Surface stress-induced self-assembly

Transcription

1 Surface stress-induced self-assembly Zhigang Suo Work with Wei Lu

2 Solid surface Liquid surface Stretched solid surface Energy per surface atom depends on stretch Stretched liquid surface Energy per surface atom is independent of stretch

3 Surface energy of an elastic solid Reference state stress-free, infinite solid Current State G = ΓA + WV A = area in reference state. V = volume in reference state

4 Surface Stress Current State G =ΓA + WV strain ε = a a a 0 0 Bulk elastic energy W = 1 Eε 2 2 Surface energy Γ = g + fε Surface energy depends on elastic strain Residual stress field in surface layers Surface stress can be either tensile or compressive f ~ ( residualstress) ( layer thickness) surface stress ~10 ( 10 N/m 2 ) ( m)= 1N/m

5 A thin foil Reference state stress-free, infinite solid 1 W = Eε 2 Γ = g + fε 2 G = hw + 2Γ (energy/area) 1 2 ( ε ) = heε + 2( g fε ) G + 2 Select strain to minimize G dg = 0 dε heε + 2 f = 0 f f σ = Eε Force balance: σh + 2 f = 0

6 Elastic sphere and cylinder 2 f σ = R 2 f σ L L =, σt R = ft R Surface stress can be either tensile or compressive Stress in the crystal can be either compressive or tensile

7 Surface stress depends on concentration Wafer Curvature Measurement f = φc f = EH2 6R

8 Translating biomolecular recognition into nanomechanics Fritz et al., Science 288, 316 (2000) Target Molecule

9 STM Scanning Tunneling Microscope Invented by Gerd Binnig, Heinrich Rohrer IBM Zurich, 1980s

10 Self-assembled, nanoscale monolayer patterns O on Cu (011) N on Cu (100) Ag+S on Ru (0001) Kern et al. Phys. Rev. Lett., 67, 855 (1991) Parker et al. Phys. Rev. B, 56, 6458 (1997) Pohl et al. Nature, 397, 238 (1999)

11 Observations Self-assembly at nanoscale Size selection Pattern selection (sometimes) Stable on annealing (seems to be)

12 A basic question: What are the Forces that drive self-assembly?

13 A familiar example oil Evolve oil water Large surface energy water Small surface energy Free energy depends on geometry: G = γa Geometry changes to reduce free energy configurational force Geometry changes by mass transport kinetic process

14 Phase Separation gc ( ) Ω > 2 gc ( ) Ω < 2 α C 0 β C C 0 C α α β matrix α α α one phase Free energy of mixing Regular solution g g( C) ( C) = ΛkT[ C ln C + ( 1 C) ln( 1 C) + ΩC( 1 C) ]

15 Phase Coarsening Time 0 Time t Long time Phase boundary energy G = ΣγL

16 Phase Refining A B Substrate A B A A B A B A B A B A Fringe elastic field Fringe elastic field Pompe, Gong, Suo, Speck, J. Appl. Phys., 74, 6012 (1993). Lu, Suo, Zeitschrift fur Metallkunde, 90, (1999)

17 Ingredients of the model Phase separation Phase coarsening Phase refining z y x Substrate

18 Phase boundary energy C b x Diffused boundary Cahn-Hilliard model freeenergy h 0 C x 2 α β

19 Elastic energy density: Free Energy G = ΓdA + WdV E W = 21+ ( ν) ε ijε ij + ν 1 2ν ε kk ( ) 2 Surface energy is a function of ( C,C,α,ε αβ ) Γ=g( C)+ h( C)C,α C,α + f ( C)ε αα g: free energy of mixing, phase separation f: surface stress, phase refining f ( C) = ψ + φc h: phase boundary energy, phase coarsening h( C) = h 0 Suo andlu, J. Mech. Phys. Solids. 48, 211 (2000).

20 Energy Variation δg = fδu α,α da + σ ij δu i, j dv g 2 + 2h0 C + φεββ δcda C δu Elastic displacement. Elastic equilibrium: fδu α,α da + σ ij δu i, j dv = 0 δc Mass relocation. Driving force for diffusion: F α = g Λ x α C 2h 0 2 C + φε ββ

21 Morphology-Elasticity Coupling dx Elastic Body f σdx f +df Boundary Conditions: σ 31 = f x 1, σ 32 = f x 2

22 Elastic Half-Space Cerruti Solution: a point force on a half space Tangential Point Force Elastic Body Linear superposition ( ε ββ = 1 ν 2 )φ πe ( x 1 ξ 1 ) C ( ) C + x 2 ξ 2 ξ 1 ξ 2 dξ ( x 1 ξ 1 ) 1 dξ ( x 2 ξ 2 ) 2 [ ] 3/2

23 Two Length Scales F α = g Λ x α C 2h 0 2 C + φε ββ C 4πl b Demixing coarsening refining x Phase Boundary thickness: b = h 0 ΛkT 1/2 h 0 ~10 19 J Λ ~ m 2 kt ~ J (T = 400 K ) b ~ 0.3nm Phase Size: l = Eh 0 φ 2 E ~ N/m 2 φ ~ 4 N/m l ~ 0.3nm 4πl ~ 4nm

24 A Diffusion Equation ε ββ = t = M 2 g 2h C + φε 2 0 Λ C ββ C 2 Phase separation Coarsening Refining ( 2 1 ν ) πe φ ( x ξ ) + ( x ξ ) 1 [ 2 2 ( ) ( ) ] x ξ + x ξ C ξ 1 C 2 2 ξ2 dξ 3/ 2 1dξ2 2 2 Substrate elasticity Regular solution g ( C) = ΛkT[ C logc + ( 1 C) log( 1 C) + ΩC( 1 C) ]

25 Numerical: Spectral Method + + C ˆ ( k 1,k 2,t)= Cx ( 1, x 2 )e ik ( 1x 1 +k 2 x 2 ) dx1 dx 2 C ˆ t = k 2 P ˆ + 2 k 4 + b l k 3 C ˆ C PC ( )= ln +Ω1 ( 2C) 1 C ( ) 1/2 k = k k 2 2

26 Simulation Results, C = 0.5 No elasticity refining With elasticity refining t = 0 t = 10 t = 10 2 t = 10 3 t = 10 4 t = 10 5 t = 10 6 Lu, Suo, J. Mech. Phys. Solids 49, 1937 (2001).

27 Simulation Results, C = 0.4 t = 0 t = 10 2 t = 10 3 t = Suo, Lu, J. Nanoparticles Res. 2, 333 (2000). W. Hong

28 Pb on Cu (111) The bright region is the Pb phase. The dark region is the Pb-Cu surface alloy. The average concentration increases from (a) to (e). Courtesy of Gary Kellogg, of Sandia National Lab.

29 Break the symmetry! Anisotropy in Substrate elasticity Phase boundary energy Surface stress

30 Anisotropic Surface Stress f 2 f 1 f 1 f 2 f 1 = φ 1 C f 2 = φ 2 C

31 φ 2 Patterns due to anisotropic surface stress (t = 2.0E5) a b c d e φ 1 a b c d e Lu and Suo, Phys. Rev. B, 65, (2002).

32 Mesophase transition by symmetry breaking φ 22 φ 11 Normalized free energy ( ) = 1 r R η Critical point ( )[ r r c ]η 2 + ν( 1 r) 2 η 4 r c = ( 1 2ν) θ φ 22 φ 11 Control parameter Order parameter φ 22 -θ φ 11 r = φ 2 /φ 1 η = sinθ Normalized free energy, R r = 0.5 r = -0.5 r = Order parameter, η r = -1

33 C( θ) Cubic Elastic Substrate = C C 44 1 sin 2 θ cos 2 θ C 11 C 11 Isotropy: C C 44 C 11 = Copper Molybdenum C C 44 C 11 = 1.61 C C 44 C 11 = 0.86

34 Copper compliant direction [100] t = 0 t = 0.1 t = 1 t = 10 t = 100 t = 500 Lu and Suo, Phys. Rev. B, In press. Gao and Suo, Mech. Phys. Solids. In press.

35 Molybdenum compliant direction [110] t = 0 t = 1 t = 10 t = 100 t = 1000 t = 5000

36 To assemble a specific, aperiodic structure, use a template.

37 Crystal Seeds t=0 t = 10 t = 100 t = 500 t = 1.0E4 t = 4.0E4

38 Guide assembly with a coarse pattern t = 0 t = 10 t = 100 t = 1000 Suo and Lu,. J. Nanoparticles Research 2, 333 (2000).

39 Effect of Initial Condition t = 0 t = 10 t = 100 t = 1000

40 Summary Size selection coarsening and refining Long-range order symmetry breaking Aperiodic structure Templating