Bicontinuous cubic morphologies in block copolymers and amphiphile/water systems: Mathematical description through the minimal surfaces

Macromolecules

Volumn 30, Issue 11, 1997, Pages 3395-3402

  Benedicto, Alto D ^a   O'Brien, David F ^a  

^a University of Arizona   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BLOCK COPOLYMERS; CALCULATIONS; MACROMOLECULES; MICROSCOPIC EXAMINATION; NEUTRON DIFFRACTION; PHASE TRANSITIONS; SURFACES; X RAY DIFFRACTION;

AMPHIPHILE WATER SYSTEM; BICONTINUOUS CUBIC MORPHOLOGY; POLYMORPHISM;

MORPHOLOGY;

EID: 0031548693     PISSN: 00249297     EISSN: None     Source Type: Journal    
DOI: 10.1021/ma9614353     Document Type: Article

References (56)

1. Odian, G. Principles of Polymerization; 3rd ed.; John Wiley & Sons, Inc.: New York, 1991.
2. Hamley, I. W.; Koppi, K. A.; Rosedale, J. H.; Bates, F. S.; Almdal, K.; Mortensen, K. Macromolecules 1993, 26, 5959-5970.
3. Hamley, I. W.; Gehlsen, M. D.; Khandpur, A. K.; Koppi, K. A.; Rosedale, J. H.; Schulz, M. F.; Bates, F. S.; Almdale, K.; Mortensen, K. J. Phys. II Fr. 1994, 4, 2161-2186.
4. Disko, M. M.; Liang, K. S.; Behal, S. K.; Roe, R. J.; Jeon, K. J. Macromolecules 1993, 26, 2983-2986.
5. Spontak, R. J.; Fung, J. C.; Braunfeld, M. B.; Sedat, J. W.; Agard, D. A.; Ashraf, A.; Smith, S. Macromolecules 1996, 29, 2850-2856.
6. Winey, K. I.; Thomas, E. L.; Fetters, L. J. Macromolecules 1992, 25, 422-428.
7. Thomas, E. L.; Alward, D. B.; Kinning, D. J.; Martin, D. C.; Handlin Jr., D. L.; Fetters, L. J. Macromolecules 1986, 19, 2197-2202.
8. Hajduk, D.; Harper, P. E.; Gruner, S. M.; Honeker, C. C.; Kim, G.; Thomas, E. L.; Fetters, L. J. Macromolecules 1994, 27, 4063-4075.
9. Seddon, J. M. Biochim. Biophys. Acta 1990, 1031, 1-69.
10. Lindbolm, G.; Rilfors, L. Biochim. Biophys. Acta 1989, 988, 221-256.
11. Longley, W.; McIntosh, T. J. Nature 1983, 303, 612-614.
12. Wennerstrom, H.; Evans, D. F. The Colloidal Domain: where Physics, Chemistry, Biology, and Technology Meet; VCH Publishers: New York, 1994.
13. Actually, it is χN, where N is the degree of polymerization.
14. Schulz, M. F.; Khandpur, A. K.; Bates, F. S.; K., A.; Mortensen, K,. Hajduk. D. A.; Gruner, S. M. Macromolecules 1996, 29, 2857-2867.
15. Bates, F. Annu. Phys. Chem. 1990, 41, 525-557.
16. Helfrich, W. Z. Naturforsch., C 1973, 28, 693.
17. Deuling, H. J.; Helfrich, W. J. Phys. (Paris) 1976, 37, 1335.
18. c is the bending rigidity constant, while λ is a Lagrange multiplier accounting for the area constraint. The free energy is also called the curvature or shape energy, i.e., the energy assigned to the shape of the object or system. The stability of the selected shapes arising from fluctuations has been dealt with elsewhere, e.g., see Helfrich's work.
19. Anderson, D. M.; Gruner, S. M.; Leibler, S. Proc. Natl. Acad. Sci. U.S.A. 1988, 85, 5364-5368.
20. Hoffman, D. Math. Intelligencer 1987, 9, 8-21.
21. Mariani, P.; Luzzati, V.; Delacroix, H. J Mol. Biol. 1988, 204, 165-189.
22. Lindblom, G. In Advances in Lipid Methodology; Christie, W. W., Ed.; Oily Press: Dundee, 1996; Vol. 3; pp 133-209.
23. Hajduk, D. A.; Harper, P. E.; Gruner, S. M.; Honeker, C. C.; Thomas, E. L.; Fetters, L. J. Macromolecules 1995, 28, 2570-2573.
24. Fontell, K. Adv. Colloid Interface Sci. 1992, 41, 127-147.
25. Briggs, J.; Caffrey, M. Biophys. J. 1994, 67, 1594-1602.
26. Frank, J. Three-Dimensional Electron Microscopy of Macromolecular Assemblies; Academic Press, Inc.: San Diego, CA, 1996.
27. Wolfram, S. Mathematics: A System for Doing Mathematics by Computer; Addison-Wesley Publishing Co.: Redwood City, CA, 1991.
28. Software Development Group, The Geometry Center Minnesota, 1993.
29. Molecular Simulations Incorporated: San Diego, CA, 1994.
30. Nitsche, J. C. C. Lectures on Minimal Surfaces, Cambridge University Press: Cambridge, England, 1989; Vol. 1.
31. Dierkes, U.; Hildebrandt, S.; Kuster, A.; Wohlrab, O. Minimal Surfaces I; Springer-Vertag: Berlin, 1992.
32. Osserman, R. A. Survey of Minimal Surfaces; Van Nostrand Reinhold Company: New York, 1969.
33. Luzsati, V.; Vargas, R.; Mariani, P.; Gulik, A.; Delacroix, H. J. Mol. Biol. 1993, 229, 540-551.
34. Spontak, R. J.; Fung, J. C.; Braunfeld, M. B.; Sedat, J. W.; Agard, D. A.; Kane. L.; Smith, S. D.; Satkowski, M. M.; Ashraf, A.; Hajdukm D. A.; Gruner, S. M. Macromolecules 1996, 96, 4494-4507.
35. Wells, A. F. Three-Dimensional Nets and Polyhedra; John Wiley & Sons: New York, 1977.
36. The first number denotes the shortest circuit, while the second number is the valence of the vertex.
37. Seddon, J. M.; Hogan, J. L.; Warrender, N. A.; Pebay-Peyroula, E. Prog. Colloid Polym. Sci. 1990, 81, 189-197.
38. Without "tearing" means that the surface remained continuous.
39. In mathematical terms, a surface without intersections is said to be embedded.
40. Schoen, A. H. NASA Tech. Note 1970, D-5541, 1-98.
41. This is closely related to page 31 of ref 35.
42. A simplified analysis goes as follows: In order for the labyrinth net (6,4) to be transformed to ( 10,3)a, the tetrahedral vertices of the former should be transformed into the trigonal vertices of the latter. Moreover, circuit 6 of the former should also be transformed into circuit 10 of the latter. This is achieved by noting that the probability of a circuit 6 expanding into a greater circuit by random "pulling apart" of the tetrahedral vertices to form two trigonal vertices is 2/3. Thus, upon such "random" pulling apart of the (6,4) labyrinth net (of the D surface), the resultant total number of vertices per circuit is = (2/3 enlargement/vertex)(6 vertex/ circuit)(1 new vertex generated/enlargement) + 6 old vertex/ circuit = 10 resultant vertex/circuit-which is the (10,3)a labyrinth (corresponding to the G surface).
43. Similar analysis as above shows that the "random" pulling apart of the (4,6) labyrinth net (of the P surface), the resultant total number of vertices per circuit is = (6/8 enlargement/ vertex)(4 vertex/circuit)(2 new vertex generated/enlargement) + 4 old vertex/circuit = 10 resultant vertex/circuit-which is the (10,3)a labyrinth (corresponding to the G surface).
44. Hyde, S. T.; Andersson, S.; Ericsson, B.; Larsson, K. Z. Kristallogr. 1984, 168, 213-219.
45. Hyde, S. T.; Anderson, S. Z. Kristallogr. 1985, 170, 225-239.
46. International Tables for Crystallography; Hahn, T., Ed.; D. Reidel Publishing Company: Dordrecht, The Netherlands, 1983; Vol. A.
47. Anderson, D. M.; Bellare, J.; Hoffman, J. T.; Hoffman, D.; Gunther, J.; Thomas, E. L. J. Colloid Interface Sci. 1992,148, 398-414.
48. As noted earlier, to unambiguously identify a particular bicontinuous cubic phase requires several slices.
49. Lee, Y. S.; Yang, J. Z.; Sisson, T. M.; Frankel, D. A.; Gleeson, J. T.; Aksay, E.; Keller, S. L.; Gruner, S. M.; O'Brien, D. F. J. Am. Chem. Soc. 1995, 117, 5573-5578.
50. Deng, Y.; Landh, T. Zool. Studies 1995, 34 Supplement 1, 175-177.
51. Landh, T. Zool. Studies 1995, 34 Supplement 1, 241-244.
52. Landh, T. FEBS Lett. 1995, 369, 13-17.
53. Frank, J. Electron Tomography; Plenum Press: New York, 1992.
54. Sawyer, L. C.; Grubb, D. T. Polymer Microscopy; Chapman and Hall, Ltd.: London, 1987.
55. von Schnering, H. G.; Nesper, R. Angew. Chem., Int. Ed. Engl. 1987, 26, 1059-1200.
56. Andersson, S.; Hyde, S. T.; Larsson, K.; Lidin, S. Chem. Rev. 1988, 88, 221-242.