Nanotobes and Spheric Vesicles Induced by Concentrating Solution

Transcription

1 The geometric elastic theory of fluid membrane vesicles with a liquid crystal model: the Helfrich model of lipid bilayers and its application to soft matter Ou-Yang Zhong-can Institute of Theoretical Physics, Chinese Academy of Sciences oy@itp.ac.cn The seminar of MLC programm, Issac Newton Institute of Mathematical Sciences, June 6, 013

2 Outline: I. Shape problem in material science: Crystals and soap bubble II. Shape of red blood cell and elastic theory of membranes in liquid crystal phases: Helfrich Model for Bilayer Vesicles III. Helfrich Model for Multi-layer Vesicles (I): Focal conic structure of smectic liquid crystal IV. Helfrich model for Multi-layer Vesicle (II): The shapes of fullerenes and carbon nanotubes V. Helfrich model for Reversible Transition between Peptide Nanotobes and Spheric Vesicles Induced by Concentrating Solution VI. Helfrich model for Icosahedral Self-Assemblies VII. Shape formation in D lipid Monolayer at air/water interface VIII. Conclusion

3 I. Shape problem in material science 1. The equilibrium shapes of crystals N. Stensen (1669): law of constant angle of crystal planes. G. Wulff (1901): construction of equilibrium crystals. It shows surfaces of crystals to be convex. F = γ( nda ) + λ dv δ F = 0 γ ( n) = (1/) Y n

4 . The shape of fluid films The soap films ---- Minimal surfaces,j. Plateau (1803) F δ F = γ da = 0, H = 0 Soap bubble ---- sphere, T. Young (1805), P.S. Laplace (1806) Δ P= P P o F =Δ P dv + γ da δ F i ΔP = 0, H = = γ 1 R There is only one solution for H=const, the sphere Alexandrov (1950 s) R

5 II. Shape of red blood cell and elastic theory of membranes in liquid crystal phases: Helfrich Model for bilayer vesicles (1973) 1. Shape of red blood cell (RBC) RBC --- the unique cell without nucleus in human body, its shape depends on the cell membranes and environment in physiology. Why the RBCs in human body are always in a rotationally symmetric and biconcave, neither convex nor spherical?

6 . The explanations of RCB shape by biomechanical scientists E. Ponder (1948) --- High deformability for transportation of oxygen in capillary blood vessels Y.C. Fung & P. Tong (1968) --- The thickness of membrane varies from region to region to regulate the biconcave shape, contradictory to observation under electron microscope (EM). L. Lopez et al (1968) --- The difference in electric charge distribution over surface. But Greer & Baker (1970) measured a uniform distribution of charge. J.R. Murphy (1965) --- Regional variations in distribution of cholesterol in membranes, but not supported by experiment (P. Seeman et al 1973)

7 Helfrich Model for Elasticity of Lipid Bilayers Derived by Liquid Crystal (LC) Curvature Elastic Energy(1973) Free energy of curvature elasticity in LC: [F.C. Frank(1958)] F = g dv LC LC cell --- Director: average orientation of long axis of LC molecules 1 ( ) ( ) ( ) glc = k11 d S0 + k d d k / k + k33 d d ( )( ) 1 ( ) ( ) ( ) k1 d d d k + k 4 d + d d : d = 0 : Achiral (Nematic LC) k 0 : Chiral (Cholesteric LC)

8 Curvature Elastic Energy of Biomembrane F = g dv = t gda 1 g = k( H + C ) 0 + kk ---- Helfrich (1973) 1 k = k11t 10 erg k = ( k + k ) t 0 0 LC 4 C = S / t---- Helfrich spontaneous curvature H-- mean curvature, K-- Gaussian curvature

9 The SCI cited number of the pioneering paper by W. Helfrich, Naturforsch C 8 (1973) 693 ( ) W. Helfrich. Elastic Properties of Lipid Bilayers: Theory and Possible Experiments, Z. Naturforsch. 8 c, [1973]

10 Helfrich Variation: General shape equation for an equilibrium vesicle: F = gda+δ P dv + λ da Δ P= Po Pi :Osmotic pressure difference λ=γ+ γ :Surface tension o i δ F = 0, Ou-Yang & Helfich (1987,1989) ΔP λh + k( H + C )(H K C H) + k H = 0 k 0 Eq. Δ P= λh 0 0 Generalized Laplace formula [S. Komura & K. Seki (1993); S. Ljunggren & J.C. Erilkssen (199)]

11 The biological function of sphere solution: Laplace Equation-- There is only one solution for H=const, the sphere but Helfrich Variation have 3 cases ΔP r + λr kcr( Cr) = Transport of Protein through membrane: Exocytosis,pinocytosis endocytosis

12 Shape equation of rotationally symmetric vesicles ρ = 0 + z z tan ψ ( ρ ) d ρ d ψ d ψ dψ 1 dψ 7sinψ cos ψ dψ ψ = ψ ψ ψ ψ ψ + 3 cos 4sin cos cos sin cos dρ dρ dρ dρ ρ dρ cos ψ d ψ C0 C0sinψ sin ψ λ sin ψ cos ψ dψ cos ψ ρ dρ ρ ρ k ρ dρ 3 ΔP λsinψ sin ψ C0 sinψ sinψ cos ψ k kρ ρ ρ ρ J.G. Hu & Ou-Yang (1993)

13 Previous Shape equation of rotationally symmetric vesicles cos d ψ sinψ cosψ dψ cos ψ dψ sin ψ ψ = + d ρ dρ ρ dρ ρ ΔPρ λsinψ sinψ + + c kc cosψ cosψ ρ 0 (I) Eq.(I): ( ) δ F ψ( ρ), dψ / dρ dρ = 0 H. Deulin & W. Helfrich(1976); J. Jenkins (1977); M. Peterson (1985); S. Sevetina, & B. Zeks (1989); L.Miao, B.Fourcarde, M.Rao, M. Wortis, & R.K.P. Zia (1991)

14 Eq.(II): ( ) δ F ψ(), s dψ()/ s ds, d ψ ()/ s ds ds = 0 J. Berndl, J. Käs, R. Lipowsky, E. Cackmann, & U. Seifert (1990); U. Seifert (1991); U. Seifert, K. Berndl, & R. Lipowsky (1991). 3 3 d ψ cos ψ d ψ dψ dψ cos 3sin cos cos sin 3 = ψ ψ + ψ ψ dρ sinψ dρ dρ dρ 3 ( + 5sin ψ) cos ψ dψ cos ψ d ψ c0 sinψ sin ψ ΔPρ dψ + cosψ + + ρ sinψ dρ ρ d ρ ρ ρ kc cosψ dρ ΔP λsinψ c0 sinψ sin ψ(1 + cos ψ) (II) 3 kc kcρ ρ ρ

15 Shape equation of rotationally symmetric vesicles ρ = 0 + z z tan ψ ( ρ ) d ρ d ψ d ψ dψ 1 dψ 7sinψ cos ψ dψ ψ = ψ ψ ψ ψ ψ + 3 cos 4sin cos cos sin cos dρ dρ dρ dρ ρ dρ cos ψ d ψ C0 C0sinψ sin ψ λ sin ψ cos ψ dψ cos ψ ρ dρ ρ ρ k ρ dρ 3 ΔP λsinψ sin ψ C0 sinψ sinψ cos ψ k kρ ρ ρ ρ J.G. Hu & Ou-Yang (1993)

16 Helfrich Variation Predict RBC Shape : Analytic solution for shape of RBC [H. Naito, M. Okuda, Z.C. Ou- Yang (1993)] sin ψ( ρ) = C ρln( ρ/ ρ ), C < 0 biconcave 0 B R0 = A/ 4π = 3.5 um, C0R0 = 1.6 = E.A. Evans, Y.C. Fung, Microvasc. Res. 4 (197) 335 The biofunction of C 0 [Ou-Yang, Hu J.G., & Liu J.X. 199] Δ ψ = ψ ψ < 0 (Electric potential of membrane) o E= Δψn/ d i 1 Golden Section

17 LC flexoelectric (piezoelectric) effect [R.B. Mayer, 1969] P = e n n = e n( H) Δ F = da P Edz 0 C = e Δψ / k d / 1 11 = 10, = 10 yields : Δψ=-15.0 e dyne k erg mv Exp: U.V. Lassen, O. Sten-Knudsen, J. Physiol 195 (1968) 681: Δψ=-14.0mV

18 1990 s textbook Molecular and Cell Biophysics R.J. Nossal & H. Lecar, (Addison-Wesley, 1991) regards W. Helfrich LC curvature elasticity model of membranes as the interpretation of RBC shape

19 Predicted torus solutins[ou-yang Z.C., 1990] 3 CR 0 < 4 R/ r =, ( ) 1 π R r Confirmed by experiments: M. Muty & D. Bensimon, PRA, 1991, 4 tori; A.S. Rudolph et al, Nature, 1991, in Phospholipid membrane; Z. Lin et al, Langmuir, 1994, in Micelles

20 Polygon Deformation Instability of spherical vesicle and myelin form k biconcave Δ P > 3 (6 C0r0 ) W. Helfrich (1973) r instability 0 k Δ P > 3 [ l( l + 1) C0r0] Ou-Yang, Helfrich (1987) r0 l =,3,4; deformations corresponding to spherical harmonics: Y ( θφ, ) lm H. Hotani, J. Mol. Biol. 178, 113 (1984)

21 Myelin Form is the Conic function Solution of Shape Equation 1 l = + iq P ( θ ) Conic function 1 + iq Zhou J.J. et al, IJMPB 15 (001)

22 Dynamics of vesicles and micro-emulsion droplets Navie-Stokes Eq.: V = 0 ρ out, in V + V V t out, in out, in out, in out, in = P + η V out, in out, in out, in Generalized Laplace Eq.: f n = 0 ΔP λh + k( H + C0)(H K C0H) + k H n =Δ σ : n σ σ out σ in Δ = ( V ) ( V ) σ out, in = ηout, in out, in + out, in αβ βα αβ

23 Theory of helical structure of tilted chiral membranes (TCM) Free energy of CLC: Free energy of TCM: F = k t cos θ d dl k sin θ cosθ τ da Geodesic torsion: τ =( C C )sinϕcosϕ g g = k d d LC In 1993, 1996 Schnur (Science, PNAS, PRL) indicated the result in agreement with observation but different from those theories proposed by de Gennes, Lubensky- Prost. 1 o δf = 0, ϕ=45 [ Ou-Yang, Liu, PRL 65, (1990) 1679; PRA 43, (1991) 686] g first approximation of Frank free energy

24 High- and Low-pitch helical structures of TCM in cholesterol crystallization [Komura, Ou-Yang, PRL 81 (1998) 473] By taking complete Franck free energy, Two TCM helices: (1) helical ribbon with parallel packing of molecules; () helical ribbon with antiparallel packing of molecules. We show the helical angle of (1) <45 o and that of () to be 1/ o φ0 = arctan cos arccos + = Exp. In cholesterol crystallization in native and model biles [D.S. Chung, et al, PNAS 90 (1993) 11341] o φ0 = 53.7 ± 0.8, φ = pal 11.3 o

25 III. Helfrich Model for Multi-layer Vesicles (I): Focal conic structures in Smectic A LC and general variation problem of surfaces J.C.C. Nitsche (1993) regarded Helfrich FM theory as the renewal of the Poisson s elastic shell theory. In his encyclopedic book on minimal surface, the Helfrich energy was generalized to the form: [ ψ( ) γ ] F = H K da The equation of δ F = 0 is: ( H K) 4H 0, ψh + ψh ψ = ψh = ψ H

26 Shape problem in SmA LC: The configuration of minimal energy: Dupin cyclides are always formed when LC cools from Isotropic phase to SmA: G. Friedel, Annls. Phys. 18 (19)

27 W. Bragg, Nature 133 (1934) 445. Why the cyclides are preferred to other geometrical structures under the preservation of the interlayer spacing? H. Naito, M. Okuda, Z.C. Ou-Yang, PRL 70 (1993) 91; PRE 5 (1995) The relieved energy of the difference in Gibbs free energy of I- SmA transition must be balanced by the curvature elastic energy of SmA layers. Assume the outward growth of a SmA layer of thickneess d: δ F = ( k d/) ( H) da+ k d KdA C A 11 5 ( ) δf = γ Hd + d K da 1 3 δ FG = g0 ( d d H + d K) da 3 δf + δf + δf = 0 C A G (1) Shape: Differential Eq. of SmA nucleus surface; () Thickness: Integral Eq. of the serface

28 The most general equation of surface variation: δf = δ Φ ( D, H, K) da= 0, D = SmA thickness Surface Eq.[PRE,1995] δf/ δa= 0 : 1 + Φ + + Φ Φ = Φ = Φ/ H, Φ = Φ/ K ( H K ) H ( HK ) K H 0 H 1 ij = i g g j 1 g ( ) ij = i KL g j g K ( ) Integral Eq. F/ D = 0, Φ/ DdA= 0 Solving both equations gives good explanation of FCD. H,K are mean and Gaussin Curvatures at inner surface D is the thickness of multi-layer vesicle

29 Detailed derivation : Geometric Methods in the Elastic Theory of Membrane in Liquid Crystal Phases Ou-Yang Zhong-Can, Liu Ji-Xing, Xie Yu-Zhang, World Scientific,

30 IV. Helfrich Model for Multi-layer Vesicle (II): The shapes of fullerenes and carbon nanotubes; Lenosky s lattice model E ( s) b = ε 1 u ij i j ( 1 nn. ) i j + ε i, j Contribution of bond angle changes Contribution from the bending of nearest-neighbor fullerene surface + ε ( nu. )(. ) i i j nj uji 3, i, j Lenosky T., Gonze X., Teter M. and Elser V., Nature, 355 (199)

31 Basic idea is same as the FCD formation in I S A LC: The relieved energy of free carbon atoms to form graphite must be balanced by the elastic energy of curved graphite. Lenosky et al, Nature 355 (199) E = ε1 uij + ε ( 1 ni nj ) + ε3 ( ni uij )( nj uji ) i < j> < i, j> < i, j> ( ε, ε, ε ) = (0.96,1.9,0.05) ev (local density approximation) Ou-Yang Z.C., Z.B. Su, C.L. Wang, PRL 78 (1997) 1 Eb = kc( H) + KK da 1 kc = ( 18 ε1+ 4 ε + 9 ε3)( a / σ) = 1.17 ev 3 K = 8ε + ε k / 6ε + 8ε + 3ε = 1.56k ( ) ( ) 1 3 C 1 3 C

32 Multi-wall fullerenes and nanotubes viewed as SmA LC F = F + F + F b A V ( ) F = E = π k / d ln( ρ / ρ ) F b b C 0 i A V = πγ ( ρ + ρ ) L 0 ( i ) F = πg ρ ρ L 0 0 Surface Eq. ----C 60, carbon ring δ F = 0 κ Curve Eq.----nanotube 3 m kss + k kτ k = 0 α k τ = const Straight tubes: k, τ = 0 Helical coils: k, τ = const 0 i, τ κ -curvature τ -torsion

33 V. Helfrich Variation for Reversible Transition between Peptide Nanotobes and Spheric Vesicles Induced by Concentrating Solution Amphiphilic Peptide Nanotubes Sufactant-like peptide nannotubes S.I.Stupp et al, Science, 004,303,135 S.G. Zhang et al, PNAS, 00,99,5355 Bola-amphiphilic peptide Synthetic peptides H.Matsui et al, J. Phys. Chem. B, 000,104,3383 D. T. Deming, Nat. Mater. 007, 6,

34 Dilution of peptide induced tube to vesicle transition through necklace-like structure a) b) 100 nm c) d) 00 nm 50 nm TEM 图 : a) 8 mg ml -1, b) 7 mg ml -1, c) 5 mg ml -1 及 d) 1 mg ml

35 Fluorescence images Tube to Spherical Vesicle transition: Joined necklace-like structures

36 Challenge: when C > CMC what happens? (CMC)

37 he self-organisation of the molecules of surfactants and lipids depends on he concentration of the lipid present in solution. Below the critical micelle concentration the lipids form a single layer on the liquid surface and are dispersed in solution. At the first critical micelle concentration (CMC-I), the lipids organize in spherical micelles, at the second critical micelle concentration (CMC-II) into elongated pipes, and at the lamellar point LM or CMC-III) into stacked lamellae of pipes. The CMC depends on the chemical composition, mainly on the ratio of the head area and the tail length. Figure 1: Structure of a Lipid.... Schematic of a micelle

38 H. Naito, M. Okuda, Z.C. Ou-Yang, PRL 70 (1993) 91; PRE 5 (1995) Focal Conic Formation in SmA LC: The relieved energy of the difference in Gibbs free energy of I-SmA transition must be balanced by the curvature elastic energy of SmA layers. Regarded Nano-structure Formation in Peptide as Focal Conic Formation in SmA LC. The focal point is what is free energy of solution-aggregate transition Assume the outward growth of a SmA layer of thickneess d: δ F = ( k d/) ( H) da+ k d KdA C A 11 5 ( ) δf = γ Hd + d K da 1 3 δ FG = g0 ( d d H + d K) da 3 δf + δf + δf = 0 C A G (1) Shape: Differential Eq. of SmA nucleus surface; () Thickness: Integral Eq. of the serface. g>

39 The free energy of solution-aggregate transition: Regarded it as the work of compressing gas 1. Estimate aggregate energy: compressing a molecule from solution state to aggregate phase with idea gas model need a work kbt ln( CA / CS ) This work is provide by the energy relieved due to aggregation. i.e., for an aggregate volume δv the aggregate bulk energy is = g δv δf V 0 g = C k T ln( C / C ) (1) 0 A B A S C concentration of aggregate phase., A CS concentration in solution

40 The Energy Variation F / d = 0 The equilibrium surface must be a Weingarten surface

41 ( 0 γ 1/ + g ) γ / γ 0k kg ( / γ + k g ) γ k g / γ

42 From Eq.(4) we find free energies for sphere and cylindrical tube: 3 3 F sphere = ( g0 d /1γ + g0 d / 4γ ) F g tube = d 0 / γ Ftube F g d 3γ sphere 0 g = C k T ln( C / C 0 A B A S ) C S C A e 3γ / C A dk B T CTVT Nanotube must transform into spherical vesicle at the Critical Tube-to -Vesicle Transition Concentration during dilution

43 Theoretical Model: Finding CTVC CTVC = C A e -3γ/C Adk B T diluting d C A thickness of tube γ tension of solution/ aggregate interface concentration of aggregate phase. Peptide nanotube concentrating Vesicle CTVC critical tube-to-vesicle concentration Reversible Transition C S concentration In solution C=molecule number pervolume Yan et al & OY, Chemistry - A European Journal, Vol. 14, (008) 43

44 Metastable necklace-like structure can be described as Delaunay surface (1841) k 5 If value is ignored, the solution of above shape equation is a Surface with H=constant. In 1884, Delaunay find beautiful way to construct such a surface with rotationally symmetry: By rolling a given conic section on a straight line in a plane, and then rotating the trace of a focus about the line, one obtains the surface

45 Delaunay Constructing: Surface of H=const with rotational symmetry a) b) f c) Tube to Spherical Vesicle transition: Through Joined necklace-like structures

46 Fluorescence images Tube to Spherical Vesicle transition: Joined necklace-like structures

47 VI.Helfrich Model for The Elastic Energy Model of Icosahedral Self-Assemblies Experimentally, virus capsids are often Icosahedral Cowpea Chlorotic Mottle virus (CCMV) Herpes Simpler Virus Cryo-TEM (Cryogenic-temperature Transmission Electron Microcopy) reconstruction of CCMV: Truncated icosahedron J. M. Fox et al., Virology 44, S. Saad et al., J.Virol. 73, Cryo-TEM reconstruction of Herpes Simpler Virus, 5 times bigger than CCMV

48 Recently, surfactant aggregates are found to be Icosahedral bilayer Why icosahedron is preferred to the other four regular polyhedra? Cryo-TEM images surfactants aggregates Sketch of aggregate structure: icosahedron with pores at each vertex M. Dubois, B. Dem e, T. Gulik-Krzywicki, J. Dedieu, C. Vautrin, S. D esert, E. Perez, and T. Zemb, Nature 411, 67 (001)

49 There are only 5 regular polyhedra Tetrahedron Cube or Hexahedron Octahedron (p,q) Schläfli symbol N 0 Vertex Number N 1 Edge Number N Face Number Dihedral Angle (3, 3) o (4, 3) (3, 4) α Dodecahedron (5, 3) Icosahedron (3, 5) (p,q) means the regular polyhedron has q regular p-gon faces around each vertex

50 Previous models focus on geometrical parking proposed by Crick and Watson A small virus must be built up by the regular aggregation of smaller asymmetrical identical sub-units The explanation predicts that the spheric virus are cubic symmetry, however, the actual virus are icosahedric symmetry Howeve the number of morphological units observed on the surface of icosahedral viruses is never 60 or a multiple of 60 and in most cases it is greater than 60 F. Crick and J. Watson, Nature 177, 473 (1956)

51 The quasi-equivalence theory of Capspar and Klug Quasi-equivalence theory: the surface is made up of quasiequivalent triangles and that these are grouped in hexamers and pentamers. k h The extension plane of an icosahedron with each face made up of 9 triangular facets D. Caspar and A. Klug, Cold Spring Harb Symp. Quant Biol. 7, 1 (196)

52 Puzzles of Capspar-Klug theory For Polyoma virus (Rayment et al., 198) and Simian Virus 40 (Liddington et al., 1991) Capspar-Klug theory predicts the above two types of virus are built up of hexagons and pentagons, however, the experiment shows that they are built up only by pentagons The above approaches focus on the packing feature of molecules in capsules without consideration of the elastic energy of the capsules

53 Our Mdel: From Lenosky s lattice model to Helfrich Curvature Elasticity Our model will estimate the elastic energy of icosahedron based on Lenosky s lattice model which describes the energy of curved graphite layers E ( s) b = ε 1 u ij i j ( 1 nn. ) i j + ε + ε ( nu. )(. ) i i j nj uji i, j 3, i, j Contribution of bond angle changes Lenosky T., Gonze X., Teter M. and Elser V., Nature, 355 (199) 333. Contribution from the bending of nearest-neighbor fullerene surface

54 Helfrich energy of smooth curved surface from Lenosky s lattice model We have derived the continuum form of the Helrrich elastic energy for a smoothly curved fullerene from Lenosky s lattice model ( s) 1 E ~ bs [ κc ( H) + κk] da For carbon nanotubes: 1 κc = (18 ε1+ 4 ε + 9 ε3 )( a / σ ) 3 κ = (8ε + ε ) κ / (6ε + 8ε + 3 ε ) 3 c 1 3 σis the occupied area per atom Z. Ou-Yang, Z. Su, and C. Wang, Phys. Rev. Lett. 78, 4055 (1997)

55 E ( s) bp The Helfrich elastic energy of { = N ε (4 4cos β) 1 polyhedra cos β + 1/ sin β + ε 4(1 ) + 5/4+ cosβ (5 / 4 + cos β ) sin β + ε3 } 4(5 / 4 + cos β ) If subunit and its nearest neighbors are in the same plane, its contributions to the elastic energy is zero, only the subunits besides each edge contribute to the elastic energy. N is molecular number in edges related to the surface area and kind of polyhedra. β = π / α / S N = N1 tan( π / p) a cosθ Li Zhou,OY, Europhys. Lett. 9,68004 (010)

56 There are only 5 regular polyhedra Tetrahedron Cube or Hexahedron Octahedron (p,q) Schläfli symbol N 0 Vertex Number N 1 Edge Number N Face Number Dihedral Angle (3, 3) o (4, 3) (3, 4) α Dodecahedron (5, 3) Icosahedron (3, 5) (p,q) means the regular polyhedron has q regular p-gon faces around each vertex

57 The elastic energy of icosahedron is the lowest S- the area of the capsid,obviously energy increases with area

58 Sphere to Icosahedron transition Elastic energy of a sphere: ( s) 1 Ebs = [ κc ( H) + κk] da= 4 π( κc + κ) We have obtained the relationship κ = ε + 10 ε /9 + ε /9 c κ / κ = ν 1 c 1 3 Elastic energy of a sphere independent on area S. Above a threshold of S icosahedron must change into sphere Elastic energy of a sphere: E = 4 π(1 + ν)(ε + 10 ε /9 + ε /9) ( s) bs 1 3 Z. Ou-Yang, Z. Su, and C. Wang, Phys. Rev. Lett. 78, 4055 (1997)

59 VII. Shape formation in D lipid Monolayer at air/water interface Observation by Brewster angle microscopy (BAM) in lipid monolayer BAM-MDC Measurement System Different phases show different reflectivity. Can image monolayer phases CCD P Barrier TL Ob θ B θ B PH P 1 HeNe laser Electrode 1 Shield A Electrometer Fig. 5 Electrode Fig. 4 PC

60 Shape formation in two-dimensional lipid domains θ No regular circle domain and Cusp appears in the boundary Boojum (or Peach)-like and Kidney-like domains Boojum is an imaginary animal, a particularly dangerous kind of snark, (Lewis Carroll, <The Hunting of the Snark>, <Alice in Wonderland>)

61 I. Introduction: Shape formation in two-dimensional lipid domains θ No compact circle but torus appears

62 Approximate Theory of Shape formation in two-dimensional lipid domains: An Artificial cut-off in calculation must invoked The energy of dipole-dipole interaction D. Andelman et al., Acad. Sci., C 301, 675 (1985); D. Andelmann et al., J. Chem. Phys. 86, 3673 (1987) [CAS]. γ Fluid phase p Solid i p o phase μ Normal to surface Δ P = p p o i θ F μ( r ) μ( r ) μ t ( l) t ( s) = da = dlds dipole 3 r r r ( l) r ( s) t ( s) = dr ( s) / ds Green's-theorem

63 Approximate Theory of Shape formation in two-dimensional lipid domains: Artificial cut-off in calculation 1 t( l) t( s) rl ( ) rs ( ) F =Δ P da+ γ ds μ dlds Δ P = g, Gibbs free energy density difference between outer 0 (fluid) and inner (solid) phases μ ----Dipole density, t dr / ds γ -----Line tension, γ = Tangential Vector of Boundary of Domain Iwamoto, Ou-Yang,Phys. Rev. Lett. 93 (004)

64 Key step: Assuming smooth boundary and using Tyler s expanding with Frenet formulas of curve F μ 1 t( s) t( s+ x) = μ ( dx) ds rs ( + x) rs ( ) dt / ds = κ ( s) m( s), Frenet formulas of a plane curve, k is dm( s) / ds = κ ( s) t ( s) curvature of boundary. Introducing artificial cutoff to prevent divergence: r ( s + x) r ( s) h 1 ( ) ( ) ( ) ( ) [ ( ) ( )] t s + x = t s + κ s m s x + κ s m s κ t s x 1 rs ( + x) = rs () + t() sx+ κ () smsx () L 11 F 1 = μ ln ds + μ L κ μ ( s) ds Negative line tension and h 96 Bending energy L---boundary length, integral dx from 0 to L induces divergence, so integral L takes from h-(cutoff ) to L, h is artificially regarded as dipole-dipole - separation distance, or the thickness of the monolayer

65 Dipole energy form L F 1 11 ln ds L μ = μ ( s) ds h + μ 96 κ Makes variation of D free energy to get shape equation being mapped into known 3D variation of membrane vesicle 3D Shape Equation D Shape Equation Δ P = g, 0 Gibbs free energy density difference between outer (fluid) and inner (solid) phases θ (PRL,1987) (PRL, 004) μ 11 λ = γ ln L, α = μ L h 96 Origin of dipole energy is due to orientational ordering of polar molecules M.Iwamoto and Z.C. Ou-Yang, PRL 93,06101 (004)

66 F =Δ P da+ λ ds+ α κ ds μ 11 λ γ L α μ h 96 = ln, = L Size and length-dependent line tension and line Curvature elastic modulus positive λ --shorten boundary, Circle-like domain. Negative----thin the shapes with increasing size, Circle Instability δ F P = Δ λκ + ακ + ακ ss = μ L 11 λ γ ln Lμ κ d he 48 = + s κ = ss d ---Shape- and Size-dependent line tension κ / ds

67 A D circle of radius 0 κ = 1/ ρ κ =+ 1/ ρ 0 0 ρ Inner phase is solid Inner phase is fluid Shape equation of circle domain: 3 ΔP( κ) = λκ ακ ( α 0) is a cubic curve Predication: 1) Two circles and two Tori; ) Only one circle; 3) No compact Circular domain for λ, ΔP

68 D Case: Shape transition of circle Iwamoto, Fei Liu, ZC Ou-Yang JCP, 15, 4701 (006) M=6 M= M=3 α M=4 Circle to m-side quasipolygon Transition

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70 135] M. Iwamoto, F. Liu and Z.C. Ou-Yang, Eur. Phys. J. E 7 (008)

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72 VIII. Conclusion The formation of nanomaterial structures, lipid bilayer vesicles, carbonnanotube, and peptide nanotubes and spherical vesicles and many other soft materials can be found from the formation of focal conic structures in Smectic-A liquid crystals with Helfrich model

73 Thanks for your attention!