Abstract.
Balanced infinite periodic minimal surface families that contain the cubic Gyroid (G), Diamond (D) and Primitive (P) surfaces are studied in terms of their global packing and local curvature properties. These properties are central to understanding the formation of mesophases in amphiphile and copolymer molecular systems. The surfaces investigated are the tetragonal, rhombohedral and hexagonal tD, tP, tG, rG, rPD and H surfaces. These non-cubic minimal surfaces furnish topology-preserving transformation pathways between the three cubic surfaces. We introduce `packing (or global) homogeneity', defined as the standard deviation Δd of the distribution of the channel diameter throughout the labyrinth, where the channel diameter d is determined from the medial surface skeleton centered within the labyrinthine domains. Curvature homogeneity is defined similarly as the standard deviation ΔK of the distribution of Gaussian curvature. All data are presented for distinct length normalisations: constant surface-to-volume ratio, constant average Gaussian curvature and constant average channel diameter. We provide first and second moments of the distribution of channel diameter for all members of these surfaces complementing curvature data from [A. Fogden, S. Hyde, Eur. Phys. J. B 7, 91 (1999)]. The cubic G and D surfaces are deep local minima of Δd along the surface families (with G more homogeneous than D), whereas the cubic P surface is an inflection point of Δd with adjacent, more homogeneous surface members. Both curvature and packing homogeneity favour the tetragonal route between G and D (via tG and tD surfaces) in preference to the rhombohedral route (via rG and rPD).
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M. Pisani, S. Bernstorff, C. Ferrero, P. Mariani, J. Phys. Chem. B 105, 3109 (2001)
L. Saturni, F. Rustichelli, G. Di Gregorio, L. Cordone, P. Mariani, Phys. Rev. E 64, 040202 (2001)
A. Squires, R. Templer, O. Ces, A. Gabke, J. Woenckhaus, J. Seddon, R. Winter, Langmuir 16, 3578 (2000)
A. Squires, R. Templer, J. Seddon, A. Gabke, J. Woenckhaus, R. Winter, T. Narayanan, S. Finet, Phys. Rev. E 72, 011502 (2005)
R.H. Templer, J.M. Seddon, N.A. Warrender, A. Syrykh, Z. Huang, R. Winter, J. Erbes, J. Phys. Chem. 102, 7251 (1998)
L. Ellison, D. Michel, F. Barmes, D. Cleaver, Phys. Rev. Lett. 97, 297801 (2006)
T.F. Bailey, M. Cordell, T. Epps, F. Bates, Macromolecules 35, 7007 (2002)
T.H. Epps, E.W. Cochran, T.S. Bailey, R.S. Waletzko, C.M. Hardy, F. Bates, Macromolecules 37, 8325 (2004)
G. Schröder, Ph.D. thesis, Australian National University, Canberra, Australia (2005)
H. Blum, J. Theor. Biol. 38, 205 (1973)
L.R. Nackman, Computer Graphics and Image Processing 20, 43 (1982)
G.E. Schröder, S.J. Ramsden, A.E. Christy, S.T. Hyde, Eur. Phys. J. B 35, 551 (2003)
A. Fogden, S. Hyde, Eur. Phys. J. B 7, 91 (1999)
D.M. Anderson, S.M. Gruner, S. Leibler, Proc. Natl. Acad. Sci. USA 85, 5364 (1988)
J. Sadoc, J. Charvolin, J. Phys. France 48, 1559 (1987)
W. Helfrich, H. Rennschuh, Landau Theory of the Lamellar-to-Cubic Phase Transition, in Colloque de Physique C7-1990 (1990), Supplément au J. Phys. France, pp. 189–195
S. Hyde, Curvature and the global structure of interfaces in surfactant-water systems, in Colloque de Physique C7-1990 (1990), Supplément au Journal de Physique, pp. 209–228
U. Schwarz, G. Gompper, Phys. Rev. E 59(5), 5528 (1999)
U. Schwarz, G. Gompper, Phys. Rev. Lett. 85(7), 1472 (2000)
P. Duesing, R. Templer, J. Seddon, Langmuir 13, 351 (1997)
S. Hyde, S. Andersson, K. Larsson, Z. Blum, T. Landh, S. Lidin, B. Ninham, The Language of Shape, 1st edn. (Elsevier Science B. V., Amsterdam, 1997)
A. Schoen, Tech. rep., NASA (1970)
G. Schröder, S. Ramsden, A. Fogden, S. Hyde, Physica A 339, 137 (2004)
W. Helfrich, Z. Naturforsch. 28c, 693 (1973)
J.N. Israelachvili, D.J. Mitchell, B.W. Ninham, J. Chem. Soc. Faraday Trans. 2 72, 1525 (1976)
D. Hilbert, Transactions of the A.M.S. 2 pp. 87–99 (1901)
M. Spivak, Comprehensive Introduction to Differential Geometry, Vol. 3 (Wilmington, DE, Publish or Perish Press, 1979)
V. Luzzati, R. Vargas, P. Mariani, A. Gulik, H. Delacroix, J. Mol. Biol. 229, 540 (1993)
H. Hasegawa, H. Tanaka, T. Hashimoto, C. Han, Macromolecules 20, 2120 (1987)
H. Hasegawa, H. Tanaka, K. Yamasaki, T. Hashimoto, Macromolecules 20, 1651 (1987)
E. Thomas, D. Anderson, C. Henkee, D. Hoffman, Nature 334, 598 (1988)
A. Fogden, S. Hyde, G. Lundberg, J. Chem. Soc. Faraday Trans 87 (7), 949 (1991)
R. Bruinsma, J. Phys. II France 2, 425 (1992)
P. Barois, S. Hyde, B. Ninham, T. Dowling, Langmuir 6, 1136 (1990)
G. Schröder-Turk, A. Fogden, S. Hyde, in preparation (2007)
G. Schröder-Turk, A. Sheppard, S. Hyde, in preperation (2007)
H. Karcher, Manuscripta Math. 62, 83 (1988)
H. Schwarz, Gesammelte Mathematische Abhandlungen. 2 Bände. (Springer, Berlin, 1890)
A. Fogden, M. Haeberlein, S. Lidin, J. Phys. I France 3, 2371 (1993)
S. Lidin, S. Larsson, J. Chem. Soc. Faraday Trans. 86 (5), 769 (1990)
A. Fogden, S. Hyde, Acta Cryst. A48, 442 (1992)
A. Fogden, S. Hyde, Acta Cryst. A48, 575 (1992)
H. Von Schnering, R. Nesper, Angew. Chem. Int. Ed. Engl. 26, 1059 (1987)
H. Von Schnering, R. Nesper, Z. Phys. B – Condensed Matter 83, 407 (1991)
M. Wohlgemuth, N. Yufa, J. Hoffman, E. Thomas, Macromolecules 34, 6083 (2001)
E. Sherbrooke, N.M. Patrikalakis, F.E. Wolter, Graphical Models and Image Processing 58(6), 574 (1996)
N. Amenta, S. Choi, R. Kolluri, Computational Geometry: Theory and Applications 19(2-3), 127 (2001)
N. Amenta, S. Choi, G. Rote, Incremental Constructions con BRIO, in SoCG'03, June 8–10, 2003, San Diego, California, USA (2003), pp. 211–219
D. Attali, A. Montanvert, Computer Vision and Image Understanding 67(3), 261 (1997)
J. Boissonnat, F. Cazals, Natural coordinates of points on a surface, in Proceedings of the 16th Annual ACM Symposium on Computational Geometry (2000), pp. 223–232
T.K. Dey, W. Zhao, Approximate medial axis as a voronoi subcomplex, in Proceedings of the seventh ACM symposium on Solid modeling and applications (ACM Press, 2002), pp. 356–366, ISBN 1-58113-506-8
J. Goldak, X. Yu, A. Knight, L. Dong, Int. J. Computational Geometry and its Applications 1, 327 (1991)
D. Sheehy, C. Armstrong, D. Robinson, IEEE Transactions on Visualization and Computer Graphics 2(1), 61 (1996)
K. Grosse-Braukmann, J. Colloid and Interface Science 187, 418 (1997)
V. Luzzati, A. Tardieu, T. Gulik-Krzywicki, E. Rivas, F. Reiss-Husson, Nature (London) 220, 485 (1968)
K. Larsson, F. Tiberg, Current Opinion in Colloid and Interface Science 9, 365 (2005)
A. Finnefrock, R. Ulrich, G. Toomes, S. Gruner, U. Wiesner, J. Am. Chem. Soc. 125, 13084 (2003)
P. Kekicheff, B. Cabane, J. Phys. France 48(9), 1571 (1987)
A. Levelut, M. Clerc, Liquid Crystals 24(1), 105 (1998)
M. O'Keeffe, B. Hyde, Crystal Structures, I. Patterns and Symmetry (Mineralogical Society of America Washington, USA, 1996)
S. Tolbert, T. Schaffer, J. Feng, P. Hansma, G. Stucky, Chemistry of Materials 9(9), 1962 (1997), http://dx.doi.org/10.1021/cm960454o
S. Tolbert, T. Schaffer, J. Feng, P. Hansma, G. Stucky, Chemistry of Materials 9(9), 1962 (1997), 10.1021/cm960454o
M. Clerc, J. Phys. II France 6, 961 (1996)
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Schröder-Turk, G., Fogden, A. & Hyde, S. Bicontinuous geometries and molecular self-assembly: comparison of local curvature and global packing variations in genus-three cubic, tetragonal and rhombohedral surfaces. Eur. Phys. J. B 54, 509–524 (2006). https://doi.org/10.1140/epjb/e2007-00025-7
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DOI: https://doi.org/10.1140/epjb/e2007-00025-7