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Dense Polymer Systems Mean Field Theory

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In polymer melts, chains are ideal and mean-field theory is good ... Self-Consistent Field Theory is a generalization to inhomogeneous systems ... – PowerPoint PPT presentation

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Title: Dense Polymer Systems Mean Field Theory


1
Dense Polymer Systems-Mean Field Theory
2
  • Two important facts about dense polymer systems
  • Flory-Huggins mean field theory of
  • polymer mixtures
  • Self-Consistent Field Theory (SCF)
  • Applications of SCF

3
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
4
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.
5
Mixture of A and B beads (disconnected for now)
lattice sites
(volume fraction of A)
Entropy
6
Energy
where
1 if site i has an A bead and 0 otherwise
has terms like
Mean Field Theory is the neglect of correlations
7
Mean field free energy density of a binary mixture
where
8
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!
9
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
10
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
11
Define linear combinations
Then
A similar trick turns the delta function into
a functional integral over a field w
12
Skipping to result
where
And
Is the partition function of a single particle in
a field w
13
Mean field theory is the saddle-point
approximation for this integral
14
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
15
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.
16
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?
17
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
18
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)
19
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

20
Example The interface between phases
Helfand and Tagami (1971)
where
21
Copolymers
www.physics.nyu.edu/ pine/research/nanocopoly.html
22
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
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