SAMSI Materials Workshop 2004

Two-Dimensional Self-assembled Patterns in Diblock Copolymers

Peko Hosoi, Hatsopoulos Microfluids Lab. MITShenda Baker, Dept. Chemistry Harvey Mudd College

Dmitriy Kogan (GS), CalTech

SAMSI Materials Workshop 2004

Experimental Setup

• Langmuir-Blodgett trough• Polystyrene-Polyethyleneoxide (PS-PEO) in Chloroform• Deposit on water• Chloroform evaporates• Lift off remaining polymer with silicon substrate• Image with atomic force microscope (AFM)

SAMSI Materials Workshop 2004

Experimental Observations

Continents( > 500 nm)

Stripes(~100 nm)

Dots (70-80 nm)

Photos by Shenda Baker and Caitlin Devereaux

All features ~ 6 nm tall

Low

High

conc

entra

tion

SAMSI Materials Workshop 2004

Polystyrene-Polyethyleneoxide (PS-PEO)

• Diblock copolymer

• Hydrophilic/hydrophobic

(CH - CH2)m - (CH2 - CH2 - O)n

……. ……..

SAMSI Materials Workshop 2004

Physical Picture

Marangoni Diffusion Evaporation Entanglement

SAMSI Materials Workshop 2004

Mathematical Model

Diffusion - Standard linear diffusion Evaporation - Mobility deceases as solvent evaporates. Multiply velocities by a mobility envelope that decreases monotonically with time. We choose Mobility ~ e-t. Marangoni - PEO acts as a surfactant thus Force = -kST c, where c is the polymer concentration. Entanglement - Two entangled polymers are considered connected by an entropic spring (non-Hookean). Integrate over pairwise interactions …

Small scales Low Reynols number and large damping. Approximate Velocity ~ Force (no inertia).

SAMSI Materials Workshop 2004

Entanglement

Pairwise entropic spring force between polymers1 (F ~ kT)

1 e.g. Neumann, Richard M., “Ideal-Chain Collapse in Biopolymers”, http://arxiv.org/abs/physics/0011067

Relaxation length ~where l = length of one monomerand N = number of monomers

€

l N

Find expected value by multiplying by the probability that two polymers interact and integrating over all possible configurations.

SAMSI Materials Workshop 2004

More Entanglement

Integrate pairwise interactions over all space to find the force at x0

due to the surrounding concentration:

€

Fentanglement (x0) = dr Fspringc(x0 + r)rdθ0

2π

∫0

∞

∫

Expand c in a Taylor series about x0:

€

Fentanglement (x) = πϕ 2c x+ 1

8ϕ 4cxxx + 18ϕ 4cxyy ...

ϕ 2cy + 18ϕ 4cyyy + 1

8ϕ 4cxxy ...

⎡

⎣ ⎢

⎤

⎦ ⎥

€

ϕ n ≡ rn0

∞

∫ Fspring r( )drwhere

SAMSI Materials Workshop 2004

Force Balance and Mass Conservation

€

v = Mobility × Force = Fsurf. tens. + Fent.

6πμRPS

= e -βt (−kST∇c +ϕ1∇c +ϕ 3∇∇ 2c)

€

c t +∇ ⋅(vc) = D∇ 2cConvection Diffusion:

€

cτ +∇ ⋅ fcutoff c πϕ 2 −σ( )∇c + π8 ϕ 4c∇∇ 2c{ } −D∇c[ ] = 0

Time rescaled; cutoff function due to “incompressibility” of PEO pancakes.

SAMSI Materials Workshop 2004

Numerical Evolution

time

conc

entra

tion

Experiment

QuickTime™ and aYUV420 codec decompressorare needed to see this picture.

SAMSI Materials Workshop 2004

Linear Stability

PDE is stable if where c0 is the initial concentration.

€

k > 2 2 πϕ 2 −σ −D /c0

πϕ 4

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1/ 2

Fastest growing wavelength:

€

λcritical = 2πkcritical

= π πϕ 4

πϕ 2 −σ −D /c0

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1/ 2

Recall is a function of initial concentration

SAMSI Materials Workshop 2004

Quantitative comparison with Experiment

Triangles and squares from linear stability calculations (two different entropic force functions)

Linear stability

SAMSI Materials Workshop 2004

Conclusions and Future Work

• Patterns are a result of competition between spreading due to Marangoni stresses and entanglement

• Quantitative agreement between model and experiment• Stripes are a “frozen” transient• Other systems display stripe dot transition e.g. bacteria

(Betterton and Brenner 2001) and micelles (Goldstein et. al. 1996), etc.

• Reduce # of approximations -- solve integro-differential equations