Morphological characterization of the diblock copolymer problem with topological computation
Our subject is the diblock copolymer problem in a three-dimensional space. Using numerical simulations, the double gyroid and orthorhombic morphologies are obtained as energy minimizers. By investigating the geometric properties of these bicontinuous morphologies, we demonstrate the underlying mecha...
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| Published in | Japan journal of industrial and applied mathematics Vol. 27; no. 2; pp. 175 - 190 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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01.09.2010
Springer Nature B.V |
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| Abstract | Our subject is the diblock copolymer problem in a three-dimensional space. Using numerical simulations, the double gyroid and orthorhombic morphologies are obtained as energy minimizers. By investigating the geometric properties of these bicontinuous morphologies, we demonstrate the underlying mechanism affecting the triply periodic energy minimizers in terms of a balanced scaling law. We also apply computational homology to their characterization during the dynamics of morphology transition. Our topological approaches detect the morphology of transient perforated layers as they transition from layers to cylinders, and the
t
−1
law of the Betti number in the phase ordering process. |
|---|---|
| AbstractList | Our subject is the diblock copolymer problem in a three-dimensional space. Using numerical simulations, the double gyroid and orthorhombic morphologies are obtained as energy minimizers. By investigating the geometric properties of these bicontinuous morphologies, we demonstrate the underlying mechanism affecting the triply periodic energy minimizers in terms of a balanced scaling law. We also apply computational homology to their characterization during the dynamics of morphology transition. Our topological approaches detect the morphology of transient perforated layers as they transition from layers to cylinders, and the
t
−1
law of the Betti number in the phase ordering process. Our subject is the diblock copolymer problem in a three-dimensional space. Using numerical simulations, the double gyroid and orthorhombic morphologies are obtained as energy minimizers. By investigating the geometric properties of these bicontinuous morphologies, we demonstrate the underlying mechanism affecting the triply periodic energy minimizers in terms of a balanced scaling law. We also apply computational homology to their characterization during the dynamics of morphology transition. Our topological approaches detect the morphology of transient perforated layers as they transition from layers to cylinders, and the t ^sup -1^ law of the Betti number in the phase ordering process.[PUBLICATION ABSTRACT] |
| Author | Teramoto, Takashi Nishiura, Yasumasa |
| Author_xml | – sequence: 1 givenname: Takashi surname: Teramoto fullname: Teramoto, Takashi email: teramoto@photon.chitose.ac.jp organization: Chitose Institute of Science and Technology – sequence: 2 givenname: Yasumasa surname: Nishiura fullname: Nishiura, Yasumasa organization: Research Institute of Electronic Science, Hokkaido University |
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| Cites_doi | 10.1016/j.actamat.2004.10.022 10.1006/jcis.1996.4720 10.1103/PhysRevE.54.5012 10.1137/S0036141098348176 10.1023/A:1025722804873 10.1090/mmono/209 10.1038/334598a0 10.1016/j.physd.2008.12.009 10.1007/b97315 10.1093/acprof:oso/9780198507840.001.0001 10.1063/1.882522 10.1088/0953-8984/15/26/101 10.1007/s00205-007-0050-z 10.4171/IFB/148 10.1103/PhysRevE.77.056704 10.1021/ma00164a028 10.1021/ma011716x 10.1021/ma00093a006 10.1080/08927020601052914 10.1143/JPSJ.71.1611 10.1103/PhysRevA.41.6763 10.1016/0167-2789(95)00005-O |
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| References | A Collection Papers of T. Hashimoto: ordered structures, order–disorder transition, and physical properties of block copolymers , Dept. of Polym. Chem., Kyoto University (1995) TeramotoT.NishiuraY.Double gyroid morphology in a gradient system with nonlocal effectsJ. Phys. Soc. Jpn.2002711611161410.1143/JPSJ.71.1611 Ishimura, N., Ishiwata, T., Skajo, T., Sakurai, T., Nagayama, M., Nara, T., Hayami, K., Furihata, D., Matsuo, T.: Computational homology and its applications, Hokkaido University Technical Report Series in Mathematics, No. 124 (in Japanese) (2007) ChenX.OshitaY.Applications of modular functions to interfacial dynamicsArch. Rat. Mech. Anal.20071861091321147.7402410.1007/s00205-007-0050-z2338353 HajdukD.A.HarperP.E.GrunerS.M.HonekerC.C.KimG.ThomasE.L.The gyroid: a new equilibrium morphology in weakly segregated diblock copolymersMacromolecules1994274063407510.1021/ma00093a006 GameiroM.MischaikowK.WannerT.Evolution of pattern complexity in the Cahn-Hilliard theory of phase separationActa Materialia20055369370410.1016/j.actamat.2004.10.022 ChoksiR.RenX.On the derivation of a density functional theory for microphase separation of diblock copolymersJ. Stat. Phys.20031131511761034.8203710.1023/A:10257228048732012976 BahianaM.OonoY.Cell dynamical system approach to block copolymersPhys. Rev. A1990416763677110.1103/PhysRevA.41.6763 KangX.RenX.Ring pattern solutions of a free boundary problem in diblock copolymer morphologyPhysica D20092386456651160.3738610.1016/j.physd.2008.12.0092590243 OhtaT.KawasakiT.Equilibrium morphology of block copolymer meltsMacromolecules1986192621263210.1021/ma00164a028 SpivakM.A Comprehensive Introduction to Differential Geometry1979BerkleyPublish or Perish ThomasE.L.AndersonD.M.HenkeeC.S.HoffmanD.Periodic area-minimizing surfaces in block copolymersNature198833459860110.1038/334598a0 KaczynskiT.MischaikowK.MrozekM.Computational Homology2004New YorkSpringer1039.55001 Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation in computational and mathematical models of microstructual evolution. In: Bullard, J.W., Kalita, R., Stoneham, M., Chen, L.-Q. (eds.) (MRS, 1998) Grosse-BrauckmannK.On gyroid interfacesJ. Colloid Interface Sci.199718741842810.1006/jcis.1996.4720 Goźd́źW.T.HołystR.Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsionsPhys. Rev. E1996545012502710.1103/PhysRevE.54.5012 RenX.WeiJ.On the multiplicity of two nonlocal variational problemsSIAM J. Math. Anal.2000319099240973.4900710.1137/S00361410983481761752422 ChoksiR.SternbergP.Periodic phase separation: the periodic Cahn-Hilliard and isoperimetric problemInterfaces Free Boundaries200683713921109.3509210.4171/IFB/1482273234 BaileyT.S.HardyC.M.EppsT.H.IIIBatesF.S.A noncubic triply periodic network morphology in poly(isoprene-b-styrene-b-ethylene oxide) triblock copolymersMacromolecules2002357007701710.1021/ma011716x Nishiura, Y.: Far-from-equilibrium dynamics, Translations of Mathematical Monographs Vol. 209, AMS (2002) TeramotoT.Nosé thermostat for the pattern formation dynamicsMol. Sim.200733717510.1080/08927020601052914 BraidesA.Γ-Convergence for Beginners2002OxfordOxford University Press0186593910.1093/acprof:oso/9780198507840.001.0001 BatesF.S.FredricksonG.H.Block copolymers—designer soft materialsPhys Today1999522323810.1063/1.882522 NonomuraM.YamadaK.OhtaT.Formation and stability of double gyroid in microphase-separated diblock copolymersJ. Phys. Condens. Matter200315L423L43010.1088/0953-8984/15/26/101 AksimentievA.FiałkowskiM.HołystR.Morphology of surfaces in polymer, surfactant, electron and reaction-diffusion systems: methods, measurements and simulationsAdv. Chem. Phys2002121143239 HagitaK.TeramotoT.Topological validation of morphology modeling by extended reverse Monte Carlo analysisPhys. Rev. E20087705670410.1103/PhysRevE.77.056704 NishiuraY.OhnishiI.Some mathematical aspects of the micro-phase separation in diblock copolymersPhysica D19958431391194.8206910.1016/0167-2789(95)00005-O1334695 Langer, J.S.: An introduction to the kinetics of first-order phase transitions, solids far from equilibrium. In: Godreche, G. (ed.). Cambridge University Press, Cambridge (1992) R. Choksi (14_CR9) 2006; 8 E.L. Thomas (14_CR28) 1988; 334 M. Bahiana (14_CR3) 1990; 41 K. Grosse-Brauckmann (14_CR13) 1997; 187 M. Gameiro (14_CR11) 2005; 53 M. Nonomura (14_CR22) 2003; 15 X. Kang (14_CR18) 2009; 238 14_CR10 T. Teramoto (14_CR26) 2007; 33 14_CR19 14_CR16 T. Teramoto (14_CR27) 2002; 71 W.T. Goźd́ź (14_CR12) 1996; 54 M. Spivak (14_CR25) 1979 14_CR1 R. Choksi (14_CR8) 2003; 113 K. Hagita (14_CR14) 2008; 77 T. Kaczynski (14_CR17) 2004 A. Braides (14_CR6) 2002 X. Ren (14_CR24) 2000; 31 D.A. Hajduk (14_CR15) 1994; 27 14_CR20 T. Ohta (14_CR23) 1986; 19 X. Chen (14_CR7) 2007; 186 A. Aksimentiev (14_CR2) 2002; 121 Y. Nishiura (14_CR21) 1995; 84 F.S. Bates (14_CR5) 1999; 52 T.S. Bailey (14_CR4) 2002; 35 |
| References_xml | – reference: Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation in computational and mathematical models of microstructual evolution. In: Bullard, J.W., Kalita, R., Stoneham, M., Chen, L.-Q. (eds.) (MRS, 1998) – reference: KaczynskiT.MischaikowK.MrozekM.Computational Homology2004New YorkSpringer1039.55001 – reference: RenX.WeiJ.On the multiplicity of two nonlocal variational problemsSIAM J. Math. Anal.2000319099240973.4900710.1137/S00361410983481761752422 – reference: TeramotoT.NishiuraY.Double gyroid morphology in a gradient system with nonlocal effectsJ. Phys. Soc. Jpn.2002711611161410.1143/JPSJ.71.1611 – reference: Ishimura, N., Ishiwata, T., Skajo, T., Sakurai, T., Nagayama, M., Nara, T., Hayami, K., Furihata, D., Matsuo, T.: Computational homology and its applications, Hokkaido University Technical Report Series in Mathematics, No. 124 (in Japanese) (2007) – reference: BraidesA.Γ-Convergence for Beginners2002OxfordOxford University Press0186593910.1093/acprof:oso/9780198507840.001.0001 – reference: NishiuraY.OhnishiI.Some mathematical aspects of the micro-phase separation in diblock copolymersPhysica D19958431391194.8206910.1016/0167-2789(95)00005-O1334695 – reference: ThomasE.L.AndersonD.M.HenkeeC.S.HoffmanD.Periodic area-minimizing surfaces in block copolymersNature198833459860110.1038/334598a0 – reference: BatesF.S.FredricksonG.H.Block copolymers—designer soft materialsPhys Today1999522323810.1063/1.882522 – reference: GameiroM.MischaikowK.WannerT.Evolution of pattern complexity in the Cahn-Hilliard theory of phase separationActa Materialia20055369370410.1016/j.actamat.2004.10.022 – reference: TeramotoT.Nosé thermostat for the pattern formation dynamicsMol. Sim.200733717510.1080/08927020601052914 – reference: AksimentievA.FiałkowskiM.HołystR.Morphology of surfaces in polymer, surfactant, electron and reaction-diffusion systems: methods, measurements and simulationsAdv. Chem. Phys2002121143239 – reference: A Collection Papers of T. Hashimoto: ordered structures, order–disorder transition, and physical properties of block copolymers , Dept. of Polym. Chem., Kyoto University (1995) – reference: ChoksiR.SternbergP.Periodic phase separation: the periodic Cahn-Hilliard and isoperimetric problemInterfaces Free Boundaries200683713921109.3509210.4171/IFB/1482273234 – reference: HajdukD.A.HarperP.E.GrunerS.M.HonekerC.C.KimG.ThomasE.L.The gyroid: a new equilibrium morphology in weakly segregated diblock copolymersMacromolecules1994274063407510.1021/ma00093a006 – reference: BaileyT.S.HardyC.M.EppsT.H.IIIBatesF.S.A noncubic triply periodic network morphology in poly(isoprene-b-styrene-b-ethylene oxide) triblock copolymersMacromolecules2002357007701710.1021/ma011716x – reference: Langer, J.S.: An introduction to the kinetics of first-order phase transitions, solids far from equilibrium. In: Godreche, G. (ed.). Cambridge University Press, Cambridge (1992) – reference: Goźd́źW.T.HołystR.Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsionsPhys. Rev. E1996545012502710.1103/PhysRevE.54.5012 – reference: Nishiura, Y.: Far-from-equilibrium dynamics, Translations of Mathematical Monographs Vol. 209, AMS (2002) – reference: BahianaM.OonoY.Cell dynamical system approach to block copolymersPhys. Rev. A1990416763677110.1103/PhysRevA.41.6763 – reference: HagitaK.TeramotoT.Topological validation of morphology modeling by extended reverse Monte Carlo analysisPhys. Rev. E20087705670410.1103/PhysRevE.77.056704 – reference: ChoksiR.RenX.On the derivation of a density functional theory for microphase separation of diblock copolymersJ. Stat. Phys.20031131511761034.8203710.1023/A:10257228048732012976 – reference: SpivakM.A Comprehensive Introduction to Differential Geometry1979BerkleyPublish or Perish – reference: Grosse-BrauckmannK.On gyroid interfacesJ. Colloid Interface Sci.199718741842810.1006/jcis.1996.4720 – reference: ChenX.OshitaY.Applications of modular functions to interfacial dynamicsArch. Rat. Mech. Anal.20071861091321147.7402410.1007/s00205-007-0050-z2338353 – reference: OhtaT.KawasakiT.Equilibrium morphology of block copolymer meltsMacromolecules1986192621263210.1021/ma00164a028 – reference: KangX.RenX.Ring pattern solutions of a free boundary problem in diblock copolymer morphologyPhysica D20092386456651160.3738610.1016/j.physd.2008.12.0092590243 – reference: NonomuraM.YamadaK.OhtaT.Formation and stability of double gyroid in microphase-separated diblock copolymersJ. Phys. Condens. Matter200315L423L43010.1088/0953-8984/15/26/101 – volume: 53 start-page: 693 year: 2005 ident: 14_CR11 publication-title: Acta Materialia doi: 10.1016/j.actamat.2004.10.022 – volume: 187 start-page: 418 year: 1997 ident: 14_CR13 publication-title: J. 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| Title | Morphological characterization of the diblock copolymer problem with topological computation |
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