Two-Dimensional Pattern Formation in Diblock Copolymers
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Transcript Two-Dimensional Pattern Formation in Diblock Copolymers
Two-Dimensional Self-assembled Patterns in
Diblock Copolymers
Peko Hosoi, Hatsopoulos Microfluids Lab. MIT
Shenda Baker, Dept. Chemistry Harvey Mudd College
Dmitriy Kogan (GS), CalTech
SAMSI Materials Workshop 2004
Experimental Setup
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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
High
All features ~ 6 nm tall
Continents
( > 500 nm)
Stripes
(~100 nm)
Low
Dots (70-80 nm)
Photos by Shenda Baker and Caitlin Devereaux
SAMSI Materials Workshop 2004
Polystyrene-Polyethyleneoxide (PS-PEO)
• Diblock copolymer
(CH - CH2)m - (CH2 - CH2 - O)n
……. bbbbbbbbbbbhhhhh
……..
• Hydrophilic/hydrophobic
SAMSI Materials Workshop 2004
Physical Picture
Marangoni
Diffusion
Evaporation
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Entanglement
Mathematical Model
Small scales \ Low Reynols number and large damping. Approximate
Velocity ~ Force (no inertia).
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-bt.
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 …
SAMSI Materials Workshop 2004
Entanglement
Pairwise entropic spring force
between polymers1 (F ~ kT)
Relaxation length ~ l N
where l = length of one monomer
and N = number of monomers
value by multiplying
Find expected
by the probability that two polymers
interact and integrating over all
possible configurations.
1 e.g.
Neumann, Richard M., “Ideal-Chain Collapse in Biopolymers”, http://arxiv.org/abs/physics/0011067
SAMSI Materials Workshop 2004
More Entanglement
Integrate pairwise interactions over all space to find the force at x0
due to the surrounding concentration:
Fentanglement (x 0 )
2
dr F
c(x 0 r)rd
spring
0
0
Expand c in a Taylor series about x0:
where
2c x 18 4 c xxx 18 4 c xyy ...
Fentanglement (x)
1
1
2c y 8 4 c yyy 8 4 c xxy ...
n
r n Fspringr dr
0
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Force Balance and Mass Conservation
Fsurf. tens. Fent.
v Mobility Force =
6RPS
= e -bt (k ST c 1c 3 2c)
Convection Diffusion:
c t (vc) D 2c
c f cutoff c 2 c 8 4 c 2c Dc 0
Time rescaled; cutoff function due to “incompressibility” of PEO pancakes.
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Numerical Evolution
concentration
Experiment
QuickTi me™ and a
YUV420 codec decompressor
are needed to see thi s pi ctur e.
time
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Linear Stability
2 D/c 0 1/ 2
PDE is stable if k 2 2
where c0 is the initial
4
concentration.
Fastest growing wavelength:
1/ 2
2
4
critical
kcritical
2 D/c 0
Recall is a function of initial concentration
SAMSI Materials Workshop 2004
Quantitative comparison with Experiment
Linear stability
Triangles and squares from
linear stability calculations
(two different entropic force
functions)
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
SAMSI Materials Workshop 2004