Jian-Wei Liu

Qiang Li

Li-Ping Sun
email: sunliping@hlju.edu.cn

Li-Hua Huo

Hui Zhao
email: zhaohui98@hlju.edu.cn

Jean-Marc Bassat

Sébastien Fourcade

Jean-Claude Grenier

Liu Jian-Wei

Li Qiang

Sun Li-Ping

Huo Li-Hua

Zhao Hui

Tuning the high temperature properties of PrSrCoO4 cathode with Cu2+ dopant for intermediate temperature solid oxide fuel cells

Keywords: solid oxide fuel cells (SOFCs), K2NiF4 structure, cathode material, electrochemical properties

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

Introduction

The A2BO4 oxides with K2NiF4-type structure have been considered as promising cathode materials for intermediate temperature solid oxide fuel cells (IT-SOFCs), due to their high oxygen diffusion and exchange coefficients, good electronic conductivity, suitable thermal expansion coefficient (TEC) for typical electrolytes and excellent electrocatalytic activity towards oxygen reduction reaction (ORR) [START_REF] Boehm | Oxygen diffusion and transport properties in non-stoichiometric Ln2-xNiO4+ oxides[END_REF][START_REF] Sun | Synthesis and performance of Sr1.5LaxMnO4 as cathode materials for intermediate temperature solid oxide fuel cell[END_REF][START_REF] Lyskov | The effect of temperature and oxygen partial pressure on the reduction mechanism in the Pr2CuO4/Ce0[END_REF][START_REF] Sayers | Evidence for the catalytic oxidation of La2NiO4+[END_REF]. A2BO4 oxides can be considered as a superstructure that composed by perovskite layer (ABO3) and rock salt layer (AO) alternatively overlapped along c-axis direction. Due to the insertion character of rock salt layer, these materials exhibit special structural flexibility: they can withstand reducing conditions via loss of oxygen, usually from equatorial planes, and accommodate excess interstitial oxygen under oxidizing conditions [START_REF] Skinner | Oxygen diffusion and surface exchange in La2-xSrxNiO4+[END_REF]. Studies have proved that the presence of rock salt block in Т-phase A2BO4 favors the high oxygen mobility in the bulk material [START_REF] Chroneos | Anisotropic oxygen diffusion in tetragonal La2NiO4+: molecular dynamics calculations[END_REF]. Compared to the traditional ABO3 perovskite oxides, these A2BO4 materials show relatively large oxygen ion conductivity and promising electrode performance. Therefore, Nickelate and copper based A2BO4 oxides with mixed ionic-electronic conducting (MIEC) properties have been thoroughly evaluated as cathode materials for IT-SOFCs [START_REF] Sharma | Design of interfaces in efficient Ln2NiO4+ (Ln=La, Pr) cathodes for SOFC applications[END_REF][START_REF] Sharma | Design of La2-xPrxNiO4+ SOFC cathodes: a compromise between electrochemical performance and thermodynamic stability[END_REF][START_REF] Inprasit | Effect of Sr substituted La2-xSrxNiO4+ (x=0, 0.2, 0.4, 0.6, 0.8) on oxygen stoichiometry and oxygen transport properties[END_REF][START_REF] Kolchina | Evaluation of Ce-doped Pr2CuO4 for potential application as a cathode material for solid oxide fuel cells[END_REF][START_REF] Kolchina | Evaluation of La1.8-xPrxSr0.2CuO4- oxides as cathode materials for IT-SOFCs[END_REF][START_REF] Chen | Insight into an unusual lanthanum effect on the oxygen reduction reaction activity of Ruddlesden-Poppertype cation-nonstoichiometric La2-xNiO4+ (x = 0-0.1)[END_REF][START_REF] Shen | Preparation and electrical properties of Ca-doped La2NiO4+ cathode materials for IT-SOFC[END_REF][START_REF] Li | Electrochemical investigation of Pr2CuO4-based composite cathode for intermediate-temperature solid oxide fuel cells[END_REF][START_REF] Sun | Study of oxygen reduction mechanism on Ag modified Sm1.8Ce0.2CuO4 cathode for solid oxide fuel cell[END_REF][START_REF] Li | Electrode properties of Sr doped La2CuO4 as new cathode material for intermediate-temperature SOFCs[END_REF][START_REF] Sun | Electrochemical performance of Nd1.93Sr0.07CuO4 nanofiber as cathode material for SOFC[END_REF][START_REF] Ishihara | Oxide ion conductivity in defect perovskite Pr2NiO4 and its application for solid oxide fuel cells[END_REF][START_REF] Zhao | Electrochemical property assessment of Pr2CuO4 submicrofiber cathode for intermediate-temperature solid oxide fuel cells[END_REF][START_REF] Sun | Characterization and electrochemical performances of Pr2CuO4 as a cathode material for intermediate temperature solid oxide fuel cells[END_REF].

In contrast to the wealth of data accumulated on nickelate and copper oxides, few studies have been reported to date on the structure and electrochemical catalytic activity of cobalt-based A2BO4 oxides. An early study in 2005 by Munnings et al.

found that the oxygen diffusion coefficient (D  ) and oxygen exchange coefficient (k  ) of La2CoO4+ were even higher than that of La2NiO4+ at T < 700 o C [START_REF] Munning | Oxygen transport in 25 the LaNiCoO4 system[END_REF]. Wang and Nie reported that Ln2CoO4 (Ln=Pr and Sm) could be applied as cathodes for SOFC via studying their thermal stability and conductive properties [START_REF] Wang | A2-A'BO4-type oxides as cathode materials for IT-SOFCs (A = Pr[END_REF]. The electrochemical properties of Co-based A2BO4 materials can be further optimized by doping aliovalent elements in A and/or B site. For instance, the transport properties of Sr 2+ doped La2-xSrxCoO4 have been systematically investigated [START_REF] Tealdi | Vacancy and interstitial oxide ion migration in heavily doped La2-xSrxCoO4[END_REF]. In another study, the effects of Sb doping on the cathode property of LaSrCo1-xSbxO4 was reported by Wang et al [START_REF] Wang | Investigation of structural and electrochemical properties of LaSrCo1-xSbxO4 (0X0.20) as potential cathode materials in intermediate-temperature solid oxide fuel cells[END_REF]. Recently, Cao Y et al. evaluated the cathode performance of Pr2-xSrxCoO4+ materials. They found that Pr0.8Sr1.2CoO4+ had the minimum polarization resistance of 0.29 . cm 2 at 700 o C [START_REF] Cao | Preparation and characterization of Pr2-xSrxCoO4+ cathode materials for IT-SOFC[END_REF]. It was reported that ABO3 perovskite with Cu 2+ at B site always exhibited low TECs [START_REF] Gao | Electrode properties of Cu-doped Bi0.5Sr0.5FeO3- cobalt-free perovskite as cathode for intermediatetemperature solid oxide fuel cells[END_REF][START_REF] Li | Electrochemical performance of novel cobalt-free perovskite SrFe0.7Cu0.3O3- cathode for intermediate temperature solid oxide fuel cells[END_REF], whereas Pr based perovskite oxides are known to exhibit high electrochemical activity, due to the redox cycles of Pr 3+ cations in the lattice [START_REF] Mazo | Thermal expansion behavior and hightemperature electrical conductivity of A2-xAx'Cu1-yCoyO4 (A=La, Pr; A'=Pr, Sr) oxides with the K2NiF4-type structure[END_REF][START_REF] Lyskov | Electrochemical characterization of Pr2CuO4 cathode for IT-SOFC[END_REF][START_REF] Zheng | Evaluation of Ln2CuO4 (Ln: La, Pr, Nd) oxides as cathode materials for IT-SOFCs[END_REF]. In this study, Cu 2+ doped PrSrCo1-xCuxO4+ are synthesized and the effects of Cu 2+ substitution on the crystal structure, high-temperature electrical conductivity, thermal expansion behavior and the electrochemical properties of the compounds are investigated. We propose the PrSrCo1-xCuxO4+ material, which combine the merits of both Cu 2+ dopants and Pr 3+ cations, would be potential cathode candidate for IT-SOFCs. (China). The GDC powders are pressed uniaxially at 220 MPa and then sintered at 1400 o C for 10 h to form dense pellet with 10 mm in diameter and 0.8mm in thickness.

Experimental

The cathode ink is prepared by mixing 100 mg PrSrCo1-xCuxO4+ powders with 4 mL terpineol containing 4% ethyl cellulose. The symmetric cell PrSrCo1-xCuxO4+/GDC/ PrSrCo1-xCuxO4+ is fabricated by painting the cathode ink on both sides of GDC pellet in symmetric configuration. The effective cathode area is 0.25 cm 2 . The cells are initially heated at 500 o C for 0.5 h to eliminate the organic binders, and then followed by calcinations at 950 o C for 3 h with a heating/cooling rate of 5 o C min -1 .

The electrolyte-supported single cell Ni-GDC|GDC| PrSrCo0.7Cu0.3O4+ is prepared for full cell performance test. NiO-GDC anode ink (NiO:GDC = 60:40 wt.%) is firstly painted on one side of GDC pellet, followed by heat-treatment at 1250 o C for 4 h in air to form anode with surface area of 0.25 cm 2 . PrSrCo0.7Cu0.3O4+ cathode is then prepared on the other side of GDC pellet in symmetrical configuration.

Characterization

The phase purity and chemical compatibility of PrSrCo1-xCuxO4+ with GDC are characterized by powder X-ray diffraction method (XRD, Bruker AXS D8). The measurements are conducted at 40 kV and 40 mA using Cu Kα radiation. The diffraction data is collected in the 2θ range of 20°-75° with scanning speed 8°/min. The unit cell parameters are analyzed by LeBail refinement method with the Fullprof package. In the refinement procedure, the 6-coefficients polynomial function and Pseudo-Voigt function are selected to fit the background and peak shape, respectively. The power output performance of the electrolyte-supported single cell is tested with the device of ProboStat30. Humidified H2 (97% H2/3% H2O) is fed to the anode chamber at a flow rate of 50 ml min -1 , while the cathode is exposed to ambient air. Ag meshes are used as current collectors for both anode and cathode sides.

Results and discussions

Phase identification

The XRD patterns of PrSrCo0.7Cu0.3O4+ are shown in Figure 1a. All the patterns are similar to the reported PrSrCoO4 material [START_REF] Hassen | Structure-Property Relationships in Pr1-xSr1+xCoO4 (0 ≤ x ≤ 0.40)[END_REF], indicating the sample is crystallized in single-phase K2NiF4-type structure with SG I4/mmm. The refinement results of the unit cell parameters are listed in Table 1. It is observed that the cell parameter c increases steadily with Cu 2+ doping. This observation can be understood, considering the Jahn-Teller effect of Cu 2+ with electronic configuration of d9. Cu 2+ cations tend to form square planar coordination. In this sense, the apical Cu-O distance is much longer than that of equatorial Cu-O distance in [CuO6] octahedron. In addition, more interstitial oxygen is generated in the rock-salt layer with Cu 2+ doping, which also causes the expansion of c-axis. The similar conclusions have been drawn in the other Cu 2+ doped Ln2NiO4 materials [START_REF] Nakamura | The effect of interstitial oxygen formation on the crystal lattice deformation in layered perovskite oxides for electrochemical devices[END_REF][START_REF] Gilev | Effect of calcium and copper/iron co-doping on defect-induced properties of La2NiO4-based materials[END_REF][START_REF] Aguadero | Evaluation of the La2Ni1-xCuxO4+δ system as SOFC cathode material with 8YSZ and LSGM as electrolytes[END_REF][START_REF] Tarutin | Cu-substituted La2NiO4+δ as oxygen electrodes for protonic ceramic electrochemical cells[END_REF]. On the other hand, the cell parameters a (b) decrease first and then increase again with the Cu 2+ doping, with an inflection point at x = 0.3. In order to maintain the charge neutrality requirement, Cu 2+ doping leads to the partial oxidation of Co 3+ to Co 4+ . According to the valence bond theory, the cobalt-oxygen bond is strengthened and the bond length along a(b)-axis direction decreases with the Cu 2+ doping. At the same time, we should also consider the size effects of doped Cu 2+ (the radius of Cu 2+ (72 pm) is larger than that of Co 3+ (63 pm) and Co 4+ (53 pm)). Therefore, the variation of cell parameters a (b) for PrSrCo1-xCuxO4+ oxides seems to be a competitive results of the doped amount of

Table 1

Cell parameters of PrSrCo1-xCuxO4+ (x = 0.0, 0.1, 0.3, 0.5).

a/b (Å) c (Å) V (Å 3 )

Table 2

Contents of non-stoichiometric oxygen (), average valences of Pr (m+) and Co (n+) ions in PrSrCo1-xCuxO4+ oxides. Since oxygen species on the cathode surface play a key role for oxygen reduction reaction, the O1s XPS spectra of PrSrCo1-xCuxO4+ are further analyzed, and the fitting results are presented in Figure 3. The peak around 528.50.4 eV corresponds to the lattice oxygen (Olattice) associating to the redox properties of metal ions [START_REF] Wang | Investigation of structural and electrochemical properties of LaSrCo1-xSbxO4 (0X0.20) as potential cathode materials in intermediate-temperature solid oxide fuel cells[END_REF][START_REF] Chen | Tailoring the activity via cobalt doping of a two-layer Ruddlesden-Popper phase cathode for intermediate temperature solid oxide fuel cells[END_REF][START_REF] Konysheva | Fluctuation of surface composition and chemical states at hetero-interface in composites comprised of a phase with perovskite structure and a phase related to the Ruddlesden-Popper family of compounds[END_REF][START_REF] Kim | X-ray photoelectron spectroscopy analysis of (Ln1-xSrx)CoO3- (Ln: Pr, Nd and Sm)[END_REF][START_REF] Ivanovskaya | Structure of SrCo0.5Fe0.5O3--based composites prepared by sol-gel and mechanochemical processes[END_REF][START_REF] Meng | Insights into the CO2 stability-performance trade-off of antimony-doped SrFeO3- perovskite cathode for solid oxide fuel cells[END_REF].

The broad peak centered around 531.20.3 eV can be assigned to the absorbed oxygen species (Oabs) on surface [START_REF] Chen | Evaluation of Ba-deficient PrBa1-xFe2O5+ oxides as cathode materials for intermediate-temperature solid oxide fuel cells[END_REF][START_REF] Chen | Tailoring the activity via cobalt doping of a two-layer Ruddlesden-Popper phase cathode for intermediate temperature solid oxide fuel cells[END_REF][START_REF] Konysheva | Fluctuation of surface composition and chemical states at hetero-interface in composites comprised of a phase with perovskite structure and a phase related to the Ruddlesden-Popper family of compounds[END_REF][START_REF] Kim | X-ray photoelectron spectroscopy analysis of (Ln1-xSrx)CoO3- (Ln: Pr, Nd and Sm)[END_REF][START_REF] Ivanovskaya | Structure of SrCo0.5Fe0.5O3--based composites prepared by sol-gel and mechanochemical processes[END_REF][START_REF] Meng | Insights into the CO2 stability-performance trade-off of antimony-doped SrFeO3- perovskite cathode for solid oxide fuel cells[END_REF]. Table 3 summarizes the area percentage of different kinds of oxygen species in PrSrCo1-xCuxO4+. Obviously, the amount of chemisorbed oxygen increases with the value of x, implying the improved ORR catalytic activity of Cu 2+ doped materials.

Table3

The fitting results of O 1s XPS spectra for PrSrCo1-xCuxO4+.

Surface oxygen exchange and Electrical conductivity relaxation

It is known that the lattice oxygen in the perovskite oxides can be released at high temperatures and leads to changes of the point defects. To get this knowledge, the thermogravimetric analysis (TGA) is performed on PrSrCo1-xCuxO4+ samples (Figure 4a). The measurement is carried out within two consecutive thermo-cycles in between room temperature up to 900 o C with a ramp of 10 o C min -1 each cycle, and the data of the second cycle is recorded. The corresponding oxygen non-stoichiometry as a function of temperature is determined from the weight change of the sample, assuming the initial oxygen content is obtained from the XPS analysis (Figure 4b). All For PrSrCoO4+ however, the weight could not reach to its initial value after the thermo-cycle. This behavior suggests that the oxygen exchange process between the oxides and gaseous oxygen is promoted for the Cu 2+ doped samples. This can be reasonably interpreted. Firstly, Cu 2+ doping promotes the oxygen surface adsorption ability of the PrSrCo1-xCuxO4+ oxides (as that proved from the O1s XPS results).

Secondly, the interstitial oxygen concentration () increases with the increase of Cu 2+ doping concentration (table 2). Diffusion of interstitial oxygen may enhance oxygen reduction reaction occurring on the cathode in the intermediate temperature range.

[ [START_REF] Takahashi | Electrode properties of the Ruddlesden-Popper series, Lan+1NinO3n+1 (n=1, 2, 3), as intermediate-temperature solid oxide fuel cells[END_REF][START_REF] Manvy | Oxygen reduction on porous Ln2NiO4+ electrodes[END_REF][START_REF] Kammer | An EIS study of La2-xSrxNiO4+ SOFC cathode[END_REF][START_REF] Lalanne | Intermediate temperature SOFC single cell test using Nd1.95NiO4+ as cathode[END_REF].

Fig. 5 shows the typical normalized conductivity relaxation transient curves for PrSrCo0.7Cu0.3O4+. The values of Dchem and kchem are obtained from fitting the normalized conductivities to the appropriate solution of Fick's second law (inset, Fig. 5). The Dchem and kchem values obtained at 750 o C in this study are 1.39  10 -4 cm 2 s -1 and 1.28  10 -2 cm s -1 respectively. For comparison, the Dchem value of 5.5 10 -4 cm 2 s -1 and kchem of 2.1 10 -2 cm s -1 at 750 o C were reported by Saher et al. for Pr2NiO4+ [START_REF] Saher | Influence of annealing at intermediate temperature on oxygen transport kinetics of Pr2NiO4+[END_REF]. Li et al. measured a Dchem and kchem values of 3.5  10 -5 cm 2 s -1 and 1.6  10 -3 cm s -1 at 750 o C for La2NiO4+ [START_REF] Li | Effects of surface coatings on the determination of Dchem and kchem in La2NiO4+ by conductivity relaxation[END_REF]. It can be seen that oxygen transport properties of PrSrCo0.7Cu0.3O4+ is comparable to those cathode materials with excellent ORR activity reported in the literatures. tightly on the dense GDC electrolyte, to form a porous cathode with thickness ~ 22 µm and no obvious delamination can be observed. This result is due to the similar TEC value of PrSrCo0.7Cu0.3O4+ (13.6 10 -6 o C) and GDC electrolyte (13.1 10 -6 o C) (inset figure 7b). It is also observed that the TEC value has been greatly reduced in the Cu 2+ doped material. charge transfer process. In the whole range of the oxygen partial pressure that measured, the value of RL is always larger than that of RH, which means that the charge transfer process on PrSrCoO4+δ cathode is the rate-limiting step. In the case of PrSrCo0.7Cu0.3O4+δ cathode, however the magnitude of n for high frequency arc is close to 1/2, which suggests this process relates to the dissociation of the adsorbed molecular oxygen and surface diffusion. In addition, the n value of low frequency arc is about 1.0, suggesting a diffusion process of gas oxygen inside the porous cathode.

The larger resistance value of RH than that of RL under various oxygen partial pressures means that the reaction rate-limiting step on PrSrCo0.7Cu0.3O4+δ cathode is the dissociation and surface diffusion of the adsorbed molecular oxygen. In summary, we found that Cu 2+ doping in PrSrCoO4+δ cathode changes the oxygen reduction mechanism from the charge transfer process to the dissociation of adsorbed molecular oxygen and surface diffusion process.

Fig. 11 presents the I-V and I-P curves of the electrolyte support single cell with the configuration of Ni-GDC|GDC|PrSrCo0.7Cu0.3O4+δ. The single cell exhibits the maximum power density of 327 mW cm -2 at 700 o C. This value is much higher than that of 201 mW cm -2 for (Pr0.9La0.1)2Ni0.74Cu0.21Ga0.05O4+ cathode [START_REF] Wang | Pr0.9La0.1)2Ni0.74Cu0.21Ga0.05O4+ as cathode material for CeO2-based intermediate-temperature solid-oxide fuel cell[END_REF] and 298 mW cm -2 for Pr2NiO4 cathode [START_REF] Yang | Preparation and electrochemical properties of strontium doped Pr2NiO4 cathode materials for intermediate-temperature solid-oxide fuel cells[END_REF] at 700 o C reported in the literatures with similar cell configuration. Accordingly, the oxygen exchange capacity and electrocatalytic performance of PrSrCoO4 cathode are successfully improved by Cu 2+ doping strategy.

PrSrCo0.7Cu0.3O4+δ is proved to be a potential cathode material for IT-SOFCs.

2. 1

 1 Materials synthesis PrSrCo1-xCuxO4+ (x = 0.0, 0.1, 0.3, 0.5) oxides are synthesized by traditional solid-state reaction method. The reactants are Pr6O11, SrCO3, Co3O4 and CuO. All of these materials are high purity (99.99, Aladdin industrial corporation, China), and used as received. According to the chemical formula, stoichiometric amount of above mentioned materials are mixed with ethanol in a planetary ball-mill for 1 h. Then the obtained mixtures are pressed into pellets under 100 MPa and reacted at 1000 o C for 12 h in air. The obtained pellets are ground, tableted and reacted again at 1000 o C for additional 12 h. This operation is repeated three times until the target products are obtained. For thermal expansion coefficient (TEC) and electrical conductivity measurements, the product powders are pressed under 100 MPa and then sintered at 1050 o C for 12 h in air, to form rectangular bars with dimensions 15 mm × 3 mm × 1 mm. The relative bulk densities of the rectangular bars are over 95% of the theoretical values that measured by Archimedes method (Sartorius balance with a density kit).2.2 Cell fabrication Ce0.9Gd0.1O1.95 (GDC) powder was obtained from Ningbo SuoFuRen Energy Co. Ltd.

  photoelectron spectrometer equipped with Al Kα (1486.6 eV) radiation source. The

Fig. 1 .

 1 Fig. 1. Experimental (black continuous line) and calculated (red line) XRD patterns,

Fig. 2 .

 2 Fig. 2. XPS spectra and fitting lines of Co 2p (a), Pr 3d (b) in PrSrCo1-xCuxO4+

  analysis of O content for each sample is carried out according to the XPS spectra and the results are also listed in Table2. The oxygen contents for all of the as-prepared samples are found to be slightly above 4.0, indicating the existence of interstitial oxygen in the material. The average valences of Pr and Co increase steadily with the substitution of Cu from x = 0.1 to x = 0.5, and the overstoichiometric oxygen content () increases from 0.02 to 0.08.

Fig. 3 .

 3 Fig. 3. O 1s core-level spectra and fitting lines of PrSrCo1-xCuxO4+ at room

Fig. 4 .

 4 Fig. 4. (a) Thermogravimetric curves of PrSrCo1-xCuxO4+ (x = 0.0  0.5); (b)

Fig. 5 . 3 . 4 Fig. 6 .

 5346 Fig. 5. Normalized conductivity transients measured on PrSrCo0.7Cu0.3O4+ cathode

Fig. 8a showsFig. 8 . 2 OP 2 OP 50 Fig. 9 .Fig. 10 .

 82250910 Fig.8ashows the impedance spectra of PrSrCo0.7Cu0.3O4+ cathode measured at

Fig. 11 .

 11 Fig. 11. Cell voltage and power density as a function of current density for

  (x = 0.0, 0.1, 0.3, 0.5) materials were synthesized by solid-state reaction method. The results show that PrSrCo1-xCuxO4+ has good chemical and thermal compatibility with GDC electrolyte. Copper doping increases the interstitial oxygen concentration and improves the electrochemical performance of the material. PrSrCo0.7Cu0.3O4+ cathode exhibits the lowest polarization resistance of 0.08 Ω cm 2 at 700 o C, and the rate determining step (RDS) of PrSrCo0.7Cu0.3O4+ cathode is dissociation of the adsorbed molecular oxygen and surface diffusion process.

  Cu2+ and the valence state of Co ions. To assess the possible reaction between PrSrCo0.7Cu0.3O4+ and GDC electrolyte, the two oxide powders are mixed in weight ratio of 1:1, and then sintered at 1000 o C for 24 h in air. The XRD patterns of the sintered mixtures are shown in Fig.1b. No additional peaks indicating new phase formation are detected from the patterns of the mixture. It is concluded that From the XPS analysis, the percentage contributions of different valence states are calculated and the average valences of Pr and Co ions are tabulated in Table2.

	PrSrCo0.7Cu0.3O4+ is chemically compatible with GDC below 1000 o C.
	3.2 X-ray photoelectron spectroscopy analysis and Oxygen non-stoichiometry in
	PrSrCo1-xCuxO4+
	775	780	785	790	795	800
	To identify the surface properties of PrSrCo1-xCuxO4+ (x = 0.0, 0.1, 0.3, 0.5) oxides,
							5 and 794.3 eV can be
	assigned to Co 3+ 2p3/2 and 2p1/2, whereas the other set of peaks with binding energies
	of 780.8 and 795.5 eV represent Co 4+ 2p3/2 and 2p1/2, respectively [36,37].
	PrSrCoO4+ The Pr 3d XPS spectra and the fitting curves are presented in Figure 2b. The peaks 2.40
			3.792(3)	12.460(3)	179.19	I4/mmm	6.59	8.74
	PrSrCo0.9Cu0.1O4+ with binding energy at 927.9/933.0 eV and 948.4/953.6 eV originate from Pr 3+ 3d5/2 1.36
			3.790(2)	12.461(3)	179.01	I4/mmm	4.01	5.58
	PrSrCo0.7Cu0.3O4+ and 3d3/2, and the doublet peaks located at 931.0 eV and 952.3 eV can be assigned to 1.36
			3.781(3)	12.478(7)	178.39	I4/mmm	6.25	7.96
	PrSrCo0.5Cu0.5O4+ Pr 4+ 3d5/2 and 3d3/2, respectively [37,38]. The fitting results indicate that Pr 3+ coexists 1.7
			3.787(6)	12.519(5)	179.55	I4/mmm	5.69	6.73
	with Pr 4+ in these samples.			1

Space group Rp (%) Rwp (%)  2 the chemical states of elemental Co, Cu, and O are measured by X-ray photoelectron spectroscopy (XPS) and normalized with C 1s peak at 284.6 eV. The strontium ions and copper ions exhibit mono-oxidation state as Sr 2+ and Cu 2+ in PrSrCo1-xCuxO4+, while Pr and Co cations display mixed valance states. The Co 2p XPS spectra and the fitting curves are presented in Figure 2a. The peaks at 779.

Acknowledgement:

The project was supported by National Natural Science Foundation of China (51872078 and 51372073).


N B Roozen

L Labelle

C Glorieux

On the unwrapping of dispersion curves in the irreducible Brillouin zone by means of a spatial Fourier transform approach

Keywords: periodic structures, Bloch-Floquet; spatial aliasing, Irreducible Brillouin Zone IBZ

Using Bloch-Floquet boundary conditions implies a periodicity in both the spatial and wavenumber domains, causing the numerically computed wavenumbers to be folded or aliased within the Brillouin zone. A methodology is presented to unwrap the wavenumber spectrum to extended Brillouin zones using a Fourier transform method in the spatial domain, thus providing new insights on the physical meaning and relative amplitudes of aliased wave components. In addition, a procedure is presented to identify out-ofplane (Lamb wave type) components, which are usually of interest for thin structures.

The proposed methodologies were applied to finite element based simulations of a 2D periodic reticulated structure consisting of interconnected rectangular struts with orthorombic symmetry. This reticulated structure was used as a toy model for the skeleton in porous media, focusing on the out-of-plane waves. Laser ultrasonic experiments were conducted to verify the numerical results.

Both the simulations and measurement results indicate that, in spite of bandgap features induced by the periodicity of the structure, the dispersion behavior of the out of plane Lamb wave modes in the considered type of structure is similar to the one of the Lamb waves of a homogeneous plate with thickness equal to the one of the struts, both for wavenumbers within and outside the Irreducible Brillouin Zone. HIGHLIGHTS -A technique is presented to unwrap dispersion curves in the irreducible Brillouin zones to the extended Brillouin zones -A method is presented to determine the relative amplitudes of the wave components at different wavenumbers that contribute to the respective eigenmode, for any frequency of interest.

-The unwrapping method is illustrated by means of data from a numerical model, supported by experimental results.

-The dispersion relationships of Lamb wave type of modes a 2D periodic reticulated frame structure resemble quite well the ones of a homogeneous plate

Introduction

Many studies have been dedicated to the dispersion relationships of periodic structures [START_REF] Chesnais | Effects of the local resonance on the wave propagation in periodic frame structures: Generalized Newtonian mechanics[END_REF][START_REF] Boutin | Large scale modulation of high frequency acoustic waves in periodic porous media[END_REF][START_REF] Baravelli | Internally resonating lattices for bandgap generation and lowfrequency vibration control[END_REF][START_REF] Boutin | Elastodynamic metasurface: Depolarization of mechanical waves and time effects[END_REF][START_REF] Sorokin | Plane wave propagation and frequency band gaps in periodic plates and cylindrical shells with and without heavy fluid loading[END_REF]. From an application point of view, periodic structures are quite common and therefore important, appearing in sandwich structures, porous materials, truss beams etc. Moreover, these structures exhibit interesting dispersion properties: because of the periodicity, frequency bandgaps can exist, within which wave propagation cannot occur (e.g. see [START_REF] Baravelli | Internally resonating lattices for bandgap generation and lowfrequency vibration control[END_REF][START_REF] Sorokin | Plane wave propagation and frequency band gaps in periodic plates and cylindrical shells with and without heavy fluid loading[END_REF]). This phenomenon has stimulated researchers and developers to design periodic metamaterials with enhanced structural and acoustical wave insulation or absorption in a frequency range of interest, e.g. to eliminate an unwanted tonal sound component or a disturbing narrow band vibration from a spectrum.

In order to numerically analyse periodic structures, a common approach is to use a Bloch-Floquet boundary condition. This approach reduces the modeling effort and computational effort drastically, without compromising accuracy. The method exploits the periodicity of the structure in the spatial domain. As a result of the space-wavenumber duality, this implies a periodicity in the wavenumber domain as well [START_REF] Hussein | Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook[END_REF].

Although this approach has been shown to be a computationally efficient way to exploit the periodicity of a structure [START_REF] Sorokin | Plane wave propagation and frequency band gaps in periodic plates and cylindrical shells with and without heavy fluid loading[END_REF][START_REF] Palermo | Extended bloch mode synthesis: Ultrafast method for the computation of complex band structures in phononic media[END_REF], application of Bloch-Floquet boundary conditions results in an ambiguity of the wavenumber, which complicates the interpretation of the computed results. As an illustration, in Kulpe et al. [START_REF] Kulpe | Blochwave expansion technique for predicting wave reflection and transmission in two-dimensional phononic crystals[END_REF], results that are reported to be unphysical can most likely be ascribed to aliasing effects that could be overcome by a dispersion curve unwrapping approach presented further on in this paper, for the case of 2D periodic reticulated structures.

Using the proposed unwrapping procedures, a study was performed on the wave propagation in 2D periodic reticulated structures. Our incentive for this study originates in the question whether the intrinsic, microscopic elastic properties of the porous frame material can be extracted from the propagation char-acteristics of elastic waves traveling along the foam structure. Hilyard and Cunningham [START_REF] Hilyard | Low density cellular plastics: Physical basis of behaviour[END_REF] derived expressions that link the global static stiffness of periodic frames to the elastic modulus of the frame material, with the porosity as the key parameter of the structure. However, in many cases, the foam material exhibits relaxation behavior, and thus a frequency dependent and complex dynamic elastic modulus. Hence, a first question is whether the dynamic behavior of macroscopic vibrations and waves in a reticulated structure depends on the microscopic material modulus and porosity in the same way as it does for the static case. Alternatively, starting from a microscopic viewpoint, one can wonder to what extent the propagation of guided waves traveling along a microscopic strut in a reticulated structure is affected by the presence of interconnections with other struts. The latter question becomes particularly pertinent when the wavelength is of the order of magnitude, or longer, than the interconnection length.

In the performed study emphasis was put on out-ofplane guided wave propagation in 2D grids, which are representative e.g. for woven textile and 2D scaffolds. In literature, many papers can be found that deal with 2D spatially periodic reticulated structures, especially for in-plane vibrations (e.g. [START_REF] Gonella | Analysis of in-plane wave propagation in hexagonal and re-entrant lattices[END_REF][START_REF] Phani | Wave propagation in two-dimensional periodic lattices[END_REF][START_REF] Leamy | Exact wave-based Bloch analysis procedure for investigating wave propagation in two-dimensional periodic lattices[END_REF][START_REF] Junyi | Measuring the band structures of periodic beams using the wave superposition method[END_REF][START_REF] Chesnais | Effects of the local resonance on the wave propagation in periodic frame structures: Generalized Newtonian mechanics[END_REF][START_REF] Boutin | Large scale modulation of high frequency acoustic waves in periodic porous media[END_REF][START_REF] Boutin | Elastodynamic metasurface: Depolarization of mechanical waves and time effects[END_REF]). The results presented here, which cover Lamb wavetype out-of-plane displacements, are complementary to the ones reported in those works.

The article is organised as follows. Section 2 starts off with a brief summary of Bloch-Floquet theory. In section 2.2 a method is described how dispersion curves in the irreducible Brillouin zones can be unwrapped to extended Brillouin zones. A spatial Fourier transform is used to determine the true wavenumber, and to classify modes according to their polarization. Section 3 presents the measurement setup. Laser excitation and detection was used for the sake of efficient and flexible excitation and detection of the out-of-plane waves in a wide range of wavelengths, ranging from the strut thickness d to a multiple of the interconnection distances L 1 and L 2 [START_REF] Labelle | Elastic characterization of polymer fibers by laser Doppler vibrometry[END_REF]. The measured wave dispersion results are compared with the ones obtained by numerical simulations. Special attention is given to the particular effects of periodicity, and to the comparison of the dispersion curves of the periodic structure with the ones of an equivalent plate, made of the same material, and with the same thickness as the struts. Conclusions are drawn in Section 4.

Dispersion and Bloch-Floquet theory

Bloch-Floquet's theorem [START_REF] Bloch | Über die Quantenmechanik der Elektronen in Kristallgittern[END_REF][START_REF] Brillouin | Wave Propagation in Periodic Structures[END_REF] expresses that the propagation of a wave from cell to cell does not depend on the cell location within the periodic lattice.

Following Hussein [START_REF] Hussein | Reduced Bloch mode expansion for periodic media band structure calculations[END_REF], the displacements of two adjacent Representative Elementary Volume's (REV's) of the lattice can be written as

u (r + d, k) = u (r, k) e i(k T d) (1)
where r = {x, y, z} is the position vector, u = {u x , u y , u z } the displacement vector, d = {d x , d y , d z } the lattice vector, k = {k x , k y , k z } the wave number vector, and i = √ -1. The real part of the wave number vector k reflects the phase velocity, while the imaginary part reflects the attenuation of the wave. By virtue of Bloch-Floquet's theorem, wave propagation through the entire lattice can be described by considering a single REV only, with obvious reductions of the computational load. Assuming a lossless medium, the discretized equations of motion of the REV can be written in matrix form as

K -ω 2 M u = f i + f e ( 2
)
where K is the dynamic stiffness matrix of the REV, M the mass matrix of the REV and ω the angular frequency. The right hand side of the equation represents the force vector acting on the REV. The force vector can be decomposed into an internal f i and an external force vector f e . Exploiting equation ( 1), a constraint relation can be devised that links the displacements of the edges at either side of the REV. The constraint relation is described by the lattice vector d, and is used to reduce the number of equations of motion. It can also be shown that the application of Bloch's procedure eliminates the internal forces f i [START_REF] Farzbod | The treatment of forces in Bloch analysis[END_REF]. The reduced equations of motion read

K -ω 2 M ũ = fe (3)
where K, M, ũ and fe are the reduced stiffness and mass matrices and the reduced displacement and external force vectors, respectively. The eigenvalue problem of the lattice structure can be formulated by setting the external forces fe to zero:

K -ω 2 M ũ = 0 (4)
Solving Eq. ( 4) for a range of wavenumber vectors k i results in sets of eigenfrequencies f i,j and eigenmodes, with respective phase velocities, real part, c i,j = 2 * π * f i,j / Re {k i }.

Selection of out-of-plane modes.

Although the unwrapping methodology that will be presented in Section 2.2 is applicable to any periodic structure, 2D or 3D, and any kind of mode, in this article eigenmodes of 2D structures will be considered. The set of eigenmodes is further narrowed down to eigenmodes with dominant out-of-plane motion, in view of allowing comparison with available experimental results on 2D grids, which yielded the out-of-plane displacements only. In this section we elaborate on an approach to select out-of-plane eigenmodes from a set of eigenmodes that result from a finite element analysis.

For the sake of simplicity and straightforward validation with an analytical model, we have started off with a solid plate to illustrate the selection of out-ofplane eigenmodes (this section) and the unwrapping approach (Section 2.2).

A REV with dimensions 7.5 × 7.5 mm in a polyamide plate of t = 1.1mm thickness, Young's modulus 1.8GP a, Poisson's ratio 0.4 and density 1200kg/m 3 was modeled by means of the FEM, applying Bloch-Floquett boundary conditions at the edges of the REV. Varying the wavenumber in one lateral direction of the REV, the eigenvalue frequencies were computed, thus obtaining the dispersion relations. The prescribed wavenumber in the x-direction was varied between 0 < k x < π/d x =418.9 rad m -1 , with d x = 7.5 mm. The wave propagation is considered in one direction only, the x-direction, which reduces the Bloch-Floquett boundary conditions to u (x + d x ) = u (x). As the plate was assumed to be lossless, only real values of the wavenumber were considered. The solutions to Eq. 4 that yielded real eigenvalues ω, for the prescribed wavenumbers k x , are shown in Figure 1. For every (ω,k) solution found by the FE eigensolver, the amplitudes of the Out-Of-Plane (OOP, in the z-direction) and In-Plane (IP, in the x-or y-direction) components of the corresponding displacement field were evaluated and the ones with an OOP/IP ratio larger than 50, were retained. For the selected OOP modes, the sign of the ratio between the u z displacements at the top and bottom surface of the plate was used to classify the modes into symmetrical (S n ) and antisymmetrical (A n ) ones, as illustrated in Figure 2.

Using this approach, the modes were sorted according to their symmetrical/antisymmetrical, and OOP/IP character, as illustrated in Figure 3. Besides the well-known symmetric (S0, red symbols) and antisymmetric (A0, blue symbols) branches of OOP Lamb modes known for a plate, other branches of OOP as well as IP modes modes are apparent (gray symbols).

Unwrapping of dispersion curves from the IBZ to extended Brillouin zones

Consider a structure that is periodic in the x-direction with spatial period d x . As a result of the space-wavenumber duality, the dispersion relation of a spatially periodic mechanical system is periodic in wave number domain as well [START_REF] Hussein | Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook[END_REF]. The dispersion curves are periodic in the k x direction, with the reciprocal lattice vector

K = 2π/d x . (5)
as period. Thus, any frequency-wavenumber pair (ω,k x ) will have a multiple of counterpart frequencywavenumber pairs

(ω, k x + Km), (6)
where m is an arbitrary integer. The part of the wavenumber domain defined by

[0, K/2] (7)
completely defines the dispersion relation of a periodic structure. This part of the wavenumber domain is called the Irreducible Brillouin Zone (IBZ). This aliasing phenomenon has the advantage that the full wave behavior of the structure is condensed in the IBZ, which is limiting the calculation effort. Using relationship 6, the dispersion curves can be replicated in the wave number domain, for all frequencies, from the IBZ to extended Brillouin zones, by adding multiples of K to the wavenumbers k x in the IBZ. However, it is a challenge to determine to what extent modes at different wavenumbers, replicated to extended Brillouin zones, contribute to eigenmodes of the frame for different frequencies.

An example of an unwrapped result for the A 0 and S 0 OOP modes in the full plate under consideration is shown in Figure 4. The values of m were chosen such that the well-known A0 and S0 dispersion curves were obtained, matching the non-aliased physical reality. Automated determination of m-values that correspond with a respective non-aliased physical solutions is not trivial, in particular in more complicated reticulated structures, where the physical nature of different modes is not known a priori. In order to address this difficulty and find appropriate m values, we have taken a step back. We considered for each wavenumber-frequency pair in the IBZ the corresponding displacement pattern in the unit cell, and periodically extended it over a large number of cells (typically about 800) in the propagation direction x, consistent with the Bloch-Floquet boundary condition, Eq. 1: nkxdx) , n = 1...800 [START_REF] Kulpe | Blochwave expansion technique for predicting wave reflection and transmission in two-dimensional phononic crystals[END_REF] Using this procedure, the respective structural mode was replicated in the spatial domain over a wide range of possible (eigenmode) wavelengths, up to the longest ones of interest. This replication process was performed for every eigenfrequency that resulted from the finite element solution that was obtained for a specific Bloch-Floquet boundary condition (i.e. for a specific prescribed wavenumber k x ). Next, we described the waveform of each structural mode as a sum of sinusoidal (Fourier) wave forms. Note that due to the 2D periodicity of the frames, with side branches repeating every distance d x , the structural modes deviate from simple sinusoidal shapes; a multitude of Fourier components are required to describe their displacement fields. The amplitudes of these wavenumber components were determined by taking a spatial Fourier transform of the periodically extended displacement pattern. The non-aliased components of the eigenmode were then identified from the computed Fourier components. Note that the above described procedure can be extended and applied to structures that are periodic in 2 or 3 dimensions.

u (x + nd x ) = u IBZ (x) e i(
To illustrate the use of a spatial Fourier transform to 'de-aliase' the computed eigenmode, we consider the 1.1 mm polyamide plate, whose dispersion behavior in the IBZ was shown in Figure 1. Eigenmodes were computed by prescribing a Bloch-Floquet wavenumber in the IBZ equal to k x = 230.4 rad/m. Figure 5A shows the dispersion curves after applying Bloch's theorem to replicate the dispersion curves in the IBZ to the extended Brillouin zones by shifting the wavenumbers by Km, for a number of m-values of interest, exploiting the periodicity in the wavenumber domain, Eq. 6. To identify which value of m constitutes the true 'non-aliased' component of the numerically computed eigenmode at a specific eigenfrequency (as obtained for a specific Bloch-Floquet boundary condition), the extended eigenmode was transformed from physical space (x) to wavenumber domain (k) by means of a spatial Fourier transform. The result of this transformation is illustrated, as an example, for an eigenfrequency of 170.4 kHz in Figure 5B. This figure clearly shows that, in this case of a simple plate, only one specific wavenumber, k x =2041 rad m -1 stands out, which corresponds to a physical, 'non-aliased' mode. Note that the IBL runs up to π/d x =418.9 rad m -1 (d x = 7.5 mm). Thus, the true wavenumber lies in the extended Brillouin zone. All other wavenumbers (corresponding to other integer values of m) can be associated with non-physical, aliased components: Figure 5A shows that they have a negligible amplitude, more than 70 dB smaller than the physical one. As expected, the dominant k, fpair ω/(2π)=170.4kHz, k=2041 rad m -1 corresponds to the A 0 mode for an infinite plate, which is the only antisymmetric physical A0 mode possible. The negligibly small peaks in the Fourier spectrum can be considered as numerical artifacts. We have verified that the true 'non-aliased' S0-mode can be recovered in a similar manner.

In a next step, the above described procedure for "de-aliasing" wavenumbers in the IBZ, by re-assigning them to their proper extended Brillouin zone value, was also applied to a simple polyamide beam with rectangular cross section (1.1mm x 1.1mm). Fig. 6 shows that the procedure leads to the proper dispersion curves, as known from the not-periodized, analytical modeling of the infinite plate and beam. For the beam, also the A 1 mode, including its 'aliased' components stands out, at frequencies above 325 kHz.

Dispersion curves of reticulated structures.

In this section, four 2D orthorombic reticulated structures, consisting of interconnected beams with square cross section, periodically arranged in a plane as shown in Figure 7, are considered. A top view of the investigated REV's, which had different values of the WH-ratio (ratio between the width and height) of the 2D, in plane cross section unit cell, are schematically represented in Figure 8. The thickness of the beams is 1.1mm.

Figure 9A shows all the modes within the IBZ, together with the replicated ones in the extended Brillouin zones (exploiting Eq. 6), for the frame structure with an width-height ratio (WH-ratio) of 1:1 (Fig. 8A, blue frame). Considering e.g. the computed eigenmode at a frequency of 159.5kHz, and a prescribed Bloch-Floquet wavenumber k x =230.4 rad m -1 , the wave components of this eigenmode were extracted by means of the spatial Fourier transform approach described in the previous section. Figure 9B shows, for this eigenmode, the spectral amplitude of the Fourier components of this mode, which spectrum has a rather limited spectral resolution. Figure 9C shows the wavenumber spectrum, as obtained from a spatial Fourier transform of the replicated displacement field over 800 unit cells, exploiting Eq. 8. Note that the larger the number of unit cells considered, the better is the wavenumber resolution. Fig. 9D shows the computed structural eigenmode. In order to get an idea on the relative magnitudes of the different wavenumber contributions, the contribution of the most dominant wavenumber, k x =1905m -1 (which equals the IBZ wavenumber k x =230rad/m plus two times the reciprocal lattice spacing K = 2π/d=839 rad/m), shown separately (blue asterisks). Interest- ingly, in this case, this contribution, as well the other largest contributing wavenumbers, are all lying out of the IBZ, in the extended Brillouin zones. Note that due to the different absolute values of the negative and positive wavenumbers, the amplitude spectrum is not symmetric around k x =0.

Figure 10 shows an overview of the k, ω magnitude plots obtained for the replicated displacement patterns of OOP antisymmetric modes, for the 3 geometries of Fig. 8, with WH-ratio values 1/1, 2/2, 1/0.5, 0.5/1. In general it is observed that an eigenmode as computed from a model of a repetitive frame structure is constituted of a plentitude of Fourier wave components of which a small number are dominant. For the grids considered in this study, the dominant Fourier wave components typically have wavenumbers that are close to the wavenumbers of a homogeneous plate with the same material properties and the same thickness.

In all cases shown in Fig. 10, the magnitudes of the Fourier components are strongest near the A 0 branch of the simple infinite plate with thickness equal to the thickness of the rectangular beams of which the frames are composed. This implies that, similarly to the case of the simple plate, also in a 2D frame geometry, the bending stiffness of the individual polyamide beams in the frame can be approximated by fitting the dominant modes to a simple plate model. In other words, the trend line through the dominant (k,ω) pairs existing in the frame dispersion plot is a good approximation for the dispersion of the main flexural mode of a 2D reticulated structure is a good approximation for the A 0 mode of a homogeneous plate with the same material properties and the same thickness. Interestingly, the correspondence is also valid for very small wavenumbers, for which the wavelength is much longer than the REV size. Contrary to the considered case of out-of-plane displacements, in this limit, and in the very limit of zero wavenumber corresponding to the case of static deformations, it has been shown that the in-plane elastic behavior of the frame strongly depends on the structural porosity (i.e. the filling fraction) [START_REF] Hilyard | Low density cellular plastics: Physical basis of behaviour[END_REF][START_REF] Chesnais | Effects of the local resonance on the wave propagation in periodic frame structures: Generalized Newtonian mechanics[END_REF][START_REF] Boutin | Large scale modulation of high frequency acoustic waves in periodic porous media[END_REF]. Fig. 10 shows that when the struts are long with respect to the strut thickness, the bandgaps are narrower (compare e.g. 10C with 10A) and the dispersion curve deviations (with respect to the classical dispersion curves of a simple plate) in their vicinity are less pronounced. This is consistent with the maximum wavenumber value for the Brillouin zone being inversely proportional with the size of the unit cell, i.e. the strut length (K = 2π/d x ). In the limit of infinite strut length, a simple beam structure would be obtained, while the bandgap features would disappear.

A) B) C) D)
Looking at the opposite case of a short strut length (Fig. 10A and Fig. 10D), the dispersion curve deformations are roughly twice as large and the bandgaps are wider. Note also that the WH-ratio 1:1 (blue) points that are not corresponding with the A0 mode belong to a torsional vibration mode of the strut (leading to an out-of-plane displacement component). Such torsional modes are visible in the following frequency ranges: 30 kHz -50 kHz; 110 kHz -130 kHz; 210 kHz -250 kHz.

Experimental analysis

Dispersion measurements were performed on a 3D printed polyamide frame consisting of rectangular beams with a cross section of 1.1mm x 1.1 mm and equal unit cell length and width of 7.5mm (WH-ratio 1), as shown in Fig. 11. The nominal material properties of the polyamide material were estimated as: Young's modulus = between 1 and 2 GPa, density = 1200 kg/m3, Poisson's ratio = 0.4. We have made use of photoacoustic excitation (see Fig. 12) of guided elastic waves by illuminating a spot on the sample with light pulses from a 10 ns Spectra Physics Quantaray Nd:YAG laser with a 10 Hz repetition rate. In this approach, illumination of the material with laser pulses results in partial optical absorption of the light energy, heating and thermal expansion. Due to the impulsive character of the induced stress, this results in elastic waves to be launched along the struts, in this specific case every 0.1 second. Although the generation bandwidth of 10 ns pulses is of the order of 100MHz, due to material damping, dynamic range limitations of the Polytec Laser Doppler Vibrometer (sensor head OFV-353 with controller in displacement mode, with a sensitivity of 50nm/V), and a finite laser spot size, the spectrum of the detected signal was limited to the 50 kHz-500 kHz range.

The phase velocity of the generated waves was determined from their spatiotemporal behavior. A scanning stage from APT Thorlabs was used to scan the exciting Nd:YAG laser beam along the sample, thus systematically varying the distance x between the excitation and detection position. Signals were recorded on an oscilloscope (Lecroy LC564A) and acquired by a Labview program for every position of the scan, resulting in a signal matrix S(x,t). A 2D Fourier transform of this matrix was taken, resulting in an amplitude map -S(k,f)-, from which k,f-pairs satisfying the dispersion relations were extracted.

Figure 13A depicts the spatiotemporal behavior of photoacoustically excited waves along the grid. Two right-running (arrival time increasing with pump-probe distance x) with different velocity (slope ∆x/∆t), and, from x=120 mm onwards, their reflections can be distinguished. The fastest wave packet, reaching the end position (x = 120mm) in about 100ms, keeps its width, implies that this is the S0-wave, with limited dispersion. The slower wave spreads out more in time with increasing detection distance, indicating A0-character.

Figure 13B shows the result of a 2D FFT, giving a color map of the amplitude spectrum versus wavenumber and frequency, (k, ω). The A0 and S0 modes of wave propagation are clearly visible, till about 350 kHz and 400kHz respectively.

A peak search along the k-axis for each frequency was performed to extract the dispersion curves k(f), shown in Figure 13C for the A0 and S0 mode. As a reference, also the analytically calculated dispersion curves of the A0 and S0 modes of an infinite polyamide plate of 1.1mm thickness are indicated by black and gray lines, respectively. Like the numerical simulations, the experimentally obtained dispersion curves for reticulated structures match the dispersion curve of the plate reasonably well. The dispersion curve of the plate, in turn, is known to be very similar to the one of the individual beams. This allows the material properties of the 2D periodic frame type of structure considered in this study, can be fitted (in approximation) from measurement data, using a homogeneous plate of the same thickness. It should be noted, however, that the 2D periodicity of the frames also causes a repetitive deviation from the ideal homogeneous plate dispersion curves, as becomes evident from Fig. 13C,D.

In Fig. 13D the experimental results of Fig. 13C, here in magenta circle marker, are compared with numerical simulation results using a FE-model of the frame structure, selecting out-of-plane modes and extending beyond the IBZ as explained in Section 2. The black dashed lines indicate the Brillouin zones.

Conclusions

Bloch-Floquet's theorem permits the dispersion relations of periodical reticulated structures to be computed with low computational effort, by exploiting the accompanying periodicity in wavenumber domain.

The application of the theorem, which allows to restrict the search of eigenmodes to the Irreducible Brillouin Zone (IBZ), can be considered as a beneficial exploitation of spatial aliasing.

In this work, we have shown that by taking a spatial Fourier transform of the Bloch-expanded displacement field, the true, non-aliased Fourier components of the constituting wave components can be determined. In addition, the relative amplitude of these components to the displacement field of the mode under consideration can be determined. Unwrapping the aliased components to extended Brillouin zones is useful to correctly classify and interpret the physical meaning of the numerical eigenmodes that result from the Bloch-Floquet model. The unwrapping methodology is applicable to any structure, 2D or 3D, and to any kind of mode.

Although wave propagation along rectangular struts in a 2D spatially periodic reticulated structure is affected by strut interconnections, the wavenumbers that have the highest amplitudes in the Fourier decomposition of the eigenmodes of the frame at different frequencies, are close the wavenumbers of waves propagating in infinite struts, and in an infinite, monolithic, "equivalent" plate that has the same thickness as the struts.

Laser ultrasonic experiments were performed on a 3D printed polyamide planar frame with grid size of 7.5 mm, and a strut thickness of 1.1 mm. Photoacoustically excited guided waves were measured by performing pump-probe distance scans. The strengths of the experimentally assessed out-of-plane wavenumber-frequency pairs, obtained by 2D Fourier transforming the spatiotemporal dependence of the experimental signals, were found to be consistent with the numerically predicted ones.

On the basis of both numerical simulations and experimental results, it can be concluded that, in spite of deviations due to the periodic structure, the dispersion curves of the out of plane modes are still very similar to the out-of-plane Lamb waves of a plate that is "equivalent " with the rectangular beam structure. As a consequence, the microscopic bending stiffness of a reticular frame can be approximated by fitting the measured dispersion curves of out-of-plane modes on the numerically obtained dispersion curves of a homogeneous, equivalent plate. This adequate simplification is very promising for porous material analysis.

In common porous materials (arranged in a 2D grid), the struts are not arranged in a periodical frame, so that there is no reason for the existence of bandgaps in the dispersion plots. This suggests that in such a disordered reticulated structure, compared to the investigated periodical grids, the dispersion curves can be expected to be even more similar to the ones of the infinitely extrapolated individual struts. The elastic properties of the microscopic strut material can thus be inferred from the dispersion behavior of out-ofplane polarized waves propagating along the macroscopic frame.

Figure 1 :

 1 Figure 1: Numerical eigenfrequencies found for prescribed wavenumbers, for a 1.1mm thick polyamide solid plate.

  of the two faces

Figure 2 :

 2 Figure 2: Schematic representation of the methodology used to distinguish A 0 and S 0 based on the (anti-) symmetry of the Out-Of-Plane displacement at the top and bottom plate interfaces.

Figure 3 :

 3 Figure 3: Eigenmode (ω,k) pairs of a 1.1mm thick polyamide solid plate found by an FE eigensolver. The OOP A 0 and S 0 mode are indicated by blue and red crosses, respectively. The IP modes are indicated by grey crosses.

Figure 4 :

 4 Figure 4: Unwrapping Bloch-Floquet OOP model results for a solid plate by means of a spatial Fourier transform approach.

Figure 5 :Figure 6 :

 56 Figure 5: Illustration of the unwrapping process using a search of a peak in a spatial Fourier transform of the displacement field of the solid plate model, for eigenmodes computed by prescribing a Bloch-Floquet wavenumber in the IBZ equal to k x = 230.4 rad/m. A): k,f dispersion plot of all eigenmodes found, replicated using Eq. 6; B): spatial Fourier transform spectrum of an eigenmode with an eigenfrequency of 170.4 kHz, after replicating the displacement pattern to 800 REV-cells (exploiting Eq. 8), showing one dominant peak at k=2041 rad m -1 .

Figure 7 :

 7 Figure 7: 3D perspective view of a grid with struts having rectangular cross section.

Figure 8 :

 8 Figure 8: Top view of the Finite Element model of the investigated frame models, which differ in their value of the WH-ratio of the unit cell, i.e., the ratio between the width and height of the unit cell.

Figure 9 :

 9 Figure 9: Illustration of the unwrapping process using a search of a peak in the spatial Fourier transform of the displacement field of the numerical 'WH-ratio 1:1' frame model, for a frequency of 159.5 kHz and a prescribed Bloch-Floquet wavenumber k x =230.4 rad/m. A): The full IBZ and extended IBZ dispersion plot. The size of the blue marker indicates the amplitude of the spatial Fourier component at a frequency of 159.5 kHz B): Spatial Fourier transform spectrum using the displacement pattern of the REV cell (i.e. one unit cell only). C): Spatial Fourier transform spectrum, after replicating the displacement pattern to 800 REV-cells (exploiting Eq. 8). D): Out of plane displacement pattern within a REV, including only the most dominant wave component of Fig. 9C (red circles), and including all wave components of Fig. 9C (blue asterisks)

Figure 10 :

 10 Figure 10: Dispersion curves of the A 0 mode. A) WHratio 1:1, 7.5mm strut length; B) WH-ratio 2:2, 15mm strut length; C) WH-ratio 2:1, 15mm strut length; D) WH-ratio 1:2, model 7.5mm strut length. The colors in this figure corresponds to the color coding in Fig. 8.

Figure 11 :

 11 Figure 11: 3D printed polyamide grid sample, with square cross section struts, 1.1 mm × 1.1 mm, a unit cell WH-ratio of 1/1, and a periodic length scale d x of 7.5 mm.

Figure 12 :

 12 Figure 12: Experimental test setup.

Figure 13 :

 13 Figure13: Regular frame structure WH-ratio 1:1, strut length 7.5mm. A) Measured spatiotemporal map of the photoacoustically generated waves. B) Measured amplitude map in wavenumber-frequency domain. C) Extracted frequency-wavenumber pairs from the measurement data with significant amplitude (blue circles) and numerically predicted dispersion relationship for an equivalent plate, revealing the presence of both A0 and S0 modes. D) Comparison between experimentally obtained dispersion curves (magenta circles) with numerically computed ones for an equivalent plate (black and gray solid curves) and for the Bloch-Floquet model of the periodic reticulated structure showing the unwrapped results (dotted blue curves), for both A0 and S0 modes.

Acknowledgement

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie RISE project PAPABUILD, grant agreement No 690970, and from Marie Sklodowska-Curie COST Action DE-NORMS, grant agreement No CA15125. The authors are grateful to the R&D team at Huntsman, Sterrebeek for fruitful discussions.