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Correlated Electron States in Coupled Graphene Double-Layer Heterostructures

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Fundamentals, synthesis, characterization and environmental applications of layered double hydroxides: a review

• Published: 26 February 2021
• Volume 19 , pages 2643–2661, ( 2021 )

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• Prabagar Samuel Jijoe 1 ,
• Shivamurthy Ravindra Yashas 1 &
• Harikaranahalli Puttaiah Shivaraju   ORCID: orcid.org/0000-0001-5125-4877 1 , 2  

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The availability of clean water and energy scarcity are rising issues in the context of rising population and industrialization, calling for advanced methods of remediation and energy production. Here, layered double hydroxides, as anionic clays, can be used and engineered as adsorbents or catalysts. Layered double hydroxides are generally highly stable, safe and recyclable. They can be filled with nanomaterials to form composites of enhanced performance. Here, we review fundamentals of layered double hydroxides, synthetic protocols, and applications to energy storage, dye degradation, organic pollutant degradation, water treatment, photoelectrochemical water splitting and carbon dioxide reduction. Composites appear competitive in terms of low-cost, tunable band-gaps and high electrical conductivity.

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3d hierarchical and porous layered double hydroxide structures: an overview of synthesis methods and applications.

Application of Layered Double Hydroxides (LDHs) in Photocatalysis

Electrochemical properties of layered double hydroxides containing 3d metal cations, data availability.

All data generated or analysed during this study are included in this published article.

Abbreviations

• Layered double hydroxides

Hydrotalcite

X-ray diffraction

Fourier transform infrared

Scanning electron microscope

X-ray fluorescence

Potential of hydrogen

Magnesium–aluminium layered double hydroxide

Advanced oxidation process

Volatile organic compounds

Nickel–iron layered double hydroxide

Oxygen evolution reaction

Ahmed DN, Naji LA, Faisal AAH et al (2020) Waste foundry sand/MgFe-layered double hydroxides composite material for efficient removal of Congo red dye from aqueous solution. Sci Rep 10:2042. https://doi.org/10.1038/s41598-020-58866-y

Article   CAS   Google Scholar  

Akpomie KG, Conradie J (2020) Banana peel as a biosorbent for the decontamination of water pollutants. Rev Environ Chem Lett 18:1085–1112. https://doi.org/10.1007/s10311-020-00995-x

Allmann R (1968) The crystal structure of pyroaurite. Acta Cryst B 24:972–977. https://doi.org/10.1107/S0567740868003511

Amr E, El Basiony NM, El-Taib Heakal F, Elkholy AE (2020) Mesoporous Ni-Zn-Fe layered double hydroxide as an efficient binder-free electrode active material for high-performance supercapacitors. J Power Sources 466:228294. https://doi.org/10.1016/j.jpowsour.2020.228294

Bagherifam S, Komarneni S, Lakzian A et al (2014) Evaluation of Zn–Al–SO 4 layered double hydroxide for the removal of arsenite and arsenate from a simulated soil solution: isotherms and kinetics. Appl Clay Sci 95:119–125. https://doi.org/10.1016/j.clay.2014.02.028

Bai S, Wang Z, Tan L et al (2020) 600 nm irradiation-induced efficient photocatalytic CO 2 reduction by ultrathin layered double hydroxide nanosheets. Ind Eng Chem Res 59:5848–5857. https://doi.org/10.1021/acs.iecr.0c00522

Bashir A, Malik LA, Ahad S et al (2019) Removal of heavy metal ions from aqueous system by ion-exchange and biosorption methods. Environ Chem Lett 17:729–754. https://doi.org/10.1007/s10311-018-00828-y

Boehm H-P, Steinle J, Vieweger C (1977) [Zn 2 Cr(Oh) 6 ] X ·2H 2 O, new layer compounds capable of anion exchange and intracrystalline swelling. Angew Chem Int Ed Engl 16:265–266. https://doi.org/10.1002/anie.197702651

Article   Google Scholar  

Boppella R, Choi CH, Moon J, Ha Kim D (2018) Spatial charge separation on strongly coupled 2D-hybrid of rGO/La 2 Ti 2 O 7 /NiFe-LDH heterostructures for highly efficient noble metal free photocatalytic hydrogen generation. Appl Catal B 239:178–186. https://doi.org/10.1016/j.apcatb.2018.07.063

Cao Y, Li G, Li X (2016) Graphene/layered double hydroxide nanocomposite: properties, synthesis, and applications. Chem Eng J 292:207–223. https://doi.org/10.1016/j.cej.2016.01.114

Carlino S, Hudson MJ (1998) A thermal decomposition study on the intercalation of tris-(oxalato) ferrate(III) trihydrate into a layered (Mg/Al) double hydroxide. Solid State Ion 110:153–161. https://doi.org/10.1016/S0167-2738(97)00500-6

Chaparadza A, Hossenlopp JM (2011) Removal of 2, 4-dichlorophenoxyacetic acid by calcined Zn–Al–Zr layered double hydroxide. J Colloid Interface Sci 363:92–97. https://doi.org/10.1016/j.jcis.2011.07.002

Chen W, Xing J, Lu Z et al (2018) Citrate-modified Mg–Al layered double hydroxides for efficient removal of lead from water. Environ Chem Lett 16:561–567. https://doi.org/10.1007/s10311-017-0692-5

Chen X, Chai H, Cao Y et al (2020a) Hierarchical CoGa layered double hydroxides grown on nickel foam as high energy density hybrid supercapacitor. Chem Eng J 381:122620. https://doi.org/10.1016/j.cej.2019.122620

Chen Z, Deng H, Zhang M et al (2020b) One-step facile synthesis of nickel–chromium layered double hydroxide nanoflakes for high-performance supercapacitors. Nanoscale Adv 2:2099–2105. https://doi.org/10.1039/D0NA00178C

Costantino U, Marmottini F, Nocchetti M, Vivani R (1998) New synthetic routes to hydrotalcite-like compounds - characterisation and properties of the obtained materials. Eur J Inorg Chem. https://doi.org/10.1002/(SICI)1099-0682(199810)1998:10%3c1439::AID-EJIC1439%3e3.0.CO;2-1

Cui PQ, Zhou HG, Li C et al (2016) Characteristics of using layered double hydroxides to reduce the VOCs from bituminous materials. Constr Build Mater 123:69–77. https://doi.org/10.1016/j.conbuildmat.2016.06.117

Daud M, Hai A, Banat F et al (2019) A review on the recent advances, challenges and future aspect of layered double hydroxides (LDH) – Containing hybrids as promising adsorbents for dyes removal. J Mol Liq 288:110989. https://doi.org/10.1016/j.molliq.2019.110989

dos Santos GEDS, Ide AH, Duarte JLS et al (2020) Adsorption of anti-inflammatory drug diclofenac by MgAl/layered double hydroxide supported on Syagrus coronata biochar. Powder Technol 364:229–240. https://doi.org/10.1016/j.powtec.2020.01.083

Ding J, Wu X, Shen X et al (2020) Synthesis and textural evolution of mesoporous Si3N4 aerogel with high specific surface area and excellent thermal insulation property via the urea assisted sol-gel technique. Chem Eng J 382:122880. https://doi.org/10.1016/j.cej.2019.122880

Dong X, Wang L, Wang D et al (2012) Layer-by-layer engineered Co–Al hydroxide nanosheets/graphene multilayer films as flexible electrode for supercapacitor. Langmuir 28:293–298. https://doi.org/10.1021/la2038685

Duan S, Ma W, Cheng Z et al (2016) Preparation of modified Mg/Al layered double hydroxide in saccharide system and its application to remove As(V) from glucose solution. Colloids Surf A 490:250–257. https://doi.org/10.1016/j.colsurfa.2015.11.060

El Rouby WM, El-Dek SI, Goher ME, Noaemy SG (2020) Efficient water decontamination using layered double hydroxide beads nanocomposites. Environ Sci Pollut Res 27:18985–19003. https://doi.org/10.1007/s11356-018-3257-7

Erickson KL, Bostrom TE, Frost RL (2005) A study of structural memory effects in synthetic hydrotalcites using environmental SEM. Mater Lett 59:226–229. https://doi.org/10.1016/j.matlet.2004.08.035

Fan X, Gao B, Wang T et al (2016) Layered double hydroxide modified WO3 nanorod arrays for enhanced photoelectrochemical water splitting. Appl Catal A 528:52–58. https://doi.org/10.1016/j.apcata.2016.09.014

Gao Z, Sasaki K, Qiu X (2018) Structural memory effect of Mg–Al and Zn–Al layered double hydroxides in the presence of different natural humic acids: process and mechanism. Langmuir 34:5386–5395. https://doi.org/10.1021/acs.langmuir.8b00059

Gao G, Zhu Z, Zheng J et al (2019) Ultrathin magnetic Mg-Al LDH photocatalyst for enhanced CO2 reduction: fabrication and mechanism. J Colloid Interface Sci 555:1–10. https://doi.org/10.1016/j.jcis.2019.07.025

Gao X, Peng Y, Guo L et al (2020) Arsenic adsorption on layered double hydroxides biochars and their amended red and calcareous soils. J Environ Manage 271:111045. https://doi.org/10.1016/j.jenvman.2020.111045

Gil JJ, Aguilar-Martínez O, Piña-Pérez Y et al (2020) Efficient ZnS–ZnO/ZnAl-LDH composite for H2 production by photocatalysis. Renew Energy 145:124–132. https://doi.org/10.1016/j.renene.2019.06.001

Gopi M, Shivaraju HP, Shahmoradi B, Maleki A (2020) Preparation and characterization of cost-effective AC/CeO2 nanocomposites for the degradation of selected industrial dyes. Appl Water Sci 10:25. https://doi.org/10.1007/s13201-019-1105-7

Gunjakar JL, Kim TW, Kim HN et al (2011) Mesoporous layer-by-layer ordered nanohybrids of layered double hydroxide and layered metal oxide: highly active visible light photocatalysts with improved chemical stability. J Am Chem Soc 133:14998–15007. https://doi.org/10.1021/ja203388r

Hao X, Tan L, Xu Y et al (2020) Engineering active Ni sites in ternary layered double hydroxide nanosheets for a highly selective photoreduction of CO 2 to CH 4 under Irradiation above 500 nm. Ind Eng Chem Res 59:3008–3015. https://doi.org/10.1021/acs.iecr.9b06464

Hashim N, Sharif SN, Hussein MZ et al (2017) Layered hydroxide anion exchanger and their applications related to pesticides: a brief review. Mat Res Innov 21(3):129–145. https://doi.org/10.1080/14328917.2016.1192717

Hassan NS, Jalil AA, Hitam CNC et al (2020) Biofuels and renewable chemicals production by catalytic pyrolysis of cellulose: a review. Environ Chem Lett 18:1625–1648. https://doi.org/10.1007/s10311-020-01040-7

Hong Y, Peng J, Zhao X et al (2019) Efficient degradation of atrazine by CoMgAl layered double oxides catalyzed peroxymonosulfate: optimization, degradation pathways and mechanism. Chem Eng J 370:354–363. https://doi.org/10.1016/j.cej.2019.03.127

Hou L, Li X, Yang Q et al (2019) Heterogeneous activation of peroxymonosulfate using Mn-Fe layered double hydroxide: performance and mechanism for organic pollutant degradation. Sci Total Environ 663:453–464. https://doi.org/10.1016/j.scitotenv.2019.01.190

Ifebajo AO, Oladipo AA, Gazi M (2019) Efficient removal of tetracycline by CoO/CuFe2O4 derived from layered double hydroxides. Environ Chem Lett 17:487–494. https://doi.org/10.1007/s10311-018-0781-0

Jo W-K, Kumar S, Tonda S (2019) N-doped C dot/CoAl-layered double hydroxide/g-C3N4 hybrid composites for efficient and selective solar-driven conversion of CO 2 into CH 4 . Compos B Eng 176:107212. https://doi.org/10.1016/j.compositesb.2019.107212

Khan SA, Bakhsh EM, Asiri AM, Khan SB (2020) Chitosan coated NiAl layered double hydroxide microsphere templated zero-valent metal NPs for environmental remediation. J Clean Prod. https://doi.org/10.1016/j.jclepro.2020.124830

Khenifi A, Derriche Z, Mousty C et al (2010) Adsorption of glyphosate and glufosinate by Ni 2 AlNO 3 layered double hydroxide. Appl Clay Sci 47:362–371. https://doi.org/10.1016/j.clay.2009.11.055

Lai Y-T, Huang Y-S, Chen C-H et al (2020) Green treatment of phosphate from wastewater using a porous bio-templated graphene oxide/MgMn-layered double hydroxide composite. iScience 23:101065. https://doi.org/10.1016/j.isci.2020.101065

Le K, Wang Z, Wang F et al (2019) Sandwich-like NiCo layered double hydroxide/reduced graphene oxide nanocomposite cathodes for high energy density asymmetric supercapacitors. Dalton Trans 48:5193–5202. https://doi.org/10.1039/C9DT00615J

Lee Y, Choi JH, Jeon HJ et al (2011) Titanium-embedded layered double hydroxides as highly efficient water oxidation photocatalysts under visible light. Energy Environ Sci 4:914–920. https://doi.org/10.1039/C0EE00285B

Li S, Wang F, Jing X et al (2012) Synthesis of layered double hydroxides from eggshells. Mater Chem Phys 132:39–43. https://doi.org/10.1016/j.matchemphys.2011.10.049

Li C, Wei M, Evans DG, Duan X (2014) Layered double hydroxide-based nanomaterials as highly efficient catalysts and adsorbents. Small 10:4469–4486. https://doi.org/10.1002/smll.201401464

Li A, Deng H, Ye C, Jiang Y (2020a) Fabrication and characterization of novel ZnAl-layered double hydroxide for the superadsorption of organic contaminants from wastewater. ACS omega 5:15152–15161. https://doi.org/10.1021/acsomega.0c01092

Li F, Sun Z, Jiang H et al (2020b) Ion-exchange synthesis of ternary FeCoNi-layered double hydroxide nanocage toward enhanced oxygen evolution reaction and supercapacitor. Energy Fuels 34:11628–11636. https://doi.org/10.1021/acs.energyfuels.0c02533

Liang H, Lin J, Jia H et al (2018) Hierarchical NiCo-LDH@NiOOH core-shell heterostructure on carbon fiber cloth as battery-like electrode for supercapacitor. J Power Sources 378:248–254. https://doi.org/10.1016/j.jpowsour.2017.12.046

Liu Y, Yu C, Che H et al (2021) Ag nanoparticles-decorated CoAl-layered double hydroxide flower-like hollow microspheres for enhanced energy storage performance. J Colloid Interface Sci 581:485–495. https://doi.org/10.1016/j.jcis.2020.08.018

Lu H, Sui M, Yuan B et al (2019) Efficient degradation of nitrobenzene by Cu-Co-Fe-LDH catalyzed peroxymonosulfate to produce hydroxyl radicals. Chem Eng J 357:140–149. https://doi.org/10.1016/j.cej.2018.09.111

Madhura L, Singh S, Kanchi S et al (2019) Nanotechnology-based water quality management for wastewater treatment. Environ Chem Lett 17:65–121. https://doi.org/10.1007/s10311-018-0778-8

Malakootian M, Shahamat YD, Kannan K, Mahdizadeh H (2020) Degradation of p-nitroaniline from aqueous solutions using ozonation/Mg-Al layered double hydroxides integrated with the sequencing batch moving bed biofilm reactor. J Taiwan Inst Chem Eng 113:241–252. https://doi.org/10.1016/j.jtice.2020.08.019

Mignani A, Ballarin B, Giorgetti M et al (2013) Heterostructure of Au nanoparticles—NiAl layered double hydroxide: electrosynthesis, characterization, and electrocatalytic properties. J Phys Chem C 117:16221–16230. https://doi.org/10.1021/jp4033782

Mishra G, Dash B, Pandey S (2018) Layered double hydroxides: a brief review from fundamentals to application as evolving biomaterials. Appl Clay Sci 153:172–186. https://doi.org/10.1016/j.clay.2017.12.021

Mohapatra L, Parida K (2016) A review on the recent progress, challenges and perspective of layered double hydroxides as promising photocatalysts. J Mater Chem A 4:10744–10766. https://doi.org/10.1039/C6TA01668E

Morel-Desrosiers N, Pisson J, Israëli Y et al (2003) Intercalation of dicarboxylate anions into a Zn–Al–Cl layered double hydroxide: microcalorimetric determination of the enthalpies of anion exchange. J Mater Chem 13:2582–2585. https://doi.org/10.1039/B303953F

Mourid EH, Lakraimi M, Benaziz L et al (2019) Wastewater treatment test by removal of the sulfamethoxazole antibiotic by a calcined layered double hydroxide. Appl Clay Sci 168:87–95. https://doi.org/10.1016/j.clay.2018.11.005

Mudhoo A, Gautam RK, Ncibi MC et al (2019) Green synthesis, activation and functionalization of adsorbents for dye sequestration. Environ Chem Lett 17:157–193. https://doi.org/10.1007/s10311-018-0784-x

Mudhoo A, Paliya S, Goswami P et al (2020) Fabrication, functionalization and performance of doped photocatalysts for dye degradation and mineralization: a review. Environ Chem Lett 18:1825–1903. https://doi.org/10.1007/s10311-020-01045-2

Nayak S, Parida KM (2016) Nanostructured CeO2/MgAl-LDH composite for visible light induced water reduction reaction. Int J Hydrog Energy 41:21166–21180. https://doi.org/10.1016/j.ijhydene.2016.08.062

Nayak S, Parida KM (2019) Deciphering Z-scheme charge transfer dynamics in heterostructure NiFe-LDH/N-rGO/g-C 3 N 4 nanocomposite for photocatalytic pollutant removal and water splitting reactions. Sci Rep 9:1–23. https://doi.org/10.1038/s41598-019-39009-4

Nayak S, Swain G, Parida K (2019) Enhanced photocatalytic activities of RhB degradation and H2 evolution from in situ formation of the electrostatic heterostructure MoS2/NiFe LDH nanocomposite through the Z-scheme mechanism via p–n heterojunctions. ACS Appl Mater Interfaces 11:20923–20942. https://doi.org/10.1021/acsami.9b06511

Olfs H-W, Torres-Dorante LO, Eckelt R, Kosslick H (2009) Comparison of different synthesis routes for Mg–Al layered double hydroxides (LDH): characterization of the structural phases and anion exchange properties. Appl Clay Sci 43:459–464. https://doi.org/10.1016/j.clay.2008.10.009

Olivera S, Chaitra K, Venkatesh K et al (2018) Cerium dioxide and composites for the removal of toxic metal ions. Environ Chem Lett 16:1233–1246. https://doi.org/10.1007/s10311-018-0747-2

Osman AI, Hefny M, Abdel Maksoud MIA et al (2020) Recent advances in carbon capture storage and utilisation technologies: a review. Environ Chem Lett. https://doi.org/10.1007/s10311-020-01133-3

Pallavi N, Shivaraju HP, Kitirote W, Behzad S (2020) Preparation of modified ZnO nanoparticles for photocatalytic degradation of chlorobenzene. Appl Water Sci. https://doi.org/10.1007/s13201-020-01228-w

Pavlovic I, González MA, Rodríguez-Rivas F et al (2013) Caprylate intercalated layered double hydroxide as adsorbent of the linuron, 2,4-DB and metamitron pesticides from aqueous solution. Appl Clay Sci 80–81:76–84. https://doi.org/10.1016/j.clay.2013.06.008

Pirkarami A, Rasouli S, Ghasemi E (2019) 3-D CdS@NiCo layered double hydroxide core-shell photoelectrocatalyst used for efficient overall water splitting. Appl Catal B 241:28–40. https://doi.org/10.1016/j.apcatb.2018.09.021

Prince J, Montoya A, Ferrat G, Valente JS (2009) Proposed general sol−gel method to prepare multimetallic layered double hydroxides: synthesis, characterization, and envisaged application. Chem Mater 21:5826–5835. https://doi.org/10.1021/cm902741c

Pungor E, Horvai G (1994) A practical guide to instrumental analysis. CRC Press, Florida

Google Scholar  

Qin Z, Chen J, Xie X et al (2020) CO 2 reforming of CH 4 to syngas over nickel-based catalysts. Environ Chem Lett 18:997–1017. https://doi.org/10.1007/s10311-020-00996-w

Ramos-Ramírez E, Tzompantzi-Morales F, Gutiérrez-Ortega N et al (2019) Photocatalytic degradation of 2,4,6-trichlorophenol by MgO–MgFe 2 O 4 derived from layered double hydroxide structures. Catalysts 9:454. https://doi.org/10.3390/catal9050454

Ramos-Ramírez E, Gutiérrez-Ortega NL, Tzompantzi-Morales F et al (2020) Photocatalytic degradation of 2,4-dichlorophenol on NiAl-mixed oxides derivatives of activated layered double hydroxides. Top Catal 63:546–563. https://doi.org/10.1007/s11244-020-01269-0

Rashidimoghaddam M, Saljooqi A, Shamspur T, Mostafavi A (2020) Constructing S-doped Ni–Co LDH intercalated with Fe3O4 heterostructure photocatalysts for enhanced pesticide degradation. New J Chem 44:15584–15592. https://doi.org/10.1039/D0NJ02772C

Ravindra YS, Puttaiah SH, Yadav S, Prabagar JS (2020) Evaluation of polymeric gC- 3 N 4 contained nonhierarchical ZnV 2 O 6 composite for energy-efficient LED assisted photocatalytic mineralization of organic pollutant. J Mater Sci Mater Electron 31:16806–16818. https://doi.org/10.1007/s10854-020-04235-4

Rives V (2001) Layered double hydroxides: present and future. Nova Publishers, NewYork

Rocha J, del Arco M, Rives V, Ulibarri MA (1999) Reconstruction of layered double hydroxides from calcined precursors: a powder XRD and 27Al MAS NMR study. J Mater Chem 9:2499–2503. https://doi.org/10.1039/a903231b

Sanati S, Rezvani Z (2019) g-C 3 N 4 nanosheet@CoAl-layered double hydroxide composites for electrochemical energy storage in supercapacitors. Chem Eng J 362:743–757. https://doi.org/10.1016/j.cej.2019.01.081

Saravanan A, Kumar PS, Vo D-VN et al (2020) Photocatalysis for removal of environmental pollutants and fuel production: a review. Environ Chem Lett. https://doi.org/10.1007/s10311-020-01077-8

Seftel EM, Popovici E, Mertens M et al (2008) Zn–Al layered double hydroxides: synthesis, characterization and photocatalytic application. Microporous Mesoporous Mater 113:296–304. https://doi.org/10.1016/j.micromeso.2007.11.029

Shahnaz T, Vishnu Priyan V, Pandian S, Narayanasamy S (2021) Use of nanocellulose extracted from grass for adsorption abatement of Ciprofloxacin and Diclofenac removal with phyto, and fish toxicity studies. Environ Pollut 268:115494. https://doi.org/10.1016/j.envpol.2020.115494

Shi M, Zhao Z, Song Y et al (2020) A novel heat-treated humic acid/MgAl-layered double hydroxide composite for efficient removal of cadmium: fabrication, performance and mechanisms. Appl Clay Sci 187:105482. https://doi.org/10.1016/j.clay.2020.105482

Shivaraju HP (2020) Application of Mg-doped TiO 2 coated buoyant clay hollow-spheres for photodegradation of organic pollutants in wastewater. Mater Today. https://doi.org/10.1016/j.matpr.2020.02.754

Shivaraju HP, Pallavi N (2017) Photocatalytic degradation of bromobenzene using novel Fe-Mn-CeO 2 /TiO 2 nano-composite under light emitting diodes. Int J Environ Health and Technol 1(3):78–85

Shivaraju HP, Muzakkira N, Shahmoradi B (2016) Photocatalytic treatment of oil and grease spills in wastewater using coated N-doped TiO 2 polyscales under sunlight as an alternative driving energy. Int J Environ Sci Technol 13:2293–2302. https://doi.org/10.1007/s13762-016-1038-8

Shivaraju HP, Midhun G, Kumar KA et al (2017) Degradation of selected industrial dyes using Mg-doped TiO 2 polyscales under natural sun light as an alternative driving energy. Appl Water Sci 7:3937–3948. https://doi.org/10.1007/s13201-017-0546-0

Shivaraju HP, Anil Kumar KM, Midhun G (2018) Demonstration of hybridized processes for waste to energy under partial photocatalytic processes. IOP Conf Ser Mater Sci Eng 377:012071. https://doi.org/10.1088/1757-899X/377/1/012071

Sirajudheen P, Karthikeyan P, Meenakshi S (2020) Mechanistic performance of organic pollutants removal from water using Zn/Al layered double hydroxides imprinted carbon composite. Surf Interfaces 20:100581. https://doi.org/10.1016/j.surfin.2020.100581

Sohrabi A, Yaftian MR, Dolatyari L et al (2020) Application of Mg–Al and Zn–Al layered double hydroxides modified with sodium dodecyl benzene sulfonate as a solid sorbent for removal of diazinon from water samples. J Iran Chem Soc 17:1411–1427. https://doi.org/10.1007/s13738-020-01866-6

Soni V, Raizada P, Kumar A et al (2021) Indium sulfide-based photocatalysts for hydrogen production and water cleaning: a review. Environ Chem Lett. https://doi.org/10.1007/s10311-020-01148-w

Suárez-Quezada M, Romero-Ortiz G, Samaniego-Benítez JE et al (2019) H 2 production by the water splitting reaction using photocatalysts derived from calcined ZnAl LDH. Fuel 240:262–269. https://doi.org/10.1016/j.fuel.2018.11.155

Sun Z, Jin L, Shi W et al (2011) Controllable photoluminescence properties of an anion-dye-intercalated layered double hydroxide by adjusting the confined environment. Langmuir 27:7113–7120. https://doi.org/10.1021/la200846j

Tao Q, Zhang Y, Zhang X et al (2006) Synthesis and characterization of layered double hydroxides with a high aspect ratio. J Solid State Chem 179:708–715. https://doi.org/10.1016/j.jssc.2005.11.023

Tao J, Yu X, Liu Q et al (2020a) Internal electric field induced S–scheme heterojunction MoS 2 /CoAl LDH for enhanced photocatalytic hydrogen evolution. J Colloid Interface Sci. https://doi.org/10.1016/j.jcis.2020.10.028

Tao X, Yang C, Huang L, Shang S (2020b) Novel plasma assisted preparation of ZnCuFeCr layered double hydroxides with improved photocatalytic performance of methyl orange degradation. Appl Surf Sci 507:145053. https://doi.org/10.1016/j.apsusc.2019.145053

Taylor HFW (1969) Segregation and cation-ordering in sjögrenite and pyroaurite. Miner Mag 37:338–342. https://doi.org/10.1180/minmag.1969.037.287.04

Theiss FL, Ayoko GA, Frost RL (2016) Synthesis of layered double hydroxides containing Mg 2+ , Zn 2+ , Ca 2+ and Al 3+ layer cations by co-precipitation methods—a review. Appl Surf Sci 383:200–213. https://doi.org/10.1016/j.apsusc.2016.04.150

Tsai K-J, Ni C-S, Chen H-Y, Huang J-H (2020) Single-walled carbon nanotubes/Ni–Co–Mn layered double hydroxide nanohybrids as electrode materials for high-performance hybrid energy storage devices. J Power Sources 454:227912. https://doi.org/10.1016/j.jpowsour.2020.227912

Wang L, Wang D, Dong XY et al (2011) Layered assembly of graphene oxide and Co–Al layered double hydroxide nanosheets as electrode materials for supercapacitors. Chem Commun 47:3556. https://doi.org/10.1039/c0cc05420h

Wang C, Ma B, Cao X et al (2018) Bridge-type interface optimization on a dual-semiconductor heterostructure toward high performance overall water splitting. J Mater Chem A 6:7871–7876. https://doi.org/10.1039/C8TA02001A

Wang K, Miao C, Liu Y et al (2020a) Vacancy enriched ultrathin TiMgAl-layered double hydroxide/graphene oxides composites as highly efficient visible-light catalysts for CO 2 reduction. Appl Catal B 270:118878. https://doi.org/10.1016/j.apcatb.2020.118878

Wang Q, Song X, Tang S, Yu L (2020b) Enhanced removal of tetrachloroethylene from aqueous solutions by biodegradation coupled with nZVI modified by layered double hydroxide. Chemosphere 243:125260. https://doi.org/10.1016/j.chemosphere.2019.125260

Wang Y, Zhou S, Zhao G et al (2020c) Fabrication of SnWO 4 /ZnFe-layered double hydroxide composites with enhanced photocatalytic degradation of methyl orange. J Mater Sci Mater Electron 31:12269–12281. https://doi.org/10.1007/s10854-020-03772-2

Wang Y, Gao Y, Zhu Z, et al (2021) Enhanced arsenic removal from aqueous solution by Fe/Mn-C layered double hydroxide composite. In: Adsorption science & technology. https://www.hindawi.com/journals/ast/2021/8891643/ . doi: https://doi.org/10.1155/2021/8891643Accessed from 25 Jan 2021

Weng B, Grice CR, Ge J et al (2018) Barium bismuth niobate double perovskite/tungsten oxide nanosheet photoanode for high-performance photoelectrochemical water splitting. Adv Energy Mater 8:1701655. https://doi.org/10.1002/aenm.201701655

Williams GR, O’Hare D (2006) Towards understanding, control and application of layered double hydroxide chemistry. J Mater Chem 16:3065. https://doi.org/10.1039/b604895a

Wu X, Wang S, Du N et al (2013) Facile synthesis of deoxycholate intercalated layered double hydroxide nanohybrids via a coassembly process. J Solid State Chem 203:181–186. https://doi.org/10.1016/j.jssc.2013.04.007

Wu H, Zhang H, Zhang W et al (2019) Preparation of magnetic polyimide@ Mg-Fe layered double hydroxides core-shell composite for effective removal of various organic contaminants from aqueous solution. Chemosphere 219:66–75

Wu Y, Gong Y, Liu J et al (2020) Constructing NiFe-LDH wrapped Cu 2 O nanocube heterostructure photocatalysts for enhanced photocatalytic dye degradation and CO 2 reduction via Z-scheme mechanism. J Alloy Compd 831:154723. https://doi.org/10.1016/j.jallcom.2020.154723

Xia S, Shao M, Zhou X et al (2015) Ti/ZnO–M x O y composites (M= Al, Cr, Fe, Ce): synthesis, characterization and application as highly efficient photocatalysts for hexachlorobenzene degradation. Phys Chem Chem Phys 17:26690–26702. https://doi.org/10.1039/C5CP04125B

Xia S, Qian M, Zhou X et al (2017) Theoretical and experimental investigation into the photocatalytic degradation of hexachlorobenzene by ZnCr layered double hydroxides with different anions. Mol Catal 435:118–127. https://doi.org/10.1016/j.mcat.2017.03.024

Xing Y, Li D-Q, Ren L-L et al (2003) Assembly and structural characteristics of supramolecular salicylate-pillared hydrotalcites. ACTA CHIMICA SINICA-CHINESE EDITION- 61:267–272

CAS   Google Scholar  

Xu M, Wei M (2018) Layered double hydroxide-based catalysts: recent advances in preparation, structure, and applications. Adv Funct Mater 28:1802943. https://doi.org/10.1002/adfm.201802943

Xu H, Zhu S, Xia M, Wang F (2021) Rapid and efficient removal of diclofenac sodium from aqueous solution via ternary core-shell CS@PANI@LDH composite: Experimental and adsorption mechanism study. J Hazard Mater 402:123815. https://doi.org/10.1016/j.jhazmat.2020.123815

Yan X, Jin Z (2020) Interface Engineering: NiAl-LDH in-situ derived NiP 2 quantum dots and Cu 3 P nanoparticles ingeniously constructed p-n heterojunction for photocatalytic hydrogen evolution. Chem Eng J. https://doi.org/10.1016/j.cej.2020.127682

Yang R, Zhou Y, Xing Y et al (2019) Synergistic coupling of CoFe-LDH arrays with NiFe-LDH nanosheet for highly efficient overall water splitting in alkaline media. Appl Catal B 253:131–139. https://doi.org/10.1016/j.apcatb.2019.04.054

Yang C, Wang L, Yu Y et al (2020) Highly efficient removal of amoxicillin from water by Mg-Al layered double hydroxide/cellulose nanocomposite beads synthesized through in-situ coprecipitation method. Int J Biol Macromol 149:93–100. https://doi.org/10.1016/j.ijbiomac.2020.01.096

Yashas SR, Shivaraju HP, Pema G et al (2020) Sonochemical synthesis of graphitic carbon nitride-manganese oxide interfaces for enhanced photocatalytic degradation of tetracycline hydrochloride. Environ Sci Pollut Res. https://doi.org/10.1007/s11356-020-10813-0

Zhang G, Zhang X, Meng Y et al (2020a) Layered double hydroxides-based photocatalysts and visible-light driven photodegradation of organic pollutants: a review. Chem Eng J 392:123684. https://doi.org/10.1016/j.cej.2019.123684

Zhang H, Xia B, Wang P et al (2020b) From waste to waste treatment: mesoporous magnetic NiFe 2 O 4 /ZnCuCr-layered double hydroxide composite for wastewater treatment. J Alloy Compd 819:153053. https://doi.org/10.1016/j.jallcom.2019.153053

Zhang J, Xia Q, Hong X et al (2020c) Synthesis of layered double hydroxides with nitrate and its adsorption properties of phosphate. Water Sci Technol 83:100–110. https://doi.org/10.2166/wst.2020.567

Zhang J, Zhu Q, Wang L et al (2020d) g-C3 3 N 4 /CoAl-LDH 2D/2D hybrid heterojunction for boosting photocatalytic hydrogen evolution. Int J Hydrog Energy 45:21331–21340. https://doi.org/10.1016/j.ijhydene.2020.05.171

Zhao Y, He J, Dai M et al (2020) Emerging CoMn-LDH@MnO 2 electrode materials assembled using nanosheets for flexible and foldable energy storage devices. J Energy Chem 45:67–73. https://doi.org/10.1016/j.jechem.2019.09.027

Zheng Q, Yang L, Song D et al (2020) High adsorption capacity of Mg–Al-modified biochar for phosphate and its potential for phosphate interception in soil. Chemosphere 259:127469. https://doi.org/10.1016/j.chemosphere.2020.127469

Zhu Y, Ren J, Yang X et al (2017) Interface engineering of 3D BiVO 4 /Fe-based layered double hydroxide core/shell nanostructures for boosting photoelectrochemical water oxidation. J Mater Chem A 5:9952–9959. https://doi.org/10.1039/C7TA02179H

Zhu Y, An S, Sun X et al (2020) Core-branched NiCo 2 S 4 @CoNi-LDH heterostructure as advanced electrode with superior energy storage performance. Chem Eng J 383:123206. https://doi.org/10.1016/j.cej.2019.123206

Zou W, Guo W, Liu X et al (2018) Anion exchange of Ni–Co layered double hydroxide (LDH) nanoarrays for a high-capacitance supercapacitor electrode: a comparison of alkali anion exchange and sulfuration. Chem Eur J 24:19309–19316. https://doi.org/10.1002/chem.201804218

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Jijoe, P.S., Yashas, S.R. & Shivaraju, H.P. Fundamentals, synthesis, characterization and environmental applications of layered double hydroxides: a review. Environ Chem Lett 19 , 2643–2661 (2021). https://doi.org/10.1007/s10311-021-01200-3

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Node-layer duality in networked systems

• Charley Presigny 1 ,
• Marie-Constance Corsi 1 &
• Fabrizio De Vico Fallani   ORCID: orcid.org/0000-0001-8035-7883 1  

Nature Communications volume  15 , Article number:  6038 ( 2024 ) Cite this article

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• Complex networks
• Network models

Real-world networks typically exhibit several aspects, or layers, of interactions among their nodes. By permuting the role of the nodes and the layers, we establish a new criterion to construct the dual of a network. This approach allows to examine connectivity from either a node-centric or layer-centric viewpoint. Through rigorous analytical methods and extensive simulations, we demonstrate that nodewise and layerwise connectivity measure different but related aspects of the same system. Leveraging node-layer duality provides complementary insights, enabling a deeper comprehension of diverse networks across social science, technology and biology. Taken together, these findings reveal previously unappreciated features of complex systems and provide a fresh tool for delving into their structure and dynamics.

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Degree difference: a simple measure to characterize structural heterogeneity in complex networks

More is different in real-world multilayer networks

Uncovering the hidden structure of small-world networks

Introduction.

Duality belongs to noticeable concepts of philosophy, social, and natural systems that refer to different, often antithetic, aspects of the same phenomenon or entity. Duality enhances our understanding of complex systems by highlighting complementary properties with important implications in theoretical studies and real-world applications. For example, in quantum physics elementary particles such as electrons can behave as both discrete elements and continuous waves. In electrical engineering, voltage-current duality allows to transform a circuit problem into an analogous one that may be easier to solve or analyze.

Complex systems are often characterized by many local interactions giving rise to emerging properties that affect the structure and dynamics of the global network 1 , 2 , 3 , 4 . Apart from a few related efforts 5 , 6 , 7 , the question of whether complex networks exhibit non-trivial forms of duality has remained poorly explored. This is partly because in the classical formalism where the nodes are the sole interacting units, it is hard to identify genuinely distinct aspects of the system 8 . Recent developments in multilayer network theory offer a unique opportunity to overcome this limitation.

In a multilayer network, nodes are connected through multiple types of interactions or relationships (layers), creating a more comprehensive representation of the system 9 , 10 , 11 , 12 , 13 . Real-world examples include transportation networks where commuters travel via different modalities and social networks where individuals exchanging information via different technological media 14 , as well as brain networks where neurons interact on different anatomical and functional scales 15 . Hence, in real-world scenarios the nodes constitute the entities of the system and the layers correspond to traits or aspects of those entities.

Many quantities from classical networks have been extended to multilayer networks to unveil non-trivial properties based on how the entities (i.e., nodes) interact 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 . Interestingly, complementary properties could be obtained by looking at how their different aspects (i.e., the layers) are interconnected, too. While recent works have started exploiting layerwise properties to characterize the system clustering 29 , 30 , ranking 31 , 32 , 33 and redundancy 34 , the relationship with the nodewise counterpart is still unclear as well as their complementary role. To fill in this critical gap, we propose a general framework that exploits the intrinsic properties of multilayer networks and offers a dual characterization of the same system.

Multilayer networks are mathematically represented by supra-adjacency matrices whose entry \({X}_{ij}^{\alpha \beta }\) contains the connectivity between the node i at layer α and the node j at layer β 11 , 14 , 28 . Here, the basic interacting unit is not just the node, but the so-called node-layer duplet 26 . By swapping the indices of the nodes and the layers, we define the dual network \({P}^{{\mathsf{T}}}XP=Y\) where P is an opportune permutation matrix 35 , 36 . Since this is a symmetric transformation it is always possible to go back to the primal version \(X=PY{P}^{{\mathsf{T}}}\) . Put simply, networks can be represented as nodes connected through layers or equivalently as layers connected through nodes (Fig.  1 ). At the global level, the information captured by the two representations is the same because node-layer duplets are representation-invariant, i.e., \({X}_{ij}^{\alpha \beta }={Y}_{\alpha \beta }^{ij}\) .

Real networks typically consist of entities (nodes) interacting across different modes or aspects (layers). The unitary element of such multilayer networks is the node-layer duplet. In the left side illustration, a duplet identifies a node i  = 1, 2, 3 and a layer α  = I, II. By opportunely permuting the indices ( \({P}^{{\mathsf{T}}}XP=Y\) ) nodes become layers (blue) and layers become nodes (red) without altering the local connectivity (right side). By construction, replica links become intralayer, intralayer links become replica, and interlayer links stay interlayer. This transformation is symmetric and it is always possible to go back to the primal version \(X=PY{P}^{{\mathsf{T}}}\) . The same system can be, therefore, equivalently represented as entities connected through aspects or aspects connected through entities. Two complementary descriptions can be then obtained depending on the representation side. In the primal nodewise (left), connectivity is integrated across aspects and one looks at how entities are interacting. In the dual layerwise (right), connectivity is integrated across entities and one rather looks instead at how their aspects are interconnected.

Locally, the information captured by the main units in the primal and dual representation is in general different. For example, let us consider the multidegree centrality as a simple intuitive measure of local connectivity 37 . In the primal nodewise representation, here referred as to \({{{{{{\mathcal{X}}}}}}}\) , the nodes are the main units of interest and integrate information across the layers. The multidegree centrality of node i reads \({k}_{{{{{{{\mathcal{X}}}}}}}}^{i}={\sum }_{j\alpha \beta }{X}_{ij}^{\alpha \beta }\) . In the dual layerwise representation \({{{{{{\mathcal{Y}}}}}}}\) , the layers become the main units and integrate information across the nodes. The multidegree centrality of layer α is \({k}_{{{{{{{\mathcal{Y}}}}}}}}^{\alpha }={\sum }_{ij\beta }{Y}_{\alpha \beta }^{ij}\) . In the following, we demonstrate how alterations to the connections within the system are differently perceived by the multidegree centralities in each representation.

Node-layer duality captures complementary network changes

To assess the impact of edge perturbations on both nodewise and layerwise representations, we introduced a model that randomly rewires an arbitrary network, with the type of link rearrangement determined by three parameters. Specifically, the likelihood of rewiring a link while i) keeping the layers unchanged ( p n o d e ), ii) keeping the nodes unchanged ( p l a y e r ), and iii) altering both the nodes and layers, akin to a “teleportation” process ( p t e l ) (Methods) . The total probability associated with the rewiring event satisfies p n o d e  +  p l a y e r  +  p t e l  = 1 (Fig.  2 a).

a Schematic illustration of the different types of edge rewiring. The type of perturbation is determined by the probability to make a link displacement while i ) keeping the layers unaltered ( p n o d e ), ii) keeping the nodes unaltered ( p l a y e r ), iii) altering both nodes and layers ( p t e l ). Dotted lines = old position, solid lines = new position. b Linear relation between layerwise and nodewise global connectivity changes. Global changes are computed as Euclidean distances ( d ) between multidegree centrality vectors. Lower slopes (higher \({d}_{{{{{{{\mathcal{X}}}}}}}}\) ) are obtained for p n o d e  >  p l a y e r . Higher slopes (higher \({d}_{{{{{{{\mathcal{Y}}}}}}}}\) ) are obtained for p l a y e r  >  p n o d e . Solid lines correspond to the theoretical formulas in the case of random networks with N  =  M  = 200, connection density q  = 0.0005, for the entire rewiring range r (color line). Scattered points correspond to synthetic random networks simulated with the same parameters. c Global connectivity changes as a function of rewiring parameters. Nodewise distances ( \({d}_{{{{{{{\mathcal{X}}}}}}}}\) ) increase linearly with p n o d e ( x -axis) and p t e l (white diagonals) but they cannot see edge displacements that keep nodes unvaried ( p l a y e r  → 1). Layerwise distances ( \({d}_{{{{{{{\mathcal{Y}}}}}}}}\) ) increase linearly with p l a y e r ( y -axis) and p t e l (white diagonals) but they are blind to edge displacements that keep layers unvaried ( p n o d e  → 1).

By using a time-discrete Markov chain approach 38 we first demonstrated that the expected value for the multidegree centrality, i.e., the sum of all the links connected to a node i (or a layer α ), after randomly rewiring a percentage r of links, can be explicitly derived from the initial multidegree centrality sequence. For sufficiently sparse and large multilayer networks, a compact form reads:

where μ stands for the average multidegree centrality (Text  S1.4 – 5) . Individual multidegree centralities increase or decrease linearly with the amount of rewiring depending on whether their value is lower or higher than the average. The magnitude of local change is further modulated by the type of rewiring, i.e., 1 −  p l a y e r  =  p n o d e  +  p t e l for nodewise and 1 −  p n o d e  =  p l a y e r  +  p t e l for layerwise (Supplementary Fig.  S1) .

Using Eq. ( 1 ), we analytically derived the expression of the global connectivity change as the Euclidean distance of the multidegree centrality vectors \({k}_{{{{{{{\mathcal{X}}}}}}}}\) and \({k}_{{{{{{{\mathcal{Y}}}}}}}}\) before and after rewiring:

where σ denotes standard deviation of the multidegree centrality (Text  S1.5 , and Supplementary Fig.  S2) . In general, both distances increase with the system size, the heterogeneity of the multidegree centrality, as well as with the type and amount of rewiring. Specifically, \({d}_{{{{{{{\mathcal{X}}}}}}}}\) increases with the probability that the rewiring alters the connected nodes ( p l a y e r  → 0), while \({d}_{{{{{{{\mathcal{Y}}}}}}}}\) increases with the probability that the rewiring alters the connected layers ( p n o d e  → 0).

From Eq. ( 2 ), we also showed that the layerwise distances scale linearly with nodewise distances by a factor that depends only on the intrinsic characteristics of the system \(c=\frac{\sqrt{M}{\sigma }_{{{{{{{\mathcal{Y}}}}}}}}}{\sqrt{N}{\sigma }_{{{{{{{\mathcal{X}}}}}}}}}\) and on the type of rewiring (Fig.  2 b, and Text  S1.7) :

Notably, perturbations that involve different nodes with layers being unaltered ( p n o d e  = 1) are not captured by layerwise distances. Similarly, perturbations involving different layers with nodes kept unvaried ( p l a y e r  = 1) cannot be captured by nodewise distances (Fig.  2 c).

The constant c requires the knowledge of the multidegree centrality standard deviations. In the case of random networks, where links are initially distributed in a random fashion across nodes and layers, standard deviations can be analytically derived in the limit of large systems and it is trivial to prove that c  = 1 (Text  S1.7) . More complex expressions can also be derived in the case of finite-size networks exhibiting power-law multidegree centrality distributions (Text  S1.8) . In practice, for many real-world networks standard deviations can always be calculated from the system under investigation.

Taken together, Eqs. ( 2 ) and ( 3 ) demonstrate the complementarity of nodewise and layerwise connectivity and establish a general form of duality in complex networks based on the structural invariance of node-layer permutation.

Dual classification of real-world multiplex networks

Since real-world multilayer networks have very different numbers of nodes and layers, we first studied the effects of different system sizes on nodewise and layerwise distances. To do so, we considered complete uniform rewirings starting from random multilayer and multiplex networks, the latter exhibiting interlayer connections solely between replica nodes. In this situation, both p n o d e and p l a y e r vanish with the network size leading to \({d}_{{{{{{{\mathcal{Y}}}}}}}}\simeq {d}_{{{{{{{\mathcal{X}}}}}}}}\propto NM\) for multilayers and \({d}_{{{{{{{\mathcal{Y}}}}}}}}\propto \sqrt{2M}N \, > \, {d}_{{{{{{{\mathcal{X}}}}}}}}\propto \sqrt{M}N\) for multiplexes (Texts  S1.7 , S1.9 ).

This means that a very different number of layers and nodes give in general lower distances in random systems. Instead, the largest distances occur when N  =  M in multilayer configurations and \(N=M+\frac{N+M}{3} \, > \, M\) in multiplex ones. Notably in random multiplexes \({d}_{Y}\propto \sqrt{2}{d}_{X}\) by construction, so that their dual layerwise representation a-priori better emphasizes differences with respect to the primal nodewise (Fig.  3 , and Supplementary Fig.  S3) .

In multilayer random networks, the largest global connectivity changes obtained after uniform rewiring occur in both nodewise and layerwise representations when the number of nodes N equal the number of layers M (black lines). However, in multiplex random networks the maximum change, as measured by the Euclidean distance ( d ) between multidegree centrality vectors, is reached when there are more nodes than layers (i.e., N  =  M  + ( N  +  M )/3, Text  S1.10) . In this plot N and M vary in a way to ensure the condition N  +  M  = 200 so that N  =  M  + 200/3. In addition, layerwise distances (orange) are by construction higher than nodewise distances (blue). These findings suggest that layerwise representations might be a-priori better candidates to spout out topological differences in multiplex networks, as compared to standard nodewise (Text  S1.7) . Solid lines correspond to the theoretical formulas for random networks with connection density q  = 0.0005, rewiring ratio r  = 0.5, and uniform rewiring probability (Text  S1.9) .

Next, we evaluated the ability of the node-layer duality to provide a better separation of different real multiplex systems, from transportation and social networks to genetic and neuronal interactions 23 , 24 , 34 , 39 , 40 , 41 , 42 , 43 . Here, we used the explicit formulas to obtain the nodewise and layerwise distances of the real networks from the completely uniform rewired counterparts. This avoided to perform heavy numerical computations for very big systems (e.g., Twitter N  = 88804, M  = 3). Finally, to get rid of possible different network size effects, we normalized the actual distances by those obtained from equivalently rewired random multiplex networks (Texts  S2 ,   S1.4 , S1.7 , and S1.9) .

Results confirm that adding the layerwise representation allows for a better separation of multiplex networks, which would be otherwise indistinguishable by only looking at nodewise distances (Fig.  4 a). Such separation is mainly due to the higher heterogeneity of the layer multidegree centralities as compared to nodewise representation (Tab  S1) . This can be visually appreciated by the different distributions in the projection plots showing how nodes contribute to layers, and viceversa (Fig.  4 b, Methods ).

a Log-log scatter plot of the nodewise ( x -axis) and layerwise ( y -axis) connectivity distances from uniformly rewired counterparts (Text  S1.8) . To avoid network-size biases, all values are further divided by the distances obtained rewiring equivalent random networks. The layer representation enables a better classification of networks that would be otherwise indistinguishable (e.g., Arxiv versus German transport highlighted by dashed circles). Networks optimally group into two clusters almost perfectly matching the spatial (violet) and non-spatial (green) nature of the systems ( k -means=2, Silhouette score = 0.70, Supplementary Fig.  S4) . b Projection plots of two representative multiplex networks (i.e., German transport and Arxiv). In the nodewise, markers correspond to nodes and gray lines to layers. In the layerwise, markers correspond to layers and gray lines to nodes (Methods). Differently from Arxiv (top), the markers in the German network (bottom) tend to accumulate on few main lines meaning that both nodes and layers tend to contribute preferentially to few components. Also, values in the layerwise tend to span larger intervals in comparison with nodewise, indicating the presence of more heterogenous multidegree centrality distributions in the layerwise representation (Text  S2) . The standard deviation of the Arxiv’s layer multidegree centrality ( σ Y  = 8931) is significantly higher than the German transport ( σ Y  = 80.39), and this is eventually reflected by the relative higher layerwise distance d Y in panel a).

Using a k -means clustering algorithm we found that networks tend to optimally separate into two subgroups (Supplementary Fig.  S4) . Those with relatively low \({d}_{{{{{{{\mathcal{X}}}}}}}}\) and \({d}_{{{{{{{\mathcal{Y}}}}}}}}\) values mostly correspond to systems spatially embedded (e.g., German transport, EUAir, C.Elegans connectome). Instead, those with relatively high distances lack of strong spatial connotations (e.g., PierreAuger, Arxiv, Twitter events). While there are few exceptions (e.g., Human microbiome), these results reflect the typical limited node degree heterogeneity of spatial networks due to environmental physical constraints 44 .

Latent brain frequency reorganization in Alzheimer’s disease

We used the node-layer duality framework to identify abnormal network signatures of Alzheimer’s disease (AD) across different brain areas and frequencies of brain activity. To do so, we considered full multilayer brain networks estimated from source-reconstructed magnetoencephalography (MEG) signals in a group of 23 AD human subjects matched with a group of 27 healthy controls HC 45 , 46 ( Methods ). Specifically, we used bispectral coherence to simultaneously infer weighted interactions between regions of interest (the nodes) and between signal frequencies (the layers) 47 .

Results indicate that AD is characterized by functional network disruptions in both nodewise and layerwise representations. In the nodewise, the brain regions implicated in the atrophy process exhibit a loss of multidegree centrality, here computed as the sum of all the weighted links connected to a node. In the layerwise, several brain frequencies within the alpha range (8−13 Hz) in the AD group present reduced multidegree centralities when compared to HC (Fig.  5 a, b).

Multilayer brain networks are inferred from source-reconstructed MEG signals using cross-frequency coupling. In the primal representation, nodes correspond to brain regions ( N  = 70) and layers to different frequency bins ( M  = 77). Both intralayer and interlayer links are provided, estimating the amount of activity interaction (Methods). a Statistical difference (Wilcoxon test, Z-score) between brain region multidegree centralities of AD patients and healthy controls (HC). b Statistiscal difference (Wilcoxon test, Z-score) between brain frequency multidegree centralities of AD patients and healthy controls (HC). c Group-averaged nodewise and layerwise distances between AD and HC for different frequency resolutions (from M  = 77 to M  = 3). The asterisk marks the number of layers ( M ≤31) for which distances become significantly different ( p  < 0.05, FDR corrected). Vertical bars denote standard deviations. d Spearman correlation is computed between the frequency multidegree centralities of AD patients and their cognitive decline scores (MMSE). Colored areas show the significant ranges obtained from a cluster-based permutation procedure for multiple correlations 63 . Darker color p  = 0.0374; lighter color p  = 0.049). The inset spots out the AD patients' MMSE as a function of the multidegree centrality at 9  H z , giving the highest significant correlation R  = 0.601 as indicated by the asterisk. Regressing curves resulting from a square fit are shown for illustrative purposes.

While these local connectivity differences are not statistically significant, we find that globally frequency-wise distances can better discriminate AD patients as compared to region-wise distances. In particular, the coarser the frequency resolution, the more significant the difference between \({d}_{{{{{{{\mathcal{Y}}}}}}}}\) and \({d}_{{{{{{{\mathcal{X}}}}}}}}\) (Fig.  5 c). This behavior does not solely result from the reduction in network size, but instead originates from the greater consistency in connectivity changes within frequencies compared to between frequencies (Supplementary Fig.  S5) .

Finally, we found that both node and layer multidegree centrality decrements are significantly associated with more severe cognitive decline of AD patients as measured by the mini-mental state examination (MMSE) 48 . In the brain space, the most predictive area is the bilateral caudal anterior cingulate cortex, a well-known hub of information processing in the brain that plays an essential role in AD pathophysiology 49 , 50 (Supplementary Table  S2) . Notably, a higher number of stronger correlations emerges in the frequency space. The most significant ones lie within the alpha frequency range, considered one of the most reliable noninvasive functional predictors of AD-related cognitive symptoms 51 , 52 (Fig.  5 d, and Supplementary Table  S3) .

Whether networks express duality, a general concept characterized by the coexistence and interplay of complementary aspects, is poorly understood. Drawing inspiration from recent advances in multilayer network theory, we propose a rigorous framework that reveals dual manifestations of the same system.

We show that edge perturbations that do not alter the local connectivity of the nodes (i.e., p l a y e r  = 1) are only visible in the dual layerwise representation. Conversely, changes that do not alter the local connectivity of the layers (i.e., p n o d e  = 1) are only visible in the primal nodewise representation. This exclusive duality can be relaxed by tuning the type of rewiring thus yielding partial information about nodewise and layerwise aspects simultaneously. For a given p t e l , the more information a particular instance gives about one, the less it will give about the other (Fig.  2 b). The continuous complementarity that emerges is not unique to node-layer duality but can also be observed in other contexts, such as the wave-particle duality relation 53 , 54 .

In general, the amount of measured information increases with the variability of the local connectivity Eq. ( 2 ). Compared to full multilayer configurations, the variability of the layerwise connectivity in multiplex random networks is by construction higher than the nodewise counterpart. This implies that layerwise representations may be more suitable for studying real-world systems for which interlayer connections are frequently lacking. Node-layer duality is not only important from a fundamental perspective but also because it would allow us to describe difficult models in terms of their simpler dual counterparts or improve problem-solving efficiency by equating primal and dual expansions. Furthermore, while these results are obtained using relatively simple network quantities and models, other features can be explored within this framework to enrich the overall characterization and result interpretation.

To demonstrate the significance of our approach, we examine the brain, which can be represented as a multilayer network spanning various scales and levels 15 . In particular, the interaction between neural oscillations at different frequencies allows for coordinating and integrating information across different brain regions and cognitive processes 55 . Understanding disruptions in cross-frequency coupling is therefore crucial to identifying biomarkers and novel treatments in neurological disorders, such as Alzheimer’s disease 56 . The node-layer duality reveals that the cognitive decline of AD patients is better predicted by dual changes among frequencies of brain activity rather than primal changes among regions. Although increasing the sample size, enriching the network information, and using alternative methods can improve the overall prediction in general 57 , 58 , our approach offers the first proof-of-concept accounting for node-layer duality in brain networks. More in general, node-layer duality translates to the ability to uncover fresh insights into modern neuroscience, such as understanding how neurons communicate via parallel frequency channels 59 and elucidating how local brain damages lead to structure-function reorganization 60 . These findings can significantly improve our models of brain functioning during cognitive or motor tasks and contribute to the identification of predictive biomarkers for brain diseases, such as neurodegeneration and stroke recovery.

We hope that the concept of node-layer duality will stimulate fresh investigations of complex interconnected systems, with broad implications in a wide range of disciplines, including network science, systems biology, and social network analysis.

Stochastic rewiring network model

Starting with a network of N nodes, M layers and L links arbitrarily distributed, the algorithm first randomly selects a proportion r ≤ L of edges to rewire. Then, for each selected edge, a new position is randomly drawn based on the relative importance of the rewiring parameters p n o d e , p l a y e r , and p t e l , which determine in a probabilistic fashion the type of edge reassignment. For example, p n o d e  = 0.3, p l a y e r  = 0.6 and p t e l  = 0.1 implies that about 30% of the selected edges will be rewired without altering the initial connected layers, 60% without altering the nodes, and 10% by altering both layers and nodes. During the process, if a new position is already occupied or if it alters the original nature of the network (e.g., multiplex or multilayer), another position is proposed following the same probabilistic rule. Note that when p n o d e  = 1, the layer multidegree centrality sequence is entirely preserved. Similarly, when p l a y e r  = 1, the node multildegree centrality sequence is conserved. More details in Text  S1.1 – 4 .

Projection plots for multilayer networks

To help understand how the nodes contribute to layers and vice versa, we adapted the method proposed by 61 to visualize partitioned networks. To do so, we first compute the contribution matrix C containing the number of links that a node i shares with a layer α . We next consider a typical truncated singular value decomposition (tSVD) C  =  U Σ V † , where Σ is a diagonal matrix containing the singular values, and U and V are respectively the left and right orthogonal matrices associated with the nodewise and layerwise representation. To visualize the nodewise contribution we project the space spanned by the first two left singular vectors, i.e., U (1)- U (2). In this 2D space, the layers are represented as lines whose direction depends on their cohesiveness, i.e., layers that share many links tend to be represented along similar directions. The nodes are instead represented as points. The more the nodes contribute to a specific layer the more they tend to be aligned to its direction. The distance of each point from the origin is finally proportional to its multidegree centrality. Similar visualization can be obtained for the layerwise contributions by projecting the space spanned by the first two right singular vectors, i.e., V (1)- V (2).

Multilayer brain network construction

Multilayer brain networks are obtained from the experimental data published in 46 . We refer to this paper for more detailed descriptions. 23 Alzheimer’s disease (AD) patients and 27 healthy age-matched control (HC) subjects, participated in the study. All participants underwent the Mini-Mental State Examination (MMSE) for global cognition 48 . For each subject, 6 minutes resting-state eyes-closed brain activity was recorded noninvasively using a whole-head MEG system with 102 magnetometers and 204 planar gradiometers (Elekta Neuromag TRIUX MEG system) at a sampling rate of 1000 Hz. Signal artifacts were removed using different techniques, including signal space separation, principal component analysis, and visual inspection. Finally, source-imaging was used to project the signals from the sensor to the source space consisting of N  = 70 regions of interest (ROI) defined by the Lausanne cortical atlas parcellation 62 . Here, we used spectral bicoherence to estimate functional connectivity between ROIs and between frequencies of brain activity 47 . Specifically, we considered M  = 77 layers corresponding to frequencies in the 2−40 Hz range with a resolution of 0.5 Hz. Other parameters were non-overlapping windows of 2s averaged according to the Welch method. We finally symmetrized the resulting supra-adjacency matrices by selecting the highest value of bicoherence between each pair of ROIs and frequencies. The resulting networks are full-multilayer consisting of both intralayer and interlayer connections, including weighted links between replica nodes.

Data availability

The brain network data generated in this study have been deposited in the Zenondo database and freely accessible and usable under the Creative Common license BY 4.0 license ( https://doi.org/10.5281/zenodo.12099874 ).

Code availability

All the code used to generate the results is freely available, documented and usable under the MIT license ( https://github.com/Inria-NERV/multilayer_duality ).

Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’ networks. Nature 393 , 440–442 (1998).

Article   ADS   CAS   PubMed   Google Scholar  

Barabási, A. L. & Albert, R. Emergence of scaling in random networks. Science 286 , 509–512 (1999).

Article   ADS   MathSciNet   PubMed   Google Scholar  

Newman, M. The structure and function of complex networks. SIAM Rev. 45 , 167–256 (2003).

Article   ADS   MathSciNet   Google Scholar  

Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D. U. Complex networks: structure and dynamics. Phys. Rep. 424 , 175–308 (2006).

Campanharo, A. S. L. O., Sirer, M. I., Malmgren, R. D., Ramos, F. M. & Amaral, L. A. N. Duality between time series and networks. PLOS ONE 6 , 1–13 (2011).

Article   Google Scholar  

Kaiser, F., Böttcher, P. C., Ronellenfitsch, H., Latora, V. & Witthaut, D. Dual communities in spatial networks. Nat. Commun. 13 , 7479 (2022).

Article   ADS   CAS   PubMed   PubMed Central   Google Scholar  

Krioukov, D. & Ostilli, M. Duality between equilibrium and growing networks. Phys. Rev. E 88 , 022808 (2013).

Article   ADS   Google Scholar  

Harary, F. et al. Graph Theory (Addison-Wesley Publishing Company, 1969).

Mucha, P. J., Richardson, T., Macon, K., Porter, M. A. & Onnela, J. P. Community structure in time-dependent, multiscale, and multiplex networks. Science 328 , 876–878 (2010).

Article   ADS   MathSciNet   CAS   PubMed   Google Scholar  

Buldyrev, S. V., Parshani, R., Paul, G., Stanley, H. E. & Havlin, S. Catastrophic cascade of failures in interdependent networks. Nature 464 , 1025–1028 (2010).

De Domenico, M. et al. Mathematical formulation of multilayer networks. Phys. Rev. X 3 , 041022 (2013).

Google Scholar  

Gómez, S. et al. Diffusion dynamics on multiplex networks. Phys. Rev. Lett. 110 , 028701 (2013).

Article   ADS   PubMed   Google Scholar  

Bianconi, G. Statistical mechanics of multiplex networks: entropy and overlap. Phys. Rev. E 87 , 062806 (2013).

Boccaletti, S. et al. The structure and dynamics of multilayer networks. Phys. Rep. 544 , 1–122 (2014).

Article   ADS   MathSciNet   CAS   PubMed   PubMed Central   Google Scholar  

Presigny, C. & De Vico Fallani, F. Colloquium: multiscale modeling of brain network organization. Rev. Mod. Phys. 94 , 031002 (2022).

Gao, J., Buldyrev, S. V., Havlin, S. & Stanley, H. E. Robustness of a network of networks. Phys. Rev. Lett. 107 , 195701 (2011).

Liu, X., Stanley, H. E. & Gao, J. Breakdown of interdependent directed networks. Proc. Natl Acad. Sci. 113 , 1138–1143 (2016).

Artime, O. & De Domenico, M. Abrupt transition due to non-local cascade propagation in multiplex systems. N. J. Phys. 22 , 093035 (2020).

Tang, L., Wu, X., Lü, J., Lu, J. & D’Souza, R. M. Master stability functions for complete, intralayer, and interlayer synchronization in multiplex networks of coupled Rössler oscillators. Phys. Rev. E 99 , 012304 (2019).

Danziger, M. M., Bonamassa, I., Boccaletti, S. & Havlin, S. Dynamic interdependence and competition in multilayer networks. Nat. Phys. 15 , 178–185 (2019).

Article   CAS   Google Scholar  

Della Rossa, F. et al. Symmetries and cluster synchronization in multilayer networks. Nat. Commun. 11 , 3179 (2020).

Article   ADS   PubMed   PubMed Central   Google Scholar  

Radicchi, F. & Arenas, A. Abrupt transition in the structural formation of interconnected networks. Nat. Phys. 9 , 717–720 (2013).

De Domenico, M., Nicosia, V., Arenas, A. & Latora, V. Structural reducibility of multilayer networks. Nat. Commun. 6 , 6864 (2015).

Cardillo, A. et al. Emergence of network features from multiplexity. Sci. Rep. 3 , 1344 (2013).

Article   CAS   PubMed   PubMed Central   Google Scholar  

Zanin, M. Can we neglect the multi-layer structure of functional networks? Phys. A: Stat. Mech. its Appl. 430 , 184–192 (2015).

Kivela, M. et al. Multilayer networks. J. Complex Netw. 2 , 203–271 (2014).

Battiston, F., Nicosia, V. & Latora, V. Structural measures for multiplex networks. Phys. Rev. E 89 , 032804 (2014).

Bianconi, G. Multilayer networks: structure and function. first edition (ed) Oxford : (Oxford University Press; 2018).

Stanley, N., Shai, S., Taylor, D. & Mucha, P. J. Clustering network layers with the strata multilayer stochastic block model. IEEE Trans. Netw. Sci. Eng. 3 , 95–105 (2016).

Article   MathSciNet   PubMed   PubMed Central   Google Scholar  

Kao, T. C. & Porter, M. A. Layer communities in multiplex networks. J. Stat. Phys. 173 , 1286–1302 (2018).

Taylor, D., Myers, S. A., Clauset, A., Porter, M. A. & Mucha, P. J. Eigenvector-based centrality measures for temporal networks. Multiscale Modeling Simul. 15 , 537–574 (2017).

Article   MathSciNet   Google Scholar  

Taylor, D., Porter, M. A. & Mucha, P. J. Tunable eigenvector-based centralities for multiplex and temporal networks. Multiscale Modeling Simul. 19 , 113–147 (2021).

Rahmede, C., Iacovacci, J., Arenas, A. & Bianconi, G. Centralities of nodes and influences of layers in large multiplex networks. J. Complex Netw. 6 , 733–752 (2018).

De Domenico, M. & Biamonte, J. Spectral entropies as information-theoretic tools for complex network comparison. Phys. Rev. X 6 , 041062 (2016).

Golub, G. H. & Van Loan, C. F. Matrix Computations 4th edn (Johns Hopkins University Press, 2013).

Kivela, M. & Porter, M. A. Isomorphisms in Multilayer Networks. IEEE Trans. Netw. Sci. Eng. 5 , 198–211 (2018).

Artime, O. et al. Multilayer Network Science: From Cells to Societies. Elements in the Structure and Dynamics of Complex Networks. (Cambridge Univ. Press, 2022)

Newman, M. et al. Networks: An Introduction. (Oxford Univ. Press, 2010).

De Domenico, M., Porter, M. A. & Arenas, A. MuxViz: a tool for multilayer analysis and visualization of networks. J. Complex Netw. 3 , 159–176 (2015).

Bergermann, K. & Stoll, M. Orientations and matrix function-based centralities in multiplex network analysis of urban public transport. Appl. Netw. Sci. 6 , 90 (2021).

De Domenico, M. & Altmann, E. G. Unraveling the origin of social bursts in collective attention. Sci. Rep. 10 , 4629 (2020).

Chami, G.F., Ahnert, S.E., Kabatereine, N.B. & Tukahebwa, E.M. Social network fragmentation and community health. Proc. Natl. Acad. Sci. USA 114 , E7425–E7431 (2017).

De Domenico, M., Lancichinetti, A., Arenas, A. & Rosvall, M. Identifying modular flows on multilayer networks reveals highly overlapping organization in interconnected systems. Phys. Rev. X 5 , 011027 (2015).

Barthélemy, M. Spatial networks. Phys. Rep. 499 , 1–101 (2011).

Desikan, R. S. et al. An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest. NeuroImage 31 , 968–980 (2006).

Article   PubMed   Google Scholar  

Guillon, J. et al. Loss of brain inter-frequency hubs in Alzheimer’s disease. Sci. Rep. 7 , 10879 (2017).

Jirsa, V. & Müller, V. Cross-frequency coupling in real and virtual brain networks. Frontiers in Computational Neuroscience 7 , https://doi.org/10.3389/fncom.2013.00078 (2013).

Folstein, M. F., Folstein, S. E. & McHugh, P. R. Mini-mental state. J. Psychiatr. Res. 12 , 189–198 (1975).

Article   CAS   PubMed   Google Scholar  

Killiany, R. J. et al. Use of structural magnetic resonance imaging to predict who will get Alzheimer’s disease. Ann. Neurol. 47 , 430–439 (2000).

Jones, B. F. et al. Differential regional atrophy of the cingulate gyrus in Alzheimer disease: a volumetric MRI study. Cereb. Cortex 16 , 1701–1708 (2005).

Babiloni, C. et al. Hippocampal volume and cortical sources of EEG alpha rhythms in mild cognitive impairment and Alzheimer disease. NeuroImage 44 , 123–135 (2009).

Smailovic, U. & Jelic, V. Neurophysiological markers of Alzheimer’s disease: quantitative EEG approach. Neurol. Ther. 8 , 37–55 (2019).

Article   PubMed   PubMed Central   Google Scholar  

Wootters, W. K. & Zurek, W. H. Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of Bohr’s principle. Phys. Rev. D. 19 , 473–484 (1979).

Zeilinger, A. Experiment and the foundations of quantum physics. Rev. Mod. Phys. 71 , S288–S297 (1999).

Canolty, R. T. & Knight, R. T. The functional role of cross-frequency coupling. Trends Cogn. Sci. 14 , 506–515 (2010).

Yakubov, B. et al. Cross-frequency coupling in psychiatric disorders: a systematic review. Neurosci. Biobehav. Rev. 138 , 104690 (2022).

Daqing, L., Kosmidis, K., Bunde, A. & Havlin, S. Dimension of spatially embedded networks. Nat. Phys. 7 , 481–484 (2011).

Ziukelis, E. T., Mak, E., Dounavi, M. E., Su, L. & O’Brien, J. T. Fractal dimension of the brain in neurodegenerative disease and dementia: a systematic review. Ageing Res. Rev. 79 , 101651 (2022).

Jensen, O. & Colgin, L. L. Cross-frequency coupling between neuronal oscillations. Trends Cogn. Sci. 11 , 267–269 (2007).

Park, H. J. & Friston, K. Structural and functional brain networks: from connections to cognition. Science 342 , 1238411 (2013).

Arenas, A., Borge-Holthoefer, J., Gómez, S. & Zamora-López, G. Optimal map of the modular structure of complex networks. N. J. Phys. 12 , 053009 (2010).

Hagmann, P. et al. Mapping the structural core of human cerebral cortex. PLOS Biol. 6 , 1–15 (2008).

Han, C. E., Yoo, S. W., Seo, S. W., Na, D. L. & Seong, J. K. Cluster-based statistics for brain connectivity in correlation with behavioral measures. PLOS ONE 8 , e72332 (2013).

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FDVF acknowledges support from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (Grant Agreement No. 864729).

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C.P. conceived the study, performed theoretical modeling and analysis, wrote the paper, and prepared the figures; M.C. performed data processing for the brain network construction; and F.D.V.F. conceived and coordinated the study, wrote the paper, and prepared the figures.

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In situ xps study of methanol oxidation over copper catalyst derived from layered double hydroxides.

Copper nanoparticles supported on alumina have been synthesized from CuAl-layered double hydroxide by heat treatment at 450°C and have been characterized by thermal analysis, X-ray diffraction, transmission electron microscopy, and X-ray photoelectron spectroscopy. Methanol oxidation at different molar ratios of the components of the reaction feed (CH 3 OH:O 2 = 1:1, 3:1, 6:1 and 1:3, Ptotal = 0.012 mbar) over the prepared CuAlO x catalyst has been studied by in situ X-ray photoelectron spectroscopy and mass-spectrometry. It was revealed that methanol oxidation takes place at temperatures higher than 250°C. The conversion of methanol as well as selectivities towards CH 2 O and CO 2 were found to depend on molar ratio of the reaction mixture components and reaction temperature. A quantitative estimation of surface composition for the CuAlO x catalyst under different experimental conditions was performed based on the deconvolution of copper Auger spectra using a linear combination of individual components. It was shown that treatment of the as-loaded CuAlO x catalyst in hydrogen at 300°C led to a partial reduction of Cu + to Cu 0 . However, there were only Cu 0 species on the surface in the CH 3 OH:O 2 = 1:1 reaction mixture at 400°C indicating formation of active sites directly during the catalytic reaction. Metallic copper was shown to be an active component for the production of formaldehyde under the conditions of methanol excess in the reaction mixture.

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“I’d actually been experiencing the postpartum hair [loss],” she said in the clip, adding that someone had recommended she started using the supplement after giving birth to her son, Bronze , in 2022.

“I’m not taking anything right now because I am pregnant, so I do plan to go back to the postpartum [supplement] once I give birth,” she continued.

Brittany, Patrick, and their kids Sterling Skye, 3, and Bronze, 1, coordinated in white and tan outfits as they announced their happy news on Instagram July 12 with a series of sweet family snaps.

However, don’t expect another pregnancy hair chop from the busy mom.

“I’m done. I’ll say that. I said three and I’m done,” Patrick said at a press conference last week .

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