메뉴 건너뛰기




Volumn 13, Issue 2, 2003, Pages 175-208

Triblock copolymer theory: Ordered ABC lamellar phase

Author keywords

; convergence; global minimizer; local minimizer; Triblock copolymer

Indexed keywords


EID: 84867955441     PISSN: 09388974     EISSN: 14321467     Source Type: Journal    
DOI: 10.1007/s00332-002-0521-1     Document Type: Article
Times cited : (38)

References (49)
  • 1
    • 84935168793 scopus 로고
    • Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids
    • S. Baldo. Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. Ann. Inst. H. Poincaŕe Anal. Non Lińeaire, 7(2):67-90, 1990.
    • (1990) Ann. Inst. H. Poincaŕe Anal. Non Lińeaire , vol.7 , Issue.2 , pp. 67-90
    • Baldo, S.1
  • 2
    • 0348046829 scopus 로고    scopus 로고
    • Block copolymers-designer soft materials
    • F. S. Bates and G. H. Fredrickson. Block copolymers-Designer soft materials. Phys. Today, 52(2):32-38, 1999. (Pubitemid 129600640)
    • (1999) Physics Today , vol.52 , Issue.2 , pp. 32-38
    • Bates, F.S.1    Fredrickson, G.H.2
  • 3
    • 0024107174 scopus 로고
    • A statistical theory of weakly charged polyelectrolytes: Fluctuations equation of state and microphase separation
    • V. Y. Borue and I. Y. Erukhimovich. A statistical theory of weakly charged polyelectrolytes: Fluctuations, equation of state and microphase separation. Macromolecules, 21(11):3240-3249, 1988.
    • (1988) Macromolecules , vol.21 , Issue.11 , pp. 3240-3249
    • Borue, V.Y.1    Erukhimovich, I.Y.2
  • 4
    • 33746012315 scopus 로고
    • Free energy of a nonuniform system. I. Interfacial free energy
    • J.W. Cahn and J. E. Hilliard. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys., 28(2):258-267, 1958.
    • (1958) J. Chem. Phys. , vol.28 , Issue.2 , pp. 258-267
    • Cahn, J.W.1    Hilliard, J.E.2
  • 5
    • 0000352072 scopus 로고
    • Electron crystalization
    • C. M. Care and N. H. March. Electron crystalization. Adv. Phys., 24(1):101-116, 1975.
    • (1975) Adv. Phys. , vol.24 , Issue.1 , pp. 101-116
    • Care, C.M.1    March, N.H.2
  • 6
    • 0000358405 scopus 로고
    • Structured phase transitions on a finite interval
    • J. Carr,M. E. Gurtin, and M. Slemrod. Structured phase transitions on a finite interval. Arch. Ratl. Mech. Anal., 86(4):317-351, 1984.
    • (1984) Arch. Ratl. Mech. Anal. , vol.86 , Issue.4 , pp. 317-351
    • Carr, J.1    Gurtin, M.E.2    Slemrod, M.3
  • 7
    • 0001635251 scopus 로고
    • Dynamics of simultaneous ordering and phase separation and effect of long-range Coulomb interactions
    • L. Q. Chen and A. G. Khachaturyan. Dynamics of simultaneous ordering and phase separation and effect of long-range Coulomb interactions. Phys. Rev. Lett., 70(10):1477-1480, 1993.
    • (1993) Phys. Rev. Lett. , vol.70 , Issue.10 , pp. 1477-1480
    • Chen, L.Q.1    Khachaturyan, A.G.2
  • 8
    • 0040927799 scopus 로고    scopus 로고
    • Multiple layered solutions of the nonlocal bistable equation
    • DOI 10.1016/S0167-2789(00)00143-3, PII S0167278900001433
    • A. Chmaj and X. Ren. Multiple layered solutions of the nonlocal bistable equation. Physica D, 147(1-2):135-154, 2000. (Pubitemid 33592761)
    • (2000) Physica D: Nonlinear Phenomena , vol.147 , Issue.1-2 , pp. 135-154
    • Chmaj, A.1    Ren, X.2
  • 9
    • 0000410819 scopus 로고    scopus 로고
    • Scaling laws in microphase separation of diblock copolymers
    • R. Choksi. Scaling laws in microphase separation of diblock copolymers. J. Nonlinear Sci., 11:223-236, 2001. (Pubitemid 33616503)
    • (2001) Journal of Nonlinear Science , vol.11 , Issue.3 , pp. 223-236
    • Choksi, R.1
  • 12
    • 0027592522 scopus 로고
    • Frustrated electronic phase separation and high-temperature superconductors
    • V. J. Emery and S. A. Kivelson. Frustrated electronic phase separation and high-temperature superconductors. Physica C, 209:597-621, 1993.
    • (1993) Physica C , vol.209 , pp. 597-621
    • Emery, V.J.1    Kivelson, S.A.2
  • 13
    • 0347797825 scopus 로고
    • Microphase separation in polymer systems: New approaches and objects
    • I. Y. Erukhimovich and A. R. Khokhlov. Microphase separation in polymer systems: New approaches and objects. Polym. Sci., 35(11):1522-1531, 1993.
    • (1993) Polym. Sci. , vol.35 , Issue.11 , pp. 1522-1531
    • Erukhimovich, I.Y.1    Khokhlov, A.R.2
  • 15
    • 0036347960 scopus 로고    scopus 로고
    • The Nishiura-Ohnishi free boundary problem in the 1D case
    • PII S0036141000372507
    • P. C. Fife and D. Hilhorst. The Nishiura-Ohnishi free boundary problem in the 1D case. SIAM J. Math. Anal., 33(3):589-606, 2001. (Pubitemid 33787276)
    • (2001) SIAM Journal on Mathematical Analysis , vol.33 , Issue.3 , pp. 589-606
    • Fife, P.C.1    Hilhorst, D.2
  • 16
    • 84975994718 scopus 로고
    • The gradient theory of phase transitions for systems with two potential wells
    • I. Fonseca and L. Tartar. The gradient theory of phase transitions for systems with two potential wells. Proc. Roy. Soc. Edinburgh Sect. A, 111(1-2):89-102, 1989.
    • (1989) Proc. Roy. Soc. Edinburgh Sect. A , vol.111 , Issue.1-2 , pp. 89-102
    • Fonseca, I.1    Tartar, L.2
  • 17
    • 0000989342 scopus 로고
    • Sulla convergenza di alcune successioni di integrali del tipo della'area
    • E. De Giorgi. Sulla convergenza di alcune successioni di integrali del tipo della'area. Rendiconti Mat., 8:277-294, 1975.
    • (1975) Rendiconti Mat. , vol.8 , pp. 277-294
    • De Giorgi, E.1
  • 19
    • 0142028339 scopus 로고
    • Theory of inhomogeneous polymers: Fundamentals of gaussian random walk model
    • E. Helfand. Theory of inhomogeneous polymers: Fundamentals of gaussian random-walk model. J. Chem. Phys., 62(3):999-1005, 1975.
    • (1975) J. Chem. Phys. , vol.62 , Issue.3 , pp. 999-1005
    • Helfand, E.1
  • 20
    • 33847799791 scopus 로고
    • Block copolymer theory. IV. Narrow interphase approximation
    • E. Helfand and Z. R. Wasserman. Block copolymer theory. IV. Narrow interphase approximation. Macromolecules, 9(6):879-888, 1976.
    • (1976) Macromolecules , vol.9 , Issue.6 , pp. 879-888
    • Helfand, E.1    Wasserman, Z.R.2
  • 21
    • 0345251822 scopus 로고
    • Block copolymer theory. V. Spherical domains
    • E. Helfand and Z. R. Wasserman. Block copolymer theory. V. Spherical domains. Macromolecules, 11(5):960-966, 1978.
    • (1978) Macromolecules , vol.11 , Issue.5 , pp. 960-966
    • Helfand, E.1    Wasserman, Z.R.2
  • 22
    • 4344600790 scopus 로고
    • Block copolymer theory. VI. Cylindrical domains
    • E. Helfand and Z. R.Wasserman. Block copolymer theory. VI. Cylindrical domains. Macromolecules, 13(4):994-998, 1980.
    • (1980) Macromolecules , vol.13 , Issue.4 , pp. 994-998
    • Helfand, E.1    Wasserman, Z.R.2
  • 23
    • 0001666953 scopus 로고    scopus 로고
    • Singular limit of a fourth order problem arising in the micro-phase separation of diblock copolymers
    • M. Henry. Singular limit of a fourth order problem arising in the micro-phase separation of diblock copolymers. Adv. Diff. Eq., 6(9):1049-1114, 2001.
    • (2001) Adv. Diff. Eq. , vol.6 , Issue.9 , pp. 1049-1114
    • Henry, M.1
  • 24
    • 0000685437 scopus 로고
    • Theory of inhomogeneous multicomponent polymer systems
    • K. M. Hong and J. Noolandi. Theory of inhomogeneous multicomponent polymer systems. Macromolecules, 14(3):727-736, 1981.
    • (1981) Macromolecules , vol.14 , Issue.3 , pp. 727-736
    • Hong, K.M.1    Noolandi, J.2
  • 25
    • 3743105736 scopus 로고
    • Theory of phase equilibria in systems containing block copolymers
    • K. M. Hong and J. Noolandi. Theory of phase equilibria in systems containing block copolymers. Macromolecules, 16(7):1083-1093, 1983.
    • (1983) Macromolecules , vol.16 , Issue.7 , pp. 1083-1093
    • Hong, K.M.1    Noolandi, J.2
  • 26
    • 84975988204 scopus 로고
    • Local minimisers and singular perturbations
    • R. Kohn and P. Sternberg. Local minimisers and singular perturbations. Proc. Roy. Soc. Edinburgh, 111A:69-84, 1989.
    • (1989) Proc. Roy. Soc. Edinburgh , vol.111 A , pp. 69-84
    • Kohn, R.1    Sternberg, P.2
  • 27
    • 33847086243 scopus 로고
    • Theory of microphase separation in block copolymers
    • L. Leibler. Theory of microphase separation in block copolymers. Macromolecules, 13(6):1602-1617, 1980.
    • (1980) Macromolecules , vol.13 , Issue.6 , pp. 1602-1617
    • Leibler, L.1
  • 28
    • 0002334568 scopus 로고
    • Domain structure of a new type near a photostimulated phase transition; Autosolitons
    • R. F. Mamin. Domain structure of a new type near a photostimulated phase transition; Autosolitons. JETP Lett., 60(1):52-56, 1994.
    • (1994) JETP Lett. , vol.60 , Issue.1 , pp. 52-56
    • Mamin, R.F.1
  • 29
    • 3342995995 scopus 로고
    • Stable and unstable phases of a diblock copolymer melt
    • M.W. Matsen and M. Schick. Stable and unstable phases of a diblock copolymer melt. Phys. Rev. Lett., 31(16):2660-2663, 1994. (Pubitemid 24975230)
    • (1994) Physical Review Letters , vol.72 , Issue.16 , pp. 2660-2663
    • Matsen, M.W.1    Schick, M.2
  • 30
    • 34250109116 scopus 로고
    • The gradient theory of phase transitions and the minimal interface criterion
    • L. Modica. The gradient theory of phase transitions and the minimal interface criterion. Arch. Ratl. Mech. Anal., 98:357-383, 1987.
    • (1987) Arch. Ratl. Mech. Anal. , vol.98 , pp. 357-383
    • Modica, L.1
  • 31
    • 0000359194 scopus 로고
    • Un esempio di-convergenza
    • 14
    • L. Modica and S. Mortola. Un esempio di-convergenza. Boll. Un. Mat. Ital. B (5), 14(1):285-299, 1977.
    • (1977) Boll. Un. Mat. Ital. B , vol.5 , Issue.1 , pp. 285-299
    • Modica, L.1    Mortola, S.2
  • 32
    • 0028767852 scopus 로고
    • Superlattice structures in morphologies of the ABC triblock copolymers
    • Y. Mogi, M. Nomura, H. Kotsuji, K. Ohnishi, Y. Matsushita, and I. Noda. Superlattice structures in morphologies of the ABC triblock copolymers. Macromolecules, 27(23):6755-6760, 1994.
    • (1994) Macromolecules , vol.27 , Issue.23 , pp. 6755-6760
    • Mogi, Y.1    Nomura, M.2    Kotsuji, H.3    Ohnishi, K.4    Matsushita, Y.5    Noda, I.6
  • 33
    • 0000390068 scopus 로고
    • Singular perturbations as a selection criterion for periodic minimizing sequences
    • S. M̈uller. Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Partial Diff. Eq., 1(2):169-204, 1993.
    • (1993) Calc. Var. Partial Diff. Eq. , vol.1 , Issue.2 , pp. 169-204
    • M̈uller, S.1
  • 34
    • 41349101280 scopus 로고    scopus 로고
    • Theory of domain patterns in systems with long-range interactions of Coulomb type
    • to appear
    • C. B. Muratov. Theory of domain patterns in systems with long-range interactions of Coulomb type. Phys. Rev. E, to appear.
    • Phys. Rev. E.
    • Muratov, C.B.1
  • 35
    • 0027668501 scopus 로고
    • Microphase separation of ABC-type triblock copolymers
    • H. Nakazawa and T. Ohta. Microphase separation of ABC-type triblock copolymers. Macromolecules, 26(20):5503-5511, 1993.
    • (1993) Macromolecules , vol.26 , Issue.20 , pp. 5503-5511
    • Nakazawa, H.1    Ohta, T.2
  • 36
    • 0002733111 scopus 로고
    • Some mathematical aspects of the microphase separation in diblock copolymers
    • Y. Nishiura and I. Ohnishi. Some mathematical aspects of the microphase separation in diblock copolymers. Physica D, 84:31-39, 1995.
    • (1995) Physica D , vol.84 , pp. 31-39
    • Nishiura, Y.1    Ohnishi, I.2
  • 37
    • 0028463028 scopus 로고
    • Microdomain structures in polyelectrolyte systems: Calculation of the phase diagrams by direct minimization of the free energy
    • I. A. Nyrkova, A. R. Khohklov, and M. Doi. Microdomain structures in polyelectrolyte systems: Calculation of the phase diagrams by direct minimization of the free energy. Macromolecules, 27(15):4220-4230, 1994.
    • (1994) Macromolecules , vol.27 , Issue.15 , pp. 4220-4230
    • Nyrkova, I.A.1    Khohklov, A.R.2    Doi, M.3
  • 38
    • 0012334026 scopus 로고    scopus 로고
    • Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term
    • I. Ohnishi, Y. Nishiura, M. Imai, and Y. Matsushita. Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term. Chaos, 9(2):329-341, 1999.
    • (1999) Chaos , vol.9 , Issue.2 , pp. 329-341
    • Ohnishi, I.1    Nishiura, Y.2    Imai, M.3    Matsushita, Y.4
  • 39
    • 0000482598 scopus 로고
    • Equilibrium morphology of block copolymer melts
    • T. Ohta and K. Kawasaki. Equilibrium morphology of block copolymer melts. Macromolecules, 19(10):2621-2632, 1986.
    • (1986) Macromolecules , vol.19 , Issue.10 , pp. 2621-2632
    • Ohta, T.1    Kawasaki, K.2
  • 40
    • 0034441979 scopus 로고    scopus 로고
    • Finite scale microstructures in nonlocal elasticity
    • (Blacksburg, VA 1999). J. Elasticity
    • X. Ren and L. Truskinovsky. Finite scale microstructures in nonlocal elasticity. In recognition of the sixtieth birthday of Roger L. Fosdick (Blacksburg, VA, 1999). J. Elasticity, 59(1-3):319-355, 2000.
    • (2000) Recognition of the Sixtieth Birthday of Roger L. Fosdick , vol.59 , Issue.1 , pp. 3319-3355
    • Ren, X.1    Truskinovsky, L.2
  • 42
    • 0013369127 scopus 로고    scopus 로고
    • On the spectra of 3-D lamellar solutions of the diblock copolymer problem
    • to appear
    • X. Ren and J.Wei. On the spectra of 3-D lamellar solutions of the diblock copolymer problem. SIAM J. Math. Anal., to appear.
    • SIAM J. Math. Anal.
    • Ren, X.1    Wei, J.2
  • 43
    • 85032184465 scopus 로고    scopus 로고
    • Triblock copolymer theory: Free energy, disordered phase and weak segregation
    • to appear
    • X. Ren and J. Wei. Triblock copolymer theory: Free energy, disordered phase and weak segregation. Physica D, to appear.
    • Physica D
    • Ren, X.1    Wei, J.2
  • 44
    • 0034347328 scopus 로고    scopus 로고
    • On the multiplicity of solutions of two nonlocal variational problems
    • X. Ren and J. Wei. On the multiplicity of solutions of two nonlocal variational problems. SIAM J. Math. Anal., 31(4):909-924, 2000.
    • (2000) SIAM J. Math. Anal. , vol.31 , Issue.4 , pp. 909-924
    • Ren, X.1    Wei, J.2
  • 45
    • 0036813973 scopus 로고    scopus 로고
    • Concentrically layered energy equilibria of the diblock copolymer problem
    • X. Ren and J. Wei. Concentrically layered energy equilibria of the diblock copolymer problem. Eur. J. Appl. Math., 13(5):479-496, 2002.
    • (2002) Eur. J. Appl. Math. , vol.13 , Issue.5 , pp. 479-496
    • Ren, X.1    Wei, J.2
  • 46
    • 34250095380 scopus 로고
    • The effect of a singular perturbation on nonconvex variational problems
    • P. Sternberg. The effect of a singular perturbation on nonconvex variational problems. Arch. Ratl. Mech. Anal., 101(3):209-260, 1988.
    • (1988) Arch. Ratl. Mech. Anal. , vol.101 , Issue.3 , pp. 209-260
    • Sternberg, P.1
  • 47
    • 0000482727 scopus 로고
    • Vector-valued local minimizers of nonconvex variational problems. Current directions in nonlinear partial differential equations (Provo, UT 1987)
    • P. Sternberg. Vector-valued local minimizers of nonconvex variational problems. Current directions in nonlinear partial differential equations (Provo, UT, 1987). Rocky Mountain J. Math., 21(2):799-807, 1991.
    • (1991) Rocky Mountain J. Math. , vol.21 , Issue.2 , pp. 799-807
    • Sternberg, P.1
  • 48
    • 84974286264 scopus 로고
    • Local minimisers of a three-phase partition problem with triple junctions
    • P. Sternberg and W. Zeimer. Local minimisers of a three-phase partition problem with triple junctions. Proc. Roy. Soc. Edinburgh Sect. A, 124(6):1059-1073, 1994.
    • (1994) Proc. Roy. Soc. Edinburgh Sect. A , vol.124 , Issue.6 , pp. 1059-1073
    • Sternberg, P.1    Zeimer, W.2
  • 49
    • 36749105143 scopus 로고
    • Variational model for micelle structure
    • F. H. Stillinger. Variational model for micelle structure. J. Chem. Phys., 78(7):4654-4661, 1983.
    • (1983) J. Chem. Phys. , vol.78 , Issue.7 , pp. 4654-4661
    • Stillinger, F.H.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.