samsi materials workshop 2004 two-dimensional self-assembled patterns in diblock copolymers peko...
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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
c on
c en
tra
t ion
SAMSI Materials Workshop 2004
Polystyrene-Polyethyleneoxide (PS-PEO)
• Diblock copolymer
• Hydrophilic/hydrophobic
(CH - CH2)m - (CH2 - CH2 - O)n
……. ……..
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+
18ϕ 4cxxx + 1
8ϕ 4cxyy ...
ϕ 2cy + 18ϕ 4cyyy + 1
8ϕ 4cxxy ...
⎡
⎣ ⎢
⎤
⎦ ⎥
€
ϕ n ≡ rn
0
∞
∫ 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∇ 2c
Convection 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 2πϕ 2 −σ −D /c0
πϕ 4
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 2
Fastest growing wavelength:
€
λcritical =2π
kcritical
= ππϕ 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