Efficient Prediction of Refractive Index and Abbe Number in Polymers Using Density Functional Theory

Lu-Kun Feng; Ai-Wei Zhang; Guo-Hua Huang; Cai-Zhen Zhu; Ming-Liang Wang; Jian Xu

doi:10.1007/s10118-025-3353-3

Your Location：

Home >

Browse articles >

Efficient Prediction of Refractive Index and Abbe Number in Polymers Using Density Functional Theory

RESEARCH ARTICLE | Updated：2025-07-26

- Efficient Prediction of Refractive Index and Abbe Number in Polymers Using Density Functional Theory
- Chinese Journal of Polymer Science Vol. 43, Issue 8, Pages: 1468-1482(2025)
- Affiliations：
  
  College of Chemistry and Environmental Engineering, Shenzhen University, Shenzhen 518055, China
- Author bio：
  
  czzhu@szu.edu.cn (C.Z.Z.)
  wangml@szu.edu.cn (M.L.W.)
- Funds：
- DOI：10.1007/s10118-025-3353-3
  CLC：
- Received：13 February 2025，
  
  Revised：24 March 2025，
  
  Accepted：08 April 2025，
  
  Published Online：04 June 2025，
  
  Published：01 August 2025
- Accepted：
Scan QR Code
Feng, L. K.; Zhang, A. W.; Huang, G. H.; Zhu, C. Z.; Wang, M. L.; Xu, J. Efficient prediction of refractive index and abbe number in polymers using density functional theory. Chinese J. Polym. Sci. 2025, 43, 1468–1482

Lu-Kun Feng, Ai-Wei Zhang, Guo-Hua Huang, et al. Efficient Prediction of Refractive Index and Abbe Number in Polymers Using Density Functional Theory[J]. Chinese journal of polymer science, 2025, 43(8): 1468-1482.
Feng, L. K.; Zhang, A. W.; Huang, G. H.; Zhu, C. Z.; Wang, M. L.; Xu, J. Efficient prediction of refractive index and abbe number in polymers using density functional theory. Chinese J. Polym. Sci. 2025, 43, 1468–1482 DOI： 10.1007/s10118-025-3353-3.

Lu-Kun Feng, Ai-Wei Zhang, Guo-Hua Huang, et al. Efficient Prediction of Refractive Index and Abbe Number in Polymers Using Density Functional Theory[J]. Chinese journal of polymer science, 2025, 43(8): 1468-1482. DOI： 10.1007/s10118-025-3353-3.

摘要

English Summary: This study employs coupled perturbed Density Functional Theory (DFT) to efficiently predict polymer refractive indices (RI) and Abbe numbers. By calculating frequency-dependent polarizability and molecular volume

and applying linear corrections based on polymer classes

the method achieves high correlation with experimental data.

Abstract

Polymer optical materials are becoming increasingly important in modern technologies owing to their unique properties. This study applies coupled perturbed density functional theory (DFT) to predict the refractive index (RI) and Abbe number of polymers. Using the Lorentz-Lorenz equation

the frequency-dependent polarizability and molecular volume were calculated to estimate RI. Wavelength-dependent RI values were used to derive the Abbe numbers. Our results show a strong correlation with experimental data

with Pearson coefficients of 0.912 for RI and 0.968 for Abbe number

enabling the introduction of linear correction functions to minimize discrepancies between theoretical predictions and experimental results. By categorizing polymers into classes such as poly(methyl methacrylate) (PMMA)-

polyethylene (PE)-

polycarbonate (PC)-

polyimide (PI)-

and polyurethane (PU)-based materials

this method enables precise predictions and reduces discrepancies using linear correction functions. This efficient and direct computational framework avoids the complexity of traditional models and offers a practical tool for the design and optimization of advanced optical materials.

关键词

Keywords

references

Higashihara, T.; U eda, M. Recent progress in high refractive index polymers. Macromolecules 2015 , 48 , 1915−1929..

Kleine, T. S.; Glass, R. S.; Lichtenberger, D. L.; Mackay, M. E.; Char, K.; Norwood, R. A.; Pyun, J. 100th anniversary of macromolecular science viewpoint: high refractive index polymers from elemental sulfur for infrared thermal imaging and optics. ACS Macro Lett . 2020, 9 , 245-259..

Liu, J.-g.; Ueda, M. High refractive index polymers: fundamental research and practical applications. J. Mater. Chem. 2009 , 19 , 8907−8919..

Suzuki, Y.; Higashihara, T.; Ando, S.; Ueda, M. Synthesis of high refractive index poly(thioether sulfone)s with high Abbe's number derived from 2,5-is(sulfanylmethyl)-l,4-dithiane. Polym. J. 2009 , 41 , 860−865..

Kim, K.-C. Effective graded refractive-index anti-reflection coating for high refractive-index polymer ophthalmic lenses. Mater. Lett. 2015 , 160 , 158−161..

Suwa, M.; Niwa, H.; Tomikawa, M. High refractive index pos itive tone photo-sensitive coating. J. Photopolym. Sci. Tec. 2006 , 19 , 275−276..

Nakamura, T.; Fujii, H.; Juni, N.; Tsutsumi, N. Enhanced coupling of light from organic electroluminescent device using diffusive particle dispersed high refractive index resin substrate. Opt. Rev. 2006 , 13 , 104−110..

Zhang, J.; Bai, T.; Liu, W.; Li, M.; Zang, Q.; Ye, C.; Sun, J. Z.; Shi, Y.; Ling, J.; Qin, A.; Tang, B. Z. All-organic polymeric materials with high refractive index and excellent transparency. Nat. Commun. 2023 , 14 , 3524..

Sanders, D. P. Advances in patterning materials for 193 nm immersion lithography. Chem. Rev. 2010 , 110 , 321−360..

Hecht, E. Optics . Pearson Education Inc., Boston, 2017 ..

Grothe, J.; Kaskel, S.; Leuteritz, A., Nanocomposites and hybrid materials. in Polymer Science: A Comprehensive Reference , Matyjaszewski, K.; Möller, M., eds. Elsevier: Amsterdam, 2012 , p. 17 7−209..

Afzal, M. A. F.; Cheng, C.; Hachmann, J. Combining first-principles and data modeling for the accurate prediction of the refractive index of organic polymers. J. Chem. Phys. 2018 , 148 , 241712..

Afzal, M. A. F.; Hachmann, J. Benchmarking DFT approaches for the calculation of polarizability inputs for refractive index predictions in organic polymers. Phys. Chem. Chem. Phys. 2019 , 21 , 4452−4460..

Mavila, S.; Sinha, J.; Hu, Y.; Podgórski, M.; Shah, P. K.; Bowman, C. N. High refractive index photopolymers by thiol–yne “click” polymerization. ACS Appl. Mater. Interfaces 2021 , 13 , 15647−15658..

Badur, T.; Dams, C.; Hampp, N. High refractive index polymers by design. Macromolecules 2018 , 51 , 4220−4228..

Watanabe, S.; Takayama, T.; Oyaizu, K. Transcending the trade-off in refractive index and Abbe number for highly refractive polymers: synergistic effect of polarizable skeletons and robust hydrogen bonds. ACS Polym. Au 2022 , 2 , 458−466..

Holder, A. J.; Ye, L.; Eick, J. D.; Chappelow, C. C. A quantum-mechanical QSAR model to predict the refractive index of polymer matrices. QSAR Comb. Sci. 2006 , 25 , 905−911..

Yu, X.; Yi, B.; Wang, X. Prediction of refractive index of vinyl polymers by using density functional theory. J. Comput. Chem. 2007 , 28 , 2336−2341..

Park, S. S.; Lee, S.; Bae, J. Y.; Hagelberg, F. Refractive indices of liquid-forming organic compounds by density functional theory. Chem. Phys. Lett. 2011 , 511 , 466−470..

Redmond, H.; Thompson, J. E. Evaluation of a quantitative structure–property relationship (QSPR) for predicting mid-visible refractive index of secondary organic aerosol (SOA). Phys. Chem. Chem. Phys. 2011 , 13 , 6872−6882..

Alexandridis, A.; Chondrodima, E.; Moutzouris, K.; Triantis, D. A neural network approach for the prediction of the refractive index based on experimental data. J. Mater. Sci. 2012 , 47 , 883−891..

Sharma, V.; Wang, C.; Lorenzini, R. G.; Ma, R.; Zhu, Q.; Sinkovits, D. W.; Pilania, G.; Oganov, A. R.; Kumar, S.; Sotzing, G. A.; Boggs, S. A.; Ramprasad, R. Rational design of all organic polymer dielectrics. Nat. Commun. 2014 , 5 , 4845..

Duchowicz, P. R.; Fioressi, S. E.; Bacelo, D. E.; Saavedra, L. M.; Toropova, A. P.; Toropov, A. A. QSPR studies on refractive indices of structurally heterogeneous polymers. Chemometr. Intell. Lab. 2015 , 140 , 86−91..

Huan, T. D.; Mannodi-Kanakkithodi, A.; Kim, C.; Sharma, V.; Pilania, G.; Ramprasad, R. A polymer dataset for accelerated property prediction and design. Sci. Data 2016 , 3 , 160012..

Mannodi-Kanakkithodi, A.; Pilania, G.; Huan, T. D.; Lookman, T.; Ramprasad, R. Machine learning strategy for accelerated design of polymer dielectrics. Sci. Rep. 2016 , 6 , 20952..

Jabeen, F.; Chen, M.; Rasulev, B.; Ossowski, M.; Boudjouk, P. Refractive indices of diverse data set of polymers: a computational QSPR based study. Comput. Mater. Sci. 2017 , 137 , 215−224..

Venkatraman, V.; Alsberg, B. K. Designing high-refractive index polymers using materials informatics. Polymers 2018 , 10 , 103..

Afzal, M. A. F.; Haghighatlari, M.; Ganesh, S. P.; Cheng, C.; Hachmann, J. Accelerated discovery of high-refractive-index polyimides via first-principles molecular modeling, virtual high-throughput screening, and data mining. J. Phys. Chem. C 2019 , 123 , 14610−14618..

Lightstone, J. P.; Chen, L.; Kim, C.; Batra, R.; Ramprasad, R. Refractive index prediction models for polymers using machine learning. J. Appl. Phys. 2020 , 127 , 215105..

Ando, S. Efficient hybrid functional and basis set functions for DFT calculation of refractive indices and Abbe numbers of organic compounds. Chem. Lett. 2018 , 47 , 1494−1497..

Lorentz, H. A. Über die Beziehungzwischen der fortpflanzungsgeschwindigkeit des lichtes der körperdichte. Ann. Phys . 1880, 245 , 641-665..

Lorenz, L. Über die refractionsconstante. Ann. Phys . 1880, 247 , 70-103..

Rice, J. E.; Handy, N. C. The calculation of frequency-dependent polarizabilities as pseudo-energy derivatives. J. Chem. Phys. 1991 , 94 , 4959−4971..

Rice, J. E.; Handy, N. C. The calculation of frequency-dependent hyperpolarizabilities including electron correlation effects. Int. J. Quantum Chem. 1992 , 43 , 91−118..

Tomasi, J.; Mennucci, B.; Cammi, R. Quantum mechanical continuum solvation models. Chem. Rev. 2005 , 105 , 2999−3094..

Scalmani, G.; Frisch, M. J. Continuous surface charge polarizable continuum models of solvation. I. General formalism. J. Chem. Phys. 2010 , 132 , 114110..

Tang, Z.; Chang, C.; Bao, F.; Tian, L.; Liu, H.; Wang, M.; Zhu, C.; Xu, J. Feasibility of predicting static dielectric constants of polymer materials: a density functional theory method. Polymers 2021 , 13 , 284..

Lin, Y.; Feng, L.; Li, Y.; Chang, C.; Zhu, C.; Wang, M.; Xu, J. Dielectric constant calculation of poly(vinylidene fluoride) based on finite field and density functional theory. Chinese J. Polym. Sci. 2024 , 42 , 655−662..

Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Petersson, G. A.; Nakatsuji, H.; Li, X.; Caricato, M.; Marenich, A. V.; Bloino, J.; Janesko, B. G.; Gomperts, R.; Mennucci, B.; Hratchian, H. P.; Ortiz, J. V.; Izmaylov, A. F.; Sonnenberg, J. L.; Williams; Ding, F.; Lipparini, F.; Egidi, F.; Goings, J.; Peng, B.; Petrone, A.; Henderson, T.; Ranasinghe, D.; Zakrzewski, V. G.; Gao, J.; Rega, N.; Zheng, G.; Liang, W.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Throssell, K.; Montgomery Jr., J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M. J.; Heyd, J. J.; Brothers, E. N.; Kudin, K. N.; Staroverov, V. N.; Keith, T. A.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A. P.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Millam, J. M.; Klene, M.; Adamo, C.; Cammi, R.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Farkas, O.; Foresman, J. B.; Fox, D. J. Gaussian 16 Rev. B.01 , Wallingford, CT, 2016.

Becke, A. D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 1988 , 38 , 3098−3100..

Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 1988 , 37 , 785−789..

Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 1993 , 98 , 5648−5652..

Foresman, J.; Frisch, A. Exploring chemistry with electronic structure methods . 2nd ed.; Gaussian Inc.: Pittsburgh, PA, 1996 ..

Kingslake, R.; Barry Johnson, R., The work of the lens designer. In lens design fundamentals , 2 nd ed.; Kingslake, R.; Barry Johnson, R., Eds. Academic Press: Boston, 2010 ; pp 1−23..

Okutsu, R.; Ando, S.; Ueda, M. Sulfur-containing poly(meth)acrylates with high refractive indices and high Abbe’s numbers. Chem. Mater. 2008 , 20 , 4017−4023..

Liu, H.; Li, G.; Cao, Y. Synthesis and polymerization of monomer DCPMA for optical resin. China Plast. Ind. 2007 , 35 , 1−3..

Minns, R. A.; Gaudiana, R. A. Design and synthesis of high refractive index polymers. II. J. Macromol. Sci. A 1992 , 29 , 19−30..

Tokushita, Y.; Furuya, S.; Nobe, S.; Nakabayashi, K.; Samitsu, S.; Mori, H. Preparation of highly transparent poly(meth)acrylates with enhanced refractive indices by radical (co)polymerization of seleno(meth)acrylates. Polymer 2021 , 237 , 124346..

Tang, Y. Molecular design, synthesis and application of high refractive index optical resin . Thesis, Beijing University of Chemical Technology, 2019 ..

El-Dessouky, H. M.; Lawrence, C. A.; Voice, A. M.; Lewis, E. L. V.; Ward, I. M. An interferometric prediction of the intrinsic optical properties for cold-drawn i PP, PTFE and PVDF fibres. J. Opt. A: Pure Appl. Opt. 2007 , 9 , 1041..

Kaino, T., Linear optical properties of organic solids. in organic molecular solids: properties and applications , 1st ed.; Jones, W., Ed. CRC Press: Boca Raton, 1997 ; pp 201−241..

Simril, V. L.; Curry, B. A. The properties of polyvinyl fluoride film. J. Appl. Polym. Sci. 1960 , 4 , 62−68..

Huo, N.; Tenhaeff, W. E. High refractive index polymer thin films by charge-transfer complexation. Macromolecules 2023 , 56 , 2113−2122..

Suri, G.; Jha, G. S.; Seshadri, G.; Khandal, R. K. Modification of low refractive index polycarbonate for high refractive index applications. Int. J. Polym. Sci. 2009 , 2009 , 836819..

Li, Z.; Wang, H.; Yan, H.; Wang, H.; Zhang, Z.; Zhang, Y.; Chu, J.; Huo, F.; Xu, F. Design and synthesis of optical biobased polycarbonates with high refractive index and low birefringence. Ind. Eng. Chem. Res. 2024 , 63 , 3975−3985..

Seto, R.; Sato, T.; Kojima, T.; Hosokawa, K.; K oyama, Y.; Konishi, G.-I.; Takata, T. 9,9′-Spirobifluorene-containing polycarbonates: transparent polymers with high refractive index and low birefringence. J. Polym. Sci. A Polym. Chem . 2010, 48 , 3658-3667..

Li, C.; Long, X.; Sun, T.; Wang, Q.; Wang, G. Study on the preparation and optical properties of co-polycarbonates based on binaphthalene and cardo structures. ChemistrySelect 2023 , 8 , e202204829..

Kato, N.; Ikeda, S.; Hirakawa, M.; Ito, H. Correlation of the Abbe number, the refractive index, and glass transition temperature to the degree of polymerization of norbornane in polycarbonate polymers. Polymers 2020 , 12 , 2484..

Yue, T. J.; Ren, B. H.; Zhang, W. J.; Lu, X. B.; Ren, W.-M.; Darensbourg, D. J. Randomly distributed sulfur atoms in the main chains of CO 2 -based polycarbonates: enhanced optical properties. Angew. Chem. Int. Ed. 2021 , 60 , 4315−4321..

Park, T.; Ali, S. N.; You, J.; Kim, B.; Kim, E. Synthesis and preparation of fluorinated polycarbonates enhancing the light absorption of fluorescent nanoparticles. J. Nanosci. Nanotechnol. 2013 , 13 , 6130−6135..

Refractive index of polymers by index. Scientific Polymer Products Inc.: https://scipoly.com/technical-library/refractive-index-of-polymers-by-index/, accessed on April 4, 2024.

Hasegawa, T.; Koyama, Y.; Seto, R.; Kojima, T.; Hosokawa, K.; Takata, T. Diphenolic 9,9-diarylfluorene trimers and derivatives possessing flexible alkylene chain spacers: synthesis of the monomers, their polymerization, and properties of the resulting polymers. Macromolecules 2010 , 43 , 131−136..

Nakazono, K.; Yamashita, C.; Ogawa, T.; Iguchi, H.; Takata, T. Synthesis and properties of pendant fluorene moiety-tethered aliphatic polycarbonates. Polym. J. 2015 , 47 , 355−361..

Chen, Z.; Yang, J. L.; Lu, X. Y.; Hu, L. F.; Cao, X. H.; Wu, G. P.; Zhang, X. H. Triethyl borane-regulated selective production of polycarbonates and cyclic carbonates for the coupling reaction of CO2 with epoxides. Polym. Chem. 2019 , 10 , 3621−3628..

Sato, S.; Ose, T.; Miyata, S.; Kanehashi, S.; Ito, H.; Matsumoto, S.; Iwai, Y.; Matsumoto, H.; Nagai, K. Relationship between the gas transport properties and the refractive in dex in high-free-volume fluorine-containing polyimide membranes. J. Appl. Polym. Sci. 2011 , 121 , 2794−2803..

Liu, J.-g.; Nakamura, Y.; Suzuki, Y.; Shibasaki, Y.; Ando, S.; Ueda, M. Highly refractive and transparent polyimides derived from 4,4‘-[m-sulfonylbis(phenylenesulfanyl) ] diphthalic anhydride and various sulfur-containing aromatic diamines. Macromolecules 2007 , 40 , 7902−7909..

Ando, S.; Watanabe, Y.; Matsuura, T. Wavelength dependence of refractive indices of polyimides in visible and rear-IR regions. Jpn. J. Appl. Phys. 2002 , 41 , 5254−5258..

Liu, J.-g.; Nakamura, Y.; Shibasaki, Y.; Ando, S.; Ueda, M. Synthesis and characterization of high refractive index polyimides derived from 4,4′-( p -phenylenedisulfanyl)dianiline and various aromatic tetracarboxylic dianhydrides. Polym. J. 2007 , 39 , 543−550..

Fukuzaki, N.; Higashihara, T.; Ando, S.; Ueda, M. Synthesis and characterization of highly refractive polyimides derived from thiophene-containing aromatic diamines and aromatic dianhydrides. Macromolecules 2010 , 43 , 1836−1843..

Tapaswi, P. K.; Choi, M. C.; Jeong, K. M.; Ando, S.; Ha, C. S. Transparent aromatic polyimides derived from thiophenyl-substituted benzidines with high refractive index and small birefringence. Macromolecules 2015 , 48 , 3462−3474..

Li, Q.; Liu, S.; Xu, M.; Pan, X.; Li, N.; Zhu, J.; Zhu, X. Selenide-containing soluble polyimides: high refractive index and redox responsiveness. Eur. Polym. J. 2020 , 122 , 109358..

Lee, C.; Seo, J.; Shul, Y.; Han, H. Optical properties of polyimide thin films. effect of chemical structure and morphology. Polym. J. 2003 , 35 , 578−585..

Huang, T. T.; Tsai, C. L.; Tateyama, S.; Kaneko, T.; Liou, G.-S. Highly transparent and flexible bio-based polyimide/TiO 2 and ZrO 2 hybrid films with tunable refractive index, Abbe number, and memory properties. Nanoscale 2016 , 8 , 12793−12802..

Tsai, C.-L.; Liou, G.-S. Highly transparent and flexible polyimide/ZrO2 nanocomposite optical films with a tunable refractive index and Abbe number. Chem. Commun. 2015 , 51 , 13523−13526..

Lopez de Pariza,X.; Fanlo, P.; Polo Fonseca, L.; Ruiz de Luzuriaga, A.; Sardon, H. Polythiourethanes: synthesis, applications, and opportunities. Prog. Polym. Sci. 2023 , 145 , 101735..

Shen, F.; Wang, K.; Yan, D.; Zhang, Z.; Guan, J.; Zhang, K.; Dai, Z. Progress on sulfur-containing resin optical material with high refractive index. Chem. World 2016 , 57 , 457−464..

Tan, J. Preparation and properties of optical polymeric materials and ZnS nanoparticles . Jiangsu University, 2013 ..

Zhang, A. Synthesis and properties of polyurethane optical plastics . Thesis, Zhejiang University, 2003 ..

Kanemura, Y.; Sasagawa, K.; Imai, M.; Suzuki, T. 1991 , US5059673A.

Shin, J. H.; Myung, J. H.; Han, H.H.; Shim, J. M. 2019 , KR101961941B1.

Jia, Y.; Shi, B.; Jin, J.; Li, J. High refractive index polythiourethane networks with high mechanical property via thiol-isocyanate click reaction. Polymer 2019 , 180 , 121746..

Zhang, Y.; Wang, Y.; Chen, Y.; Yang, Z.; Chen, M.; Qin, Z. High-refractive index polythiourethane resin based on 2,3-bis((2-mercaptoethyl) thio)-1-propanethiol and 1,3-bis(isocyanantomethyl) cyclohexane using tertiary amine catalyst. J. Appl. Polym. Sci. 2021 , 138 , 50278..

Hong, S.-M.; Kim, O. Y.; Hwang, S.-H. Chemistry of polythiols and their industrial applications. Materials 2024 , 17 , 1343..

Chen, Y.; Qin, Z.; Tang, G.; Wei, L.; Du, H.; Du, W. Balancing optical property and enhancing stability for high-refractive index polythiourethane with assistance of cubic thiol-functionalized silsesquioxanes. ACS Appl. Polym. Mater. 2021 , 3 , 153−161..

Li, Q.; Zhou, H.; Wicks, D. A.; Hoyle, C. E.; Magers, D. H.; McAlexander, H. R. Comparison of small molecule and polymeric urethanes, thiourethanes, and dithiourethanes: hydrogen bonding and thermal, physical, and mechanical properties. Macromolecules 2009 , 42 , 1824−1833..

Shin, J.; Matsushima, H.; Comer, C. M.; Bowman, C. N.; Hoyle, C. E. Thiol-isocyanate-ene ternary networks by sequential and simultaneous thiol click reactions. Chem. Mater. 2010 , 22 , 2616−2625..

Okubo, T.; Kohmoto, S.; Yamamoto, M. Synthesis, characterization, and optical properties of polymers comprising 1,4-dithiane-2,5-bis(thiomethyl) group. J. Appl. Polym. Sci. 1998 , 68 , 1791−1799..

Teruyuki, N.; Koju, O.; Tohru, M. 1992 , US5084545A.

Views

Downloads

CSCD

Alert me when the article has been cited

Submit

Tools

Publicity Resources

No data

Related Author

No data

Related Institution

No data

⁰