Sandra Pioquinto-García

Juana María Rosas

Margarita Loredo-Cancino

Sylvain Giraudet

Eduardo Soto-Regalado

Pasiano Rivas-García
email: pasiano.rivasgr@uanl.edu.mx

Nancy Elizabeth Dávila-Guzmán

Environmental assessment of metal-organic framework DUT-4 synthesis and its application for siloxane removal

Keywords: Adsorption, Life Cycle Assessment, Metal-organic frameworks, Biogas, Octamethylcyclotetrasiloxane, Kinetics of adsorption

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

INTRODUCTION

Renewable energies have been the focus of attention among countries due to the continuous increases in the price of fossil fuels and the need for environmental care. Biogas is a green energy source that is used as a cleaner transportation fuel [START_REF] Yusup | Emerging Technologies for Biofuels Production[END_REF]. Basically, the biogas is a mixture of methane (35-70 %), carbon dioxide (15-50 %), and small amounts of undesirable compounds (<2 %) such as H 2 S (~1560 ppmv), NH 3 (~91 ppmv), CO (~416 ppmv), H 2 (~367 ppmv), moisture, and volatile compounds [START_REF] Bomboí | Pretratamiento del biogás procedente de la digestión anaerobia de lodos de EDARs para su posterior valorización energética[END_REF][START_REF] Santos-Clotas | Efficient removal of siloxanes and volatile organic compounds from sewage biogas by an anoxic biotrickling filter supplemented with activated carbon[END_REF][START_REF] Pertiwiningrum | Heating value enhancement by biogas purification using natural zeolite and rice straw-based biochar[END_REF][START_REF] Santos-Clotas | Sewage biogas efficient purification by means of lignocellulosic waste-based activated carbons[END_REF].

Among the volatile compounds found in biogas, siloxanes are of great interest given their adverse effects on combustion systems. After combustion, the siloxanes decompose into two new products: formaldehyde and orthosilicic acid [START_REF] Álvarez-Flórez | Analysis of damage caused by siloxanes in stationary reciprocating internal combustion engines operating with landfill gas[END_REF]; these products form a soft paste that contributes to abrasion, metal friction, and failure of mechanical elements of the combustion system.

Siloxane removal from biogas is a priority for advancing the energy recovery potential of biogas.

Current technologies for siloxane removal from biogas include adsorption, absorption, cryogenics, separation by membranes, and biological processes [START_REF] Ruiling | Research progress of siloxane removal from biogas[END_REF][START_REF] Shen | A review on removal of siloxanes from biogas: with a special focus on volatile methylsiloxanes[END_REF][START_REF] Wang | Recent advances in technologies for the removal of volatile methylsiloxanes: A case in biogas purification process[END_REF][START_REF] Tansel | Managing siloxanes in biogas-to-energy facilities: Economic comparison of pre-vs post-combustion practices[END_REF]. Nevertheless, the adsorption process with activated carbon (AC) is the most used siloxane removal technique due to its relatively low acquisition cost [START_REF] Tansel | Managing siloxanes in biogas-to-energy facilities: Economic comparison of pre-vs post-combustion practices[END_REF][START_REF] Tran | Adsorption of linear and cyclic siloxanes on activated carbons for biogas purification: Sorbents regenerability[END_REF][START_REF] Yang | Online removal of volatile siloxanes in solid-state anaerobic digester biogas using a biofilter and an activated carbon filter[END_REF] and efficiency (90-99 %), achieving siloxanes concentration up to 0.1 mg/m 3 in the biogas [START_REF] Wang | Recent advances in technologies for the removal of volatile methylsiloxanes: A case in biogas purification process[END_REF]. However, the main drawback of using AC is the high regeneration temperature (>400 ºC), causing high operation costs. Accordingly, new J o u r n a l P r e -p r o o f adsorbent materials as metal-organic frameworks (MOF) have been recently developed to overcome the shortcomings of several conventional adsorption processes.

MOF are porous materials formed by metal centers linked with organic ligands (coordination compounds). Their type of permanent pore and cavity depends on the geometric characteristics of the metallic units and the organic ligands of the MOF. Due to high porosity and surface area, easy modification, thermal stability, and regeneration [START_REF] Gándara | Metal-organic frameworks: nuevos materiales con espacios llenos de posibilidades[END_REF], MOF have been widely used in catalysis, gas adsorption [START_REF] Nguyen | A Series of Metal-Organic Frameworks for Selective CO 2 Capture and Catalytic Oxidative Carboxylation of Olefins[END_REF][START_REF] Kumar | Metal organic frameworks as potent treatment media for odorants and volatiles in air[END_REF], optics, magnetism [START_REF] Gándara | Metal-organic frameworks: nuevos materiales con espacios llenos de posibilidades[END_REF], and chemical sensors [START_REF] Rudd | Highly Efficient Luminescent Metal-Organic Framework for the Simultaneous Detection and Removal of Heavy Metals from Water[END_REF]. Recently, Mito-oka et al. [START_REF] Mito-Oka | Siloxane D4 capture by hydrophobic microporous materials[END_REF] employed the MOF developed by Dresden University Technology 4 (DUT-4) and conventional AC for siloxane adsorption. The results showed that the DUT-4 has high selectivity and uptake even in wet conditions compared to AC performance. The DUT-4 removal efficiency at 30 ºC was higher than 90 %, and the AC removal efficiency was 70 % at 500 min of operation. The DUT-4 removal efficiency was attributed to its strong hydrophobicity and optimal chemical interactions with siloxane octamethylcyclotetrasiloxane (D4). Secondly, Gargiulo et al. [START_REF] Gargiulo | Chromium-based MIL-101 metal organic framework as a fully regenerable D4 adsorbent for biogas purification[END_REF] demonstrated the high affinity of the MOF MIL-101(Cr) for the removal of siloxane D4 as well as the complete regeneration of this adsorbent at moderated temperature (150 ºC). The authors indicated that contrary to AC and silica gel (SG), D4 removal by MIL-101(Cr) does not cause polymerization of D4 on the adsorbent surface, which allowed its reusability.

Although MOF have been pointed out as "green materials" for sustainable processes, the environmental impact of their synthesis has been barely evaluated. The Life Cycle Assessment (LCA) is a methodology that can determine how green a material is [START_REF] Tufvesson | Life cycle assessment in green chemistry: overview of key parameters and methodoligical concerns[END_REF]. For example, LCA has been used by Loya-González et al. [START_REF] Loya-González | Optimal activated carbon production from corn pericarp: A life cycle assessment approach[END_REF] and Sepúlveda-Cervantes et al. [START_REF] Sepúlveda-Cervantes | Technical-environmental optimisation of the activated carbon production of an agroindustrial waste by means response surface and life cycle J o u r n a l P r e -p r o o f assessment[END_REF] to determine the impacts of some adsorbent materials such as AC synthesized from biomass waste. In these works, the impregnation stage of the AC production was found to be the largest contributor to J o u r n a l P r e -p r o o f environmental damage, owed to the high energy consumption and the use of chemicals such as ZnCl 2 and KOH. Grande et al. [START_REF] Grande | Life-cycle assessment as a tool for eco-design of metal-organic frameworks (MOFs)[END_REF] evaluated the environmental impact of the MOF CPO-27-Ni synthesized by four different routes. They found that the main contributors to the environmental impact of CPO-27-Ni were the synthesis and cleaning stages, mainly associated with the toxicity of tetrahydrofuran and dimethylformamide. When water was used in the synthesis or cleaning stages, the environmental impact was reduced from 1136.2 to 12.3 kg CO 2 eq per kg of MOF.

To the authors' knowledge, there is no literature available that addresses the environmental impact assessment of the synthesis of DUT-4, and this is of particular interest because DUT-4 is a promising material for cleaner biogas production. Therefore, this work aims to evaluate the environmental impact of the MOF DUT-4 through the LCA methodology, proposing alternatives to reduce its environmental impact. In addition, a new techno-environmental impact factor is introduced to choose the most suitable synthesis route of MOF for adsorption applications.

MATERIALS AND METHODS

DUT-4 synthesis routes

Synthesis of DUT-4(DMF). The synthesis of DUT-4(DMF) is based on the procedure described by Senkovska et al. [START_REF] Senkovska | New highly porous aluminium based metal-organic frameworks: Al(OH)(ndc) (ndc=2,6naphthalene dicarboxylate) and Al(OH)(bpdc) (bpdc=4,4′-biphenyl dicarboxylate)[END_REF]. A solution of 0.26 g 2,6-naphthalene dicarboxylic acid in 30 mL of N,N-dimethylformamide (DMF), and 0.52 g nonahydrate aluminum nitrate was mixed in a Teflon liner. The liner was placed into a 250 mL autoclave and heated at 120 ºC for 24 h. Afterwards, the autoclave was allowed to cool to room temperature before it was opened. The product was washed three times with 20 mL portions of DMF and centrifuged for 2.5 min at 10,000 rpm between each washing step. The final product was dried in an oven at 120 ºC for 4 h.

J o u r n a l P r e -p r o o f

Synthesis of DUT-4(DCM). The synthesis of DUT-4(DCM) was carried out by including a solvent exchange step in the synthesis of DUT-4(DMF). After the washes with DMF, the product was washed three times with 20 mL of dichloromethane (DCM) for 20 min and centrifuged for 2.5 min at 10,000 rpm between each wash. The final product was dried in an oven at 120 ºC for 4 h.

Synthesis of DUT-4(H).

In the synthesis of DUT-4(H), a second solvent exchange stage was included in the DUT-4(DMF) procedure synthesis. After the solvent exchange of DCM, the powder was washed three times with portions of 20 mL of hexane for 20 min. Then, the product was centrifuged and dried in the same manner as in the synthesis of DUT-4(DMF).

Synthesis of DUT-4(W).

This synthesis was carried out using deionized water instead of DMF to contribute to a greener production. A solution of 0.26 g 2,6-naphthalene dicarboxylic acid in 30 mL of water and 0.52 g nonahydrate aluminum nitrate were mixed in a Teflon liner. The liner was placed into a 250 mL autoclave at 120 ºC for 24 h. The autoclave was cooled to room temperature before it was opened. The content was centrifuged for 2.5 min at 10,000 rpm, then washed three times with portions of 20 mL of water. Finally, the product was dried in an oven at 120 ºC for 4 h.

Characterization techniques of the adsorbents

Physicochemical characterization. X-ray diffraction patterns (XRD) were recorded on a X'Pert PRO MPD diffractometer (PANalytical, Germany) using Cu-Kα radiation (40 mA and 45 kV).

The relative crystallinity was determined by the area under the curve of the crystalline peaks (after baseline correction). The specific surface area (A s ) of DUT-4 was obtained by N 2 physisorption at 77 K using automatic equipment ASAP 2020 (Micromeritics, USA), after out-J o u r n a l P r e -p r o o f gassing the DUT-4 at 150 ºC for 8 h. Solid-state 27 Al MAS NMR spectra were recorded at room temperature in an AVANCE III HD 600 MHz (Bruker, Germany) NMR spectrometer using triple resonance DVT probe of 2.5 mm at a spinning rate of 20kHz, with a length pulse of 0.27 s and a 1 s delay with 1H decoupling and summing up 2000 scans. Scanning electron microscopy (SEM) images were obtained using a JSM-6490LV microscope (JEOL, USA) with acceleration voltage from 0.3 to 30 kV and thermionic electron gun with W filament. FTIR spectra were obtained from a Tensor 27 (Bruker, Germany) using a Golden Gate Single Reflection diamond ATR cell and a standard spectral resolution of 4 cm -1 in the spectral range of 4000-500 cm -1 .

Adsorption experiments. The adsorption experiments were carried out at a 450 mg/Nm 3 D4 initial concentration, 100 mg of adsorbent material, and room temperature. Flasks were stirred at 700 rpm for 6-9 h to ensure equilibrium. During the adsorption experiments, the samples were analyzed for D4 concentration by gas chromatography. Each sample was injected into a Nexis GC-2030 gas chromatograph (Shimadzu, Japan) with a flame ionization detector (FID) to measure the concentration of siloxane D4. The SH-Rxi-5ms Shimadzu capillary column (15 m x 0.25 mm inner diameter x 0.25 µm film thickness) was used. The inlet oven and detector temperatures were set at 150, 110, and 250 °C, respectively. Helium was used as carrier gas at a constant flow rate of 1.5 mL/min and 83.3 kPa. The injected volume was 0.5 mL with split ratio 5.0; the duration of the method was 1.4 min. The adsorption capacity was determined by mass balance as follows:

𝑞 = (C o -𝐶)V m Eq. (1)

J o u r n a l P r e -p r o o f

where q is the adsorption capacity (mg/g), C o is the D4 initial concentration (mg/Nm 3 ), C is the D4 concentration over time (mg/Nm 3 ), V is the volume (m 3 ), and m is the adsorbent mass (g).

Life cycle assessment implementation

Goal and scope definition of the product system. The LCA of the DUT-4 production process on a laboratory scale was carried out as per ISO 14044 [24]. The system boundary for DUT-4 includes five stages: mixing, synthesis, cleaning, solvent exchange, and drying (Figure 1). In this analysis, the environmental burdens associated with the laboratory equipment construction used in the production of DUT-4 were not considered for two reasons: on the one hand, because it has been reported that its contribution to the environmental impact is less than 2% [START_REF]ecoQuery -Login[END_REF], and on the other, to maintain consistency with other LCA studies on the production of adsorbent materials These scenarios are represented in the system boundaries, as shown in Figure 1. The functional unit (FU) was defined as the production of 1 g of DUT-4 produced on a laboratory scale. The FU was established on a mass basis, according to other LCA studies of adsorbents materials production (e.g., activated carbon and MOF-74 [START_REF] Sepúlveda-Cervantes | Technical-environmental optimisation of the activated carbon production of an agroindustrial waste by means response surface and life cycle J o u r n a l P r e -p r o o f assessment[END_REF], [START_REF] Grande | Life-cycle assessment as a tool for eco-design of metal-organic frameworks (MOFs)[END_REF]). All experimental values are based on at least quadruplicate measurements with a coefficient of variation less than 4%.

J o u r n a l P r e -p r o o f

The system inputs considered in this study include chemical reagents and energy demands (in the form of electricity). The LCI for these resources was taken from the Ecoinvent v3.3 database [START_REF] Lara | A comprehensive hazardous waste management program in a Chemistry School at a Mexican university[END_REF] (Table S3). The energy consumption in the synthesis stage was calculated with an energy balance that considers sensible heat and Teflon liner and autoclave reactor heating (Eq. S1). The cleaning and solvent exchange stages include the separation of the solvent by centrifugation. The energy consumption in these stages was calculated by the rotational kinetic energy equation (described in detail in Section S.2). Finally, the energy consumption in the drying stage was obtained by calculating the heat energy required to remove the solvent per unit mass of DUT-4 [START_REF]Life Cycle Assessment[END_REF] (a detailed explanation is provided in Section S.3).

The corresponding system outputs were the final MOF DUT-4 product and emissions related to solvent evaporation in the drying stage. The solvent emissions were estimated by the differences The wastewater generated in the cleaning stage was disposed of as per our university's hazardous waste management program [START_REF] Huijbregts | ReCiPe2016: a harmonised life cycle impact assessment method at midpoint and endpoint level[END_REF]. Therefore, the environmental impacts of the final disposal of this waste were not considered in the LCI, as reported by Loya-González et al. [START_REF] Loya-González | Optimal activated carbon production from corn pericarp: A life cycle assessment approach[END_REF] and Sepúlveda-Cervantes et al. [START_REF] Sepúlveda-Cervantes | Technical-environmental optimisation of the activated carbon production of an agroindustrial waste by means response surface and life cycle J o u r n a l P r e -p r o o f assessment[END_REF].

Life Cycle Impact Assessment. The model used in the Life Cycle Impact Assessment (LCIA)

was the ReCiPe 2016 method [START_REF] Glover | Gas Adsorption in Metal-Organic Frameworks: Fundamentals and Applications, First[END_REF], considering midpoint and endpoint indicators. The LCIA was implemented through the LCA software SimaPro 7.3.3 (PRe Consultants, Amers-foort, the Netherlands).

RESULTS AND DISCUSSIONS

Physicochemical characterization

The FTIR spectrum of DUT-4(DMF), DUT-4(DCM), DUT-4(H), and DUT-4(W) showed similar adsorption bands (Figure 2). The band in the range 1693-1668 cm -1 in each of the spectra corresponded to the stretching vibration of ((C=O)) of the carbonyl group, which is indicative of the coordination of the linker to the metal ion [START_REF] Glover | Gas Adsorption in Metal-Organic Frameworks: Fundamentals and Applications, First[END_REF]. The band in each spectrum at 1614-1598 cm -1 corresponded to asymmetric stretching vibrations of the carboxylate group ( as (C-O)) [START_REF]Porous aluminum-based DUT metal-organic frameworks for the transformation of CO2 into cyclic carbonates: A computationally supported study[END_REF];

the bands in the range 1434-1415 cm -1 corresponded to symmetric stretching vibrations of the carboxylate group ( s (C-O)), and the peaks observed at 989-985 cm -1 are related to the bending of the bridging groups (O-H) [START_REF] Embrechts | In situ Raman and FTIR spectroscopic study on the formation of the isomers MIL-68(Al) and MIL-53(Al)[END_REF]; the bands between 792 and 783 cm -1 in each of the spectra corresponded to flexions into the aromatic plane [START_REF]Infrared Spectra of Some Common Functional Groups[END_REF], [START_REF] Callejas | Tablas de Espectroscopía Infrarroja[END_REF], and the bands between 586-561 cm -1 in the four spectra were associated with the stretching vibrations of the aluminum atom with Solid-state 27 Al MAS NMR spectroscopy was used to determine the local chemical environment and the coordination state of the metal sites. Here, the 27 Al MAS NMR spectra for DUT-4(DMF), DUT-4(DCM), DUT-4(H), and DUT-4(W) were obtained (Figure 3). The primary resonating signal of all the materials studied corresponded to 6-fold coordinated aluminum sites (Al [6] ) [START_REF] Rosales-Sosa | High Elastic Moduli of a 54Al2O3-46Ta2O5 Glass Fabricated via Containerless Processing[END_REF], [START_REF] Benito | Carboxylate-intercalated layered double hydroxides aged under microwave-hydrothermal treatment[END_REF], with three overlapped signals in the chemical shift range between -20 to 10 ppm as reported by Volkringer et al. [START_REF] Volkringer | The use of aluminium and others p elements (gallium, indium) for the generation of MOF-type materials[END_REF] for the Al-MIL-100 structure. According to Vyalikh et al. [START_REF] Vyalikh | Hydrogen bonds and local symmetry in the crystal structure of gibbsite: Hydrogen bonds and local symmetry in structure of gibbsite[END_REF], differences in aluminum sites are due to coordination with distinct types of OH groups;

i.e., each aluminum site is coordinated to different OH-groups that participate in intralayer and interlayer hydrogen bonds, causing changes in the chemical environment. Altogether, the NMR spectra for DUT-4 demonstrate the presence of octahedral aluminum sites in the MOF structure.

J o u r n a l P r e -p r o o f and DUT-4(H), respectively. These results suggest that the use of low surface tension solvents instead of DMF improves the crystallinity of the MOF and prevents structural collapse [START_REF] Ma | Rapid Guest Exchange and Ultra-Low Surface Tension Solvents Optimize Metal-Organic Framework Activation[END_REF]. On the other hand, the hydrothermal synthesis of DUT-4(W) allowed to obtain a structure with 89.89 % crystallinity but non-isostructural to DUT-4 (Figure 4). The higher intensity peaks of the DUT-4(W) phase were 16.39º and 27.74º, which are similar to the organic linker 2,6-naphthalene dicarboxylic acid [START_REF] Kaduk | Structures of 2,6-disubstituted naphthalenes[END_REF]. It has been reported that water molecules can prevent the coordination between the organic linkers and the ion atoms by hydrogen bonding interactions with the organic linkers [START_REF] Zhang | Solvent determines the formation and properties of metal-organic framework[END_REF].

J o u r n a l P r e -p r o o f The specific surface area of the synthesized DUT-4(DMF) had a similar value (1348 m 2 /g) to that reported by Senkovska et al. [START_REF] Senkovska | New highly porous aluminium based metal-organic frameworks: Al(OH)(ndc) (ndc=2,6naphthalene dicarboxylate) and Al(OH)(bpdc) (bpdc=4,4′-biphenyl dicarboxylate)[END_REF] (1308 m 2 /g). According to Howarth et al. [START_REF] Howarth | Best Practices for the Synthesis, Activation, and Characterization of Metal-Organic Frameworks[END_REF], the solvent exchange with low surface tension solvents helps to achieve the highest possible surface area and porosity. Accordingly, a solvent exchange of DMF for CH 2 Cl 2 was carried out to obtain the DUT-4(DCM) material, although a slight reduction of 12 % of the A s was observed (1184 m 2 /g).

To overcome this reduction on the A s , DUT-4(DCM) was subjected to a second solvent exchange process where DCM was exchanged for C 6 H 14 , and the as-synthesized material was named DUT-4(H). The specific surface area of DUT-4(H) (1335 m 2 /g) was very similar to that obtained for DUT-4(DMF). Thus, high specific surface areas were obtained by solvothermal synthesis, irrespective of the solvent exchange. The total pore volumes for DUT-4(DMF), DUT-4(DCM), and DUT-4(H) were 1.71, 1.48, and 1.46 cm 3 /g, respectively. A hydrothermal synthesis was carried out to use a greener solvent such as water instead of the toxic DMF solvent.

J o u r n a l P r e -p r o o f 14 However, a low specific surface area was obtained (1.78 m 2 /g), as well as low total pore volume (0.003 cm 3 /g). Usually, water-based MOF has low porosity in comparison with MOF obtained by solvothermal synthesis [START_REF] Duan | Water-based routes for synthesis of metal-organic frameworks: A review[END_REF].

The particle size of the materials obtained by solvothermal synthesis depended on the solvent used in the drying stage. When low surface tension solvents were used (DCM, 29.5 and H, 18.8 dyn/cm), a small particle size was obtained. For instance, DUT-4(DCM) and DUT-4(H) particle size was lower than 500 nm. Conversely, when a high surface tension solvent such as DMF (37.9 dyn/cm) was used, the particle size of DUT-4(DMF) was greater than 1 µm (Figure 5). Similar

J o u r n a l P r e -p r o o f

In summary, according to the FTIR and NMR analysis, the spectra of the materials obtained by solvothermal and hydrothermal synthesis show the characteristic absorption bands of the functional groups of the organic linker and the chemical shifts related to the six-fold coordinated Al, respectively. The four synthesized materials had a degree of crystallinity of about 80 %, but only the materials obtained by solvothermal synthesis were isostructural to DUT-4. In addition, high specific surface areas with values higher than 1100 m 2 /g were obtained by solvothermal synthesis. Meanwhile, the hydrothermal synthesis generated a low specific surface area and low total pore volume caused by a high surface tension solvent. Similarly, the solvent used on the drying stage affected the particle size of the studied materials. That is, low surface tension solvent leads to a small particle size. The trend on the particle size was DUT-4(H)DUT-4(DCM) <DUT-4(DMF) <DUT-4(W) which is similar to the trend of the surface tension of solvents: C 6 H 14 (18.8 dyn/cm) <Cl 2 CH 4 (29.5 dyn/cm) <DMF (37.9 dyn/cm) <H 2 O (72.8 dyn/cm).

Adsorption. The adsorption experiments were carried out at an initial concentration of 450 mg/Nm 3 of siloxane D4 at room temperature. The kinetic adsorption of the DUT-4 materials was rapid, reaching equilibrium at about 4 h (Figure 6). The equilibrium adsorption capacity for DUT-4(DMF), DUT-4(DCM), and DUT-4(H) was 9.80 mg/g, which is 1.4 times greater than the adsorption capacity of AC (A s 1240 m 2 /g) reported in our previous work under the same experimental conditions [START_REF] García | Siloxane removal for biogas purification by low cost mineral adsorbent[END_REF]. The siloxane adsorption during the first hour was higher for DUT-4(DCM) than for DUT-4(H), which is related to the adsorption kinetics rate. Thus, a simple approach was made to quantify the adsorption kinetics rate by calculating the slope of the first experimental points (from zero to one hour). According to the experimental data, the adsorption rate followed the trend of DUT-4(DCM) <DUT-4(DMF) <DUT-4(H) with values of 7.09, 6.54, J o u r n a l P r e -p r o o f and 5.08 mg/g•h, respectively. On the other hand, the adsorption capacity of DUT-4(W) was 1.80 mg/g, reaching equilibrium in 2 h, and was five times less than the adsorption capacity of the rest of DUT-4 adsorbents, associated with its low specific surface area (1.78 m 2 /g). Life Cycle Inventory of DUT-4 production. The input and output flows in the production of DUT-4 are detailed in Table 1. All experimental data were reported as mean values from quadruplicate measurements with a coefficient of variation less than 4%. According to these data, the use of chemical reagents to produce DUT-4 generated the highest environmental impact with a single score between 88.6 and 245 mPt (79.1 and 91.1 % of total environmental damage),

J o u r n a l P r e -p r o o f while the electrical energy consumption generated a single score between 2.12 and 24.7 mPt, except for Scenario DUT-4(W), where the electricity consumption generated 31.6 mPt while the use of chemical reagents caused 0.66 mPt. Thus, the production of chemical reagents comprises the activities with the most significant environmental damage. Similar results were reported by Loya-González et al. [START_REF] Loya-González | Optimal activated carbon production from corn pericarp: A life cycle assessment approach[END_REF] and Sepúlveda-Cervantes et al. [START_REF] Sepúlveda-Cervantes | Technical-environmental optimisation of the activated carbon production of an agroindustrial waste by means response surface and life cycle J o u r n a l P r e -p r o o f assessment[END_REF] who found that, for the single score indicator (determined through the ReCiPe 2008 method [START_REF]ReCiPe | PRé Sustainability[END_REF]), the production of chemical inputs represents between 93 and 83-93 % of the total environmental damage in the production of activated carbon from agro-industrial waste, respectively. Furthermore, Hjaila et al. [START_REF] Hjaila | Environmental impact associated with activated carbon preparation from olive-waste cake via life cycle assessment[END_REF] found that the production of phosphoric acid contributes in more than 90 % to the midpoint indicators of human toxicity and terrestrial acidification (evaluated through the CML 2 Baseline 2000 method) for the production of activated carbon from olive-waste cakes. J o u r n a l P r e -p r o o f 20 Scenario DUT-4(DCM) has similar behavior to Scenario DUT-4(H), but a lower environmental impact (211.07 mPt) since it uses 29 % and 7 % less DMF and DCM solvents, respectively, and hexane is not used (Table 1). The production of DUT-4 only uses DMF as the solvent, and its environmental impact is 112 mPt (less than half of Scenario DUT-4(H)). The solvent used determines the environmental profile of these adsorbents. Similar results were found by Grande et al. [START_REF] Grande | Life-cycle assessment as a tool for eco-design of metal-organic frameworks (MOFs)[END_REF], who evaluated the LCA of the MOF CPO-27, concluding that solvents used in the synthesis and the cleaning stages have the most significant environmental damage. The authors reduced the environmental impact up to 100 times through water-based synthesis since water is non-toxic, stable, cheap, and readily available [START_REF] Duan | Water-based routes for synthesis of metal-organic frameworks: A review[END_REF].

In Scenario DUT-4(W), deionized water as the solvent cause minimum environmental damage, 32 mPt. In this Scenario, the electricity used in the synthesis and cleaning stages (Table 1)

accounts for 97 % of the environmental indicator. México's energy primary source for electricity production is 87 % fossil sources (62 % crude oil, 20 % natural gas, 4 % coal, and the remaining condensates) [START_REF] Tang | Principles of green chemistry: PRODUCTIVELY[END_REF].

As shown in Figure 8, the main environmental impact category indicator in all Scenarios of DUT-4 production is fossil depletion (10.71-109.14 mPt); this is mainly due to solvent production. In terms of fossil depletion midpoint indicator, the solvent used in each Scenario has impacts of 1.86x10 -4 , 0.28, 0.53, and 0.75 kg oil eq for Scenarios DUT-4(W), DUT-4(DMF), DUT-4(DCM), and DUT-4(H), respectively; and an electricity consumption 0.08 kg oil eq for all scenarios (Table S5).

J o u r n a l P r e -p r o o f S4), mostly associated with carbon dioxide and methane emission. As in the case of fossil depletion indicator, solvent production is the activity with the most incidence in GHG emissions, 7.23x10 - 4 , 0.45, 1.50, and 1.83 kg CO 2 eq for Scenarios DUT-4(W), DUT-4(DMF), DUT-4(DCM), and DUT-4(H), respectively, while electricity consumption generated 0.24 kg CO 2 eq (Table S5).

The human toxicity indicator in Figure 8 Similar results were obtained by Loya-González et al. [START_REF] Loya-González | Optimal activated carbon production from corn pericarp: A life cycle assessment approach[END_REF] and Sepúlveda Cervántes et al. [START_REF] Sepúlveda-Cervantes | Technical-environmental optimisation of the activated carbon production of an agroindustrial waste by means response surface and life cycle J o u r n a l P r e -p r o o f assessment[END_REF],

where the leading environmental indicators were fossil depletion (55 and 42 %, respectively), climate change (33 and 36 %, respectively), and particulate matter formation (11 and 10 %, respectively).

It is possible to observe that the substitution of DMF solvent for water during the synthesis of DUT-4(DMF)-Scenario DUT-4(W)-decreases environmental damage, but this substitution changed the physicochemical properties of the original MOF, resulting in low siloxane adsorption capacity. This matter should be analyzed from a techno-environmental perspective.

A techno-environmental impact factor-environmental impact/adsorption capacity ratio-was calculated for the four different DUT-4 production scenarios (right axis in Figure 8). The DUT-4(H) adsorbent production has the highest significant environmental impact/adsorption capacity ratio among all four DUT-4 scenarios. In contrast, the DUT-4(DMF) production scenario has the lowest environmental impact/adsorption capacity ratio. These results can be explained by taking into account that DUT-4(DMF) and DUT-4(H) have the same adsorption capacity, but the latter has a higher environmental impact caused by the two solvent exchange stages. Instead, the DUT-4(W) production scenario with a green solvent has a higher environmental impact/adsorption capacity ratio than DUT-4(DMF) production. This result can be explained considering that DUT-4(W) has the lowest siloxane adsorption capacity (1.8 mg/g); thus, to achieve similar adsorption J o u r n a l P r e -p r o o f than the rest (9.80 mg/g), it is necessary to produce five times more of this material. This fact could lead to economic savings to foster greater sustainability in removing siloxanes from biogas. Therefore, from an environmental and technical point of view, DUT-4(DMF) is the best scenario to produce the siloxane adsorbent since the ratio obtained was 53, 42, and 63 % lower than DUT-4(DCM), DUT-4(H), and DUT-4(W), respectively.

CONCLUSIONS

In this study, the synthesis of DUT-4 was carried out by four different routes to improve its physicochemical characteristics for higher siloxane D4 adsorption. In addition, the environmental impacts of the different routes were evaluated by the life cycle assessment methodology.

According to the physicochemical analysis, only the solvothermal synthesis produced materials isostructural to DUT-4 and favored high specific surface areas with values higher than 1100 m 2 /g. The experimental data demonstrated DUT-4 to be a suitable candidate for use as an adsorbent to remove siloxane from gaseous streams such as biogas. The life cycle assessment emphasizes that using DMF, DCM, and hexane increases environmental damage, mainly in the indicators of fossil depletion, climate change, and human toxicity. Therefore, the synthesis of DUT-4(W) was the one with the lowest environmental impact. However, DUT-4(W) presented the lowest siloxane adsorption capacity compared to DUT-4(DMF), DUT-4(DCM), and DUT4(H). Finally, based on the environmental impact/adsorption capacity ratio, Scenario DUT-4(DMF) production is the most convenient since lower adsorbent mass is required to remove siloxane, which reduces the environmental impact of the adsorption process.

J o u r n a l P r e -p r o o f

Highlights:

 The solvothermal synthesis produced materials isostructural to DUT-4 with high surface areas.  DUT-4 is suitable for the adsorption of siloxane from gaseous streams such as biogas.  Cleaning and solvent exchange stages have the highest environmental impact according to LCA.  The hydrothermal synthesis of DUT-4 has the lowest environmental impact and adsorption capacity.  The synthesis of DUT-4(DFM) has the lowest environmental impact/adsorption capacity ratio.

J o u r n a l P r e -p r o o f

  on a laboratory scale (Sepúlveda-Cervantes et al., 2018; Loya-Gonzalez et al., 2019; Luo et al., 2021). Each of the different DUT-4 synthesis strategies was considered as study scenarios:  Scenarios DUT-4(DMF) and DUT-4(W): include the stages of mixing, synthesis, cleaning, and drying.  Scenarios DUT-4(DCM) and DUT-4(H): consider mixing, synthesis, cleaning, solvent exchange, and drying.

Figure 1 .

 1 Figure 1. System boundaries for the production of DUT-4.

  r n a l P r e -p r o o f between the weights of DUT-4 samples before and after being dried in an oven at 120 °C for 4 h.

J o u r

  n a l P r e -p r o o f oxygen atoms ((Al-O))[START_REF] Hoffman | Elucidating the Vibrational Fingerprint of the Flexible Metal-Organic Framework MIL-53(Al) Using a Combined Experimental/Computational Approach[END_REF]. Additionally, in the DUT-4(W) spectrum, the band at 1294 cm-1 was associated with the ((C-O)) stretching vibration, and the bands at 921-914 cm -1 were due to the ((O-H)) bend[START_REF]IR: carboxylic acids[END_REF].

Figure 2 .

 2 Figure 2. FTIR/ATR spectra for DUT-4(DMF), DUT-4(DCM), DUT-4(H), and DUT-4(W).

Figure 3 .

 3 Figure 3. 27 Al MAS NMR Spectra for DUT-4(DMF), DUT-4(DCM), DUT-4(H), and DUT-

Figure 4 .

 4 Figure 4. XRD pattern obtained from DUT-4(DMF), DUT-4(DCM), DUT-4(H), and DUT-4(W)

  results were obtained for the hydrothermal synthesis (H 2 O, 72.8 dyn/cm), where particles of different sizes exceeding 3.25 µm were found. The formation of larger agglomerates during drying of the DUT-4(DMF) and DUT-4(W) was due to the capillary forces overcoming the repulsive forces between the particles; therefore, the particles came closer to each other[START_REF] Maskara | Agglomeration during the Drying of Fine Silica Powders, Part II: The Role of Particle Solubility[END_REF][START_REF] Lim | Understanding and preventing agglomeration in a filter drying process[END_REF].

Figure 5 .

 5 Figure 5. SEM image for a) DUT-4(DMF), b) DUT-4(DCM), c) DUT-4(H), and d) DUT-4(W).

Figure 6 .

 6 Figure 6. Kinetics adsorption for DUT-4(DMF), DUT-4(DCM), DUT-4(H), and DUT-4(W) at the initial concentration of D4 of 450 mg/Nm 3 and room temperature. Error bars represent

Environmental impacts of DUT- 4

 4 production. The environmental impact assessment per FU for each scenario of DUT-4 production is illustrated in Figure7. Scenario DUT-4(H) has the highest environmental impact score at 269 mPt (single score) due to the environmental impact in the cleaning and solvent exchange stages where three types of solvents were used: DMF, DCM, J o u r n a l P r e -p r o o f and hexane. The use of these solvents also increases the cleaning stage's energy demand (Table1) because the use of each solvent requires centrifugation. Altogether, solvent production contributes to 245 mPt in Scenario DUT-4(H), where DMF production represents 61.5 % of the environmental impact score, while DCM and hexane solvent production accounts for 33.3 %, and 5.2 % of the impact score, respectively. The use of these three solvents in Scenario DUT-4(H) led it to be the leader in all the environmental impact indicators (FigureS2). Due to the fact that same values of adsorption capacities (9.8 mg/g) were obtained by the three adsorbents DUT-4(DMF), DUT-4(DCM), and DUT-4(H), it is possible to propose a cleaner and greener DUT-4 production by avoiding additional solvents such as DCM and H. In this manner, principles 3, 4, and 5 of green chemistry are fulfilled[START_REF] Guinee | Handbook on Life Cycle Assessment. Operational Guide to the ISO Standards[END_REF].

Figure 7 .

 7 Figure 7. Environmental impact of each scenario of DUT-4 production through ReCiPe 2016

Figure 8 .

 8 Figure 8. LCA for different DUT-4 production scenarios through endpoint using SimaPro

Table 1 .

 1 Inventory data for DUT-4 production.

	FLOW

FU: functional unit.

  emitting selenium into the air accounts for 3.1-4.4 %, while the electricity consumption generates <2.6 % in each scenario. The particulate matter formation indicator (2.59-16.72 mPt) comes mainly from the air emissions in solvent production: PM 2.5, sulfur

	dioxide (0.64-5.41 mPt), and nitrogen oxides (0.32-4.30 mPt).
	ranges from 7.48-58.70 mPt, equivalent to 0.01-0.42

kg 1,4-DB eq. Similarly, solvent use is the main activity of this environmental damage. For solvent production, pouring barium into water represents 81.4-84.7 % of this indicator, and J o u r n a l P r e -p r o o f

ACKNOWLEDGMENT

We would like to gratefully acknowledge Facultad de Ciencias Quimicas, UANL (PAICYT-UANL Project No IT1297-20) and Consejo Nacional de Ciencia y Tecnologia de Mexico (Scholarship number 781451). As well as the financial support received from the Consejo Nacional de Ciencia y Tecnologia (CONACyT), under project CB-2015-254294. Sandra Pioquinto is grateful for the support of Melissa Itzel Sánchez Alvarado and Alan A. Rico-Barragán.

ASSOCIATED CONTENT

Supporting Information

PDF files containing details of the calculation of the energy balance, estimation methods, environmental impact categories, description of the indicators of fossil depletion, and climate change.

AUTHOR INFORMATION

Corresponding Author

* Nancy Elizabeth Dávila-Guzmán

Phone number: 52-8183294000, ext. 3475 Fax number: 52-8183294000, ext. 6282 E-mail: nancy.davilagz@uanl.edu.mx * Pasiano Rivas García E-mail: pasiano.rivasgr@uanl.edu.mx

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Notes

The authors declare no competing financial interest.


J G Caputo

I Danaila

C Tain

An Abelian Higgs model of pulsed field magnetization in superconductors

Pulsed field magnetization leads to trapped magnetic field persistent for long times.We present a one-dimensional model of the interaction between an electromagnetic wave and a superconducting slab based on the Maxwell-Ginzburg-Landau (Abelian Higgs) theory. We first derive the model starting from a Lagrangian coupling the electromagnetic field with the Ginzburg-Landau potential for the superconductor. Then we explore numerically its capabilities by applying a Gaussian vector potential pulse and monitoring usual quantities such as the modulus and the phase of the order parameter. We also introduce defects in the computational domain. We show that the presence of defects enhances the remanent vector potential and diminishes the modulus of the order parameter, in agreement with existing experiments.

Introduction

High temperature superconductors have great advantages for energy applications because of their zero electrical resistance and relatively low cooling costs. Important devices are cryo-magnets capable to trap a large magnetic field inside a material cooled below a critical temperature. These systems have many applications such as Maglev trains, motors, wind-mills, etc. Figure 1 illustrates a Pulsed Field magnetization (PFM) setup where a cooled superconductor is submitted to a short pulse (a few µs) of magnetic field. Figure 2 shows a typical experimental result with a trapped field that is shown to last for days, even weeks [1].

This multi-physics aspect makes the problem difficult to tackle theoretically when flux motion, mechanical and thermal time scales are considered. To model PFM, a common pattern is to couple Maxwell's equations, a constitutive law, the heat equation and the equations of elasticity [3]. This type of model permits to reproduce experimental results but does not allow to understand the micro/macroscopic mechanism that enables a superconductor to trap a magnetic field in a few milliseconds.

In this work, we address precisely this question. First, we note that Maxwell's equations and the standard constitutive laws (E/J power-law) cannot explain the presence of a stationary field after the passage of the applied electromagnetic pulse. Then, to refine the theoretical approach, we couple Maxwell's equations for the vector potential with the Ginzburg-Landau equations. We keep the wave like character of the model (Lorentz invariance). This is the Abelian Higgs model. We study a one-dimensional configuration where a superconducting slab is submitted to an electromagnetic pulse. We derive the equations of motion and interface conditions using a Lagrangian formalism. The numerical model deals with a pulse longer than the computational domain and absorbing boundary conditions. Numerical simulations show very little trapping in a clean sample without defects and important trapping when defects are present. In the latter case, we observe flux jumps.

The article is organized as follows: in section 2, we recall the constitutive equations used for modeling superconductors. In section 3 we introduce the idea of a non trivial fixed point of the system of equations. In section 4, we derive the equations for the one-dimensional system. In section 5 we present the numerical method and in section 6 we show the results.

The model Consider Maxwell's equations [4]

∇ • E = ρ 0 , (1)
∇ × E = - ∂B ∂t , (2)
∇ • B = 0, (3)
c 2 ∇ × B = J 0 + ∂E ∂t , (4)
where E, B are the electric and magnetic fields, respectively. ρ is a static charge density, J is the current (moving charge) density and c is the velocity of light. We model a superconducting material. Hence, no static charges are present and we assume ρ = 0. We introduce the vector potential A, such that

B = ∇ × A. (5)
Then Eq. ( 3) is satisfied and Eq. ( 2) implies

E = - ∂A ∂t . (6)
We also use the London gauge

∇ • A = 0. (7)
Then, the last Maxwell equation ( 4) can be written as

∂ 2 A ∂t 2 -c 2 ∆A = J 0 . (8)
The main problem in this approach is to find an expression for J. A first choice is the London approximation [START_REF] Duzer | Principles of Superconductive Devices and Circuits[END_REF]:

J = - 0 c 2 λ 2 A, ( 9
)
where λ is the London characteristic length. Another expression, frequently used for large fields is a power law [START_REF] Rhyner | [END_REF]

E = J n , ( 10
)
where E is the amplitude of E. This can be refined to obtain the Bean-Kim model [7]:

J = J 0 B B + B 0 tanh( E E c ), ( 11
)
where B is the magnitude of B and J 0 , B 0 , E c are constants.

Existence of a non trivial fixed point

Experiments on magnetic flux jumps [2] report that there is a threshold of the applied magnetic field above which the sample switches from a transient to a permanent magnetization. In [2], the trapped field is presented as a function of the radius, for different applied fields B from 1.44 T to 2.69 T, and shows a clear jump for B > 1.57 T. The trapped field thus produced is observed to remain stable for very long times, much longer than the duration of the initial pulse. This indicates that the system switches from one type of state close to zero to another non-nul state. Dynamically, this means that there should exists a non trivial fixed point in the coupled equations ( 8),( 9) or ( 8), (11). Equations ( 8) and ( 9), or ( 8) and (11), do not support non trivial fixed points. This is straightforward to obtain for [START_REF] Nielsen | [END_REF]. For (11), we rewrite the system (8),( 11) using ( 5) and ( 7)

∂A ∂t = C, (12)
∂C ∂t = c 2 ∆A - 1 0 J 0 B B + B 0 tanh( ∂A ∂t E c ), ( 13
)
where C is an auxiliary vector. Assume all quantities are scalars. When searching a fixed point, we assume ∂A ∂t = ∂C ∂t = 0 and find that the only existing solution is A = 0.

The conclusion of this simple analysis is that the constitutive equations ( 9),( 10), (11) do not explain why a trapped field remains after the passage of the pulse. To address this issue, we start from first principles and use the full Ginzburg-Landau functional to describe the superconductor.

The Maxwell-Ginzburg-Landau model

As assumed above, the electromagnetic component of the problem is described by the vector potential. The superconductivity is represented by the complex order parameter ψ. The Ginzburg-Landau free energy in SI units can be written as [START_REF] Tinkham | Introduction to Superconductivity[END_REF]

F (A, ψ) = -α|ψ| 2 + β 2 |ψ| 4 + 1 4m |(ih∇ + 2eA)ψ| 2 , ( 14
)
where m is the electron mass and e is the electron charge. The α coefficient is usually taken as α 0 (T -T c ) where α 0 > 0; here we assume T < T c so we use -α.

Most studies of time dependent superconductivity assume that the system relaxes to an equilibrium. The dynamics is typically of gradient type. The effect we are describing takes the system out of equilibrium, hence we need a relativistic extension of the theory of superconductivity. This is provided by the Abelian Higgs model [START_REF] Nielsen | [END_REF].

For simplicity, we start with a one-dimensional formulation of the problem. In the spirit of the study [10], we write a Lagrangian for Maxwell's equations for the single component A of the vector potential and the Ginzburg-Landau potential

L(A, ∂A ∂t , ∂A ∂x , ψ, ∂ψ ∂t , ∂ψ ∂x ) = 1 2µ 0 c 2 ∂A ∂t 2 - 1 2µ 0 ∂A ∂x 2 + I(x) α|ψ| 2 - β 2 |ψ| 4 + h2 4mc 2 ∂ψ ∂t 2 - 1 4m |(ih∇ + 2eA)ψ| 2 , ( 15
)
where I(x) is the indicator function of the superconductor. We introduced a time dependence of ψ following the Abelian Higgs model [START_REF] Nielsen | [END_REF]. We use the subscripts t , x to indicate partial derivatives. The | ∂ψ ∂t | 2 term corresponds to a relativistic generalization of the Ginzburg-Landau theory of superconductors. It describes a fast and out of equilibrium rearrangement of the order parameter.

We introduce the characteristic lengths ξ, λ and their ratio κ

ξ 2 = h2 4mα , λ 2 = mβ 2e 2 µ 0 α , κ = λ ξ . (16)
Following [11] we normalize the main variables as

x = λx , t = λ c t , ψ = α β ψ , A = h 2eξ A . ( 17
)
Plugging these expressions into the Lagrangian L, we obtain

L = ∂A ∂t 2 - ∂A ∂x 2 + I(x) 1 κ 2 ∂ψ ∂t 2 - 1 κ 2 ∂ψ ∂x 2 + i 1 κ A ψ ∂ψ * ∂x -ψ * ∂ψ ∂x +|ψ| 2 (1 -A 2 ) - 1 2 |ψ| 4 . ( 18
)
The Euler-Lagrange equations yield the final system including the coupling conditions at interfaces x = ±L (see details in Appendix A)

∂ 2 A ∂t 2 - ∂ 2 A ∂x 2 = I(x) i 1 2κ ψ ∂ψ * ∂x -ψ * ∂ψ ∂x -A|ψ| 2 , (19)
∂ 2 ψ ∂t 2 - ∂ 2 ψ ∂x 2 = -iκ ∂A ∂x ψ + 2A ∂ψ ∂x + κ 2 ψ(1 -|ψ| 2 -A 2 ) = 0, x ∈ [-L, L], (20)
-iAψ + 1 κ ∂ψ ∂x = 0, x = ±L. (21)
Note that the scale of variation of ψ is 1/κ. For large values of κ giant vortex states are expected [12].

Numerical model

Partial differential equations ( 19),(20) were solved using an ODE solver for the time advancement and finite differences for the space discretization.

We study how an electromagnetic pulse scatters off a superconducting layer. It is therefore important to prevent any out-going wave to bounce off the edge of the computational domain and interact again with the layer. To prevent this effect, we use absorbing boundary conditions

∂A ∂t = - ∂A ∂x , ∂A ∂t = ∂A ∂x
at the left and right sides of the computational domain, respectively (see Fig. 3).

Figure 3. Schematics of the computational domain.

Defects

To trap magnetic flux inside the superconductor, we also study the effect of including defects in the model. We speculate that in the superconductor without defects, as soon as the field recedes, the order parameter returns to its original value and cannot sustain any long term magnetization.

Defects can be geometrically modelled as a wedge placed at the surface of the superconducting sample [11]. This is adapted to a 2D modeling of the and such a geometrical defect favors the penetration of vortices inside the sample. Another type of defect is a material inhomogeneity where the superconductivity breaks down at specific locations inside the slab. In practice, this can be obtained by bombarding the sample with heavy ions [2]. Our one-dimensional model then incorporates a function s(x) in the α term of the free energy (14). This term varies from 1 (superconducting) to -1 (non superconducting), as in [13].

The precise form of the defect is

s(x) = n d i=1 H(x -ix d ), ( 22
)
where H is the square function

H(ξ) = -1 if ξ ∈ [-w d 2 , w d 2 ]
, H(ξ) = 1 otherwise. The equations including this type of defect are

∂ 2 A ∂t 2 - ∂ 2 A ∂x 2 = I(x) i 1 2κ ψ ∂ψ * ∂x -ψ * ∂ψ ∂x -A|ψ| 2 , (23)
∂ 2 ψ ∂t 2 - ∂ 2 ψ ∂x 2 = -iκ ∂A ∂x ψ + 2A ∂ψ ∂x + κ 2 ψ s(x) -|ψ| 2 -A 2 , (24)
-iAψ + 1 κ ∂ψ ∂x = 0, x = ±L. (25)
To model a pulse A 0 longer than the computational domain, we use the following transformation:

A = A + A 0 (x -t), (26)
where A 0 (x) = a 0 exp(-x 2 2w 0 ). Since A 0 (x -t) satisfies the wave equation with speed c = 1, Eqs. ( 23) and (24) become

∂ 2 A ∂t 2 - ∂ 2 A ∂x 2 = I(x) i 1 2κ ψ ∂ψ * ∂x -ψ * ∂ψ ∂x -(A + A 0 )|ψ| 2 , ( 27
)
∂ 2 ψ ∂t 2 - ∂ 2 ψ ∂x 2 = -iκ ∂A ∂x ψ + 2(A + A 0 ) ∂ψ ∂x + κ 2 ψ s(x) -|ψ| 2 -(A + A 0 ) 2 , ( 28
) -i(A + A 0 )ψ + 1 κ ∂ψ ∂x = 0, x = ±L. ( 29
)

Numerical results

In all the runs presented in this section, the computational domain is [-15, 15] and the superconductor extent is [-11.5, 11.5]. The time step is dt = 10 -3 and the space step dx = 7.69 10 -3 . Most runs were performed with pulses hitting the slab from both directions. The typical pulse position and width are x 0 = -200 and w = 50, respectively.

Modulus and phase

A preliminary analysis of equations could be useful to understand numerical results. To this purpose we write the equations using modulus and phase of ψ:

ψ = ρe iθ . ( 30
)
Then Eqs. ( 19)-(20) become

∂ 2 A ∂t 2 - ∂ 2 A ∂x 2 = I(x)ρ 2 1 κ ∂θ ∂x -A , (31)
∂ 2 ρ ∂t 2 - ∂ 2 ρ ∂x 2 = ρ ∂θ ∂t 2 - ∂θ ∂x 2 + 2κAρ ∂θ ∂x + κ 2 ρ(1 -ρ 2 -A 2 ), (32)
∂ 2 θ ∂t 2 - ∂ 2 θ ∂x 2 = 2 ρ ∂θ ∂x ∂ρ ∂x - ∂θ ∂t ∂ρ ∂t -κ ∂A ∂x -2κA 1 ρ ∂ρ ∂x . ( 33
)
The interface condition at x = ±L is

∂ρ ∂x = 0, ∂θ ∂x = κA. ( 34
)
When analysing the system (31), (32), (33) we notice that a first approximation of the trapped field is ∂θ ∂x = κA = 0. The corresponding full solution is

A = A ∞ , ρ = 1, ∂θ ∂x = κA ∞ . ( 35
)
6.2. Effect of amplitude a 0 Figure 4 shows the time evolution of the averaged A inside the superconducting strip for a 0 = 0.2, 1, 3 and 10. One notes that the trapped A is very small for large amplitudes, a 0 = 3. Therefore to trap A we need defects that will lock the phase.

Influence of defects

Figure 5 shows snapshots of the modulus and phase of ψ(x, t) for t = 4416, 4608 and 4800 for an incident pulse of amplitude a 0 = 0.2, with defects spaced by 0.05 and without defects. The left panel shows the modulus of the order parameter ρ(x, t) for both cases. Note that without defects ρ equals one (continuous line, blue online) and does not evolve. Similarly, the phase θ remains flat at 0.

Figure 6 shows the importance of defects to obtain a non-zero vector potential.

Influence of defect spacing

Figure 7 shows the influence of the density of defects on the modulus of the order parameter. More precisely, the average of |ψ| both on space range [-L, L] and time range [0, 40000] is a decreasing function of the density of defects. We defined this density as the ratio between the number of nodes where s(x) = -1 and the ones where s(x) = +1. In

Conclusion

We presented a one-dimensional model describing the interaction of an electromagnetic pulse with a superconducting slab; it is based on the Abelian Higgs model. The novelty of this approach is to introduce second time derivatives in the equations for both vector potential and order parameter. The main features of our numerical method are the use of absorbing boundary conditions and an applied pulse whose support is longer than the computational domain.

The equations of motion and boundary conditions are derived from a Lagrangian via Euler-Lagrange equations. Numerical results indicate that the absence of defect leads to a vector potential close to zero. Moreover, the space and time average of the modulus of the order parameter is a decreasing function of the density of defects: this is consistent with the fact that defects reduce superconductivity by pinning vortices and, as a consequence, lower the modulus of the order parameter.

In conclusion, using this one-dimensional model, we retrieve the usual behaviour of superconductors. Future work will consider 2D configurations. The equation for A is where δ(x) is the Dirac distribution corresponding to the x derivative of the characteristic function I(x).

∂ 2 A ∂t 2 - ∂ 2 A ∂x 2 = I(x) i 1
To obtain boundary conditions we integrate (A.1) over one of the edges of the domain. For example, for x = -L we obtain Taking the limit → 0 and assuming the bracket in the integral is bounded, we recover the following standard boundary condition [START_REF] Duzer | Principles of Superconductive Devices and Circuits[END_REF] at the edge of a superconductor:

-iAψ + 1 κ ∂ψ ∂x = 0. (A.3)

Figure 1 .

 1 Figure 1. Pulse field magnetization experimental set up [2].

Figure 2 .

 2 Figure 2. Magnetic cartography of trapped field ina GdBCO sample magnetised with the flux cooling method at CRISMAT (Caen, France).

Figure 4 .

 4 Figure 4. Time evolution of the averaged A inside the superconducting strip for four amplitudes of the applied pulse: a 0 = 0.2, 1, 3 and 10.

Figure 5 .

 5 Figure 5. Snapshots of the modulus ρ(x, t) (left panel) and the phase θ(x, t) (right panel) for t = 4416, 4608 and 4800. The incident pulse has amplitude a 0 = 0.2.

Fig. 8

 8 the phase gradient and trapped A are approximately 3 • 10 -3 for x d = 0.05, while they raise to values around 1.8 for x d = 0.01. Moreover, ∂θ ∂x is negative for x d = 0.05 and positive for x d = 0.01. In addition ρ(x, t) for x d = 0.01 becomes rather small and seems to oscillate periodically.

Figure 6 .

 6 Figure 6. Time evolution of the averaged A (left) and the phase θ (right) considering the presence or the absence of defects.

Figure 7 .

 7 Figure 7. Space and time average of |ψ| as a function of the density of defects (%).

Figure 8 .

 8 Figure 8. Snapshots of the phase θ(x, t) (top), modulus ρ(x, t) (middle) and A(x, t) (bottom) for t = 7680, 8640 and 9600 for a defect spacing x d = 0.05 (left panels) and 0.01 (right panels). The incident pulse has an amplitude a 0 = 3 and the defect spacing is w d = 10 -3 .

-ψ * ψ x -A|ψ| 2 .

 2 Taking the limit → 0 and assuming bounded variations of the integrands, we obtain that ∂A ∂x is continuous at the interface x = -L. The second equation gives

Acknowledgments

The present work was performed using computing resources of CRIANN (Normandy, France). The authors thank Nikos Flytzanis, Michikazu Kobayashi and Mads Peter Soerensen for useful comments. They also thank Pierre Bernstein and Jacques Noudem for sharing their experimental results.