Adhesion of membranes and filaments on patterned surfaces

Transcription

1 Adhesion of membranes and filaments on patterned surfaces Olivier Pierre-Louis CNRS / Oxford Theoretical Physics 1 Keble Road, Oxford OX1 3NP, UK olivier.pierre-louis@physics.ox.ac.uk Spring 2008

2 Nanotube on ripples: Derike et al, NanoLetters (2001). Graphene on rough SiO 2 : E.D. Williams et al, NanoLetters (2007); scale-bar 2nm.

3 L. Tan et al PNAS (2003); scale-bar 10µm Simple (fluid) lipid membrane on a non-flat surface? M. Abkarian A. Viallat, Biophys J. (2005); D = 130µm

4 Bico, Marzolin, Quéré, Europhys. Lett (1999). Wenzel to Cassie-Baxter transition Analogous transisiton for soft objects?

5 1D model: Filament or membrane on ripples Fakir Carpet Crenellated Sinusoidal Saw tooth Wavelength λ Amplitude ǫ

6 Total energy E = ds [ ] C 2 κ(s)2 + σ + V (r(s)). outside solid V = 0, surface V = γ, inside V = +. i.e. Deformations l eq Adhesion energy γ Bending rigidity C Tension σ Deformations l eq Swain and Andelman, PRE eq l l eq

7 Equilibrium equations I: δe = 0 κ>0 κ<0 s Free parts C ss κ + C 2 κ3 σκ = 0.

8 Equilibrium equations II: δe = 0 B F (BC1) F B (BC2) (BC3) Boundary conditions BC1 where κ eq = (2γ/C) 1/2 BC2 BC3 κ F = κ B κ eq, κ B κ eq κ F κ B + κ eq. κ + = κ, and s κ + s κ,

9 Small slope approximation x h 1, λ ǫ κ xx h d 2 xxxx h xx h = 0 h(x) = A 1 e x/d + A 2 e x/d + A 3 x + A 4 d = (C/σ) 1/2 cutoff length d h(x) h 0 1 S(x) x

10 Constructing solutions from n-bridges n=1 n=2 n= All possible solutions Non-overlapping reduces the number of possible states

11 Bridges for various surfaces n=1 n=2 n= λ 3 x A single family of bridges for: sinusoidal more for other surfaces fakir carpet and

12 Energy of a state φ n : number of n-bridges divided by the total number of bridges Φ = {φ n ;n = 1, 2,3..} n=1 φ n = 1 Energy density G[Φ] = σ + γg[φ,α, β] Superposition formula g[φ, α,β] = ( ) 1 nφ n g n [α,β] nφ n. n=1 n=1 g n energ. dens. of np state np: φ n = 1, φ m = 0, n m States with same Φ have same energ. dens. Adhesion energy G[Φ] G F = γg[φ,α,β] floating state: G F = σ Ground state either Floating or periodic

13 Natural parameters β = λ d β 1 tension dominates β 1 curvature dominates α = ( ) 1/2 κeq κ g geometrical curvature κ g = 4π 2 ǫ/λ 2 α 1 Floating state α 1 Membrane follows patterns σ C λ β ε γ 0 α

14 Fakir carpet surface n=3 λ 3 x 3 x n = [λ n (n 1)λ]/2 0 = ζ + ζ cosh[ βx n λ Energy density of np sinh[ βx n λ β cosh[ βx 4π 2 α 2 n λ ] sinh[ βx n λ ] βx n λ + (n 1)β 2 ] + (n 1)β 2 ] cosh[ (n 1)β 2 ] gn[α,β] f = 1 n ( 1 + 2x n λ ) + 2 nβ [s 1 + ζc 1 ] 2 b + 2 [ (1 + ζ 2 ) s 2 nβ 2 + ζc 2 + ζx ] n d 2ζc 1 2ζ 2 s 1 c p = cosh[pβx n /λ] 1, s p = sinh[pβx n /λ], b = coth[(n 1)β/2]..

15 Fakir-carpet surface 0,5 1 1,5 α β n 8α F 1P 3P 2P 4P np, 0 σ C λ γ ε α β

16 Decimation of states n m α > α [β] n m n m α < α [β] n m α n m [β] > α n m+1 [β] α n m [β] = α m n [β] α n m [β] > α n m+p [β] But no total order, e.g. α 5 5 [0] < α 4 9 [0], and α 5 5 [1] > α 4 9 [1] β-dependent decimation order Non-sticky F ground state at α < α [β]

17 Fakir-carpet surface α 0,5 1 1,5 10 F β α3 α α 2 2 α 1 2 8α2 α3 3 α 1 8 8α α 1 1 1P 3P 2P 4P np, n 8 σ C λ β ε γ 0 α

18 Sinusoidal surface. 30 F α 1 1,5 2 2,5 1P β 10 0P α 0,8 0,82 0,84 0,86 0,88 0,9 F β n np 1 0P P P 0 1 4P σ C λ β ε γ 0 α

19 Small slope approx C 0 BC1 σ(1 + cos θ) = γ where θ contact angle σ γ, and x h s 1. θ Self-consistent limit for sinusoidal but not for Fakir Carpet and Crenellated. Wenzel to Cassie-Baxter transition for sinusoidal

20 Small slope approx σ 0 ǫκ eq 1 ε

21 Orders of magnitude Graphene C = 0.9eV, γ 6meVÅ 2, σ 0 ǫκ eq 1. ǫ = 1nm, and λ = 10nm, α 2 larger than 100nm follow A. Incze, A. Pasturel and P. Peyla, Phys.Rev.B (2004) Oxygen adhesion tunes the bending rigidity: 12.5% oxygen C = 40eV Peyla et al ǫκ eq < 1. ǫ = 1nm, and λ = 10nm, α = 0.6. Oxygen adhesion scan tranisiton region

22 Orders of magnitude lipid membranes(swain and Andelman) C = J, and σ = Jm 2 γ = Jm 2, l eq = 3nm Choosing ǫ 50nm l eq, λ 500nm ǫ, we obtain α 1.5 and β 5 Nanotubes σ 0, C = 20eV.nm, and γ 1eV.nm 1 Choosing ǫ = 5nm, λ = 50nm we obtain α 2 (Nevertheless ǫκ eq 1)

23 Conclusion (1) infinite staircases of np ground-states; (2) complex decimation sequences; (3) non-sticky surfaces with F groundstate. Other surfaces crenellated saw tooth more bridges, but similar behavior: Results (1), (2), and (3) are also obtained for crenellated and saw-tooth patterns suggesting that these features should occur for a wider range of pattern shapes.