Skeletons in the Labyrinth

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1 Gerd Schröder-Turk (with Stephen Hyde, Andrew Fogden, Stuart Ramsden) Skeletons in the Labyrinth Medial surfaces and off-surface properties of triply periodic minimal surfaces of cubic and non-cubic symmetries Darmstadt, 29 August 2010

3 Non-cubic TPMS G Cubic TPMS: G, D, P, I-WP, C(Y), F, S, Families of non-cubic TPMS: td, tp, tg, rpd, rg, H, Transition pathways between cubic G, D, P

4 rpd a rhombohedral TPMS family 0.51 (departure point to rg family) 1/sqrt(2) (Cubic P surface) sqrt(2) (Cubic D surface) minimal bicontinuous embedded not isometric A. Fogden & S.T. Hyde, EPJB, 7, (1999) r

5 Skeletons, Graphs and Domain Sizes Minimal Surface Entwined Graphs Medial Surfaces Source : epinet Definition of Point-wise Domain Size Packing frustration in copolymers

Medial Surface Shrink labyrinth to its 2D skeleton collapsing parallel surfaces skeletal graph = spine medial surface = spine plus rib cage Amenta,

6 Medial Surface Shrink labyrinth to its 2D skeleton collapsing parallel surfaces skeletal graph = spine medial surface = spine plus rib cage Amenta, Choi, Kolluri, Comp. Geom: Th & Appl 19 (2-3), 127 (2001) Gyroid with line graph Medial Surface Schroeder-Turk, Ramsden, Christy, Hyde, EPJ B 35, 551 (2003)

7 Formal definition of Medial Surface Reduce objects to their symmetric axes or skeletons is the locus of centres of maximal spheres is the set of points with 2 or more nearest boundarypoints. p S : ms( p) = p + d(p)n( p) d(p) gives thickness/depth of labyrinth at point p. First description of MS in context of biological shape description and in 2D: Blum, Biological shape and visual sciences, J. Theo. Biol. 38, (1973)

8 Some simple examples

9 P G D G Schröder et al, Eur.Phys.J.B 35, (2003)

10 Computing the Medial Surface as a subset of the Voronoi diagram Triangulated Object Delaunay Triangulation Voronoi Diagram Medial Surface (Amenta et al, 1998)

11 2D Medial Surface Skeleton Medial surface maximally distant to minimal surface Line graphs contained in medial surface for P, D, G Shrink 2D Medial Surface to 1D skeletal graph?

12 Skeletal graph on MS Distance contours d(p) on Medial Surface Skeletal graph = ridge line of distance profile Connect saddles of d to maxima Maximally distant line network with topology of labyrinth

13 Connect saddles to maxima of radius function Maximum Distance on circle Skeletal graph on MS Saddle MS with distance contours Distance on circle Angle

14 Medial Surface Parallel surfaces collapsed to 2D skeleton Maximally distant points to minimal surface Skeletal graph Reduce medial surface to 1D graph Traces lines furthest away from TPMS interesting graphs in non-cubic TPMS?

of distance Edge midpoints = widest points Schröder,

15 Nodes not always widest points Cubic Gyroid Cubic Diamond d(p) Nodes = widest points Nodes = saddles of distance Edge midpoints = widest points Schröder, Ramsden, Christy, Hyde, Eur.Phys.J.B 35, (2003)

16 The centered graph has curved edges Transition from 6- to 4-nodes Nodes not defined by symmetry R3c rpd P (6-coordinated) D (4-coordinated) [Schröder-Turk, Sheppard, Hyde, in prep.]

17 Changes in Coordination number 3-coordinated 4-coordinated

18 Alternative 3 4 transition in tg family tg = retain 4-fold symmetry of cubic Gyroid, but not the 3-fold symmetry. 3-nodes split 3-nodes normal points, mid-edge asymmetric 4-nodes relax 4-node

19 Medial Surfaces and Skeletal Graphs TPMS Medial Surface Skeletal graph Symmetry alone not sufficient to define graph Centeredness of medial surface 4-coordinated to 6-coordinated transition in rpd Alternative 4- to 3-coordinated transition in tg Schröder, Ramsden, Christy, Hyde, Eur.Phys.J.B 35, (2003) [ Schröder-Turk, Sheppard, Hyde, in prep.

20 Part II : Packing frustration in self-assembly processes and what we can learn about them from non-cubic minimal surfaces!

21 Self-Assembly in Lipid-Water Systems hydrophilic head group hydrophobic tail Channel diameter: ~ nm Cubic Q230 and Q214 phases [Luzatti, 1960s]

22 AB-copolymers : Ia3d core-shell Gyroid ( solvent evaporation) A B A and B are immiscible. They would like to phase-segregate!. much Like oil and water Strong covalent bond Macroscopic phase-separation impossible because the strong bond holds A s and B s together! bulk phase of A separated from a bulk phase of B

23 Covalent bonds cannot be broken, hence Lamellar Micro-phase separation Hexagonal (sphere/cylinder) L Bicontinuous

Gyroid Phase in Hard Pears Shown are half the

24 Gyroid Phase in Hard Pears Shown are half the pears those in 1 domain Also shown (in green): Gyroid Medial Surface similar to skeletal graphs details later hard pears Box with fixed volume V Periodic boundary conditions Ellison, Cleaver et al, PRL 97, (2006)

25 Identical particles assemble into homogeneous (point-wise similar) structures How can Medial Surface help us understand Ubiquitous P, D and G structures are cubic Why cubic symmetry? Why (in co-polymers) only the Gyroid? Transitions D G in lipid

26 Point-wise Domain Size from Medial Surface p S : ms( p) = p + d(p)n( p) d(p) gives thickness/depth of labyrinth at point p.

Energetics for monodisperse strongly-segregated co-polymer Gyroid phases Thickness variations of structure stretching penalty of chains Minimise interface between moieties A, B interface (parallel

27 Energetics for monodisperse strongly-segregated co-polymer Gyroid phases Thickness variations of structure stretching penalty of chains Minimise interface between moieties A, B interface (parallel to) minimal surface Polymer coils have to fill space one end on TPMS, the other on MS Coils incur entropic penalty for stretching/squashing (d-<d>)^2 (frustration) AB copolymer Coreshell Gyroid search for structure with least amount of thickness variations ( Homogeneity )

28 Is there a labyrinth of constant thickness? P(d) r Circle or sphere r Parallel planes r cylinder r d these are perfectly homogeneous shapes for elliptic, planar and parabolic geometries! NO: a labyrinth with constant MS distance (or constant curvature*) does not exist P(d) d * Hilbert knew that already

29 Optimal labyrinthine shape? P analyse thickness variations of minimal surfaces! P(d) Start with three candidate structures (of cubic symmetry): D Primitive, Diamond and Gyroid Gyroid best! Schroeder,Ramsden,Christy&Hyde, Eur.Phys.J.B 35, (2003) G d

30 Homogeneity of Self-Assembly Systems Typical molecular shape V, a, l preferred curvature and domain thickness Harmonic approximation of an energy that penalises 1. Deviations from a preferred curvature (bending) 2. Deviations from a preferred radius (stretching) Borrows from Helfrich and concept of preferred molecular shape (assuming minimal surface interfaces and H0=0) Study fluctuations

31 Non-cubic TPMS G Cubic TPMS: G, D, P, I-WP, C(Y), F, S, Families of non-cubic TPMS: td, tp, tg, rpd, rg, H, Transition pathways between cubic G, D, P

32 Is the most homogeneous TPMS cubic? Investigate all (*) Minimal Surface families of tetragonal & rhombohedral symmetry Plot std dev of Gaussian curvature histogram as function of free surface parameter H rg G rpd P D td tg tp D G P G D P [(*) of the regular class, see Fogden&Hyde, Acta Cryst A, 1992] Curvature fluctuations minimal for cubic TPMS

33 Cubic cases minimise curvature and thickness fluctuations Curvature fluctuations H rg G rpd P td D tg G tp P Schroeder,Fogden& Hyde, Eur.Phys.J.B (2006) Channel thickness variations

34 Why do bicontinuous soft-matter structures have cubic symmetry? Because they are more homogeneous! Minimal variations of channel diameter and Gaussian curvature

35 Cubic-To-Cubic Transitions in Lipids Polysaccharid-induced Mezzenga et al, Langmuir 21 (14), 6165 (2005) Pressure-induced Squires et al, Phys. Rev. E 72, (2005) Pisani et al, J. Phys. Chem. B 105 (15), 3109 (2001) Conn et al, Langmuir 24 (6), 2331 (2008) What is transition pathway?

36 Transition structures D G D P G D G Tetragonal pathway Rhombohedral pathway [Schroeder-Turk, Fogden, Hyde, European Physical Journal B, 54, 509 (2006)]

37 H rg G rpd P D td tg D G tp P Curvature fluctuations G Tetragonal and rhombohedral pathways Tetragonal transition has smaller energy barrier D P Channel thickness variations Schroeder,Fogden& Hyde, Eur.Phys.J.B (2006) P G D D G P

38 Which is the optimal transition pathway between D and G? Tetragonal symmetry! Minimal energy barrier (homogeneity) Resolve symmetry of transition structure by time-resolved scattering?

39 Isotropic hexagonal TPMS Hexagonal TPMS family with symmetry P6m2 one free parameter A G D c P [Source: Brakke] For which A is H surface most homogeneous? For which A is H surface most isotropic? a c/a(a)

40 Channel radius variations Curvature variations Value for which curvature Value for which channel radius variations are minimised variations are minimised

41 Unit tensor Equal surface density for x, y and z direction Isotropic Isotropic c/a very different from Hexagonal closed-packed spheres Min/Max Eigenvalue ratio of surface tensor

42 Distribution of surface normals Surface integral K N(r) K K N(r) N(r) r Origin with w(n ) = K δ(n -N(r)) Identical eigenvalues (isotropic) = S w(n ) N N Unit sphere Surface area of that part of K that has surface normal N Different eigenvalues (not isotropic)

43 Unit tensor Equal surface density for x, y and z direction Isotropic Isotropic c/a very different from Hexagonal closed-packed spheres Min/Max Eigenvalue ratio of surface tensor

44 Isotropic Hexagonal bicontinuous structure? Hexagonal TPMS with single free parameter A. The values for A with (a) smallest curvature variations (b) smallest channel radius variations (c) most isotropic distribution of surface directions are very close to each other. Likely geometry for a bicontinuous hexagonal lipid phase

45 Could isotropic H-surface be misidentified as cubic TPMS? Simulated scattering pattern of simple biphasic H surface Schröder-Turk, Varslot, Di Campo, Kapfer, Mickel, in preparation (2010)

46 Conclusions Medial Surfaces as Centered Skeletons TPMS Medial Surface Line Graph Homogeneity and Packing frustration Non-cubic minimal surfaces and their homogeneity: Fogden & Hyde, EPJB, 7, (1999) Schröder-Turk, Fogden, Hyde, Eur. Phys. J.B 54, (2006) Medial Surface construction and homogeneity Gyroid versus Diamond, Primitive: Schröder-Turk, Ramsden,Christy, Hyde, Eur.Phys.J.B 35, (2003) Isotropy analysis of general phases using Minkowski curvature integrals: Schröder-Turk et al, Tensorial Minkowski Valuations and Anisotropy, in preparation (2010)

Monte Carlo simulations of confined hard sphere fluids

Physics Procedia 3 (2010) 1493 1497 www.elsevier.com/locate/procedia Monte Carlo simulations of confined hard sphere fluids G.E. Schröder-Turk, S. Kuczera, D. Hörndlein, F. Krach, T. Hiester, K. Mecke