Course M6 – Lecture 3
21/1/2004 (JAE)
Course M6 – Lecture 3
21/1/2004
Mixtures of polymers
Polymers in solution and
polymer blends
Dr James Elliott
3.1 Introduction
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Polymer mixtures are an important part of many industrial
processing applications:
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We will also spend some time interpreting the binary
phase diagram predicted by Flory-Huggins theory, and
looking at demixing processes
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Polymer synthesis
Fibre spinning
Membrane formation
Nucleation and growth
Spinodal decomposition
Finally, we will review the use of solubility parameters,
and see how to calculate χ parameters
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Course M6 – Lecture 3
21/1/2004 (JAE)
3.2.1 FH free energy for polymer blend
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Last lecture, we introduced the FH free energy for a
polymer/solvent mixture
For a blend of two polymers, this now looks like :
Fmix Φ1
Φ
=
ln Φ1 + 2 ln Φ2 + χΦ1Φ2 per site
kBT M1
M2
or, writing Φ1 = Φ, Φ2 = 1 – Φ, and M1 = M2 = M :
Fmix Φ
(1 − Φ)
= ln Φ +
ln(1 − Φ) + χΦ(1 − Φ) per site
kBT M
M
3.2.2 FH free energy for polymer blend
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As χ (or equivalently 1/T) is varied, the shape of the free
energy curve changes
At χ = 0.5/M, there is a
critical point where
coexistence of two separate
phases is favoured
Concentrations of the
phases determined by the
double tangent construction
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Course M6 – Lecture 3
21/1/2004 (JAE)
3.3.1 FH free energy and phase diagrams
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The resulting phase diagram can be calculated, and also
includes information about kinetics of demixing
Mχ
3.3.2 FH free energy and phase diagrams
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Note that for coexistence χ ≥ 0.5/M, so the slightest
degree of unfavourable interactions between the two
polymers will cause phase separation
Polymers are generally immiscible
This is because the entropy of mixing is greatly reduced
compared to mixing of simple solutions as polymers are
1D objects not point-like particles
Immiscibility of polymers has implications for industrial
processes such as joining, as polymer-polymer interfaces
tend to be very weak
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Course M6 – Lecture 3
21/1/2004 (JAE)
3.4 Aside on polymer interfaces
w
Ei ≈ χkBTNi = kBT
⇒w≈
aK
χ
w ≈ aK Ni0.5
w = 10-30 Å
3.5.1 Nucleation vs. spinodal decomposition
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Fundamentally different kinetic mechanisms of demixng
Nucleation and growth
Spinodal decomposition
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Course M6 – Lecture 3
21/1/2004 (JAE)
3.5.2 Nucleation vs. spinodal decomposition
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Very characteristic microstructures can be observed in
polymer blends which have separated by the two means
Nucleation and growth
Spinodal decomposition
3.6 Coarsening during spinodal decomposition
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Linearised approximation breaks down leading to
increasing length scale of phase separation
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Course M6 – Lecture 3
21/1/2004 (JAE)
3.7 Corrections to FH theory
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Of course, FH is only a mean-field approximation
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In practice, there are a number of complicating factors
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The interaction parameter is concentration dependent
χ = a + bΦ + cΦ2
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Effects of molecular weight polydispersity
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Effects of compressibility and thermal expansions
3.8 Upper critical solution temperature (UCST)
If the enthalpic interactions
are disfavourable, the
mixture will exhibit an upper
critical solution
temperature
Immiscible at lower T due to
disfavourable enthalpic
interactions
Miscible at higher T due to
reduced enthalpic
interactions
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Course M6 – Lecture 3
21/1/2004 (JAE)
3.9 Dependence of UCST on molecular weight
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Polystyrene and polybutylene blends as a function of PS
molecular weight
M(PB) = 2350
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M(PS) = 2250
•
M(PS) = 3500
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M(PS) = 5200
3.10 Lower critical solution temperature (LCST)
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Returning to the phase diagram, non-FH dependence of
χ on temperature leads to critical solution temperatures
Two phase
Polystyrene and poly(vinyl
methylene) exhibit a lower
critical solution
temperature
Miscible at lower T due to
favourable enthalpic
interactions.
One phase
Immiscible at higher T due to
free volume differences
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Course M6 – Lecture 3
21/1/2004 (JAE)
3.11 UCST and LCST phase diagram types
3.12.1 Solubility parameter approach
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Miscibility can be estimated by using solubility
parameters, which are tabulated for many different
polymers and solvents
For most (non-polar) solvents, the enthalpic contribution
to the χ parameter can be written
χH =
Vm
(δA − δB )2
RT
where δA, δB are the solubility parameters of the solvent
and polymer, representing the cohesive energy densities.
δ = (∆H vap Vm ) 2
1
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Course M6 – Lecture 3
21/1/2004 (JAE)
3.12.2 Solubility parameter approach
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The total χ parameter is then
χ=
z
z
z
Vm
(δA − δB )2 + 0.34
RT
For polar solvents, need to add correction for the
electrostatic couplings between solvent and polymer
From tables of δ (or measurements) can then estimate χ
and predict phase diagram
Can also deduce χ from molecular simulations
3.13 Solubility of polymers in solvents
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Course M6 – Lecture 3
21/1/2004 (JAE)
3.14 Solubility params of solvents & polymers
3.15 Determining the χ parameter
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Direct measurement of heat of mixing
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Measurement of the partial vapour pressure
(µ
1
z
− µ10 ) / RT = ln Φ1 + (1 − 1 / N 2 )Φ2 + χΦ22
Scattering
SC (q)−1 = {Φ1N1SD (R12q2 )} + {Φ2 N 2 SD (R22q2 )} − 2χ
−1
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−1
χ is concentration dependent (χS constant)
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Course M6 – Lecture 3
21/1/2004 (JAE)
3.16.1 Calculation of χ from simulation
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The following slides demonstrate a molecular simulation
of the mixing behaviour between a polymer and a
solvent, in this case polystyrene and cyclohexane.
The simulation is used to calculate the parameters of the
Flory-Huggins model, i.e. the interactions energies and
the coordination number.
Polystyrene is represented by one monomer unit.
3.16.2 Simulation of the interaction energy
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Many different constellations of the two molecules are
generated (avoiding close contacts between cyclohexane
and the styrene head and tail) and the interaction energy
of each pair is calculated.
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Course M6 – Lecture 3
21/1/2004 (JAE)
3.16.3 Simulation of the interaction energy
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A typical energy distribution obtained from the simulation
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The simulation is carried out at a range of temperatures,
and the resulting energy curve is fitted by a second order
polynomial. This results in a temperature dependent
Flory-Huggins interaction parameter.
3.16.4 Simulation of the coordination number
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Many clusters are generated to determine how many
solvent molecules that can be packed around a monomer
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Course M6 – Lecture 3
21/1/2004 (JAE)
3.16.5 Simulation of the coordination number
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The average coordination number for each of a number
of clusters generated by the simulation.
3.16.6 Phase diagram prediction
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The Flory-Huggins free energy and resulting phasediagram as determined from the simulation.
For this system, there is an upper critical solution
temperature which agrees well with experiment.
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Course M6 – Lecture 3
21/1/2004 (JAE)
Lecture 3 summary
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In this lecture, we discussed the mixing of polymers
We started by reviewing Flory-Huggins theory and its
predictions for the phase diagram of polymer mixtures, in
particular the transition to coexistence
In general, polymers are immiscible unless there are
strongly favourable enthalpic interactions
The temperature dependence of the χ parameter gives
rise to a wide variety of phase diagrams including those
with upper and lower critical solution temperatures
We briefly discussed the two kinetic mechanisms of
demixing: nucleation and spinodal decomposition
Finally, we looked at measuring and calculating χ
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