根據自洽均場理論計算發現，在非對稱嵌段共聚物的相圖內有一狹小區間為最球狀密堆積相。在傳統最密堆積存在兩種排列方式，分別為面心立方堆積及六方最密堆積，然而在非對稱嵌段共聚物系統中，並未有實驗證明何者為最穩定的相。本論文中，我們使用了非對稱組成之聚乙二醇及聚丁烯二酸雙段嵌段共聚物為實驗材料，並且使用X-ray小角度散射發現經過升溫移除熱履歷後的降溫過程形成穩定的六方最密堆積排列。此外，我們也發現在從無序結構到有序結構的轉換過程中遵守奧斯瓦爾德規則中的亞歷山大-麥格塔轉換，也就是先形成體心立方推積後才形成六方最密堆積。本次研究的實驗結果也與近年針對非對稱嵌段共聚物的相圖自由能計算的結果相符，也間接指出了由高分子系統組成的軟求系統與其他硬球系統的熱力學穩定相有所差異。

Self-consistent field theory (SCFT) predicted the existence of a close-packed sphere (CPS) phase over a narrow window in the phase diagram of conformationally symmetric block copolymers (bcps). It however remains unclear whether face-centered cubic (FCC) and hexagonal close-packed (HCP) lattice represents the more stable close-packed lattice of the spherical micelles formed by neat bcp in the quiescent melt. Here we visit this problem by exploring the stable close-packed lattice of a conformationally symmetric poly(ethylene oxide)-block-poly(1,2-butadiene) (PEO-b-PB). We disclosed that HCP structure eventually formed in the ordered phase upon cooling from the micellar liquid phase. The micelle ordering was found to follow the Ostwald’s step rule of the Alexander-McTague type, where a metastable BCC phase first developed followed by transformation into the stable HCP structure. The higher thermodynamic stability of HCP relative to that of FCC was consistent with the prediction of a SCFT calculation by combining the spectral method with Anderson mixing, and it demonstrated the generic difference between soft colloid and hard colloid in selecting their thermodynamically stable close-packed lattice.

Abstract i
摘要 ii
Chapter 1 Introduction 1
1-1 A Brief Description of Block Copolymer 1
1-2 Phase Behavior of Block Copolymers 4
1-3 Sphere Morphology of Block Copolymer 12
1-3-1 The diblock foam model (DFM) for spherical domain 12
1-3-2 BCC and CPS phase of block copolymers 14
1-3-3 The stability FCC versus that of HCP Phase 15
1-3-4 Frank Kasper phase in block copolymer 19
1-4 The Ostwald’s Step Rule 24
1-5 Motivation and Objective of Research 27
Chapter 2 Experimentals 30
2-1 Materials 30
2-2 Sample Preparation 30
2-3 Small Angle X-ray Scattering(SAXS) Measurement 31
Chapter 3 Result and Discussion 32
3-1 The Discovery of HCP Phase in PEO-b-PB 32
3-2 Order-Disordered Transition Temperature 36
3-3 Time-resolved SAXS Profiles and Formation of HCP Phase via the Ostwald’s Step Rule 37
3-3 Relative Thermodynamic Stabilities of CPS and BCC 40
Chapter 4 Conclusion 50
Reference 51

1. Bates, F. S.; Fredrickson, G. H., BLOCK COPOLYMER THERMODYNAMICS - THEORY AND EXPERIMENT. Annual Review of Physical Chemistry 1990, 41, 525-557.
2. Holden, G., Thermoplastic elastomers. In Rubber technology, Springer: 1987; pp 465-481.
3. Stoykovich, M. P.; Müller, M.; Kim, S. O.; Solak, H. H.; Edwards, E. W.; De Pablo, J. J.; Nealey, P. F., Directed assembly of block copolymer blends into nonregular device-oriented structures. Science 2005, 308 (5727), 1442-1446.
4. Li, H.-W.; Huck, W. T., Ordered block-copolymer assembly using nanoimprint lithography. Nano Lett. 2004, 4 (9), 1633-1636.
5. Martin, C. R., Nanomaterials: a membrane-based synthetic approach. Science 1994, 266 (5193), 1961-1966.
6. Helfand, E., BLOCK COPOLYMER THEORY .3. STATISTICAL-MECHANICS OF MICRODOMAIN STRUCTURE. Macromolecules 1975, 8 (4), 552-556.
7. Helfand, E.; Wasserman, Z. R., BLOCK COPOLYMER THEORY .4. NARROW INTERPHASE APPROXIMATION. Macromolecules 1976, 9 (6), 879-888.
8. Helfand, E.; Wasserman, Z. R., BLOCK COPOLYMER THEORY .5. SPHERICAL DOMAINS. Macromolecules 1978, 11 (5), 960-966.
9. Leibler, L., Theory of microphase separation in block copolymers. Macromolecules 1980, 13 (6), 1602-1617.
10. Fredrickson, G. H.; Helfand, E., FLUCTUATION EFFECTS IN THE THEORY OF MICROPHASE SEPARATION IN BLOCK COPOLYMERS. J. Chem. Phys. 1987, 87 (1), 697-705.
11. Matsen, M. W.; Bates, F. S., Unifying weak- and strong-segregation block copolymer theories. Macromolecules 1996, 29 (4), 1091-1098.
12. Matsen, M. W.; Bates, F. S., Block copolymer microstructures in the intermediate-segregation regime. J. Chem. Phys. 1997, 106 (6), 2436-2448.
13. Grason, G. M., The packing of soft materials: Molecular asymmetry, geometric frustration and optimal lattices in block copolymer melts. Physics reports 2006, 433 (1), 1-64.
14. Sakamoto, N.; Hashimoto, T., Order-disorder transition of low molecular weight polystyrene-block-polyisoprene. 1. SAXS analysis of two characteristic temperatures. Macromolecules 1995, 28 (20), 6825-6834.
15. Thomas, E. L.; Alward, D. B.; Kinning, D. J.; Martin, D. C.; Handlin Jr, D. L.; Fetters, L. J., Ordered bicontinuous double-diamond structure of star block copolymers: a new equilibrium microdomain morphology. Macromolecules 1986, 19 (8), 2197-2202.
16. Hajduk, D. A.; Harper, P. E.; Gruner, S. M.; Honeker, C. C.; Kim, G.; Thomas, E. L.; Fetters, L. J., The gyroid: a new equilibrium morphology in weakly segregated diblock copolymers. Macromolecules 1994, 27 (15), 4063-4075.
17. Tyler, C. A.; Morse, D. C., Orthorhombic Fddd network in triblock and diblock copolymer melts. Phys. Rev. Lett. 2005, 94 (20), 4.
18. Vukovic, I.; ten Brinke, G.; Loos, K., Hexagonally Perforated Layer Morphology in PS-b-P4VP(PDP) Supramolecules. Macromolecules 2012, 45 (23), 9409-9418.
19. Xie, N.; Li, W.; Qiu, F.; Shi, A.-C. J. A. M. L., σ phase formed in conformationally asymmetric AB-type block copolymers. 2014, 3 (9), 906-910.
20. Lee, S.; Bluemle, M. J.; Bates, F. S., Discovery of a Frank-Kasper σ phase in sphere-forming block copolymer melts. Science 2010, 330 (6002), 349-353.
21. Chen, C.-K.; Hsueh, H.-Y.; Chiang, Y.-W.; Ho, R.-M.; Akasaka, S.; Hasegawa, H., Single helix to double gyroid in chiral block copolymers. Macromolecules 2010, 43 (20), 8637-8644.

22. Zhao, W.; Russell, T. P.; Grason, G. M., Chirality in block copolymer melts: Mesoscopic helicity from intersegment twist. Phys. Rev. Lett. 2013, 110 (5), 058301.
23. Goodfellow, B. W.; Yu, Y.; Bosoy, C. A.; Smilgies, D.-M.; Korgel, B. A., The role of ligand packing frustration in body-centered cubic (bcc) superlattices of colloidal nanocrystals. The journal of physical chemistry letters 2015, 6 (13), 2406-2412.
24. Semenov, A. N., THEORY OF BLOCK-COPOLYMER INTERFACES IN THE STRONG SEGREGATION LIMIT. Macromolecules 1993, 26 (24), 6617-6621.
25. Pusey, P.; Van Megen, W.; Bartlett, P.; Ackerson, B.; Rarity, J.; Underwood, S., Structure of crystals of hard colloidal spheres. Phys. Rev. Lett. 1989, 63 (25), 2753.
26. Kratky, K. W., Stability of fcc and hcp hard-sphere crystals. Chemical Physics 1981, 57 (1-2), 167-174.
27. Bolhuis, P. G.; Frenkel, D.; Mau, S.-C.; Huse, D. A., Entropy difference between crystal phases. Nature 1997, 388 (6639), 235-236.
28. Mau, S.-C.; Huse, D. A., Stacking entropy of hard-sphere crystals. Physical Review E 1999, 59 (4), 4396.
29. Matsen, M. W., Fast and accurate SCFT calculations for periodic block-copolymer morphologies using the spectral method with Anderson mixing. European Physical Journal E 2009, 30 (4), 361-369.
30. Brazovskii, S., Phase transition of an isotropic system to a nonuniform state. Soviet Journal of Experimental and Theoretical Physics 1975, 41, 85.
31. Matsen, M. W., Phase behavior of block copolymer/homopolymer blends. Macromolecules 1995, 28 (17), 5765-5773.
32. McConnell, G. A.; Gast, A. P.; Huang, J. S.; Smith, S. D., Disorder-order transitions in soft sphere polymer micelles. Phys. Rev. Lett. 1993, 71 (13), 2102.
33. Lodge, T. P.; Pudil, B.; Hanley, K. J., The full phase behavior for block copolymers in solvents of varying selectivity. Macromolecules 2002, 35 (12), 4707-4717.
34. Huang, Y.-Y.; Hsu, J.-Y.; Chen, H.-L.; Hashimoto, T., Existence of fcc-packed spherical micelles in diblock copolymer melt. Macromolecules 2007, 40 (3), 406-409.
35. Chen, L.-T.; Chen, C.-Y.; Chen, H.-L., FCC or HCP: The stable close-packed lattice of crystallographically equivalent spherical micelles in block copolymer/homopolymer blend. Polymer 2019, 169, 131-137.
36. Zhang, J. W.; Bates, F. S., Dodecagonal Quasicrystalline Morphology in a Poly(styrene-b-isoprene-b-styrene-b-ethylene oxide) Tetrablock Terpolymer. J. Am. Chem. Soc. 2012, 134 (18), 7636-7639.
37. Su, Z.; Hsu, C.-H.; Gong, Z.; Feng, X.; Huang, J.; Zhang, R.; Wang, Y.; Mao, J.; Wesdemiotis, C.; Li, T., Identification of a Frank–Kasper Z phase from shape amphiphile self-assembly. Nature chemistry 2019, 1-7.
38. Liu, M.; Li, W.; Qiu, F.; Shi, A.-C. J. S. M., Stability of the Frank–Kasper σ-phase in BABC linear tetrablock terpolymers. 2016, 12 (30), 6412-6421.
39. Takagi, H.; Hashimoto, R.; Igarashi, N.; Kishimoto, S.; Yamamoto, K. J. J. o. P. C. M., Frank–Kasper σ phase in polybutadiene-poly (ε-caprolactone) diblock copolymer/polybutadiene blends. 2017, 29 (20), 204002.
40. Schulze, M. W.; Lewis III, R. M.; Lettow, J. H.; Hickey, R. J.; Gillard, T. M.; Hillmyer, M. A.; Bates, F. S. J. P. r. l., Conformational asymmetry and quasicrystal approximants in linear diblock copolymers. 2017, 118 (20), 207801.
41. Reddy, A.; Buckley, M. B.; Arora, A.; Bates, F. S.; Dorfman, K. D.; Grason, G. M., Stable Frank–Kasper phases of self-assembled, soft matter spheres. Proceedings of the National Academy of Sciences 2018, 115 (41), 10233-10238.
42. Ostwald, W., Studies on the formation and change of solid matter. Z. phys. Chem. 1897, 22, 289-302.
43. Bang, J.; Lodge, T. P., Long-lived metastable bcc phase during ordering of micelles. Phys. Rev. Lett. 2004, 93 (24), 245701.
44. Nishiyama, Z., Martensitic transformation. Elsevier: 2012.
45. Masuda-Jindo, K.; Nishitani, S.; Van Hung, V., hcp-bcc structural phase transformation of titanium: Analytic model calculations. Physical Review B 2004, 70 (18), 184122.
46. Liu, Y.; Nie, H.; Bansil, R.; Steinhart, M.; Bang, J.; Lodge, T. P., Kinetics of disorder-to-fcc phase transition via an intermediate bcc state. Physical Review E 2006, 73 (6), 061803.
47. Huang, Y.-Y.; Chen, H.-L.; Hashimoto, T., Face-centered cubic lattice of spherical micelles in block copolymer/homopolymer blends. Macromolecules 2003, 36 (3), 764-770.
48. Huang, Y.-Y.; Hsu, J.-Y.; Chen, H.-L.; Hashimoto, T., Precursor-Driven Bcc− Fcc Order− Order Transition of Sphere-Forming Block
49. Imaizumi, K.; Ono, T.; Kota, T.; Okamoto, S.; Sakurai, S., Transformation of cubic symmetry for spherical microdomains from face-centred to body-centred cubic upon uniaxial elongation in an elastomeric triblock copolymer. Journal of applied crystallography 2003, 36 (4), 976-981.
50. Alexander, S.; McTague, J., Should all crystals be bcc? Landau theory of solidification and crystal nucleation. Phys. Rev. Lett. 1978, 41 (10), 702.
51. Auer, S.; Frenkel, D., Prediction of absolute crystal-nucleation rate in hard-sphere colloids. Nature 2001, 409 (6823), 1020-1023.
52. Tan, P.; Xu, N.; Xu, L., Visualizing kinetic pathways of homogeneous nucleation in colloidal crystallization. Nature Physics 2014, 10 (1), 73-79.
53. Ziherl, P.; Kamien, R. D., Maximizing entropy by minimizing area: Towards a new principle of self-organization. ACS Publications: 2001.