Making Prussian blue analogues nanoparticles luminescent: effect of the luminophore confinement over the properties. Multifunctional molecular materials are systems combining several physical or chemical properties. [START_REF] Ouahab | Multifunctional molecular materials[END_REF] Such smart systems have attracted a great deal of attention not only from a fundamental point of view, but also due to their future technological applications. During the last two decades, advances have been made by combining magnetic properties with optical or nonlinear activity, 2 conductivity, 3 porosity, 4 electron switching or ferroelectricity. [5][6] Moreover and despite the frequent emission quenching activated by paramagnetic transitions metal ions, many efforts have been recently devoted to the synthesis of molecule-based magnets exhibiting photoluminescence.

7-9

Another aspect of multifunctional molecular materials, which has attracted a high interest in the recent decade, consists in their controlling design at the nanoscale. In this line of thought, two families of multifunctional molecular materials, well known as Metal-Organic Frameworks (MOF), 10,11 and Prussian Blue Analogues (PBA) nano-objects, have been developed. The latter constitutes an exciting class of inorganic nanosystems [12][13] made from transition metal ions assembled by cyano ligands into a 3D porous structure of general formula A x M[M'(CN) 6 ] y  z (A is a monovalent cation, M and M' are transition metal ions,  represents the cyanometallate vacancies). In addition to the advantages relative to their molecular nature, such as optical transparency, low density or soft synthetic routes; they present numerous assets: (i) a versatility of chemical compositions giving rise to various properties (magnetism, optics, conductivity) that may be adjusted by the careful choice of the metal ions; (ii) a porous structure that may be used to encapsulate various species; (iii) their chemical robustness towards various media making them particularly adapted for future applications. Surprisingly, although efforts have been conducted to render these nanosized PBA multifunctional, mainly for their possible use in various applications, 13 combination with luminescent probes remains almost unexplored. The association of magnetism and luminescence in nanosized PBA could be highly interesting both from a fundamental point of view to study the synergy between the two properties as for potential applications in sensors, biology and electronics to name a few.

To our knowledge, the only examples reported in the literature consist in Mn-doped Prussian Blue (PB) nanoparticles whose surface was functionalized with the fluorescent avidin or core@shell made of polypyrrole@PB nanoparticles. [14][15] On the other hand, while the adsorption of guest luminophores is one of the most powerful strategies to render MOF luminescent, [16][17][18][19] it has never been applied to PBA systems. In this sense, we report here the post-synthetic functionalization of magnetic

K + /Ni 2+ /[Cr(CN) 6 ]
3 PBA nanoparticles by a 2-aminoanthracene (AA) luminophore to yield a bifunctional magneto-luminescent nanosystem. Both, magnetic and photoluminescent properties were investigated and combined with molecular simulations to determine the adsorption mechanism.

Our strategy to implement luminescence in magnetic PBA nanoparticles takes advantage of the great flexibility of this coordination framework and uses a two-step approach consisting in the synthesis of the surface free

K + /Ni 2+ /[Cr(CN) 6 ]
3 nanoparticles and their post-synthetic functionalization with AA. This PBA was chosen owing to its interesting magnetic properties (a relatively high Curie temperature around 65 K) and the absence of absorption bands in the visible region of the electronic spectrum to avoid the quenching of the guest's luminescence. On the other hand, the AA luminophore was selected due to: i) the presence of a donor amine function that allows its anchorage via the formation of hydrogen or coordination bonds to the Ni 2+ metal ions located close to the cyanometallate vacancies; ii) its suitable molecular size that can fit into cavities generated by the hexacyanochromate vacancies; iii) its green colour emission upon various excitation wavelengths ranging from 300 to 450 nm, 20 which makes it widely used in applications, such as reagents and assay kits. Subsequently, the obtained nanoparticles were dispersed in an ethanol solution containing the AA luminophore and further reacted for 24 h to give K 0.04 Ni[Cr(CN) 6 ] 0.60 @(AA) 0.18 •3.8H 2 O (1@AA) as a brown-yellow solid (Fig. 1b). The resulting functionalized nanoparticles were thoroughly washed with ethanol and next dialyzed to fully remove any physisorbed AA moieties.

The pristine nanoparticles 1 exhibit the characteristic structural features of the bulk PBA, namely a stretching cyanide ν(Cr III -CN-Ni II ) band located at 2173 cm -1 in the FT-IR spectra (Fig. S1), as well as its Powder X-Ray Diffraction Pattern showing the typical fcc structure with a lattice parameter a = 10.44 Å (Fig. S2). The crystallite size obtained from the Scherrer law is equal to 55 nm (d 200 reflection) and is in good accordance with the mean diameter of 65 ± 6 nm of the cubic nanoparticles determined by Transmission Electronic Microscopy (TEM) (Fig. S3). The post-synthetic functionalization of K + /Ni 2+ /[Cr III (CN) 6 ] 3 giving 1@AA is confirmed by the FT-IR spectrum, which shows the typical vibrations bands of the luminophore in the 600-1700 cm 1 region (Fig. S1). Elemental and thermogravimetric (TGA) analyses (Fig. S4) allow the determination of the AA and water content giving the formula 1@AA as K 0.04 Ni[Cr(CN) 6 ] 0.60 @(AA) 0.18 •3.8H 2 O. This corresponds to a loading capacity (defined as the amount of AA divided by the total nanoparticle weight) of 11.9 wt%. Since PBA incorporate both, coordinated and zeolitic water molecules, the slight difference in the water content for 1 and 1@AA may originate from either, the adsorption of AA, as observed for Cs + adsorption, 22 or from different number of zeolitic water molecules before and after functionalization. Yet, TGA indicates that the decomposition of the fluorophore in 1@AA occurs in the temperature range 325-400°C, slightly greater that of the PBA (300-325°C). Remarkably, this remains much higher than the decomposition temperature of AA at 200°C, suggesting that in 1@AA, the luminophore AA is efficiently protected within the porosity. Additional characterisations show that: i) 1@AA retain their initial structure, shape and size (Fig. 1d, Fig. S1-S2); ii) unreacted AA crystallites are not present at the particles surface (Fig. S5); iii) 1 and 1@AA have an identical zeta-potential, which indicates a similar surface state (Table S1). All these data suggest that AA is localized within the porosity of the PBA rather than anchored at the nanoparticles' surface. Additionally, the saturation capacity obtained by Monte Carlo simulations taking the hypothesis of the AA loading in an empty structure is estimated at 4.50 AA molecules per unit cell, which is higher than the experimental value of 0.72. This corresponds to a loading capacity of 74.4 wt% to compare with the experimental one of 11.9 wt%. However, this theoretical capacity was obtained without taking into account the presence of solvent molecules, which are able to chemically interact with unsaturated metal centers (CUS) and enters in competition with AA. In this line, molecular simulations indicate that the ethanol molecules fixed on the CUS prevent the adsorption of AA. Since AA molecules are located in the large micropores formed by vacancies and distributed randomly, the observed discrepancy between theoretical and experimental capacity could be explained by diffusion issues of AA upon adsorption (Fig. S7). Moreover, in order to provide a deeper insight in the mechanism of interaction between AA and the PBA network, complementary calculations have been performed to estimate the interaction energy taken the hypothesis of the AA adsorption by physisorption (since Universal Force Field is unable to predict the formation of a covalent interaction). Firstly, the enthalpy corresponding to a physisorption at low coverage (1 molecule per cell) is estimated at 21 kcal.mol 1 , while the chemisorption involving a coordination bond between AA and Ni 2+ gives 27 kcal.mol 1 .

Such difference of 6 kcal.mol 1 in the enthalpies between the two mechanisms suggests a favourable interaction by coordinative bonds between AA and the PBA. obtained by Monte-Carlo (physisorption), but it should be mentioned that interaction between NH 2 and cyano-groups can be observed. In contrast, by considering chemisorption, a shorter distance of 1.79 Å between AA and Ni 2+ is found. The magnetic properties investigations on 1@AA reveal the typical behaviour closed to the one observed for the bulk

K + /Ni 2+ /[Cr III (CN) 6 ]
3 PBA. It is in good accordance with the size of the considered nanoparticles, which are much larger than the single domain critical size estimated at around 22 nm.

23

Hence, the magnetic irreversibility was studied by applying the Zero Field-Cooled (ZFC)-Field Cooled (FC) procedure. The ZFC curve of 1@AA exhibits a peak with a maximum at 40 K, while the FC curve increases upon cooling (Fig. 2a). From the ZFC, the Curie temperature, T c , is estimated at 65 K, which remains unchanged upon functionalization (Fig. S8). The field dependence of the magnetization measured at 2.5 K shows a saturation value of 3.4 N in accordance with the ferromagnetic interaction between Ni 2+ and Cr 3+ ions and a hysteresis loop with a small coercive field of 110 Oe confirming the blocking of the magnetization (Fig. 2b, Fig. S9). 24 In spite of the low T c , the strong paramagnetism of 1@AA allows its interaction with a magnetic field even at room temperature as shown in Fig. 2c. The photoluminescence of 1@AA was investigated in details. Whereas 1 is optically inactive, 1@AA displays a green emission under excitation by a UV-lamp (Fig. 1c). Moreover, the nanoparticles could be re-dispersed in various solvents, such as water and ethanol, retaining their absorption (Fig. S10) and emission properties (Figs. S11,S12). Apart from minor blue-shift (e.g., 5 nm, at 300 K and 16 K, Figs. 3,S13), the solid-state emission spectrum of 1@AA resembles to what is observed for AA. The broad asymmetric emission band, centred at 500 nm, is ascribed to the AA fluorescence, 25 being excited from the UV to the blue spectral regions (290-435 nm, Fig. 3 and Fig. S14-S15). The slight blue-shift suggests an interaction between AA and the coordination framework when the chromophore is adsorbed. This is reinforced by the comparison of the excitation solid-state spectra for AA and 1@AA. Indeed, whereas the excited singlet states absorption dominates the AA spectra (vertical lines, Fig. 3), 26 a low-energy band at 435 nm is the main excitation channel of the 1@AA emission. This component, also seen in the excitation spectra of AA at 455 nm, is strongly dependent of the chromophore's concentration, as Fig. S12 shows for AA and 1@AA in ethanolic suspensions. This dependence is a strong argument favouring the modification in the environment of AA molecules (through - or intermolecular hydrogen bonds) upon adsorption within PBA. The possibility of this band be ascribed to a T n T 1 absorption (where T 1 and T n are excited triplet states), similar to what was reported for anthracene after protonation, 27 is discharged because here there is no sign of protonation. To go further in the photoluminescence characterization of 1@AA, the temperature dependence of the steady-state emission (Figs. S16-S20) and the thermal dependence of the emission lifetime (Figs. S21-S23) were studied. The emission integrated intensities of AA and 1@AA reveal a similar decrease as the temperature is raised from 16 to 300 K (Fig. S20), suggesting a comparable non-radiative transition probability for both samples indicating that the PBA does not induce extra-non-radiative pathways. The decay curves of both samples are well described by a single exponential function suggesting a homogeneous distribution of AA molecules. The lifetime () values (in the 10 9 s timescale) are in good agreement with a singlet-related emission. Despite an analogous trend with the temperature increase, we note a larger decrease for 1@AA. Thus, at 20 K, the value of 16.9 ± 0.2 ns is larger than that of AA of 14.7 ± 0.1 ns, whereas at 300 K the lifetime values decrease to 7.3 ± 0.1 and 8.8 ± 0.5 ns, for 1@AA and AA, respectively (Fig. S23). Such differences may be rationalised attending to the fact that the experimental transition probability may be expressed as

-1 = nr -1 + r -1
, where  nr and  r represent the non-radiative and radiative lifetimes, respectively, and that at 12 K  -1  r -1

. Thus, the greater lowtemperature  value of 1@AA indicates a decrease in the radiative transition probability caused by the adsorption of AA.

Nonetheless, the decrease in  as the temperature increases suggests the presence of competing thermally Energy / x10 3 cm -1 activated non-radiative mechanisms, being most severe for 1@AA, compared with AA. Further arguments supporting this conclusion are found by the absolute emission quantum yield (q). Despite the low q values measured for both samples, we note a larger one for AA (0.02  0.01 excited within 274-430 nm) with respect to that found for 1@AA that lies below the detection limits of our experimental set-up (< 0.01). The lifetime thermal dependence was rationalised using the classical Mott-Seitz model. [28][START_REF] Bunzli | Handbook on the Physics and Chemistry of Rare Earths: Including Actinides[END_REF] In short, the excited state lifetime is given by (T)= 0 /(1+exp(E/k B T), where  0 is the excited state lifetime value at T = 0 K,  the ratio between the non-radiative and radiative probabilities of the deactivation channel at T = 0 K, and ΔE the activation energy of the thermal quenching process. From the data best fit (Fig. S22), similar ΔE and α values were found for AA/1@AA (804 ± 98 / 758 ± 72 cm 1 and 41 ± 14 / 48 ± 17, respectively). The obtained ΔE values are in good agreement with the experimental energy difference between the high-wavelength component in the excitation spectra and the low-lying singlet emitting state (Fig. S13). In our experimental conditions, no noticeable changes in the photoluminescence features were detected below the magnetic ordering temperature.

In conclusion, we have shown that magnetic PBA nanoparticles could be easily post-functionalized by adsorption of an amino-luminophore. AA could be strongly adsorbed within the pores created by cyanometallate vacancies giving an exceptional thermal stability of the resulting functionalized nano-objects. In-depth analysis indicates a significant modification of the luminescence properties caused by the interaction between the AA luminophore and PBA framework while the long-range magnetic ordering is preserved affording bifunctional molecule-based nanomagnets. The proposed simple methodology could be extended to various functional guests and different PBA or others functional nano-objects in order to design multifunctional systems at the nanoscale.

  To confirm this hypothesis, molecular simulations (Monte Carlo and classical force field-based calculations) were performed (ESI). The pores size distribution (Fig. S6) shows the occurrence of two different porisity in the PBA network: tetrahedral sites of the cfc structure (pore diameter close to 3.5 Å) and larger elongated pores formed by cyanometallates vacancies with dimensions of 8 Å  8 Å  11 Å, suitable to accommodate AA (10 Å  5 Å  1 Å).

Figure 1 .

 1 Figure 1. a) Scheme of the synthetic strategy used in the post-synthetic functionalization of PBA nanoparticles; b) Photographs of 1 and 1@AA; c) Photographs of 1 and 1@AA under excitation with an UV lamp at 365 nm; d) TEM image of 1@AA. Inset: corresponding histogram of the size distribution. The solid line represents the fit with a Gaussian function.

  Fig.S7illustrates this interaction between NH 2 from AA and Ni 2+ of 2.39 Å

Figure 2 .

 2 Figure2. a) ZFC/FC curves performed for 1@AA with an applied magnetic field of 100 Oe ; b) Field dependence of the magnetization at 2.5 K for 1@AA; c) Photograph 1@AA interacting with a magnet at room temperature.

Figure 3 .

 3 Figure3. Low temperature (16 K) excitation and emission spectra monitored around 500 nm and excited at 365 nm, respectively for powdered AA and 1@AA.

nanoparticles by the 2-aminoanthracene luminophore to yield a bifunctional magneto-luminescent nanosystem. The photoluminescence properties of the fluorophore are found modified by confinement effet upon adsorsption, while the magnetic behavior of PBA is preserved.

	Ekaterina Mamontova, Salles, d Yannick Guari, a Luis D. Carlos a Jérôme Long*, b and Joulia Larionova a Rute A. S. Ferreira, a b Alexandre M. P. Botas,	b Fabrice
	We report in this communication the post-synthetic
	functionalization of K + /Ni 2+ /[Cr(CN) 6 ] 3 Prussian Blue Analogue
	(PBA)

Acknowledgements

The authors thank the University of Montpellier, CNRS and PAC of ICGM. The Portugal-France bilateral action, PESSOA Program (Hubert Curien) "Magneto-optical Multifunctional Nanoparticles" (40773PE) is also acknowledged. This work was developed within the scope of the project CICECOAveiro Institute of Materials, POCI-01-0145-FEDER-007679 (FCT Ref. UID/CTM/50011/2013) financed by national funds through the FCT/MEC and when appropriate co-financed by FEDER under the PT2020 Partnership Agreement. A.M.P. Botas thanks FCT for a PhD fellowship (SFRH/BD/104789/2014).

Notes and references

Conflicts of interest

There are no conflicts of interest to declare.


Jérôme Long

Aleksei O Tolpygin

Anton V Cherkasov

Konstantin A Lyssenko

Yannick Guari

Joulia Larionova

Alexander A Trifonov

Single-Molecule Magnet Behavior in Dy 3+ Half-Sandwich Complexes based on ene-diamido and Cp* Ligands

ABSTRACT:

We report in this communication the synthesis, structure and magnetic investigations of two new half-sandwich complexes [Dy(DAD)Cp*(THF)]

(1) and [Li(THF) 3 ][Dy(DAD)Cp*Cl] (2) (Cp* = C 5 Me 5 , DAD = [2,6-iPr 2 C 6 H 3 N-CMeCMe-NC 6 H 3 iPr2-2,6). Both compounds exhibit zero-field SMM behavior but distinct relaxation dynamics originating from difference in the arrangement of Cp* and DAD ligands. The anisotropic barrier for 1 is found one order of magnitude greater than for 2.

Coordination complexes exhibiting slow relaxation of their magnetization associated with magnetic bistability show tremendous potentialities for information storage or spin-based computing. 1 Since the report of such effect more than two decades ago, 2 the chemical design and resulting properties of such Single-Molecule Magnets (SMMs) have been greatly enhanced by taking advantage of the fascinating properties of lanthanide ions. 1b, 3 In 4f complexes, the appropriate combination between a specific lanthanide ion and surrounding ligands generates a crystal-field that may allow the appearance of an anisotropic barrier, , that separates two opposite directions of the magnetic moment (± m J ). Naturally, increasing the crystal-field splitting should result in larger anisotropic barriers and blocking temperatures (which could be the temperature at which or relaxation time is equal to 100 s or alternatively as the highest temperature at which an hystereis loop is observed). [START_REF] Gatteschi | Molecular Nanomagnets[END_REF] Such systems with largely separated ± m J states could be engineered based on simple electrostatic considerations that exploit the optimum stabilization the electronic density of the 4f ions. Thus, lanthanide ions such as Dy 3+ exhibiting oblate electronic density are efficiently stabilized by axial crystal-field that maximizes the splitting of the m J levels and stabilizes the highest m J = ± 15/2 states, while minimizing the Quantum Tunneling of the Magnetization (QTM). However, such scenario is frequently altered by the existence of additional spinlattice relaxation processes (Raman and direct) which create underbarrier relaxation paths. 1b, 3, 5 One possible strategy to obtain large axial crystal-field relies on the use of coordination 6 or organometallic chemistry ligands 7 that reduce the lanthanide coordination number while simultaneously affording an efficient stabilization of the 4f electronic density. In this regards, major advances have been recently achieved with Dy 3+ metallocenium complex based on cyclopentadienyl (Cp R ) ligands 8 that show exceptional magnetic hysteresis features that could overcome liquid nitrogen's temperature. [START_REF] Guo | Magnetic hysteresis up to 80 kelvin in a dysprosium metallocene single-molecule magnet[END_REF] In this last example, the heteroleptic sandwich complex [Dy(Cp iPr5 )(Cp*)][B(C 6 F 5 ) 4 ] was found to exhibit large anisotropy thanks to a synergy between short Dy-Cp* /iPr5 distances and large Cp-Dy-Cp angle. Additionally, these impressive relaxation dynamics appear to be not solely related to the targeted coordination environment but also to the reduced molecular vibrations (metalligands vibrational modes) imposed by the rigid Cp ligands. 5, 8a, 8b, 9 It appears therefore essential to gain deeper understanding on the parameters affecting such spin-phonon coupling.

In this sense, we have recently reported the use of doubly reduced diazabutadiene ligands for the design of lanthanide SMMs. 10 These anionic ligands, acting as both, n and -electron donors, benefit from a large steric and electronic tunability that could be taken as an advantage to create highly axial crystal-field. In our previous studies, the homoleptic Li(DME) 3 [Dy(DAD) 2 ] complex (DAD = [2,6-iPr 2 C 6 H 3 N-CHCH-NC 6 H 3 iPr 2 -2,6]) exhibit a genuine SMM behaviour. 10b Yet, despite a significant crystal-field splitting generated by these ligands, Raman relaxation was found to dominate the relaxation dynamics. Aiming at reducing this spin-phonon coupling while increasing the magnetic anisotropy, we propose here to associate bulky DAD-type ligands, that are known to generate short Dy-N distances, with rigid Cp * moiety in order to design heteroleptic half-sandwich dysprosium complexes.

Scheme 1: Synthesis of 1 and 2.

The synthetic strategy to design the targeted heteroleptic halfsandwich-type complexes [Dy(DAD)Cp * ] relies on the cascade of reactions of anhydrous DyCl 3 with [DADK 2 (THF) n ] and alkali metal cyclopentadienide MCp* (M = Li, K) (Scheme 1). Using KCp* leads to the formation of a salt-free complex [Dy(DAD)Cp*(THF)] (1), while in the case of LiCp*, atecomplex [Li(THF) 3 (μ 2 -Cl)Dy(DAD)Cp*] (2) was isolated (see ESI for details). The X-ray diffraction studies indicate that both compounds crystallize in the P2 1 /c space group with a unique complex within the asymmetric unit (Table S1). Compound 1 is isostructural to the ytterbium complex we recently obtained. [START_REF] Selikhov | Base-Free Lanthanoidocenes(II) Coordinated by Bulky Pentabenzylcyclopentadienyl Ligands[END_REF] The coordination environments of the Dy 3+ in 1 and 2 are composed of one Cp* ligand, one dianionic 2:² ene-diamido DAD 2 ligand and one coordinated THF molecule (for 1) or μ 2 -ClLi(THF) 3 fragment (for 2). The Dy-C Cp bond lengths in 1 are ranging from 2.612(2) to 2.727(1) Å, the Dy-Cp Centr distance is of 2.391(2) Å (Table S2). The Dy-N bond lengths are slightly different and equal to 2.189(2) and 2.202(2) Å. With respect to 1, the Dy-Cp Centr distance in 2 (2.367(2) Å) is slightly shorter, while one Dy-N bond lengths is longer (2.198(2) and 2.205(2) Å). The Dy-N bond lengths in 1 and 2 are noticeably shorter than the related values measured for eight-coordinate Dy 3+ complexes and correspond to covalent Dy-N bonds. 12 The short distances between the Dy 3+ ions and the carbon atoms of the NCCN moiety in 1 (2.837(2), 2.845(2) Å) and 2 (2.764(2), 2.769(2) Å) are indicative of ²coordination of the CC bonds to the Dy 3+ ion. However, these distances are longer than in the previously reported [Li(DME) 3 ][Dy(DAD) 2 ] (2.672(3)-2.712(5) Å). 10b The Dy-O distance in 1 is equal to 2.362(2) Å. In 2, the Cl  ligand μ 2 -bridges [Li(THF) 3 ] + and Dy 3+ ions. The Dy-Cl distance is 2.6403(6) Å and the Li-Cl one is of 2.339(4) Å. Taking into account the two centroids defined by the DAD 2 (NCCN) and Cp* ligands, bite angles of 142.0(2)° and 139.6(2)° are found for 1 and 2 respectively. The shortest intermolecular Dy 3+ -Dy 3+ distance measured in 1 (9.543(2) Å) proved to be noticeably shorter compared to that in 2 (11.413(2) Å) (Fig. S1).

The magnetic properties of 1 and 2 were investigated in both, static and dynamic modes. The detailed dc measurements could be found in the ESI and reveal differences that may originate from dissimilar crystal-field splitting.

The occurrence of slow relaxation of the magnetization was investigated by alternating current (ac) measurements. Under a zero-dc field, the compounds exhibit a clear frequency dependence in the in-phase (') and in the out-of-phase (") components of the magnetic susceptibility (Fig. 2, Fig. S4-S5) indicating a SMM behavior. While for 1 a maximum could be discerned, 2 exhibits a very broad signal without a clear maximum of ". The Cole-Cole plots (Fig. S6) reveal the presence of two relaxation processes for 1. These data could be fitted with the sum of two modified Debye functions [START_REF] Guo | Relaxation dynamics of dysprosium(iii) single molecule magnets[END_REF] for 1 (Table S3) leading to large values (i.e. 0.6) of the  parameter at low temperature for the main relaxation process but such treatment yields to unrealistic fitting parameters for 2. Fig. 2. Frequency dependence of the out-of phase (") susceptibilities for 1 and 2 under a zero dc-field.

Insights into the dynamics of the relaxation of the magnetization could be obtained from the temperature dependence of the relaxation time, . The broad maximum for 2 prevent extracting relevant values of . The ln vs. T 1 plot for 1 related to the main relaxation process deviates from the linearity before becoming temperature independent at low temperature, indicating the presence of QTM (Fig. 3, Fig. S7). Consequently, the overall data range could be modeled using the following equation:

 1 =  0 1 exp(/kT) + CT n +  1
QTM (Eq. 1). [START_REF] Meihaus | Influence of pyrazolate vs Nheterocyclic carbene ligands on the slow magnetic relaxation of homoleptic trischelate lanthanide(III) and uranium(III) complexes[END_REF] The first term accounts for a thermally activated process, while the second and third ones stand for two-phonon Raman and QTM, respectively. In order to avoid over-parameterization, n was fixed to different values until getting the best fitting coefficient. The obtained fit parameters (Table 1) point out that a relaxation involving these three different processes. The low value of the n exponent as observed in others Dy/Cp systems 8a, 8b, 9 may indicate the presence of optical phonons 15

*fixed parameter

In order to reduce the observed QTM, the field dependence of the ac susceptibilities was studied. An increase of the relaxation time is observed with optimum fields found at 2000 and 1000 Oe for 1 (15 K) and 2 (2 K), respectively (Fig. S8). The field dependence of  could be modeled with the equation  1 = DH 4 T + B 1 /(1+B 2 H²) + K (Eq. 2), for which the first term accounts for the direct process (for Kramers-ion), the second one for the QTM and the K constant accounts for the field-independent Raman and thermally activated processes (Fig. S9, Table S4). The frequency dependence of the ac susceptibilities measured under these dc fields confirms the shortcut of the QTM. While a broad maximum of " could be still observed for 1, applying a dc field totally modifies the frequency dependence of " for 2 (Fig. S10). Thus, although a plateau is observed at low frequencies (also confirmed by the Cole-Cole plots, Fig. S11, Table S6), a clear maximum could be now discerned allowing the extraction of the relaxation time. The temperature dependence of the relaxation time (Fig. 3) for both complexes could be fitted with:  1 =  0 1 exp(/kT) + CT n + AT (Eq. 3) (Fig. 3, Fig. S7, Table 1) in which the third term accounts for the direct process.  is found one order of magnitude lower for 2 with respect to 1 confirming an important difference in the crystal-field spitting as previously evidenced from the dc measurements. Moreover, although no direct process could be evidenced for 1, it contributes greatly to the overall relaxation for 2, while the Raman process is found inoperative;

The magnetic analysis indicates a dramatic variation in the dynamic of slow relaxation of the magnetization for 1 and 2 despite a closely related structure. To get further insights, magnetostructural correlations could be achieved by evaluation of the orientation of the anisotropic axes of the ground Kramers doublet using the MAGELLAN [START_REF] Chilton | An electrostatic model for the determination of magnetic anisotropy in dysprosium complexes[END_REF] software based on electrostatic considerations. As expected, the orientation of the anisotropic axes for both complexes are mainly imposed by both, Cp* and DAD-Me ligands (Fig. S12). Hence, a deviation of less than 15° with the anisotropic axis is observed between the centroids of DAD-Me and Cp* ligands (Table S7). Moreover, the THF or chloride in 1 and 2, respectively, define a mean angle close to 90° with the anisotropic axis, inducing a transverse component that explains the QTM. Surprisingly, substitution of a THF by a negatively charged chloride does not induce a significant tilt of the anisotropic axis for 2. This could be imputed to the rather long Dy-Cl distance of 2.640 Å to compare with the shorter Dy-O one of 2.363 Å, counterbalancing the effect of the chloride negative charge. On the other hand, 1 exhibits the shortest Dy 3+-Dy 3+ distance. It should therefore exhibit the highest tunneling rate caused by dipolar interactions which is not experimentally observed. Hence, the observed differences in the magnetic properties may be rather explained by subtle differences in the arrangement of DAD-Me and Cp* ligands. Recent studies on dysprosium metallocene complexes [Dy(Cp R ) 2 ] + clearly point out two critical parameters that affect the magnetic axiality in these systems: the Cp-Dy-Cp angle and Dy-C distances. Maximizing the anisotropy requires achieving large bite angles, while shortening the Dy-C bonds. [17][18][19][20] Despite the fact that the Dy-C are found slightly shorter in 2, the greater bite angle Cp*-Dy-DAD associated with the reduced Dy-N distances for 1 may explain its larger crystal-field splitting and greater anisotropic barrier. Such dominant influence of the DAD ligand is also in line with its two-fold greater negative charge carried in comparison with Cp*.

Furthermore, the magnetic properties of 1 may be favorably compared with those obtained in our previously reported homoleptic complex [Li(DME) 3 ][Ln(DAD) 2 ]. 10b In this latter, the DAD-Dy-DAD angle was closer to the linearity (172°), but significantly larger Dy-N distances of 2.221-2.246 Å were found. In addition, the magnetization was found to mainly relax through a Raman process. Consequently, incorporation of rigid Cp* moieties in association with DAD ligands permits to increase the axiality while reducing the Raman relaxation.

In summary, we have reported in this article two different heteroleptic Dy 3+ complexes based on DAD and Cp* ligands. Although they differ by the nature of an additional coordinated species (THF vs. Cl  ), both compounds exhibit distinct slow relaxation of the magnetization dynamics caused by slight changes in the Dy-N distances and Cp*-Dy-DAD angles. Thanks to the great tunability of both, DAD and Cp ligands, extension to other heteroleptic complexes exhibiting shorter Dy-N distances and greater bite angles may be viewed as a promising route to enhance the magnetic relaxation properties in lanthanide SMM.

ASSOCIATED CONTENT Supporting Information

The Supporting Information is available free of charge on the ACS Publications website. Experimental procedures, additional structural and magnetic data could be found in the PDF.

Fig. 1 .

 1 Fig. 1. Structure of 1 (top) and 2 (bottom). Color code: orange, Dy; red, O; grey, C; green, Cl; light blue, Li. Hydrogen atoms have been omitted for clarity.

Fig. 3 .

 3 Fig. 3. Temperature dependence of , for 1 and 2. The solid lines correspond to the fit with Eq. 1or Eq. 3.

Table 1 :

 1 Fit parameters of the temperature dependence of the relaxation time for 1 and 2.

	Compound	 (cm 1 )	0 (s)	n*	C (s 1 .K m )	A (s 1 .K 1 )
	1 (0 Oe)	254 ± 17	(6 ± 5)  10 10	3	0.008 ± 0.005	-
	1 (2000 Oe)	206 ± 10	(10 ± 4)  10 9	3	0.007± 0.003	-
	2 (1000 Oe)	20 ± 4	(5 ± 3)  10 9	-	-	281 ± 139

AUTHOR INFORMATION

Corresponding Author *E-mail: jerome.long@umontpellier.fr *E-mail: trif@iomc.ras.ru

Author Contributions ‡All the authors contributed equally.

Notes

The authors declare no competing financial interests.

ACKNOWLEDGMENT

The financial support of the Russian Science Foundation is highly acknowledged (Project № 17-73-30036). The X-ray study of 1 has been carried out in the framework of the Russian state assignment (Тheme № 44.2, Reg. № АААА-А16-116122110053-1) using the equipment of The Analytical Center of IOMC RAS. The French authors thank the University of Montpellier, CNRS and PAC of ICGM.


Elsa Bayart

Ilya Svetlizky

Jay Fineberg

Fracture Mechanics determine the lengths of interface ruptures that mediate frictional motion

come

Fracture mechanics determine the lengths of interface ruptures that mediate frictional motion

Introductory paragraph:

The transition from static to sliding friction is mediated by rapid interfacial ruptures [1][START_REF] Xia | Laboratory Earthquakes: The Sub-Raleigh-to-Supershear Rupture Transition[END_REF][START_REF] Baumberger | Self-healing slip pulses and the friction of gelatin gels[END_REF][START_REF] Ben-David | The Dynamics of the Onset of Frictional Slip[END_REF][START_REF] Passelègue | From Sub-Rayleigh to Supershear Ruptures During Stick-Slip Experiments on Crustal Rocks[END_REF] propagating through the solid contacts forming a frictional interface [START_REF] Bowden | The Friction and Lubrication of Solids[END_REF] . While propagating, these ruptures correspond to true shear cracks [START_REF] Svetlizky | Classical shear cracks drive the onset of dry frictional motion[END_REF] . Frictional sliding is only initiated when a rupture traverses the entire interface 1 , however, arrested ruptures can occur at applied shears far below the transition to frictional motion [START_REF] Rubinstein | Dynamics of precursors to frictional sliding[END_REF][START_REF] Maegawa | Precursors of global slip in a longitudinal line contact under non-uniform normal loading[END_REF][START_REF] Katano | Novel Friction Law for the Static Friction Force based on Local Precursor Slipping[END_REF][11][12][13][14][START_REF] Kammer | Linear Elastic Fracture Mechanics Predicts the Propagation Distance of Frictional Slip[END_REF][START_REF] Taloni | Scalar model for frictional precursors dynamics[END_REF][START_REF] Bar-Sinai | Velocitystrengthening friction significantly affects interfacial dynamics, strength and dissipation[END_REF] . Here we show, by measuring the real contact area and strain fields near rough frictional interfaces, that fracture mechanics quantitatively describe rupture arrest and therefore determine the onset of overall frictional sliding. Our measurements reveal both the local dissipation and the global elastic energy released by the rupture. The balance of these quantities entirely determines rupture lengths, whether finite or system-wide. These results suggest a new fracture mechanics based paradigm for describing frictional motion and shed light on the selection [START_REF] Ampuero | Properties of Dynamic Earthquake Ruptures With Heterogeneous Stress Drop[END_REF][19][START_REF] Lapusta | Nucleation and early seismic propagation of small and large events in a crustal earthquake model[END_REF][START_REF] Olson | The deterministic nature of earthquake rupture[END_REF] of an earthquake's magnitude.

Main text:

A frictional interface is formed by the interlocked solid asperities of rough surfaces in contact, whose area is much smaller than the nominal one [START_REF] Bowden | The Friction and Lubrication of Solids[END_REF] . Failure of the asperities via rupture fronts is the fundamental mechanism responsible for the transition from static to sliding frictional motion 1,[START_REF] Ben-David | The Dynamics of the Onset of Frictional Slip[END_REF] . Ruptures can propagate well before the onset of global sliding, and then arrest before spanning the entire interface. Rupture arrest can, for example, result from inhomogeneous stress distributions along the interface [START_REF] Rubinstein | Dynamics of precursors to frictional sliding[END_REF][START_REF] Maegawa | Precursors of global slip in a longitudinal line contact under non-uniform normal loading[END_REF][START_REF] Katano | Novel Friction Law for the Static Friction Force based on Local Precursor Slipping[END_REF] . As no overall motion of the contacting bodies is induced by such events, they are often called precursors to sliding motion. Arrested events are analogous to earthquakes, which are dynamic ruptures of finite extent within pre-existing natural faults; the boundary between contacting tectonic plates [START_REF] Scholz | The mechanics of earthquakes and faulting[END_REF][START_REF] Ben-Zion | Dynamic ruptures in recent models of earthquake faults[END_REF] . Predicting the length of precursory ruptures is, therefore, closely related to the question of what determines the size of an earthquake [START_REF] Ampuero | Properties of Dynamic Earthquake Ruptures With Heterogeneous Stress Drop[END_REF][19][START_REF] Lapusta | Nucleation and early seismic propagation of small and large events in a crustal earthquake model[END_REF][START_REF] Olson | The deterministic nature of earthquake rupture[END_REF] .

Since their initial discovery [START_REF] Rubinstein | Dynamics of precursors to frictional sliding[END_REF] , a rich variety of models has been dedicated to the dynamics of precursory ruptures in frictional systems. Aimed at reproducing nucleation and arrest, these include minimalistic 1D models [START_REF] Maegawa | Precursors of global slip in a longitudinal line contact under non-uniform normal loading[END_REF] , discrete contacts descriptions 11,13,[START_REF] Taloni | Scalar model for frictional precursors dynamics[END_REF] , rate-and-state friction laws [START_REF] Bar-Sinai | Velocitystrengthening friction significantly affects interfacial dynamics, strength and dissipation[END_REF] and fracture mechanics 12 . These models are able to reproduce the existence of arrested ruptures but they provide no explicit predictions of where and how arrest occurs in real systems. Kammer et al. [START_REF] Kammer | Linear Elastic Fracture Mechanics Predicts the Propagation Distance of Frictional Slip[END_REF] , in recent theoretical work, explained the available data 8 by explicitly demonstrating how fracture mechanics can be used to predict rupture arrest. Here we describe new experiments that confirm these theoretical predictions and show that this general framework indeed enables us to understand the selection of rupture length for any system geometries and loading conditions.

Recent experiments have shown that the strain fields driving ruptures along frictional interfaces are described 7 by the Linear Elastic Fracture Mechanics (LEFM) [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] . In this context, a precursory event is an arrested crack 12,[START_REF] Kammer | Linear Elastic Fracture Mechanics Predicts the Propagation Distance of Frictional Slip[END_REF][START_REF] Ampuero | Properties of Dynamic Earthquake Ruptures With Heterogeneous Stress Drop[END_REF] . In fracture mechanics, crack arrest is defined by the Griffith criterion [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] , a crack arrests when the amount of energy flowing to its tip becomes smaller than the fracture energy Γ, the dissipated energy per unit crack advance. Following the theoretical approach outlined in earlier studies [START_REF] Kammer | Linear Elastic Fracture Mechanics Predicts the Propagation Distance of Frictional Slip[END_REF][START_REF] Ampuero | Properties of Dynamic Earthquake Ruptures With Heterogeneous Stress Drop[END_REF] we will experimentally verify that this criterion is fulfilled by spontaneously arrested frictional ruptures. These results provide new insights on the predictability of both frictional processes and earthquakes dynamics.

We examine the frictional sliding of two poly(methyl-methacrylate) (PMMA) blocks whose contact interface is flat to within a few µm. We define 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧 as, respectively, the rupture propagation, normal loading and sample thickness directions. We use two different sample geometries (Fig. 1a). The asymmetric geometry is formed by blocks of different thicknesses (6 mm top block, 30 mm bottom block) and dimensions with a 3 µm r.m.s surface roughness along the interface. The symmetric geometry is made of blocks having the same dimensions whose interface is formed by two optically flat surfaces. PMMA has strain rate dependent Young's modulus 3< 𝐸𝐸 <5.6 GPa and Poisson ratio 𝜈𝜈 = 1/3. The Rayleigh wave speed is 𝑐𝑐 𝑅𝑅 =1237±10 m/s (plane stress) and 𝑐𝑐 𝑅𝑅 =1255±10 m/s (plane strain) (Methods). The blocks are pressed together with an externally imposed normal force 𝐹𝐹 𝑁𝑁 , ranging from 1,500-7,000 N. After applying 𝐹𝐹 𝑁𝑁 , shear forces 𝐹𝐹 𝑆𝑆 are applied either at a single-point or more uniformly via a sliding stage (Fig. 1a), giving rise to a variety of inhomogeneous distributions of normal and shear stresses along the interface (see examples in Fig. 1b). Throughout each experiment, continuous parallel measurements of the two-dimensional strain tensor 𝜀𝜀 𝑖𝑖𝑖𝑖 (𝑡𝑡) are performed at 15 to 18 locations along and 3.5 mm above the interface at a rate of 10 6 samples/sec (Fig. 1a). From the strain measurements we obtain stresses, while taking into account material viscoelasticity (Methods). At the same time, we measured the real contact area 𝐴𝐴(𝑥𝑥, 𝑡𝑡) at 1,280 × 8 (𝑥𝑥 × 𝑦𝑦) spatial locations along the interface at 580,000 frames per second by an optical method based on total internal reflection 1,4,7 (for details see Svetlizky et al. [START_REF] Svetlizky | Classical shear cracks drive the onset of dry frictional motion[END_REF] ). Under these stress conditions, we observe a succession of arrested ruptures of increasing length (Fig. 1c), well before 𝐹𝐹 𝑆𝑆 reaches the threshold for overall stick-slip motion of the blocks (Fig. 1d). The transition to the overall motion of the contacting blocks happens only when these ruptures span the entire interface (event VI).

The variations of the stresses at the tip of each propagating rupture are quantitatively described by the singular fields predicted by LEFM for shear (mode II) cracks [START_REF] Svetlizky | Classical shear cracks drive the onset of dry frictional motion[END_REF] (Fig. 2a).

The singular term of the stress field has the form [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] :

Δ𝜎𝜎 𝑖𝑖𝑖𝑖 (𝑟𝑟, 𝜃𝜃) = 𝐾𝐾 𝐼𝐼𝐼𝐼 (𝑣𝑣) √2𝜋𝜋𝑟𝑟 Σ 𝑖𝑖𝑖𝑖 𝐼𝐼𝐼𝐼 (𝜃𝜃, 𝑣𝑣) (1)
in polar coordinates with respect to the crack tip, where Σ 𝑖𝑖𝑖𝑖 𝐼𝐼𝐼𝐼 (𝜃𝜃, 𝑣𝑣) is a known universal angular function and 𝐾𝐾 𝐼𝐼𝐼𝐼 is the stress intensity factor. Δ𝜎𝜎 𝑖𝑖𝑖𝑖 expresses the stress changes [START_REF] Palmer | The Growth of Slip Surfaces in the Progressive Failure of Over-Consolidated Clay[END_REF] between the initially applied and residual stresses along the frictional crack faces. In the framework of LEFM no generality is lost by adding constant values, 𝜎𝜎 𝑥𝑥𝑥𝑥 0 , 𝜎𝜎 𝑦𝑦𝑦𝑦 0 and 𝜎𝜎 𝑥𝑥𝑦𝑦 𝑟𝑟𝑟𝑟𝑟𝑟 (Fig. 2a) to Eq. 1 (Methods). For a known rupture velocity 𝑣𝑣, the amplitude of the singular term, i.e. the stress intensity factor 𝐾𝐾 𝐼𝐼𝐼𝐼 , is related to the energy release rate 𝐺𝐺, defined as the flux of elastic energy per unit extension of a crack's tip, by:

G = 𝛼𝛼 𝐸𝐸 𝑓𝑓 𝐼𝐼𝐼𝐼 (𝑣𝑣)𝐾𝐾 𝐼𝐼𝐼𝐼 2 (𝑙𝑙, 𝑣𝑣) 𝑣𝑣→0 �⎯� 𝐺𝐺 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠 (𝑙𝑙) (2)
where 𝑓𝑓 𝐼𝐼𝐼𝐼 (𝑣𝑣) is a universal function, whose value is nearly unity for low rupture velocities 𝑣𝑣. The coefficient is 𝛼𝛼 = 1 for the plane stress conditions used in the symmetric system while 𝛼𝛼 = (1 -𝜈𝜈 2 ) for the plane strain conditions used in the asymmetric one [START_REF] Svetlizky | Classical shear cracks drive the onset of dry frictional motion[END_REF] . Following Eq. 1, fitting the three stress components (see, e.g. Fig. 2a) provides a dynamic measurement 7 of 𝐾𝐾 𝐼𝐼𝐼𝐼 .

During rupture propagation, 𝐺𝐺 = Γ. Using the value of 𝐾𝐾 𝐼𝐼𝐼𝐼 in Eq. 2, we can therefore calculate the fracture energy Γ, which has been shown to be roughly velocityindependent [START_REF] Svetlizky | Classical shear cracks drive the onset of dry frictional motion[END_REF] . As Fig. 2b shows, we find that Γ linearly depends on the mean normal stress, < 𝜎𝜎 𝑦𝑦𝑦𝑦 >. If we assume that the real contact area, 𝐴𝐴, entirely determines Γ, this result is in accordance with both the Bowden and Tabor picture [START_REF] Bowden | The Friction and Lubrication of Solids[END_REF] (where 𝐴𝐴 ∝ 𝐹𝐹 𝑁𝑁 ) and the suggestion [START_REF] Svetlizky | Classical shear cracks drive the onset of dry frictional motion[END_REF] that Γ effectively measures the number of contacts that have to be broken in order for a crack to propagate.

Can a frictional rupture arrest be described by the fracture mechanics based criterion for crack arrest? If so, according to Eq. 2, upon rupture arrest, 𝐺𝐺 → 𝐺𝐺 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠 (𝑙𝑙) = 𝛼𝛼 𝐾𝐾 𝐼𝐼𝐼𝐼 2 (𝑙𝑙,𝑣𝑣=0) 𝐸𝐸 and arrest will occur [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] when 𝐺𝐺 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠 (𝑙𝑙) ≤ Γ. In general, 𝐾𝐾 𝐼𝐼𝐼𝐼 (𝑙𝑙, 𝑣𝑣 = 0) ≡ 𝐾𝐾 𝐼𝐼𝐼𝐼 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠 (𝑙𝑙), can be explicitly calculated [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] providing 𝐺𝐺 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠 (𝑙𝑙). For our sample geometry [START_REF] Kammer | Linear Elastic Fracture Mechanics Predicts the Propagation Distance of Frictional Slip[END_REF][START_REF] Tada | The stress analysis of cracks handbook[END_REF] (Methods):

𝐾𝐾 𝐼𝐼𝐼𝐼 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠 (𝑙𝑙) = 2 √𝜋𝜋𝑙𝑙 � Δ𝜏𝜏(𝑠𝑠)𝐹𝐹(𝑠𝑠/𝑙𝑙) �1 -(𝑠𝑠/𝑙𝑙) 2 𝑙𝑙 0 𝑑𝑑𝑠𝑠 (3)
where, 𝐹𝐹(𝑠𝑠 𝑙𝑙 ⁄ ) = 1 + 0.3(1 -(𝑠𝑠/𝑙𝑙) 5/4 ) and the stress drop for each event is defined as Δ𝜏𝜏(𝑥𝑥) = 𝜎𝜎 𝑥𝑥𝑦𝑦 0 (𝑥𝑥) -𝜎𝜎 𝑥𝑥𝑦𝑦 𝑟𝑟𝑟𝑟𝑟𝑟 (𝑥𝑥); the stress drop from the initial shear stress 𝜎𝜎 𝑥𝑥𝑦𝑦 0 (𝑥𝑥) to the residual stress 𝜎𝜎 𝑥𝑥𝑦𝑦 𝑟𝑟𝑟𝑟𝑟𝑟 at each point (Fig. 2a).

To predict the arrest location using Eq. 3, we need to measure both 𝜎𝜎 𝑥𝑥𝑦𝑦 0 (𝑥𝑥) and 𝜎𝜎 𝑥𝑥𝑦𝑦 𝑟𝑟𝑟𝑟𝑟𝑟 (𝑥𝑥). Figs. 2c and 2d demonstrate that both quantities vary significantly in space. By definition, 𝜎𝜎 𝑥𝑥𝑦𝑦 𝑟𝑟𝑟𝑟𝑟𝑟 (x) is not defined beyond the rupture arrest location. Except very near 𝑥𝑥 = 0, where applied torques produce measurable effects [START_REF] Ben-David | Static Friction Coefficient Is Not a Material Constant[END_REF] , we find that, within a given experiment, 𝜎𝜎 𝑥𝑥𝑦𝑦 𝑟𝑟𝑟𝑟𝑟𝑟 (𝑥𝑥) is invariant from one event to another (Fig. 2d). After accounting for the edge effects (Methods), we therefore use 𝜎𝜎 𝑥𝑥𝑦𝑦 𝑟𝑟𝑟𝑟𝑟𝑟 (𝑥𝑥) from the first system-wide event, as characteristic of the interface. We can now measure Δ𝜏𝜏(𝑥𝑥) for each event at each point 𝑥𝑥 (Fig. 3a). As stresses are measured slightly above the interface, we improve the accuracy of Δ𝜏𝜏(𝑥𝑥) at the interface by accounting for stress gradients (Methods).

In Fig. 3a we present the computed value of 𝐾𝐾 𝐼𝐼𝐼𝐼 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠 (𝑙𝑙) (see Eq. 3) in successive events for the experiment described in Fig. 1. Predicted arrest locations, predicted, for each event are the locations where 𝐺𝐺 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠 = 𝛤𝛤, where Γ is determined by < 𝜎𝜎 𝑦𝑦𝑦𝑦 >. In Fig. 3b we compare predicted to the measured arrest length, measured, obtained from the contact area measurements. predicted agrees well with measured. The same procedure is applied for 11 other experiments, each includes from 4 to 9 precursory events. We performed each experiment under different loading conditions for a wide range of Γ (Fig. 4). It is clear that all of the predicted arrest lengths are in excellent agreement with the measured lengths. The variety of stress distributions and values of Γ that are used emphasize the generality of this result. Note that this framework naturally incorporates the effects of local stress variations. The dependence of the arrest location with a heterogeneous distribution of stresses is not trivial; a given mean value of Δ𝜏𝜏(𝑥𝑥) can yield large variations in the predicted rupture arrest location, since the expression that determines 𝐺𝐺 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠 (𝑙𝑙) generally includes a singular weight function [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] (see, e.g. Eq. 3). For example, stress fluctuations more strongly affect the arrest if located near the arrest point.

Our results provide clear evidence that frictional rupture is really a fracture process that can be quantitatively described by fracture mechanics. The concepts presented here suggest a completely different paradigm for understanding friction than the classical picture, which is based on the balance of local forces (stresses). We have shown that rupture arrest is governed by energy balance precisely as described by the Griffith criterion; balancing the global (system-wide) release of elastic energy, that is embodied in the integral nature of Eq. 3, and the local dissipative properties of the interface governed by Γ.

While in our experiments we considered relatively constant values of Γ, energy balance is also valid when Γ varies along the interface. Locally high values of Γ, can, for instance, act as a barrier to propagation. What can cause spatial variations of Γ? We have shown that Γ ∝ 𝜎𝜎 𝑦𝑦𝑦𝑦 as the normal stress governs the geometrical size of the real contact area.

Additionally, Γ could be affected by varying interface properties, e.g. interfaces incorporating different materials, chemical treatments, or pressure-induced phase transitions.

We have shown that applying fracture mechanics to friction has fundamental predictive power; knowledge of the prospective stress release at each point along the interface will tell us where a frictional rupture will arrest and moreover, when ruptures will traverse the entire interface and precipitate frictional sliding. The initial stress profile along the interface coupled with knowledge of the residual stress, entirely determines the eventual rupture length. Two caveats currently impede making specific predictions;

(1) knowledge of the residual stresses along the interface prior to an upcoming event and (2) knowledge of when rupture nucleation will occur. The first of these might well be addressed by knowledge of the normal stress profile along the interface. Residual stresses are the manifestation of the non-broken contacts that sustain the normal load at the frictional interface. In preliminary work, we observe a strong correlation between residual shear stress and local normal stress, suggesting a local dynamic friction coefficient, but additional study is required to cement this relationship. Rupture nucleation, or the onset of friction, is a more delicate point. Previous work has shown [START_REF] Passelègue | From Sub-Rayleigh to Supershear Ruptures During Stick-Slip Experiments on Crustal Rocks[END_REF][START_REF] Ben-David | Static Friction Coefficient Is Not a Material Constant[END_REF] that characteristic static friction coefficients that govern the onset of frictional motion do not exist. The Griffith criterion could, as in rupture arrest, predict rupture nucleation.

The Griffith criterion, however, can only be applied when a well-defined crack tip exists.

In the case of rupture arrest, a singular tip obviously exists prior to arrest [START_REF] Svetlizky | Classical shear cracks drive the onset of dry frictional motion[END_REF] . Prior to nucleation, however, no initially sharp crack exists; a sharp initial crack must either be created or be dynamically formed. This enigmatic nucleation stage is still the subject of much active research [START_REF] Lapusta | Nucleation and early seismic propagation of small and large events in a crustal earthquake model[END_REF][START_REF] Ben-Zion | Dynamic ruptures in recent models of earthquake faults[END_REF][START_REF] Yang | Dynamics of static friction between steel and silicon[END_REF][29][30] .

While the results described above are generally relevant to every frictional interface, they are especially important to the particular question of earthquake arrest.

Understanding what determines an earthquake's spatial extent is a central unresolved issue [START_REF] Ampuero | Properties of Dynamic Earthquake Ruptures With Heterogeneous Stress Drop[END_REF][19][START_REF] Lapusta | Nucleation and early seismic propagation of small and large events in a crustal earthquake model[END_REF][START_REF] Olson | The deterministic nature of earthquake rupture[END_REF] ; either the rupture size is selected during the nucleation process or an earthquake arrests only when encountering a sufficiently high barrier. We have shown that the selection of the rupture length is deterministic; balancing the global stress release with local dissipation. Our results enable us to understand both viewpoints.

When Γ doesn't vary significantly, the initial stress profile will wholly govern rupture arrest. For example, a large drop of stress near the nucleation zone can control the arrest location. On the other hand, the existence of a sufficiently high local value of Γ could precipitate rupture arrest by overcoming the dominance of the stress profile.

To conclude, it is far from trivial that fracture mechanics necessarily govern rupture arrest in experiments. A text book shear crack is in many ways different than frictional ruptures in a real experimental system which include significant normal stress variations, non-constant initial and residual shear stresses and effects of frictional dissipation along the crack faces. Despite these numerous and substantial complexities, fracture mechanics still provide a fundamental description of friction, which is surprisingly accurate. 3) along 𝑥𝑥 for each event from I to VI. Red circles denote the predicted location of the arrest, 𝑥𝑥 = predicted, as determined by 𝐺𝐺 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠 (𝑙𝑙) = Γ (dashed line). Note that event VI is not arrested, being the first system-wide sliding event. b, Comparison of the measured arrested rupture lengths, measured, as determined from the contact area measurements to the predicted lengths, predicted computed in a. The dashed line has a slope of 1.

References:

Fig. 1 .

 1 Fig. 1. setup and definition of spontaneously arrested interfacial ruptures. a,

Fig. 2 .

 2 Fig. 2. Determination of fracture energy, pre-stress and residual stress. a, The fracture

Fig. 3 .

 3 Fig. 3. Prediction of rupture arrest locations. a, (upper panel), The shear stress drop, 𝛥𝛥𝜏𝜏(𝑥𝑥) = 𝜎𝜎 𝑥𝑥𝑦𝑦 0 (𝑥𝑥) -𝜎𝜎 𝑥𝑥𝑦𝑦 𝑟𝑟𝑟𝑟𝑟𝑟 (𝑥𝑥), along the interface for each of the rupture events presented

Fig. 4 .

 4 Fig. 4. Comparison of the predicted and measured rupture lengths. Precursor lengths

observed in laboratory experiments. Geophys. Res. Lett. 38, L21307 (2011).

Acknowledgments We acknowledge support from the James S. McDonnell Fund, the European Research Council (Grant No. 267256) and the Israel Science Foundation (Grant 76/11). E.B. acknowledges support from the Lady Davis Trust. We thank G. Cohen for comments.

Author contributions E.B. and I.S. performed the measurements. All authors contributed to the analysis and writing the manuscript.

Author information

The authors declare no competing financial interests.

Correspondence and requests for materials should be addressed to J.F.

(jay@mail.huji.ac.il)