Dense Polymer Systems Mean Field Theory

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Dense Polymer Systems-Mean Field Theory
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• Two important facts about dense polymer systems
• Flory-Huggins mean field theory of
• polymer mixtures
• Self-Consistent Field Theory (SCF)
• Applications of SCF

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Polymer melts - two key facts

• Florys Theorem
• chains in a melt are ideal

2. Mean field theory becomes valid in the limit
of high density.
Ginzburg parameter
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Two generations of mean field theory of polymers

• Flory-Huggins Theory (1940s)
• Self-Consistent Field Theory (1970s,)
• (able to deal with spatial inhomogeneity and more)

Both approaches will be described in the context
of a mixture of two kinds of polymer.
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Mixture of A and B beads (disconnected for now)
lattice sites
(volume fraction of A)
Entropy
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Energy
where
1 if site i has an A bead and 0 otherwise
has terms like
Mean Field Theory is the neglect of correlations
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Mean field free energy density of a binary mixture
where
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Flory-Huggins model of a mixture of polymers
Each A polymer has NA beads
Each B polymer has NB beads
Only change is in the translational entropy
It is very easy for polymers to phase separate!
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Modern approach (de Gennes, Edwards)
Again, lets start with a mixture of disconnected
beads.
nA beads of type A, nB beads of type B Each with
volume v0
Densities
(similarly for B)
Interaction between beads
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Partition function
The functional delta function expresses the fact
that there is polymer everywhere
Now we will re-express both parts of this
integral in terms of functional integrals over
fields
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Define linear combinations
Then
A similar trick turns the delta function into
a functional integral over a field w
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Skipping to result
where
And
Is the partition function of a single particle in
a field w
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Mean field theory is the saddle-point
approximation for this integral
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We can get the free energy from this
The quality of the saddle point approximation is
determined by the large factor in the exponent.
Ginzburg parameter
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Why self-consistent field theory?
Saddle-point equation then gives the fields in
terms of the densities.
Typically the equations are solved
numerically, by iteration.
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What about polymers instead of disconnected
beads??
The difference is that Qw is now the partition
function of a single polymer in a field w(r).
How do we get this partition function?
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Ideal chains in a field w(r)
We will use a continuous ideal chain desribed
by a function
Where s ranges from 0 to N continuously
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The benefit of using continuous ideal chains
Where the propagator q is given by solving
the diffusion equation in field w
This is the basis for using analogies
with quantum mechanics (e.g. ground-state
dominance approximation)
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If we look for spatially uniform solutions of
this model, we will get the Flory-Huggins free
energy.
However, the self-consistent field approach is
much more general

• Spatial inhomogeneity
• (phase boundaries, confinement effects)
• Different architectures of polymers

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Example The interface between phases
Helfand and Tagami (1971)
where
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Copolymers
www.physics.nyu.edu/ pine/research/nanocopoly.html
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Summary

• In polymer melts, chains are ideal and mean-field
  theory is good
• Flory-Huggins A simple mean field theory, but
  only for uniform systems
• Self-Consistent Field Theory is a generalization
  to inhomogeneous systems