E Ikonnikov

Marco Paolino

J C Garcia-Alvarez

Y Orozco-Gonzalez

C Granados

A Röder

J Léonard

M Olivucci

S Haacke

O Kornilov
email: kornilov@mbi-berlin.de

J Leonard

S Gozem
email: sgozem@gsu.edu

Photoelectron Spectroscopy of Oppositely Charged Molecular Switches in the Aqueous Phase: Theory and Experiment

Keywords: photoelectron spectroscopy, electronic structure, molecular environment, biomimetic photoswitches

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tronic structure of molecules. While originally a gas-phase or solid-state surface technique, developments in experimental technologies such as liquid microjets [START_REF] Winter | Photoemission from liquid aqueous solutions[END_REF][START_REF] Seidel | Photoelectron spectroscopy meets aqueous solution: studies from a vacuum liquid microjet[END_REF] and ambient pressure instruments [START_REF] Salmeron | Ambient pressure photoelectron spectroscopy: A new tool for surface science and nanotechnology[END_REF] have extended XPS to the solution phase, where many chemical and biological systems reside. The past decade has seen several applications of static XPS to biologicallyrelevant organic molecules [START_REF] Seidel | Anion photoelectron spectroscopy of protein chromophores[END_REF][5][START_REF] Tau | Liquid-microjet photoelectron spectroscopy of the green fluorescent protein chromophore[END_REF] as well as time-resolved XPS to small inorganic and organic molecules [START_REF] Ojeda | Charge-transfer and impulsive electronic-to-vibrational energy conversion in ferricyanide: ultrafast photoelectron and transient infrared studies[END_REF][START_REF] Engel | Light-induced relaxation dynamics of the ferricyanide ion revisited by ultrafast XUV photoelectron spectroscopy[END_REF][START_REF] Hummert | Femtosecond Extreme Ultraviolet Photoelectron Spectroscopy of Organic Molecules in Aqueous Solution[END_REF][START_REF] Longetti | Photoemission from non-polar aromatic molecules in the gas and liquid phase[END_REF][START_REF] Titov | Electronic relaxation of aqueous aminoazobenzenes studied by time-resolved photoelectron spectroscopy and surface hopping TDDFT dynamics calculations[END_REF][START_REF] Orimo | Ultrafast Electronic Relaxation in 6-Methyluracil and 5-Fluorouracil in Isolated and Aqueous Conditions: Substituent and Solvent Effects[END_REF] Experimentally, the interpretation of solution-phase XPS is complicated by several factors. The well-resolved vibronic peaks readily observed in gas-phase spectra are broadened, shifted, and merged together in solution, resulting in broad bands that are more difficult to connect to theoretical quantities such as vertical binding energies, adiabatic binding energies, and Franck-Condon factors. Scattering and gas-liquid interface effects also complicate the interpretation of XPS spectra. Predicting solution-phase binding energies from first principles can also be challenging; quantum chemical computations must treat the initial and ionized states of the system in a balanced way and capture both short-and long-range solute-solvent interactions either through a large (or periodic) solvent box, [START_REF] Pederson | DFT-based QM/MM with particle-mesh Ewald for direct, long-range electrostatic embedding[END_REF][START_REF] Tazhigulov | Free energies of redox half-reactions from firstprinciples calculations[END_REF][START_REF] Warshel | Simulating electrostatic energies in proteins: Perspectives and some recent studies of pKas, redox, and other crucial functional properties[END_REF][START_REF] Jacobson | A one-electron model for the aqueous electron that includes many-body electron-water polarization: Bulk equilibrium structure, vertical electron binding energy, and optical absorption spectrum[END_REF][START_REF] Ghosh | First-principle protocol for calculating ionization energies and redox potentials of solvated molecules and ions: theory and application to aqueous phenol and phenolate[END_REF][START_REF] Ghosh | Effect of solvation on the vertical ionization energy of thymine: From microhydration to bulk[END_REF] non-equilibrium continuum solvation, [START_REF] Herbert | Dielectric continuum methods for quantum chemistry[END_REF][20][START_REF] Cammi | Nonequilibrium solvation theory for the polarizable continuum model: a new formulation at the SCF level with application to the case of the frequency-dependent linear electric response function[END_REF][START_REF] Aguilar | Nonequilibrium solvation: An ab initio quantum-mechanical method in the continuum cavity model approximation[END_REF][START_REF] Mennucci | Excited states and solvatochromic shifts within a nonequilibrium solvation approach: A new formulation of the integral equation formalism method at the self-consistent field, configuration interaction, and multiconfiguration self-consistent field level[END_REF][START_REF] Coons | Quantum chemistry in arbitrary dielectric environments: Theory and implementation of nonequilibrium Poisson boundary conditions and application to compute vertical ionization energies at the air/water interface[END_REF] or a mix of explicit and implicit solvation. [START_REF] Coons | Quantum chemistry in arbitrary dielectric environments: Theory and implementation of nonequilibrium Poisson boundary conditions and application to compute vertical ionization energies at the air/water interface[END_REF][START_REF] Coons | The hydrated electron at the surface of neat liquid water appears to be indistinguishable from the bulk species[END_REF][START_REF] Tazhigulov | Polarizable embedding for simulating redox potentials of biomolecules[END_REF] Here we report for the first time the experimental XPS spectra and calculated electron binding energies of two molecular switches. The first switch is the methoxy N-alkylindanylidene-pyrrolinium (dMe-MeO-NAIP, from now on referred to as NAIP). This switch is demethylated on C5 of the pyrroline compared to the previously investigated compound of ref. 27. The second switch is para-hydroxydimethylindanylidene-oxopyrrolidine (p-HDIOP).

These molecules, shown in Fig. 1, are biomimetic switches designed to mimic the ultrafast photo-isomerization of the retinal protonated Schiff base (rPSB) chromophore of rhodopsin [START_REF] Sinicropi | An artificial molecular switch that mimics the visual pigment and completes its photocycle in picoseconds[END_REF][START_REF] Gueye | Engineering the vibrational coherence of vision into a synthetic molecular device[END_REF] or the anionic chromophore of the green fluorescent protein, [START_REF] Paolino | Design, synthesis, and dynamics of a green fluorescent protein fluorophore mimic with an ultrafast switching function[END_REF] respectively. Photo-isomerisation of these switches occurs on sub-picosecond time scales in methanol, within 300 fs for NAIP [START_REF] Dunkelberger | Photoisomerization and relaxation dynamics of a structurally modified biomimetic photoswitch[END_REF] and 400 fs for (p-HDIOP), [START_REF] Paolino | Design, synthesis, and dynamics of a green fluorescent protein fluorophore mimic with an ultrafast switching function[END_REF] but with a relatively low quantum efficiency in the 20% range. NMR analysis has shown that before illumination, NAIP is predominantly (> 95%) in the E form, [START_REF] Gueye | Engineering the vibrational coherence of vision into a synthetic molecular device[END_REF] and p-HDIOP also adopts the E isomer in the ground state. [START_REF] Paolino | Design, synthesis, and dynamics of a green fluorescent protein fluorophore mimic with an ultrafast switching function[END_REF] For the purpose of this study, we consider them as model benchmark systems for studying the effect of solvation on electronic binding energies. Importantly, NAIP and p-HDIOP are structurally similar, but they are oppositely charged. NAIP is positively charged, becoming a 2+ ion upon photoionization, while p-HDIOP is negatively charged, becoming neutral upon photodetachment. Because of this difference, their electron binding energies are very different in the gas phase. However, the experiments reported in this manuscript find that this difference is reduced to just 1.1 eV in the aqueous phase due to solvation effects. This poses a challenge to computations, which must capture the large and opposite solvent effect on the binding energies of the two molecules.

Computational Methods

Vertical binding energies (E v ) were first computed using three different electronic structure methods, introduced as M1, M2, and M3 below. The goal of using three methods is not to compare their accuracy, but rather to check whether solvation effects computed for those three different methods are generally reproduced.

For each method we also report adiabatic binding energies (E a ee ). Fig. 2 schematically presents the definitions of E v , E a ee , and E a 00 . E v is a difference between the final (ionized) state energy and the initial state energy both computed at the equilibrium geometry of the initial state. E a ee is a difference between the final (ionized) state energy and the initial state energy each computed at the equilibrium geometry of the corresponding state. E a 00 are adiabatic energies that account for the zero-point vibrational energy (ZPVE) correction for each state obtained from frequency calculations. In the case of M1, 0-0 binding energies (E a 00 ) are also computed.

Figure 2: Schematic representation of vertical (E v ), adiabatic (E a ee ), and 0-0 (E a 00 ) binding energies. ∆Q represents the geometric displacement of the equilibrium structure of the ionized molecule compared to its initial state.

Method 1 (M1): Gas-phase geometries and frequencies were obtained using the PBE0 density functional [START_REF] Adamo | Toward reliable density functional methods without adjustable parameters: The PBE0 model[END_REF][START_REF] Ernzerhof | Assessment of the Perdew-Burke-Ernzerhof exchangecorrelation functional[END_REF] and 6-311+G* basis set. E v and E a ee were computed using the ∆SCF approach. E a 00 was computed with ∆SCF after taking the sum of electronic and zero-point vibrational energies for each of the initial and final states at their respective equilibrium geometries. M1 calculations were carried out with Q-Chem 5.4. [START_REF] Epifanovsky | Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package[END_REF] Method 2 (M2): Gas-phase geometries were optimized using the ωB97X-D density functional [START_REF] Chai | Long-range corrected hybrid density functionals with damped atom-atom dispersion corrections[END_REF] and cc-pVDZ correlation-consistent basis set. [START_REF] Dunning | Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen[END_REF] E v and E a ee were computed using equation-of-motion ionization-potential coupled-cluster with single and double excitations (EOM-IP-CCSD) and the cc-pVDZ basis set. EOM-IP-CCSD computes ionized states from a coupled cluster reference wave function using an electron-annihilating excitation operator. [START_REF] Stanton | Analytic energy derivatives for ionized states described by the equation-of-motion coupled cluster method[END_REF][START_REF] Krylov | Equation-of-motion coupled-cluster methods for open-shell and electronically excited species: The hitchhiker's guide to Fock space[END_REF] To make the calculations tractable, the single-precision implementation [START_REF] Pokhilko | Double precision is not needed for manybody calculations: Emergent conventional wisdom[END_REF] of EOM-IP-CCSD and Frozen Natural Orbital (FNO) truncation [START_REF] Landau | Frozen natural orbitals for ionized states within equation-of-motion coupled-cluster formalism[END_REF] were employed. M2 calculations were carried out with Q-Chem 5.4. [START_REF] Epifanovsky | Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package[END_REF] Method 3 (M3): E v and E a ee were computed with the second-order perturbation theory corrected complete-active-space self-consistent field (CASPT2//CASSCF) method. [START_REF] Andersson | Second-order perturbation theory with a CASSCF reference function[END_REF][START_REF] Andersson | Second-order perturbation theory with a complete active space self-consistent field reference function[END_REF] The CASPT2//CASSCF approach has been regularly employed to study NAIP, p-HDIOP, and rPSB, and has been shown to benefit from a cancellation of errors that makes it suitable for modeling the excitation energies and photo-isomerization mechanism of those systems. [START_REF] Paolino | Design, synthesis, and dynamics of a green fluorescent protein fluorophore mimic with an ultrafast switching function[END_REF][42][43][44][45] Methods M1 and M2 or similar protocols have been used in multiple instances for computing electron binding energies, with benchmarks indicating typical errors of 0.1-0.2 eV relative to experiments for molecules that do not have strong electron correlation. [START_REF] Tau | Liquid-microjet photoelectron spectroscopy of the green fluorescent protein chromophore[END_REF][START_REF] Ghosh | First-principle protocol for calculating ionization energies and redox potentials of solvated molecules and ions: theory and application to aqueous phenol and phenolate[END_REF][46][47][48][START_REF] Bravaya | On the photodetachment from the green fluorescent protein chromophore[END_REF][START_REF] Bravaya | Four bases score a run: Ab initio calculations quantify a cooperative effect of H-bonding and π-stacking on the ionization energy of adenine in the AATT tetramer[END_REF][START_REF] Woodhouse | A photoelectron imaging study of the deprotonated GFP chromophore anion and RNA fluorescent tags[END_REF] CASPT2 and related multi-reference methods have also been used to compute binding energies accurately in systems ranging from small metal clusters [START_REF] Hendrickx | Adiabatic electron affinities of ScC2 and ScC3 evaluated by a multiconfigurational approach[END_REF][START_REF] Clima | Interpretation of the photoelectron spectra of FeS2-by a multiconfiguration computational approach[END_REF] to other biologically relevant molecules. [START_REF] Boichenko | Accurate Vertical Electron Detachment Energies and Multiphoton Resonant Photoelectron Spectra of Biochromophore Anions in Aqueous Solution[END_REF]55 Vertical and adiabatic binding energies were computed using ∆E's computed with CASPT2.

Geometry optimizations were carried with CASSCF and the 6-31G* basis set, while energies of the initial and final (ionized) states were computed using CASPT2 single point calculations with the ANO-L-VDZP basis set. [START_REF] Widmark | Density matrix averaged atomic natural orbital (ANO) basis sets for correlated molecular wave functions: I. First row atoms[END_REF] For both molecular switches, an active space of 12 electrons in 11 orbitals was used for the initial state and 11 electrons in 11 orbitals for the ionized state. The CASSCF orbital with single occupancy in the ionized state are shown in Fig. 3. In both E -NAIP and E-p-HDIOP, the ionization occurs from π orbitals, whether in the gas phase or in solution. No state-averaging was employed for the CASSCF wave function, and a single-state CASPT2 was used for each of the initial and ionized states.

The Cholesky decomposition was used for CASPT2. 57 M3 calculations were carried out in OpenMolcas version 22.10. [START_REF] Aquilante | Modern quantum chemistry with [Open] Molcas[END_REF] The effect of solvation on the computed electron binding energies was computed using both implicit and explicit solvent models. For M1, the E v , E a ee , and E a 00 calculations were repeated using the conductor-like polarizable continuum model (C-PCM) to account for the effect of the water solvent dielectric. [START_REF] Barone | Quantum calculation of molecular energies and energy gradients in solution by a conductor solvent model[END_REF][START_REF] Cossi | Energies, structures, and electronic properties of molecules in solution with the C-PCM solvation model[END_REF] By default, these models allow both the fast (e.g., polarization) and slow (e.g., orientational) components of solvent response to relax and are labeled equilibrium PCM (e-PCM). e-PCM is not suitable for computing vertical binding energies (E v ), where only the fast component of the solvent polarization remains in equilibrium with the sudden ionization but the slow component is effectively frozen. [START_REF] Herbert | Dielectric continuum methods for quantum chemistry[END_REF] Therefore, we also computed E v using a non-equilibrium formulation of PCM (ne-PCM) using the Marcus-Brady-Carr state-specific approach. 20,21,62-64

For M2, EOM-IP-CCSD ne-PCM calculations were computed using a zeroth-order treatment that uses solvent-polarized molecular orbitals for the EOM-IP-CCSD calculations. [START_REF] Mewes | Experimental benchmark data and systematic evaluation of two a posteriori, polarizablecontinuum corrections for vertical excitation energies in solution[END_REF] Explicit solvent calculations were performed using the Average Solvent Electrostatic Configuration and Free Energy Gradient (ASEC-FEG) quantum mechanical / molecular mechanical (QM/MM) approach. The theoretical details of this method have been described

previously. [START_REF] Coutinho | An efficient statistically converged average configuration for solvent effects[END_REF][START_REF] Georg | Electronic properties of water in liquid environment. A sequential QM/MM study using the free energy gradient method[END_REF][START_REF] Bistafa | Combining ab initio multiconfigurational and Free Energy Gradient methods to study the π-π* excited state structure and properties of uracil in water[END_REF][START_REF] Franco | Elucidating the structure of merocyanine dyes with the ASEC-FEG method. Phenol blue in solution[END_REF][START_REF] Orozco-Gonzalez | An average solvent electrostatic configuration protocol for Qm/mm free energy optimization: Implementation and application to rhodopsin systems[END_REF][START_REF] Nikolaev | Free Energy Computation for an Isomerizing Chromophore in a Molecular Cavity via the Average Solvent Electrostatic Configuration Model: Applications in Rhodopsin and Rhodopsin-Mimicking Systems[END_REF][START_REF] Iyer | MOLCAS 7: the next generation[END_REF][72][73] The details of the model set-up and methodology is presented in the Supporting Information. Briefly, the quantum chemical optimization and energy calculations are performed iteratively in the field of a time-averaged environment of the solution obtained from MD snapshots. Water molecules and counterions (Na + in the case of p-HDIOP and Cl -in the case of NAIP) are included explicitly in the MD and ASEC-FEG QM/MM calculations. Their presence not only ensures a globally neutral system before ionization, but also accounts for the electrostatic interaction between the solute and the counterion in an aver-aged way. 72 The ASEC-FEG approach is related to other methods that simulate QM energies in a conformationally sampled solvent environment such as ASEP/MD, 74 MESS-E,, 75 and QM-NBB. 76 MD simulations were carried out using GROMACS 2022.3, 77,78 while QM/MM calculations were carried out using the OpenMolcas 79 and Tinker 6.3 80 interface, [START_REF] Ferré | Probing the rhodopsin cavity with reduced retinal models at the CASPT2//CASSCF/AMBER level of theory[END_REF] treating the molecular switches at the QM level of theory and the ASEC solvent and counterion at the MM level of theory. The QM subsystem was optimized at the CASSCF/6-31G* level of theory. As with the gas-phase calculations, CASSCF employed a 12 electron in 11 orbital active space and no state averaging.

For each of p-HDIOP and NAIP, the ASEC configurations (taken from the last iteration of ASEC-FEG protocol) were exported to Q-Chem, and PBE0/6-311+G* ∆SCF and EOM-IP-CCSD/cc-pVDZ binding energies were computed in that ASEC charge environment.

Test calculations were also carried out where solution ions (Na + for p-HDIOP and Cl - for NAIP) were deleted for each step of the ASEC-FEG protocol to compute the CASTP2 ionization energies in the absence of ions. The results are shown in the Results section as M3-no ion.

Radial Distribution Functions (RDFs) for the counterions were computed with periodic boundary conditions using the gmx rdf tool in GROMACS. 77,78 The cumulative number (CN), a normalized integral of the RDF that, in this case, indicates the probability of finding the counterion within distance r, was also generated.

Experimental Methods

The two molecular switches were prepared and characterized according to procedures reported in the literature (see ref. 82 for NAIP and ref. 29 for p-HDIOP).

The spectroscopy experiments were performed at the XUV monochromator beamline described previously. [START_REF] Hummert | Femtosecond Extreme Ultraviolet Photoelectron Spectroscopy of Organic Molecules in Aqueous Solution[END_REF][START_REF] Titov | Electronic relaxation of aqueous aminoazobenzenes studied by time-resolved photoelectron spectroscopy and surface hopping TDDFT dynamics calculations[END_REF][START_REF] Eckstein | Dynamics of N2 Dissociation upon Inner-Valence Ionization by Wavelength-Selected XUV Pulses[END_REF] In short, XUV light is produced by high-order harmonic generation (HHG) driven by a Ti:Sa femtosecond multi-mJ laser source with the central wavelength of It is worth emphasizing that direct comparison of the computed and experimentally determined binding energies is complicated by elastic and inelastic scattering of the photoelectrons by the solvent. Scattering arises from a complex interplay of multiple factors, including solubility of the switches (related to probing depth), energy of the ionizing radiation, solvent, and angle of detection of the electrons. [START_REF] Tau | Liquid-microjet photoelectron spectroscopy of the green fluorescent protein chromophore[END_REF][START_REF] Michaud | Cross sections for low-energy (1-100 eV) electron elastic and inelastic scattering in amorphous ice[END_REF][START_REF] Luckhaus | Genuine binding energy of the hydrated electron[END_REF][START_REF] Dupuy | 30 Core-Level Photoelectron Angular Distributions at the Liquid-Vapor Interface[END_REF][START_REF] Gozem | Probing the electronic structure of bulk water at the molecular length scale with angle-resolved photoelectron spectroscopy[END_REF] It can be argued that inelastic scattering may only allow higher energy electrons to escape the solvent and therefore give a higher apparent binding energy than the true one. However, given that the photoelectron spectra can be fitted accurately using Gaussian curves in Fig. 4, this suggests that scattering is not significant enough to distort the spectra. [START_REF] Tau | Liquid-microjet photoelectron spectroscopy of the green fluorescent protein chromophore[END_REF] In any case, it is expected that the effect of electron loss is on the same order of magnitude as the experimental uncertainty at the ionizing radiation energy employed (23.4 eV, 16-17 eV above the ionization/detachment threshold for the two switches). Another factor is the sensitivity of the binding energy to the the degree of solvation of the molecular switches; the switches near the air-water interface may have different binding energies compared to the bulk. However, a recent computational study indicates that the binding energy of ions is relatively insensitive to a bulk versus interfacial solvation. [START_REF] Paul | Probing Interfacial Effects on Ionization Energies: The Surprising Banality of Anion-Water Hydrogen Bonding at the Air/Water Interface[END_REF] Finally, the computational methods employed here also have limitations; while implicit solvation may miss specific details of the solvent-solute interactions, the ASEC-FEG approach used here treats the solvent with a fixed-charge force field and misses solvent polarization effects. The errors of the fits are on the order of 1% of the values, but systematic uncertainties (e.g. from the bandwidth of the XUV light) may be larger. From experience we conservatively estimate them to be on the order of 0.2-0.3 eV.

The anionic p-HDIOP solute requires a basic solution of the otherwise neutral chromophore. Therefore 100 mM of KOH was added to the water solvent, which also contained 10 mM of NaCl. This solution was used both for sample preparation and for reference measurements. The p-HDIOP sample thus contained 1.15 mM of p-HDIOP, 100 mM of KOH, 10 mM of NaCl in pure water. Other parameters of the measurement correspond to those of NAIP, as described above. The photoelectron spectra of p-HDIOP are shown in Fig. 4b).

Two bands are again observed in the signal of the solute. The two-Gaussian fit yields the central band positions of 6.0 eV and 8.4 eV with the FWHM of 1.1 eV and 2.0 eV, respectively. The solvent spectrum in Fig. 4b) is different from that of Fig. 4a) due to the weak OH -band observed in the range of 7-8 eV binding energies. This signal is removed by the solvent signal subtraction.

The remainder of this manuscript will focus on the first electron binding energy. The results of the gas-phase, PCM, and ASEC ab initio calculations for the three electronic structure methods are summarized in Table 1 along with the experimental results. 2 E v = Vertical binding energy, E a ee = Adiabatic binding energy, E a 00 = 0-0 binding energy. 3 e-PCM = equilibrium PCM; ne-PCM = non-equilibrium PCM; ASEC = Average solvent electrostatic configuration.

In the gas-phase calculations, NAIP and p-HDIOP have binding energies that differ by ca. 8 eV. This is not surprising considering their opposite charges; p-HDIOP is negatively charged and has a low photodetachment energy in the gas phase. NAIP is positively charged and its ionization to the 2+ charged state requires significantly more energy.

Upon solvation, the negatively charged p-HDIOP is stabilized more than the neutral final state, which increases its binding energy relative to the gas phase. Conversely, solvation of NAIP is likely to have a larger stabilizing effect on the final (2+) state compared to the initial (1+) state, reducing the binding energy. Solvation therefore is expected to have a large and opposite effect on the binding energies of the two molecular switches. Indeed, the experimental results and the calculations in Table 1 indicate that accounting for the solvent environment brings the binding energies of NAIP and p-HDIOP much closer together from 8 eV in the gas phase to ca. 1 eV in the aqueous phase.

Before discussing the effect of solvation, we look at the gas phase calculations to determine a suitable protocol for computing binding energies. The three electronic structure methods give mostly consistent vertical and adiabatic binding energies in the gas phase. Methods M1 and M3 in particular agree with each other to within 0.1-0.2 eV in all cases. M2 underestimates the binding energy of p-HDIOP compared to the other two methods. This can be attributed to the missing diffuse basis functions in the EOM-IP-CCSD/cc-pVDZ calculations. Diffuse basis functions are important for accurately determining the energy of the negatively charged initial state of p-HDIOP. Accounting for those diffuse functions should increase the binding energy of p-HDIOP.

Comparing E-and Z-isomers of NAIP and p-HDIOP. Previous studies have established that the E-stereoisomer is more stable than Z-for both NAIP and p-HDIOP. [START_REF] Gueye | Engineering the vibrational coherence of vision into a synthetic molecular device[END_REF][START_REF] Paolino | Design, synthesis, and dynamics of a green fluorescent protein fluorophore mimic with an ultrafast switching function[END_REF] We computed E v and E a ee for all four systems (E-NAIP, Z-NAIP, E-p-HDIOP, and Z-p-HDIOP). The computations show that binding energies for E-and Z-isomers for each of NAIP and p-HDIOP are close (within 0.1 eV for NAIP, and within 0.2 eV for p-HDIOP).

Therefore, the stereoisomers cannot be resolved experimentally using photoelectron spectroscopy. The remainder of the discussion will focus on the E-NAIP and E-p-HDIOP stereoisomers.

Comparing vertical (E v ), adiabatic (E a ee ), and 0-0 (E a 00 ) binding energies. E a ee was computed using methods M1, M2, and M3 in the gas phase. In all cases, the effect of geometry relaxation of the ionized state reduced the binding energy is limited to under 0.2 eV compared to E v . We accounted for the effect of ZPVE on adiabatic excitation energies for method M1. The ZPVEs are similar for the initial and ionized states (within 0.05 eV)

and therefore give E a 00 energies that are similar to E a ee .

Of the three quantities (E v , E a ee , and E a 00 ), only E a 00 has a direct connection to an experimental quantity. In a well-resolved spectrum for a vibrationally cold molecule, E a 00 corresponds to the lowest energy transition and marks the onset of the spectral band. The other two quantities are theoretical, although E v can be (and often is) connected to the energy where the absorption or emission signal is at a maximum for that band. [START_REF] Gozem | The ezSpectra suite: An easy-to-use toolkit for spectroscopy modeling[END_REF] Given the similarity of calculated E v and E a 00 for each of p-HDIOP and NAIP, it appears that neither switch undergoes significant structural rearrangement upon ionization. This is in contrast to the behavior of these switches in the first singlet excited state, which quickly leads to a strong relaxation along bond-length alternation and out-of-plane deformation modes. [START_REF] Sinicropi | An artificial molecular switch that mimics the visual pigment and completes its photocycle in picoseconds[END_REF][START_REF] Gueye | Engineering the vibrational coherence of vision into a synthetic molecular device[END_REF][START_REF] Schapiro | Initial excited-state dynamics of an N-alkylated indanylidene-pyrroline (NAIP) rhodopsin analog[END_REF][START_REF] Pooler | Effect of charge-transfer enhancement on the efficiency and rotary mechanism of an oxindole-based molecular motor[END_REF][START_REF] Léonard | Mechanistic origin of the vibrational coherence accompanying the photoreaction of biomimetic molecular switches[END_REF] The remainder of the discussion will therefore focus on vertical binding energies (E v ), which will be compared to the experimentally determined central binding energies from the first bands in Fig. 4. Comparing solvation models. The E v and ∆E v results from Table 1 are plotted in Fig. 5. The binding energies of NAIP and p-HDIOP differ by ca. 8 eV in the gas phase.

Using an equilibrium implicit PCM model decreases this ∆E v to 1.5 eV for M1, which is comparable to the experimental difference of binding energies ∆eBE value of 1.1 eV. Using a polarizable continuum model therefore largely captures the effect of solvation on those two oppositely charged molecules. However, equilibrium PCM calculations underestimate the absolute binding energies of both switches relative to the experimental ones by over 1 eV. This is not surprising; e-PCM fully relaxes the solvent environment in the ionized state, which is an incorrect representation of the solvent response to sudden ionization. [START_REF] Herbert | Dielectric continuum methods for quantum chemistry[END_REF] The ionized state energy is therefore overstabilized and the binding energy underestimated.

This underestimation of the binding energy is largely resolved by using a non-equilibrium PCM approach. ne-PCM calculations using both M1 and M2 give binding energies that are in better agreement with those obtained experimentally (within 0.5 eV). The computed ∆E v s for M1 using ne-PCM is also slightly improved, at 1.4 eV compared to the experimental 1.1 eV. For M2, the ∆E v is overestimated, at 1.8 eV, but this could be explained again by the missing diffuse basis functions, which leads to an underestimation of the p-HDIOP binding energy.

ASEC-FEG QM/MM explicit solvent calculations further reduce the computed ∆E v to 0.8-1.4 eV, in good agreement with the experimental ∆eBE. The result from M1 is in particularly good agreement, at 1.0 eV, while M2 overestimates ∆E v (due to the basis set) and M3 underestimates ∆E v . Nonetheless, all three models are within 0.3 eV of the experimental ∆IE. On the other hand, while the ASEC-FEG QM/MM calculations give a ∆E v that is in good agreement with the experiment, all three methods (M1, M2, and M3) overestimate the absolute binding energies of both NAIP and p-HDIOP by over 0.5 eV. This is on the order of magnitude of the effect of solvent polarization, which can exceed 0.5 eV, [START_REF] Tazhigulov | Polarizable embedding for simulating redox potentials of biomolecules[END_REF][START_REF] Barrozo | To be or not to be a molecular ion: the role of the solvent in photoionization of arginine[END_REF][START_REF] Sadybekov | Coupled-cluster based approach for core-level states in condensed phase: Theory and application to different protonated forms of aqueous glycine[END_REF] and is missing in the ASEC fixed charge model used here.

The effect of the counterion. Although both ne-PCM and ASEC-FEG give binding energies that are in reasonably good agreement with the experiment, it would be interesting to look at one of the aspects where the ASEC and PCM solvation models differ; ASEC includes not only explicit solvent, but also explicit counterions. The ASEC configuration samples the distribution of those ions from MD simulation. Fig. 6 shows the radial distribution functions (RDF) and cumulative number (CN) of the counterion relative to the center of mass of the switches computed from one of the MD simulations. From this distribution of charges, it is possible to estimate the effect of the ions using Coulomb's law. Assuming a centro-symmetric potential interacting with a point charge of magnitude e, and in the absence of a dielectric medium:

V = - 1 4πϵ 0 ρ(r) r dτ (1)
The ρ(r) term can be represented using the CN plots in Fig. 6. Integrating the expression above using these CN plots gives an energy of 0.52 eV for the interaction between Na + ions and the negatively charged p-HDIOP, and 0.60 eV for the interaction between Cl -ions and positively charged NAIP. Considering that this charge interaction will stabilize the 2+ ionized state more than the 1+ initial state in NAIP, but will instead stabilize the -1 initial state in p-HDIOP more than the neutral state, the direct effect of counterions on ∆E v is over 1 eV. Indeed, we find that simply deleting the counterions and repeating the QM/MM calculations increases the computed ∆E v from 0.8 eV to 2.1 eV (M3-no ion in Table 1).

Of course, deleting the counterions does not give a realistic representation of the effect they have on the binding energy. The calculations only account for a minimal number of counterions of the solute, while experimental solutions are more complex and contain NaCl and KOH for experimental purposes. Furher, water molecules are arranged in such a way to screen the counterions and will rearrange in the absence of these counterion. If we account for the relative permittivity of water (78.4 at ambient conditions), the effect of the counterions on the binding energy of the molecular switches would be substantially diminished. However, NAIP and p-HDIOP are not point charges in a uniform dielectric field, and the counterion distribution is certainly not centro-symmetric. Therefore, counterions in the ASEC-FEG model may contribute to long-range electrostatic solute-counterion interactions that are missing in the PCM models, leading to a slightly lower computed ∆eBE. Such long-range effects were found to be important in QM/MM calculations of proteins such as rhodopsins and flavoprotein photoreceptors, 72,[START_REF] Pedraza-González | ARM: automatic rhodopsin modeling with chromophore cavity generation, ionization state selection, and external counterion placement[END_REF] where counterions are distributed near the surface of the protein. More realistic models of the solute-ion interactions will be addressed in future work.

In conclusion, we have presented experimental XPS and computed binding energies of a positively charged molecule, NAIP, and a negatively charged molecule, p-HDIOP. These photo-switches pose a challenge to electronic structure calculations, which need to account for large and opposite solvation effects through suitable solvation models. Experimentally, the difference in electron binding energy (∆eBE) between p-HDIOP and NAIP is 1.1 eV, compared to ca. 8 eV difference predicted in the gas phase. Non-equilibrium implicit (ne-PCM) solvation models capture a large part of the solvent effect and give both absolute and relative binding energies that are in good agreement with the experiments. Equilibrium

Figure 1 :

 1 Figure 1: Structures of E -NAIP and E-p-HDIOP.

Figure 3 :

 3 Figure 3: CASSCF molecular orbital having single occupancy for E -NAIP (left) and E-p-HDIOP (right) in the ionized state. The orbital CASSCF occupancies are between 1.01 -1.02. The figures were generated using Molden.59

Figure 4 :

 4 Figure 4: a) Photoelectron spectrum of NAIP. The sample and solvent spectra are shown as orange and blue lines, respectively. The difference spectrum (green open circles) shows two bands and is well described by two Gaussian profiles (green dashed curve). The individual Gaussians are shown with black dashed lines. The central binding energies of the two bands are 7.1 eV and 9.0 eV. b) Photoelectron spectrum of p-HDIOP. The central binding energies of the two bands are 6.0 eV and 8.4 eV.

A 10 mM

 10 NaCl solution was prepared and used both for the NAIP sample preparation and for reference solvent measurements performed before and after the sample measurements, which is important for background suppression as discussed below. The NAIP sample thus contained 1.35 mM concentration of NAIP molecules, 10 mM of NaCl in pure water. The total volume of the sample solution was 15 ml, which was sufficient for 15 minutes of signal accumulation at the flow rate of 0.4 ml/min after accounting for the volume of the sample delivery system. All the flasks and sample supply capillaries were covered by aluminium foil to prevent the compound isomerization under the laboratory light. The reference solvent measurements were accumulated for the same amount of time. The measured photoelectron spectrum of NAIP is shown in Fig.4a) as an orange line together with the reference solvent spectrum (blue curve), plotted as a function of the binding energy (eBE, determined using the photon energy minus kinetic energy of the detected electrons). The open circles show the difference spectrum resulting from subtraction of the solvent signal. The green dashed line represents a fit line composed of two Gaussian profiles, which describe well the two bands observed in the range of 5 to 9 eV. The central binding energies of the two bands are 7.1 eV and 9.0 eV. The full widths at half maximum (FWHM) are 1.0 eV and 2.0 eV, respectively.

Figure 5 :

 5 Figure 5: Top. Computed E v s for E -NAIP (red) and E-p-HDIOP (blue) using the computational methods and solvation models as shown in Table 1. Bottom. A plot of the difference (green) in the E v for E -NAIP and E-p-HDIOP (∆E v ) as a function of the method and solvation model. The horizontal dashed lines indicate the experimental values.

Figure 6 :

 6 Figure 6: Top. Radial distribution functions (RDF) and cumulative number (CN) of the counterion for E -NAIP (A) and Z-p-HDIOP (B). The RDF and CN for each of the two systems were computed from a 4 ns MD simulation. The CN converges to 1 at a large radius because there is only one counterion, and therefore represents the probability of finding the counterion at a given value of r. The numbers at the top indicate the probability of finding the counterion in each 10 Å shell around the center of mass of the molecule, multiplied by the sign of the charge. Bottom. A schematic representation of an ASEC configuration that shows how the counterion is distributed. The blue circle represents the water solvent ASEC configuration, which is explicitly included in the QM/MM calculations for up to 20 Å. The surrounding concentric circles represent the counterions that interact at a long range with the solute. The color and shading of each shell relates to the total charge within that spherical shell, based on the CNs computed from an MD simulation.

  (e-PCM) models reproduce the relative binding energies of the two switches relatively well (∆eBE=1.5 eV) but give underestimated binding energies by over 1 eV. ASEC-FEG explicit solvent models gives a ∆E v of 0.8-1.4 eV, depending on the method, in closer agreement to the experimental ∆eBE. An advantage of the ASEC-FEG explicit solvent model is that it captures long-range electrostatic interactions with a time-averaged solvent and counterion environment. However, it misses fast polarization effects of the nearby solvent molecules, which are better captured by ne-PCM solvent models. The results presented in this work will serve as a solid basis for time-resolved XPS experiments, which are being performed in our laboratory. and characterization of a dipole moment switch driven by Z/E photoisomerization. Journal of the American Chemical Society 2010, 132, 9310-9319. (43) Gozem, S.; Huntress, M.; Schapiro, I.; Lindh, R.; Granovsky, A. A.; Angeli, C.; Olivucci, M. Dynamic electron correlation effects on the ground state potential energy surface of a retinal chromophore model. Journal of chemical theory and computation 2012, 8, 4069-4080. (44) Gozem, S.; Melaccio, F.; Luk, H.; Rinaldi, S.; Olivucci, M. Learning from photobiology how to design molecular devices using a computer. Chemical Society Reviews 2014, 43, 4019-4036. (45) Gozem, S.; Luk, H. L.; Schapiro, I.; Olivucci, M. Theory and simulation of the ultrafast double-bond isomerization of biological chromophores. Chemical reviews 2017, 117, 13502-13565. (46) Gallandi, L.; Marom, N.; Rinke, P.; Körzdörfer, T. Accurate ionization potentials and electron affinities of acceptor molecules II: non-empirically tuned long-range corrected hybrid functionals. Journal of chemical theory and computation 2016, 12, 605-614. (47) Kanchanakungwankul, S.; Truhlar, D. G. Examination of how well long-range-corrected density functionals satisfy the ionization energy theorem. Journal of Chemical Theory and Computation 2021, 17, 4823-4830. (48) Pieniazek, P. A.; Arnstein, S. A.; Bradforth, S. E.; Krylov, A. I.; Sherrill, C. D. Benchmark full configuration interaction and equation-of-motion coupled-cluster model with single and double substitutions for ionized systems results for prototypical charge transfer systems: Noncovalent ionized dimers. The Journal of chemical physics 2007, 127, 164110.

Table 1 :

 1 Computed binding energies of NAIP and p-HDIOP

	Method 1	Energy 2 Solvation 3 E -NAIP (eV) Z -NAIP (eV) E-p-HDIOP (eV) Z-p-HDIOP (eV) ∆(E -NAIP -E-p-HDIOP)
	M1	E v	None	10.6	10.6	2.8	2.6	7.8
	M2	E v	None	10.4	10.5	2.2	2.0	8.2
	M3	E v	None	10.7	10.8	2.8	2.7	7.9
	M1	E a ee	None	10.5	10.5	2.7	2.5	7.8
	M2	E a ee	None	10.3	10.3	2.1	1.9	8.1
	M3	E a ee	None	10.6	10.6	2.7	2.6	7.9
	M1	E a 00	None	10.5	10.5	2.7	2.5	7.8
	M1	E v	e-PCM	6.0	6.0	4.5	4.5	1.5
	M1	E v	ne-PCM	6.9	6.9	5.5	5.5	1.4
	M2	E v	ne-PCM	7.6	7.6	5.8	5.8	1.8
	M1	E a ee	e-PCM	5.9	5.9	4.4	4.4	1.5
	M1	E a 00	e-PCM	5.9	5.9	4.5	4.4	1.4
	M1	E v	ASEC	7.7	7.7	6.7	6.7	1.0
	M2	E v	ASEC	7.7	7.6	6.3	6.3	1.4
	M3	E v	ASEC	7.8	7.8	7.0	7.0	0.8
	M3-no ion E v	ASEC	8.4	8.4	6.3	6.4	2.1
	Experiment			7.1	-	6.0	-	1.1

1 M1 = ∆SCF, PBE0/6-311+G*; M2 = EOM-IP-CCSD//ωB97X-D/cc-pVDZ ; M3 = CASPT2/ANO-L-VDZP//CASSCF/6-31G*.

Acknowledgement

This material is based upon work supported by the National Science Foundation (NSF) under Grant CHE-2047667 (S.G.). We also acknowledge NSF XSEDE for computational resources from Research Allocation CHE180027 and the Advanced Research Computing Technology and Innovation Core (ARCTIC) resources at Georgia State University's Research Solutions, made available by the National Science Foundation Major Research Instrumentation (MRI) grant number CNS-1920024. J.C.GA acknowledges the Molecular Basis of Disease Program fellowship from Georgia State University. I.E., C.G and O.K. acknowledge the support of the

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Supporting Information Available

The supporting information includes details of the QM/MM calculations, tables of computed absolute and vertical binding energies, and optimized coordinates for the ground and ionized states of the molecular switches.


Martin Poncelet

Christelle Combescure

Fabien Amiot
email: fabien.amiot@femto-st.fr

Detecting bifurcations in 2D periodic metamaterials from images

Keywords: Metamaterials, Instabilities, Imaging

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Architectured materials are composite materials in which the macroscopic properties emerge from the local organisation of constitutive materials in space at the mesoscopic scale. When optimised to improve the stiffness-to-weight ratio, buckling at the mesoscopic scale becomes a possible mode of deformation of these materials [START_REF] Papka | Biaxial crushing of honeycombs: -Part 1: Experiments[END_REF]. While many studies are focused on the numerical prediction of this buckling [START_REF] Triantafyllidis | Onset of failure in aluminum honeycombs under general in-plane loading[END_REF][START_REF] Ohno | Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation[END_REF][START_REF] Saiki | Flower patterns appearing on a honeycomb structure and their bifurcation mechanism[END_REF], there are fewer experimental studies available to validate these numerical results [START_REF] Papka | Biaxial crushing of honeycombs: -Part 1: Experiments[END_REF][START_REF] Scarpa | Elastic buckling of hexagonal chiral cell honeycombs[END_REF]. The comparison criterion between numerical and experimental results is the onset of instability [START_REF] Combescure | Deformation Patterns and their Stability in Finitely Strained Circular Cell Honeycombs[END_REF] traditionally identified using the global loading force maximum. However, this detection might be hindered, experimentally, by various defects in real systems which may locally trigger the instabilities [START_REF] Papka | Biaxial crushing of honeycombs: -Part 1: Experiments[END_REF].

The use of full-field kinematic measurements -e.g. Digital Image Correlation (DIC) -for architectured materials is not new. Different techniques have been developed to consider the fact that a large portion of the observed area is void and hence doesn't contain kinematic information. For DIC, one may circumvent the problem by adding a stiffness-free matter [START_REF] Poncelet | An experimental evidence of the failure of Cauchy elasticity for the overall modeling of a non-centro-symmetric lattice under static loading[END_REF], use multiscale approaches [START_REF] Agnelli | Systematic two-scale image analysis of extreme deformations in soft architectured sheets[END_REF][START_REF] Hild | Multiscale DIC applied to Pantographic Structures[END_REF], choose relevant element types (like beam in the case of lattice materials [START_REF] Somera | On the effective elasticity of quasi-periodic lattice materials : From microscopic foundations to experimental validation[END_REF]), use high resolution cameras and a mesh based on the mesoarchitecture [START_REF] Rouwane | Architecture-Driven Digital Image Correlation Technique (ADDICT) for the measurement of sub-cellular kinematic fields in speckle-free cellular materials[END_REF], etc. However all these techniques while providing many information, require non-negligible expertise and do not detect buckling, thus implying further data-processing. This contribution focuses on a simpler way to exploit images to detect buckling.

Materials and methods

Material

The specimen is a parallelepiped rectangle of hexagonal honeycomb, containing 11 × 11 8-mmdiameter holes, with 2-mm menisci. The complete specimen is 111 × 105 mm 2 . The out-ofplane thickness (24 mm) prevents global out-of-plane buckling.

The constitutive material is a low stiffness silicon rubber (Smooth-SilTM 950 by Smooth-On, initial tangent Young modulus of 1.9 MPa), displaying high elongation at break (320%) and very low viscosity. The specimen is cast in a 3d-printed mold. This technique enables an accurate geometry of the specimen, offers a wider range of material properties and ensures an extremely low porosity of the constitutive material.

Biaxial loading and imaging

Bicompaction tests are performed with a machine following the principle proposed by [START_REF] Hambly | A new true triaxial apparatus[END_REF] offering the advantage of continuous boundary conditions by use of flat platens contrary to [START_REF] Shan | Harnessing multiple folding mechanisms in soft periodic structures for tunable control of elastic waves[END_REF] which rely on slotted platens. It also allows different strain ratios, i.e. both uniaxial and equibiaxial loadings.

The setup (Fig. 1) is instrumented with 8 load sensors to assess the force applied by each platens. The measured normal and tangential loadings are globally denoted as T(s, n), with s spanning the {1…8} range, and n denoting the image index. In order to describe the loading through a unique scalar, the results are displayed as a function of a global loading parameter g(n) defined such that T (s, n) ≃ g(n) × f (s). This decomposition is obtained as the leading term of the singular value decomposition (SVD) of T(s,n) [START_REF] Press | Numerical recipes in C++[END_REF].

Displacement-controlled equibiaxial compression tests are done at a constant velocity and thus constant macroscopic strain rate (typically below 0.1 %/s). The images P(x,y,n) are obtained with a Digital Single Lens Reflex camera (70D, Canon) equipped with a 60-mm fixed focal lens (Canon), and are 3648 × 5472 pixels in size. x and y denote the position in the image's plane.

Proposed image processing

A region of interest (ROI) P ( x , y , n) covering the whole sample is defined out of the images P ( x , y , n) and windowed. Its boundaries are updated using the measured platens displacements, thus compensating for uniform strains. The Fourier transform P (ξ , η , n) of P ( x , y , n) reads P (ξ , η , n)= ∫ P ( x , y , n) exp (-2 iπ ( x ξ+ y η)) dxdy

(1)

P is computed using a fast Fourier transform (FFT). As P is real-valued, ‖ P (ξ ,η , n)‖ is symmetric with respect to the ξ and η axes, so that only one quarter of the spectrum is displayed in the following.

Figure 2: Magnitude of the Fourier spectra of the ROI in the undeformed state ‖ P (ξ ,η , n=1)‖ (left) and after bifurcation ‖ P (ξ ,η , n=6 1)‖ (right), see the global loading parameter g(n) (center).

As presented in Fig. 2 (left), ‖ P (ξ ,η , n=1)‖ displays a well defined structure corresponding to the symmetry of the material in its initial configuration. Indeed, since the initial undeformed geometry of the sample is composed of hexagonal tiling of circular holes, the hexagonal symmetry can be observed on the plot of the Fourier spectra magnitude. For instance, the two peaks A and C are separated by a 60° angle with respect to the origin. A global loading force maximum is observed at a time corresponding to image n=57, indicating the onset of the structural instability (Fig. 2, center). As can be seen from Fig. 2 (right), the magnitude of Fourier spectra obtained for the bifurcated structure shortly after that global loading force maximum (image n=61) displays additional, somehow blurry, peaks (see for instance the peak located around (17,3)). Indeed, generic bifurcation breaks symmetry [15] and this symmetry breaking would result in a different arrangement of the peaks in the Fourier transform.

It is therefore proposed to track the instabilities of the structure by defining the indicator I AB (n) such that

I AB (n )= ∫ AB ❑ ‖ P (ξ , η ,n )‖d ξ d η ‖ P ( A , n)‖+‖ P ( B , n)‖ (2)
I AB (n) refers to the rectangular spectral domain AB, delimited by the user-defined peaks A and B (see Fig. 2). This spectral region has to be initially free of any peak and is thus denoted as an empty spectral region (ESR). The indicator (2) thus depends on both the chosen ROI and ESR, and any other ESR, corresponding to any other bifurcated state, could be chosen in Fig 2a.

Results

The proposed analysis is first exemplified using a ROI which extends over the full sample for validation purpose. The detected buckling is then said global. The analysis is then performed using ROIs which cover only part of the sample, thus experimentally defining local buckling.

At the global scale

The indicator I AB defined by Eq. 2 is computed for the ESR delimited by the two peaks A and B indicated on Fig. 2 (left), and for a ROI covering the full sample. It measures the visibility of a peak appearing in the ESR. This indicator is displayed together with the global loading parameter as a function of the image index in Fig. 3. (n). Each line corresponds to a particular sub-region, and they share the same color in Fig. 4.

Discussion

The I AB indicator remains almost constant throughout the specimen loading and suddenly increases when approaching the instability, as it is detected from the global loading maximum. The excellent agreement between the two indicators validates the proposed image-based indicator for the detection of the onset of instabilities in periodic architectured materials. It should further be highlighted that the detection of the instability via the global loading parameter g(n) corresponds to the maximum rate of I AB with respect to the applied strain (around I AB ≃ 1). Fig. 4 further clearly shows that buckling does not appear in a uniform manner across the sample. The local bifurcation indicator clearly increases from I AB ≃ 0.6, well before the global instability is detected. This demonstrates the need for such a local bifurcation indicator: it allows to resolve, both in time and space, the transition to the bifurcated state. It further allows to discriminate buckling initiation (as detected by the local indicator) from the global (catastrophic) buckling of the structure.

Conclusion

The proposed technique, based on an image processing indicator, has been validated with respect to classical global loading maximum criterion. This image processing technique making only use of FFT, it is simply coded and computationally efficient [16]. As a consequence, it could be used to provide feedback to the machine. It can also provide additional information on a possible non-uniform or premature buckling in the specimen that would not necessarily lead to a drastic change in the global loading indicator. Used as such or combined with a global loading measurement, our proposed technique is a new tool to better understand the local-toglobal buckling transition in architectured materials.

Figure 1 :

 1 Figure 1: Experimental setup: testing machine (the wires holding the machine horizontally are in dark red) (left) and field of view of the camera (right).

Figure 3 :

 3 Figure 3: Global bifurcation indicator for the AB zone IAB(n) and the global loading parameter g(n) as a function of the image index n.4.2. At local scaleThe initial ROI is divided into 3 × 3 equal, non-overlapping sub-regions. The same computations have been performed independently considering the individual sub-regions as ROIs and keeping the same AB domain as ESR. The resulting indicators are denoted as local in the following.

Figure 4 :

 4 Figure 4: Localization of the nine sub regions (left) and local bifurcation indicator for AB zone iAB(n) as a function of the corresponding global bifurcation indicator IAB(n) (right).

Fig. 4

 4 Fig. 4 displays the local indicator i AB (n) computed on the 9 sub-regions as a function of the global indicator I AB (n). Each line corresponds to a particular sub-region, and they share the same color in Fig.4.

Acknowledgements

This work (M.P., C.C.) benefited from the research project ANR-19-CE08-0005 (project Max-Oasis) funded by the French National Research Agency (ANR).