Download 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
SAMSI Materials Workshop 2004
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 Fspringr dr
0
SAMSI Materials Workshop 2004
Force Balance and Mass Conservation
Fsurf. tens.  Fent.
v  Mobility  Force =
6RPS
= e -bt (k ST c  1c   3 2c)
Convection Diffusion:

c t    (vc)  D 2c


c    f cutoff c  2   c  8  4 c 2c Dc  0

Time rescaled; cutoff function due to “incompressibility” of PEO pancakes.
SAMSI Materials Workshop 2004
Numerical Evolution
concentration
Experiment
QuickTi me™ and a
YUV420 codec decompressor
are needed to see thi s pi ctur e.
time
SAMSI Materials Workshop 2004
Linear Stability
 2    D/c 0 
PDE is stable if k  2 2
 where c0 is the initial
 4


concentration.
1/ 2
Fastest growing wavelength:



2
 4
critical 
  

kcritical
 2    D/c 0 
1/ 2
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