Polymerization in a Smectic Liquid Crystal

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Polymerization in a Smectic Liquid Crystal
Diffusion in a Periodic Medium with Random Sinks
Brian Camley
Department of Physics, University of Colorado,
Boulder
Advisor Leo Radzihovsky
Committee Leo Radzihovsky, John Cumalat, Manuel
Lladser
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Motivation Ferroelectric liquid crystal-polymer
composites
FLC-polymer hybrids have exciting possibilities!

• Basic soft condensed matter elasticity, phase
  transitions, hydrodynamics
• Applications plastic LC displays, electronic
  paper
• Relation to other physics electrons in
  disordered systems, membrane interactions

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Polymer-LC composite
No monomer
(Clark Bowman and Guymon)
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Polymer-LC composite
Unpolymerized monomer
(Clark Bowman and Guymon)
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Polymer-LC composite
Polymer network
Polymerization location (sink)
(Clark Bowman and Guymon)
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X-Ray scattering is strange
Increasing time
X-Ray scattering from smectic during
polymerization (Noel Clark group).
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Spacing controls x-ray scattering
Smectic Spacing
X-Ray Scattering
a
Mean spacing controls peak position
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X-Ray scattering is strange
Increasing time
X-Ray scattering from smectic during
polymerization (Noel Clark group).
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Important questions

• What fixes the monomer concentration decay and
  the development of heterogeneity?
• Why is the X-ray peak width non-monotonic in
  time?
• How does the monomer distort the smectic?
• How does the smectic affect the monomer
  concentration?

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Smectic distorts due to monomer
Does the local layer spacing follows monomer
concentration?
High monomer concentration
Low monomer concentration
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Smectic distorts due to monomer
We minimize smectic elastic energy
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Radzihovsky-Clark model
Polymerization location (sink)
Diffusing monomer
Monomers diffuse until they encounter a
polymerization location, or sink.
(monomer concentration)
(frozen sink concentration)
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Equivalent problem
A random walker (monomer) moves on a lattice with
static traps (polymerization locations)
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Numerical Results
n 1
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Numerical Results
n 20
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Numerical Results
n 100
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Numerical Results
n 200
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Crossover time
We expect most of the monomer to be trapped when
(average volume a particle explores) (volume
per sink)
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Crossover Time
Cant see crossover!
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Crossover Time
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Scaling theory predicts crossover
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Less successful in two dimensions
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When does the walker die?
Survival probability
where Sn is a random variable indicating the
number of unique sites visited on the nth step
Cubic lattice with trap concentration s
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Calculating Fn
We can write the survival as a cumulant expansion
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Calculating Fn
These are known for d 1
(Montroll and Weiss 1965, Hughes 1995)
This gives the survival probability!
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Variance in Fn
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Predictions of random sink theory
tc
Time
Time
Monomer concentration decreases
Variance non-monotonic in time
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Predictions of random sink theory
Increasing time
Analytical
Experimental
We can predict qualitative behavior very well
with simple calculations
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Exact Solution (d1)
The one-dimensional problem is 'nice'
The concentration at a point is only affected by
the nearest sinks!
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Exact Solution (d1)
We extend the Grassberger-Procaccia solution to
get the particle survival exactly
Probability of L being trap-free
Particle in a box eigenvalues
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Exact Solution (d1)
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Diffusion in a periodic medium
Monomers are limited by smectic layers
This leads to anisotropic diffusion (slower in z
direction)
z
Smectic periodic potential
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Diffusion in a periodic medium
Normally, walker moves within layer
This changes the number of unique sites visited,
and we can calculate this!
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Anisotropic Hopping
Normally, in two dimensions,
(Montroll and Weiss, 1965)
We showed that with hopping probability ?,
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Diffusion in a periodic medium
Hopping probability h
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Spatial variation
We can calculate the correlation function using a
consistent thresholding model
1. We approximate the solution with
superimposition of single-sink solutions
sink
sink
3. We require consistency to fix sink radius
4. Conditional probability gives us the
correlation function
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Consistency
Whats the mean monomer concentration as a
function of time for this model?
2R
There are only two concentrations zero and one
The mean concentration is the probability that a
given location is non-zero!
If this is equal to F, we can fix the square well
size!
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The correlation function
In this model, is
nonzero only if the region around the origin and
around x are both free of traps
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The correlation function
How much volume needs to be free of traps for
C(x)C(0) to be nonzero?
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The correlation function
Predicts exponential decay to uncorrelated value
This predicts results in agreement with
one-dimensional Monte Carlo experiments!
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Conclusions

• Model accurately captures time evolution of
  smectic X-ray peak
• Decrease in monomer concentration with
  polymerization leads to shift in X-ray peak
• Spatial heterogeneity in monomer concentration
  leads to X-ray peak with width non-monotonic in
  time
• Greatest peak width at

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Many open questions

• Exact solution to two-sink problem in 3D
• What fixes sink distribution n(r)?
• Effects of slow motion of sinks
• Effect of long-range smectic response on X-ray
  peak

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Acknowledgements

• Leo Radzihovsky
• John Cumalat
• Manuel Lladser
• Noel Clark
• Julia Santos
• Ferroelectric Liquid Crystal MRC