). (3)
Here, we have replaced the quasi-random phases induced by the kinetic term p2 /2 by true random phases α(p). Usually, they are taken from a uniform distribution within [-π, π] [START_REF] Grempel | Quantum dynamics of a nonintegrable system[END_REF][START_REF] Birkho | Proof of the ergodic theorem[END_REF]. In our study, we crucially consider smaller intervals [-W, W ], where W ≤ π is the disorder strength. The motivation for choosing local interactions in p is now clear: First, it transposes to p-space the usual case where disorder and interactions are local in position (see e.g. [4]). Second, the associated quantum map Eq.( 3) allows to evolve systems of large sizes N ∼ 10 5 up to large times t ∼ 10 6 at low numerical cost. Without interaction, localization occurs in p-space and CBS and CFS take place in θ-space [START_REF] Lemarié | Coherent backscattering and forwardscattering peaks in the quantum kicked rotor[END_REF]. Starting from an initial rotor angle θ 0 , ψ + 0 (p) = Following the analogy with thermalization and condensation predicted in disordered nonlinear systems [4], we consider the eect of interactions on these peaks. Hereafter, we set the GP interaction and disorder strengths to g = 0.1 and W = 0.4 respectively and we study the role of the kick strength K and initial rotor angle θ 0 on the dynamics of the system. Depending on K and θ 0 , we nd 3 dierent dynamical behaviors: (i) the system quickly adopts a uniform θ-distribution n ∞ (θ) = 1 ("innite temperature" featureless state); (ii) the system transits through a long-lived metastable state characterized by a nontrivial θ-distribution n eq (θ) centered at θ = 0 and related to the thermal RJ distribution [START_REF] Baudin | Classical rayleigh-jeans condensation of light waves: Observation and thermody-namic characterization[END_REF] before eventually going to n ∞ (θ); and (iii) the system enters a condensation regime and develops a peak structure at θ = 0 on top of n eq (θ) before, again, eventually going to n ∞ (θ). Crucially, the life time of the metastable state can be made exponentially large, allowing for the observation of thermalization or condensation.

Characteristic time scales. The dierent characteristic time scales are illustrated in Fig. 1. First, the CBS and CFS peaks quickly decay over a characteristic time τ g ∝ g -1 (see Supplementary Material (SM) and [5,6]), signaling that the GP interaction term is indeed wiping out disorder-induced interference eects.

Meanwhile, the GP term redistributes the energy over the dierent Floquet modes and the system reaches a socalled prethermalization plateau after some equilibration time τ eq ∼ g -2 (see SM and [4]) where it stabilizes in a metastable state with distribution n eq (θ). The system gets eventually boiled, around time τ boil , to the innite temperature state n ∞ (θ) = 1. The lower panel of Fig. 1 clearly shows the equilibration, metastable and heating regimes by plotting the variance σ 2 θ (t) = θ 2 -θ 2 of n(θ, t) as a function of time (σ 2 θ (t → ∞) = π 2 /3 for n ∞ (θ) = 1). Hereafter, we focus on the parameter sector τ g τ H and τ eq τ B , τ g . In this case, interference eects are negligible and multiple scattering randomizes the wave much faster than the system equilibrates.

Floquet-Boltzmann kinetic equation. The noninteracting system is characterized by the Floquet .

(𝑎𝑎) (𝑏𝑏)
Hamiltonian H F ≡ H(t) -i∂ t . Its linear modes |φ α (t) solve H F |φ α (t) = ε α |φ α (t) , ε α being called a quasi- energy. H(t)
and |φ α (t) have the same time-periodicity and ε α is dened modulo the driving frequency Ω = 2π

here.

The support of the quasi-energy spectrum is

ε α ∈ [-E D , E D ] with E D = K + W for E D ≤ π.
Expanding the state of the interacting system over the Floquet modes,

|ψ(t) = 1 √ N N α=1 c α (t)|φ α (t) ,
f ε (t) ≡ 1 N N α=1 δ(ε -ε α )|c α (t)| 2 /ν(ε), where ν(ε) = 1 N α δ(ε -ε α ) is the density of states of the linear kicked rotor. It satises π -π dε ν(ε)f ε (t) = 1 since α |c α (t)| 2 = N .
In the limit τ g τ H and τ eq τ B , the θ-distribution at time t ∈ N reads [4]:

n(θ, t) ≈ π -π dε A ε (θ)f ε (t), (4)
where

A ε (θ) = 1 N α δ(ε -ε α )|φ α (θ, 0)| 2 is the spectral function.
For time-independent systems, a kinetic equation for f ε (t) can be derived under certain approximations [1,[START_REF] Wang | Wave-turbulence origin of the instability of anderson localization against many-body interactions[END_REF][START_REF] Nazarenko | Wave turbulence[END_REF],

∂ t f ε = 4πg 2 dε 2 dε 3 dε 4 R(ε, ε 2 , ε 3 , ε 4 ) ν(ε 2 )ν(ε 3 )ν(ε 4 ) [f ε f ε3 f ε4 + f ε2 f ε3 f ε4 -f ε f ε2 f ε3 -f ε f ε2 f ε4 ]. (5)
The 4-wave mixing collision kernel R imposes energy conservation for each energy-exchange process (ε + ε 2 = ε 3 + ε 4 ). We can generalize this description to Floquet systems by expanding the time-periodic Floquet modes in Fourier series φ (m) [ [START_REF] Bilitewski | Scattering theory for oquet-bloch states[END_REF][START_REF] Genske | Floquetboltzmann equation for periodically driven fermi systems[END_REF], thereby incorporating the so-called Umklapp processes [START_REF] Kittel | Introduction to solid state physics[END_REF] describing inter-band transitions:

α (p) = 1 0 dt φ α (p, t) exp(-2iπmt) (𝑎𝑎) (𝑏𝑏)
R = m∈Z R (m) (ε, ε 2 , ε 3 , ε 4 )δ(ε+ε 2 -ε 3 -ε 4 +2πm), (6)
see SM and [2,4]. The kernels R (m) relate to an overlap between four Fourier Floquet mode amplitudes.

Prethermalization. Following this approximate approach, the collision kernel (6) predicts a suppression of heating when Umklapp processes m = 0 are forbidden. This condition is met when the Floquet bandwidth

2E D is suciently narrow, E D = K + W < π/2.
In this case, we eectively recover the kinetic equation ( 5) of time-independent systems and the total linear contribution of the quasi-energy per mode of the system [7], is conserved. An equilibrium distribution is then obtained by canceling the bracketed term in Eq. ( 5). One nds the RJ distribution

E tot = π -π dε ε ν(ε) f ε (t), see
f RJ (ε) = T ε-µ [1, 2, 4]
where T and µ play respectively the role of temperature and chemical potential. Note τ eq t τ boil . These observations conrm the Floquet-Boltzman approach. However, at times t > ∼ τ boil , we observe heating: f ε (t), n(θ, t), E tot and σ 2 θ vary rapidly with time (see Figs. 1,4 and SM). Note that f ε (t) can still be tted by a RJ distribution, with time-dependent T (t) → +∞, µ(t) → -∞ and T (t)/µ(t) → -1 when t → ∞ (see SM). While Eqs. ( 5)-( 6 Phase diagram. We check the Onsager-Penrose criterion for condensation [10,11] by computing the time coarse-grained 1-body density matrix given by θ|ρ (1)

that µ ≤ -E D since f RJ (ε) > 0.
(t)|θ = 1 ∆t + 1 t+∆t t =t ψ * (θ , t )ψ(θ, t ) (7)
with ∆t = 30. This coarse-graining emulates a mixed state from the pure state ψ(θ, t) [11,[START_REF] Góral | Thermodynamics of an interacting trapped bose-einstein gas in the classical eld approximation[END_REF]. Writing ρ (1) (t) = n P n (t)|Φ n (t) Φ n (t)|, normalization implies n P n (t) = 1 and P n represents the occupation probability of the mode |Φ n . Sorting the probabilities by descending order P 1 ≥ P 2 ≥ P 3 ≥ •••, a 'macroscopic' probability P 1 P 2 signals condensation of the system into the eigenmode |Φ 1 . In the left panel of Fig. 4, we plot the population gap ∆P = P 1 -P 2 obtained at t = 10 4 in the (K, θ 0 ) plane. We see that condensation occurs below the threshold E D = K + W = π/2 (i.e. K below the black dashed line in Fig. 4a) where the approximate Eqs. ( 5)-( 6) predict a suppression of heating. Furthermore, as mentioned above, condensation indeed occurs (𝑎𝑎) (𝒃𝒃) for θ 0 below the θ c 0 (K) critical line (red dots in Fig. 4a).

Saliently, the duration of this prethermal plateau can be made exponentially large, as shown in Fig. 4b where we plot τ boil (K, θ 0 ) (see SM). This key observation should allow for an observation of RJ prethermalization and wave condensation in experiments with Floquet systems.

Note that we have veried that the measured quantities did not show signicant nite-size eects (see SM).

Conclusion.

We have studied the nonequilibrium dynamics of a disordered Floquet system subjected to a nonlinear Gross-Pitaevskii interaction. When Floquet quasi-energy bands are suciently narrow and well separated, inter-band transitions are forbidden and heating is strongly suppressed. This condition generalizes to disordered Floquet system the fast-driving condition for clean Floquet systems [8,9]. It allows the system to reach a prethermal plateau where it stays for an exponentiallylong time before heating processes ultimately boil it to an innite-temperature featureless state. In the prethermal plateau, low-energy physics takes place in the form of Rayleigh-Jeans prethermalization, and wave condensation at low quasi-energies.

Our predictions are based on a variant of the kicked rotor already realized with ultracold atoms and could therefore be tested in such experiments. It was argued that GP interaction in p is a good description of the spatial interaction at weak nonlinearities [START_REF] Gligori¢ | Interactions destroy dynamical localization with strong and weak chaos[END_REF][START_REF] Lellouch | Dynamics of the mean-eld-interacting quantum kicked rotor[END_REF]. Narrow Floquet bands can be experimentally achieved by working in the vicinity of quantum resonances [START_REF] Izrailev | Quantum resonance for a rotator in a nonlinear periodic eld[END_REF][START_REF] Wimberger | Quantum resonances and decoherence for δ-kicked atoms[END_REF]. Very recently, the many-body kicked rotor has been the subject of experimental [START_REF] Cao | Prethermal dynamical localization and the emergence of chaos in a kicked interacting quantum gas[END_REF] and theoretical [START_REF] Vuatelet | Eective thermalization of a many-body dynamically localized bose gas[END_REF] studies on prethermalization. Our study proposes new regimes and dierent characterizations of this physics. The Gross-Pitaevskii equation also appears in nonlinear optics.

Experiments with disordered photonic lattices [98100] could also address the prethermalization properties we have discussed. Future work could address superuid or turbulent transport in such Floquet systems [START_REF] Albert | Breakdown of the superuidity of a matter wave in a random environment[END_REF][START_REF] Paul | Anderson localization of a weakly interacting one-dimensional bose gas[END_REF].

φ α (p, t) = φ α (p, t + 1) [H(t) -i∂ t ]φ α (p, t) = ε α φ α (p, t) p∈Z φ * α (p, t)φ β (p, t) = δ αβ (S1)
with a continuum of quasi-energies ε α ∈ [-π, π]. In our numerical simulations, we have however considered a nite-size momentum basis set restricted to N states, -N/2 ≤ p ≤ N/2 -1, with periodic boundary conditions. The Hilbert space of the system having dimension N , we also need to consider N linear Floquet modes φ α (p, t) (1 ≤ α ≤ N ) and expand ψ(p, t) over these N Floquet modes

ψ(p, t) = 1 √ N N α=1 c α (t) φ α (p, t), (S2)
with the updated nomalisation conditions

N/2-1 p=-N/2 φ * α (p, t)φ β (p, t) = δ αβ N/2-1 p=-N/2 |ψ ( p, t)| 2 = 1. (S3)
In turn, these two conditions imply the normalisation condition N α=1 |c α | 2 = N . From the nonlinear Schrödinger equation (1) of the manuscript, it is easy to see that the modal coecients c α satisfy the following equation of motion

i dc α dt = ε α c α + g βγδ W αβγδ c * β c γ c δ , (S4)
featuring the 4-point Floquet modes correlator

W αβγδ (t) = 1 N p φ * α (p, t)φ * β (p, t)φ γ (p, t)φ δ (p, t). ( S5
)
At this stage, Eqs.(S4)-(S5) are exactly Eqs.(148-149) in [S1] and Eqs. (6)(7) in [S2] with the salient dierence that our correlator is time-dependent and, most importantly, time-periodic W αβγδ (t + 1) = W αβγδ (t) since the Floquet modes themselves are time-periodic. We thus expand W in Fourier series:

W αβγδ (t) = m∈Z W (m) αβγδ e -2iπmt . (S6)
Now, changing variables c α → cα = c α exp(iε α t) and dropping the tilde to ease the notation, Eq.(S4) rewrites

i dc α dt = g mβγδ e i(εα+ε β -εγ -ε δ +2πm)t W (m) αβγδ c * β c γ c δ . ( S7
)
At this point, one can follow the usual steps of the derivation of the irreversible kinetic equation by using the random phase approximation developed in [S3] and by taking the continuum limit (large N assumption) as detailed in pages 51-57 and Appendix A7 of [S1] and in [S2]. By doing so in our case, the discrete sums become integrals over quasi-energies between -π and π and introduce products of the disorder-averaged density of states per unit volume ν( ) = 1 N α δ(ε -ε α ). After tedious calculations, one gets Eqs. (4)(5) in the main text with R (m) = |W (m) | 2 . Do note that the large-N limits of sums are obtained through:

1 N θ (• • •) → π -π dθ 2π (• • •) 1 N α (• • •) → π -π dε ν(ε) (• • •) (S8)
A remark is in order here. In the usual wave turbulence (WT) approach [S1], the free Hamiltonian does not generate any randomness in the evolution of the system: One starts with an initial state with static random phases and uses averages over these initial random phases to derive the kinetic equation. Here, we instead start with a perfectly phase-coherent initial state and it is the subsequent dynamics generated by the disordered Floquet evolution operator that brings in phase randomness. In this case, as argued in [S4] for weakly nonlinear spatially disordered systems, one has just to replace the averages over the static initial random phases in the WT approach by disorder conguration averages (the number of disorder congurations used in our numerical simulations is denoted by N d ).

This prescription is at least valid as long as the Boltzmann time τ B associated to disorder is much smaller that the equilibration time τ eq associated to the nonlinear couplings. In other words, disorder quickly isotropizes the system before nonlinearity starts to signicantly modify the energy distribution.

It is crucial to note that the Floquet kinetic equation does conserve particle number but does not conserve necessarily the total initial quasi-energy. Indeed, the δ(ε α + ε β -ε γ -ε δ + 2πm) term coming along with each collision kernel R (m) shows that Floquet systems exhibit collision processes satisfying energy conservation up to nonzero multiples of the driving frequency Ω = 2π (in Eq. ( 1) the kicking period is T = 1) in addition to the strictly resonant energy transfer term m = 0. For periodically-driven systems, these m = 0 processes are the equivalent of Umklapp scattering processes in space-periodic systems, where "colliding" wavevectors are folded back to the rst Brillouin zone through appropriate reciprocal lattice vector translations. Note that these Umklapp processes heat and boil the system. In this approximate description, it is only when these m = 0 processes are forbidden (i.e. when E D < π/2) that the total energy of the system is conserved, in which case the system can reach and stay in the Rayleigh-Jeans (RJ) thermal equilibrium and, if appropriate conditions are met, undergo the classical wave condensation scenario (see below).

However, the heating observed at long times t τ boil for E D < π/2 goes beyond this Floquet kinetic description.

SCALING OF THE CBS AND CFS DECAY TIME AND EQUILIBRATION TIME WITH INTERACTION STRENGTH

As seen in Fig. 1 of the main text, when nonlinearity is switched on, the CBS and CFS peaks quickly die o over some characteristic decay time τ g for the chosen system parameters. Meanwhile, the nonlinearity term also redistributes the energy over dierent Floquet modes and the system reaches either the RJ distribution or wave condensation after a larger time scale τ eq . Then the system may stay at the prethermal plateau until τ boil . For our system with an interaction term local in p, it was argued that the nonlinear time scale τ g corresponds to the interaction energy stored in a localization volume, hence τ g ∝ g -1 [S5] (see however [S6] where τ g ∝ g -2 is predicted at very small g), and the equilibration time τ eq ∝ g -2 [S4]. In Fig. S1 left panel, we characterize τ eq by the time at which the θ-variance σ 2 θ (t) reaches a given threshold value (indicated by the horizontal black line in Fig. S1) between its initial value and its quasi-stationary value in the prethermal plateau. On the other hand, we dene τ g by tting the early time dynamics of the CBS and CFS peaks by an exponential decay n(±θ 0 , t) ∼ n(±θ 0 , t = 0) exp(-t/τ g ).

In Fig. S2, we have plotted the extracted inverse decay rates τ -1 g and τ -1 eq as a function of g and g 2 . As one can see, the agreement with the conjectured scaling relations, τ g ∝ g -1 and τ eq ∝ g -2 , is pretty good. Note however that the data presented in this paper correspond to values of the nonlinear interaction strength signicantly larger than those considered in [S6]. Data (not shown) for τ g with g ∈ [10 -3 , 10 -2 ] seem compatible with the prediction τ g ∝ g -2 of [S6]. On the other hand, Fig. S1 right panel shows that the curves of σ 2 θ (t) for dierent values of g all collapse onto each other in the transition regime between the initial stage and the prethermal plateau when plotted as a function of tg 2 . This again validates τ eq ∝ g -2 .

10 0 10 1 10 2

VARIATION OF THE TOTAL QUASI-ENERGY WITH INITIAL POSITION

The linear contribution to the total quasi-energy per mode of the system, E tot = π -π dε ε ν(ε) f ε (t), see [S7], depends on the initial rotor angle θ 0 through the initial state ψ(p, t = 0) = 1 √ N exp(-ipθ 0 ). The other system parameters being xed, low values of θ 0 correspond to low quasi-energies E tot . Indeed, for our system, θ corresponds to wave-vector k in spatially disordered systems. It is worth mentioning that E tot is symmetric in θ 0 because the kicked rotor Hamiltonian is symmetric in θ.

In the right panel of Fig. S3, we plot E tot as a function of θ 0 at 3 dierent times (t = 0, t = 10 3 and t = 10 4 ) and for 2 dierent kick strengths (K = 1 and K = 1.3). The interaction and disorder strengths are xed at g = 0.1, W = 0.4. For K = 1.3, the Umklapp processes m = 0 are present (E D = K + W ≥ π/2) and E tot increases fast with time. On the other hand, the Umklapp processes m = 0 are suppressed for K = 1 (E D = K + W < π/2) and E tot is almost conserved and independent of time in the range t ∈ [0, 10 4 ]. Note that E tot at t = 0 reads

E tot (θ 0 , t = 0) = π -π dε ε A ε (θ 0 ) (S10)
(t = 0) = φ * α (θ 0 , t = 0) and ν(ε)f ε (t = 0) = 1 N N α=1 δ(ε -ε α )|c α (t = 0)| 2 = 1 N N α=1 δ(ε -ε α )|φ α (θ 0 , t = 0)| 2 is the spectral function A ε (θ 0 ).
Moreover, we observe that the quasi-energy distribution f ε (t) can be tted by a Rayleigh-Jeans distribution not only in the prethermal regime but also in the heating regime t > ∼ τ boil . In the left and middle panels of Fig. S3, we show the extracted temperature T (t) and chemical potential µ(t) at dierent times for the same values of K as in the right panel. Both T and µ stay essentially constant if the system resides in the prethermal plateau τ eq t τ boil (case K = 1 in Fig. S3 where E D = K + W < π/2). On the contrary, in the heating regime (K = 1.3 in Fig. S3) T (t) → +∞ and µ(t) → -∞, while T (t)/µ(t) → -1 when t → ∞. In other words, the system reaches an innite temperature state with at distributions f ε (t) and n(θ, t) when t τ boil .

DETERMINATION OF τ boil FROM THE DYNAMICS OF THE SPATIAL DISTRIBUTION

As discussed in the manuscript, after a time scale τ eq , the system reaches a quasi-stationary state well described by a Rayleigh-Jeans thermal distribution or showing signatures of wave condensation, up to a characteristic time τ boil after which the system heats up rapidly. Here we demonstrate that the duration of the prethermal plateau (τ boil -τ eq ) scales exponentially with the parameter K. To extract τ boil , we plot σ 2 θ (t) and its logarithmic derivative as a function of log 10 t, see Figure S4. There are two clear peaks in the logarithmic derivative curve: The rst one corresponds to τ eq (see Fig. S1 for a more precise numerical determination) while the second one gives τ boil . One can then study the dependence of (τ boil -τ eq ) on the dierent parameters of the system, in particular K.

The upper panels of Fig. S5 show that all curves σ 2 θ (t) obtained for dierent values of K almost collapse onto each other up to a timescale ∼ 10 2 . This agrees with the expectation that τ eq ∼ g -2 (g = 0.1 in the plot) does not strongly depend on K. The lower panels of Fig. S5 show τ boil as a function of K. For θ 0 = 1.09 (lower left panel), we observe that τ boil scales exponentially with K, validating a very long prethermal plateau in our system (τ boil -τ eq ) ≈ τ boil when E D = K +W < ∼ π/2. Moreover, when θ 0 is suciently small (as shown in the lower right panel where θ 0 = 0.01), τ boil decreases an order of magnitude faster with K after the heating threshold E D = π/2 given by the Floquet kinetic equation, Eqs ( 5)-( 6) in the main text, is crossed. This is a clear signature of the heating eect generated by the Umklapp terms m = 0.

The prethermal properties that we observe in our disordered Floquet system can be seen as a generalization of those of clean Floquet systems in the fast-driving regime [S8, S9]. In fact, changing the driving frequency would amount in our case to multiplying W and g by a certain factor, which is also equivalent to changing the value of K. ∼ 1024 collapse onto each other, implying that τ boil does not suer from signicant system size eects. The system parameters are K = 1, W = 0.4, θ0 = 0.32 and g = 0.1. Fig. S6, left panel. We see that ∆P remains almost constant as the system size N increases, at least up to N = 10 5 . This suggests that wave condensation survives in the thermodynamic limit N → ∞.

We have also studied the system size dependence of τ boil . As shown in the right panel of Fig. S6, the curves for the θ-variance σ 2 θ (t) as a function of time for increasing system size collapse onto each other for N > ∼ 1024 which indicates that τ boil does not suer from strong nite size eects.

Figure 1 .

 1 Figure 1. Nonequilibrium dynamics of the nonlinear disordered Floquet system (1). (a) θ-distribution n(θ, t) obtained when interaction is switched on after dynamical localization, signalled by the CBS-CFS peaks at ±θ0, is achieved (see text).

  (b) The variance σ 2 θ (t) of n(θ, t) clearly shows the 3 dynamical regimes. The parameters are K = 1.0, W = 0.4, g = 0.1, θ0 = 1.05, N = 1024 and N d = 1200. ψ(θ, t) = p ψ(p, t) e ipθ where momentum states are labeled by integers p ∈ Z. Our normalization reads π -π |ψ(θ, t)| 2 dθ/(2π) = p |ψ(p, t)| 2 = 1. In our numerical simulations, we have considered a nite-size momentum basis set to |p| ≤ N/2 with periodic boundary conditions in p-space, and an atomic density equal to unity, i.e. N a = N . Dening ψ ± n (•) = ψ(•, t = n±0 + ), the Floquet dynamics associated to Eq. (1) is obtained by iterating the nonlinear map

1 √N

 1 exp(-ipθ 0 ), a CBS peak appears at -θ 0 on top of a diusive background over the Boltzmann transport time τ B . The CFS peak emerges at θ 0 when dynamical localization sets in, i.e. after the Heisenberg time τ H . Denoting disorder average by (• • •) and the number of disorder congurations considered by N d , the angle distribution n(θ, t) = |ψ(θ, t)| 2 becomes stationary at times t τ H and consists of twin CBS and CFS peaks over a diusive background.

Figure 2 .

 2 Figure 2. Prethermal state obtained at t = 10 4 for large θ0 = 1.05 and K + W < π/2. (a) Quasi-energy distribution fS(ε) (upper blue dots) and density of states ν(ε) (lower green dots). The Rayleigh-Jeans distribution fRJ (ε) with T = 1.17 and µ = -1.58 (red dashed line) ts very well the data. Inset: time dependence of T and µ. (b) θ-distribution nS(θ) (blue dotted curve) and neq(θ) Eq. (4) (red dashed curve) corresponding to fRJ (ε). The parameters are K = 1.0, W = 0.4, g = 0.1, N = 1024 and N d = 1200.

  the term gN |ψ(p)| 2 in Eq. (3) redistributes populations among the Floquet modes. The central quantity is therefore the quasi-energy distribution

Figure 3 .

 3 Figure 3. Wave condensed state obtained at t = 10 4 for small θ0 = 0.32 and K + W < π/2. (a) fS(ε) (upper blue dots) is compared to the best t fRJ (ε) (red dashed line) obtained at temperature T = 0.23 and computed at the maximal value µ = -ED. fRJ (ε) fails to reproduce the data at low ε. Lower green dots represent ν(ε). Inset: time dependence of the population gap ∆P of the 1-body density matrix. (b) nS(θ) (blue dotted curve) is the sum of the Rayleigh-Jeans component neq(θ) (red dashed curve) and a condensed component nc(θ) (orange curve). Its bimodal structure is typical of condensation (see text). The parameters are K = 1.0, W = 0.4, g = 0.1, N = 1024 and N d = 1200.

Fig. 2 (

 2 Fig.2(a)shows the disorder-averaged quasi-energy distribution f S (ε) obtained numerically by iterating Eq. (3) for t = 10 4 periods for the parameters used in Fig.1(E D = 1.4 < π/2). As one can see, f S (ε) is very well tted by f RJ (ε) with µ = -1.58 and T = 1.17. The corresponding n S (θ) is shown in Fig.2(b). It agrees very well with the predicted n eq (θ) obtained from f RJ (ε) using Eq. (4). Moreover, both T and µ, shown in the inset of Fig.2(a), stay essentially constant for long times

Fig. 3 ,

 3 Fig.3, the quasi-stationary n S (θ) deviates largely from n eq (θ) = TE D -E D dε 2π A(ε,θ) ε+E Din the vicinity of θ = 0, which corresponds precisely to the location of the lowest energy states. It can be decomposed into thermal and condensed components, leading to a bimodal θ-distribution reminiscent of the celebrated Bose-Einstein condensation signature[START_REF] Anderson | Observation of boseeinstein condensation in a dilute atomic vapor[END_REF][START_REF] Davis | Bose-einstein condensation in a gas of sodium atoms[END_REF], n S (θ) = n c (θ) + n eq (θ). To our knowledge, wave condensation happening in the prethermal plateau of a Floquet system had never been observed before.

Figure 4 .

 4 Figure 4. Phase diagram of the system in the (K, θ0) plane. (a) Population gap ∆P of the 1-body density matrix at t = 10 4 . The black dashed line represents the heating threshold ED = K + W = π/2 of Eq. (5)-(6). The red dots show the predicted onset θ c 0 (K) for wave condensation (the line is a guide to the eye). (b) Heating time scale τ boil after which the innite-temperature state is reached. The parameters are W = 0.4 and g = 0.1, N = 1024 and N d = 400.

  Figure S1. θ-variance σ 2 θ (t) as a function of time for dierent values of the Gross-Pitaevskii strength g. Left panel: σ 2 θ (t) is plotted against t for dierent g. Right panel: the same σ 2 θ (t) is now plotted against tg 2 for dierent g. All curves collapse in the intermediate region. The horizontal black line shows the threshold value σ 2 θ = 1.65 used to extract τeq for dierent g (see text). The parameters are K = 1, W = 0.4, θ0 = 1.09 N = 1024 and N d = 500.

Figure S2 .

 S2 Figure S2. Dependence of the characteristic CBS and CFS decay time τg and of the equilibration time τeq on the Gross-Pitaevskii strength g. Left panel: plot of τ -1 g versus g for K = 1, W = 0.4, θ0 = 1.57, N = 1024 and N d = 500. τg is extracted from an exponential t of the time decay of the CBS and CFS peaks. The red dashed line is a linear t to the data. Right panel: Equilibration time τeq versus g 2 for K = 1, W = 0.4, θ0 = 1.09 N = 1024 and N d = 500. τeq is dened by the time at which the θ-variance σ 2 θ (t) reaches the threshold value σ 2 θ = 1.65. The orange dashed line is a linear t to the data.

Figure S3 .

 S3 Figure S3. Time dependence of the temperature T (left panel) and chemical potential µ (middle panel) obtained for the initial state θ0 = 1.05 at K = 1 (blue) and K = 1.3 (red). Both T and µ are extracted by tting fε(t) with the Rayleigh-Jeans distribution. Right panel: Graph of the total quasi-energy per mode Etot as a function of θ0 at 3 dierent times t for the same two K values. For all panels, the other system parameters are xed at g = 0.1, W = 0.4, N = 1024 and N d = 1000 disorder congurations. With W = 0.4, the Umklapp processes m = 0 are active for K = 1.3 (ED = K + W > π/2) and suppressed for K = 1.0 (ED = K + W < π/2). As one can see, there is almost no heating in the absence of Umklapp processes: Both T , µ and Etot stay essentially the same in the considered time range t ∈ [0, 10 4 ]. The situation is markedly dierent when Umklapp processes are present: Both T , µ and Etot change signicantly.

Figure S6 .

 S6 FigureS6. Left panel: Population gap ∆P of the time coarse-grained 1-body density matrix, obtained at t = 10 4 for dierent system sizes N . ∆P stays constant up to N = 10 5 suggesting that wave condensation is robust in the thermodynamic limit N → ∞. Right panel: θ-variance σ 2 θ (t) as a function of time for dierent values of N . The curves for dierent N > ∼ 1024 collapse onto each other, implying that τ boil does not suer from signicant system size eects. The system parameters are K = 1, W = 0.4, θ0 = 0.32 and g = 0.1.

  Figure S1. θ-variance σ 2 θ (t) as a function of time for dierent values of the Gross-Pitaevskii strength g. Left panel: σ 2 θ (t) is plotted against t for dierent g. Right panel: the same σ 2 θ (t) is now plotted against tg 2 for dierent g. All curves collapse in the intermediate region. The horizontal black line shows the threshold value σ 2 θ = 1.65 used to extract τeq for dierent g (see text)

	2 (t)	1.3 1.6 1.9	t	10 3 2 (t)	1.3 1.6 1.9	tg 2 10 2 10 0 10 2	g=0.05 g=0.06 g=0.07 g=0.08 g=0.09 g=0.10 g=0.11 g=0.12 g=0.13 g=0.14 g=0.15

We thank M. Albert, J. Billy, N. Cherroret, D. Delande, O. Giraud, D. Guéry-Odelin, N. Macé and D. Ullmo for interesting discussions. This study has been supported by the French National Research Agency (ANR) under projects COCOA ANR-17-CE30-0024, MANYLOK ANR-18-CE30-0017 and GLADYS ANR-19-CE30-0013, the EUR grant NanoX No. ANR-17-EURE-0009 in the framework of the Programme des Investissements d'Avenir, and by the Singapore Ministry of Education Academic Research Fund Tier I (WBS No. R-144-000-437-114). Computational resources were provided by the facilities of Calcul en Midi-Pyrénées (CALMIP) and the National Supercomputing Centre (NSCC), Singapore.

SYSTEM SIZE DEPENDENCE OF THE ONSAGER-PENROSE CRITERION FOR CONDENSATION AND OF τ boil

In the main text, we have checked the Onsager-Penrose criterion for condensation [S10, S11] by computing and diagonalizing the time coarse-grained 1-body density matrix θ|ρ (1) (t)|θ , see Eq. ( 7), for N = 1024. Condensation is signalled by the existence of a "macroscopic" population gap between the largest and second largest eigenvalues of ρ (1) (t). Here, we consider this population gap ∆P = (P 1 -P 2 ), obtained at t = 10 4 , for dierent system sizes N , see