Transcript Concentrated Polymer Solutions

Concentrated Polymer Solutions
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Application and Importance of
Concentrated Polymer Solutions
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What does mean?
•
•
•
•
A dilute solution
Theta condition
Semi-dilute
Concentrated and so On..
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What we expect from a concentrated
solution?
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The phase
separation
……
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What are the requirements for phase
separation.
• The required conditions in terms of
thermodynamical properties of the system.
• The effect of temperature.
• The effect of molecular weight.
• The effect of concentration.
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Solubility, The temperature and
the Mw.
Phase diagrams for
polystyrene fractions in
cyclohexane. Circles
and solid lines,
experimental.
Theoretical curves are
shown for two of the
fractions. The viscosityaverage
molecular weights are
PSA, 43,600; PSB,
89,000; PSC, 250,000;
PSD, 1,270,000 g/mol
Tc=The
critical
temperature
is the highest
temperature
of phase
separation.
The lower molecular weight species will tend to remain in solution
at a temperature where the higher molecular weights phase-separate
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The criteria for concentration based on
chain situation in solution
Relationships of polymer chains in solution at different concentration regions. (a)
Dilute solution regime, where C < Cov. (b) The transition regions, where C = Cov. (c)
Semi-dilute regime, C > Cov. Note overlap of chain portions in space.
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A review on chain conformation
n links, each of length
l, joined in a linear sequence with no
restrictions on the angles between
successive
bonds
 is the bond angle between
atoms, and  is the
conformation angle
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• Finally by considering the bond angle =109 
and all other constraints such as excluded
volumes (lesser total number of
conformations) and so on:
The characteristic ratio C∞ = r 2/ l2n varies from
about 5 to about 10, depending on the foliage
present on the individual chains.
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Typical values of C∞
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Segmental lengths in Chains
• Kuhn Segment Length
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The Kuhn Segment, b
• The Kuhn segment length, b, depends on the chain’s end-to-end
distance under Flory -conditions, or its equivalent in the
unoriented, amorphous bulk state, r,
r 2  C nl 2  nb 2
 C  b
2
l
2
• For flexible polymers, the Kuhn segment size varies between 6 and
12 mers , having a value of eight mers for polystyrene, and 6 for
poly(methyl methacrylate).
• The Kuhn segment also expresses the idea of how far one must
travel along a chain until all memory of the starting direction is
lost, similar to the axial correlation distance.
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UCST
one phase
bimodal curves
temperature
temperature
Two phase
Two phase
spinodal curves
composition
upper critical solution temperature (UCST)
LCST
one phase
composition
lower critical solution temperature (LCST)
Most of the polymer solutions exhibit either UCST or LCST behavior with a few
exceptions that exhibit both.
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Phase Diagram
A= Entropic Part
B= Enthalpic Part
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Phase diagram
At Theta Condition
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Lever Rule
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Lever Rule; Free Energy and Stability
Metastable
Stable
fraction fa of the volume of the material having composition Фα
(and fraction fβ = 1 —fα having composition Фβ,
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Phase diagram
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Phase diagram and chain dimensions
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Excluded Volume
• Is the difference in repulsion and attraction
interactions between the chain segments:
V= the effect of repulsion forces on chain structure
- attraction
forces affects chain
structure
So
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V > 0 if repulsion >attraction => extended chain
V< 0 if attraction > repulsion => collapsed chain
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Phase diagram chain dimension and
the excluded volume
The thermal blob size is the length scale at which excluded volume
becomes important.
1For v ≈ b3, the thermal blob is the size of a monomer
(ξT≈ b) and the chain is fully swollen in an athermal solvent .
2For v ≈ — b3, the thermal blob is again the size of a monomer
(ξT≈ b) and the chain is fully collapsed in a non-solvent.
3For |v| < b3N-1/2, the thermal blob is larger than the chain size
(ξT > R0) and the chain is nearly ideal.
4For b3N-1/2 < |v| < b3 the thermal blob is between the
monomer size and the chain size, with either intermediate
swelling in a good solvent [v > 0] or intermediate collapse in a
poor solvent [v<0].
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Chain Dimensions
From Fractal
dimension
• Exclude volume and the chain dimension
– Excluded volume repulsion v>0 (good Solvent)
– Excluded volume attraction v<0 (poor solvent)
Dense
Package
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Good, Poor, Theta and Athermal
ν= 3/5
ν= 1/2
ν= 1/3
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At Theta Condition
At the θ-temperature, the chains have nearly ideal
conformations at all concentrations:
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At Theta Condition
The concentration increases moving from left
to right. At T=θ, there is a special
concentration that equals the concentration
inside the pervaded volume of the coil. This is
the overlap concentration for θ –solvent.
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At Theta Condition
The temperature at which chains begin to
either swell (above θ) or collapse (below θ):
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Poor Solvent
The highest point on the binodal line is the
critical point with critical composition
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Poor Solvent
The highest point on the binodal line is the
critical point with critical composition
Phase diagrams for polystyrenes in
cyclohexane, M = 43 600 gmol-1 (open
circles), M = 89000 gmol"1 (filled
circles), M = 250 000 gmol-1 (open
squares), M= 1 270000 gmoP1 (filled
squares),
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Poor Solvent
The size of the globules is proportional to the one-third
power of the number of monomers in them:
Supernatant
phase
Precipitant
phase
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Poor Solvent
The size of the globules is proportional to the one-third
power of the number of monomers in them:
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Good Solvent
Overlapping
concentration
in good solvent
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What happens with the chain swelling in good solvents
(due to the excluded volume) above the overlap
concentration?
• Important concept is that of the screening of
excluded volume interactions in the concentrated
solutions (Flory, Edwards):
• as the chain concentration increases in the region
*
   , the coil swelling gradually diminishes and
finally it vanishes in the melt (i.e. coils are ideal in
the melt -Flory theorem) .
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Polymer coil dimensions in semidilute
solutions: example of scaling arguments
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Scaling Theory
• Go for it….
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Scaling Law And The
Polymer Solvent Diagram
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The Polymer-Solvent Diagram
v= 1-2
Semidilute
= volume fraction,
p = C /6,
N = number of
bonds per chain, vc
2 = cross-over from
swollen to ideal.
Marginal
p3/2N-1/2
Excluded
Volume=v
dilute
Concentrated
Ideal
0
1
-N-1/2
Phase
Separation
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cr

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Macroscopic Phase Separation
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Macroscopic Phase Separation
The typical dependence of the Flory-Huggins free energy on the polymer
volume fraction in the solution
:

2
1
F
Spinodal
points
1
Ф
Binodal
points
This dependence contains both convex and concave parts.
F
F ps
Fh
1

2
Ф
Convex part of the function F(Ф): no
macroscopic phase separation.
Free energy of the solution separated into
two phases with    and   
1
Free energy of homogeneous solution at
2
  
F
Fh
F ps
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1

2
Ф
Concave part of the dependence F(Ф):
macroscopic phase separation into two
phases.
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Phase separation
by Spinodal
mechanisim
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Phase
separation
by
Nucleation
and Growth
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Some Review Slides
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Conditions for the phase separation (minimum possible free energy) are determined
from common tangent straight line - binodal curve. Conditions for the absolute
stability of homogeneous phase at a given concentration are determined from the
positions of inflexion points - spinodal curve.
 2 F  2  0
1
1

 2  0
N
1 
or
 
1 1
1




2  N
1  
This dependence is shown in the figure:

1
1

;
2
N
1

N
c 
c
c
1 2
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c
Ф
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Phase diagram with binodal and spinodal

Spinodal
Single
globules
Binodal
c
1 2
c
Ф
Conclusions:
• Macroscopic phase separation takes place at the quality of solvent only
slightly poorer than the - solvent.
1
1

2
N
• The critical point for macroscopic phase separation corresponds to the
dilute enough solution.
c 
1
N
• The region of isolated globules in solution corresponds to very low polymer
c 
concentrations, especially at the values of
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
significantly larger than1 2.
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• The precipitant phase close enough to the - point is very diluted.
• For different values of N the binodal curves (boundaries of the phase
separation region) have the form:

N3
N2
N1
c
N1  N 2  N 3
1 2
c
Ф
With the increase of N the critical temperature becomes closer to the point, and the critical concentration becomes lower.
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Method of fractional precipitation for polydisperse polymer solution:
when the quality of solvent is becoming poorer or polymer concentration
increases in the dilute enough range at first the most high-molecular
fraction precipitates, then the next fraction, etc…; polymers with lower
molecular weights require more significant increase in and
c to
precipitate. In this way polymer fractionation is achieved.
Reverse method is called the method of fractional dissolution: when one
moves from the region of insolubility to the region of partial solubility at
first the fractions with the lowest values of M are dissolved.
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• What is the connection of the Flory-Huggins parameter  and the
temperature T ? Within the framework of the lattice model
   kT  in the experimental variables T, c the phase diagram has the
form shown in the figure, i.e. the poor solvent region corresponds to T  
T

c
Such situation is called upper critical solution temperature (UCST) - critical
point is “on the top” of the phase separation region.
Examples: poly(styrene) in cyclohexane (around 35 C), poly(isobutylene) in
benzene, acetylcellulose in chlorophorm.
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However, due to the complicated renormalization of polymer-polymer
interactions due to the solvent, sometimes 
increases with the increase of T.
Then the T, c phase diagram has the form shown in the figure below, i.e. the
poor solvent region corresponds to T   .
T

Such situation is called lower critical
solution temperature (LCST)-critical
point is “on the bottom” of the phase
separation region.
c
Examples: poly(oxyethylene) in water, methylcellulose in water, in general most of the water-based solutions. The reason: increase of the so-called
hydrophobic interactions with the temperature (organic polymers contaminate
network of hydrogen in water and water molecules become less mobile
(solvated), i.e. they lose entropy - this unfavorable entropic factor for polymerwater contacts is more important at high temperatures).
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• Suppose that the polymer with UCST is glassy without solvent in this
range of temperatures. Then the situation is similar to that shown in the
figure below:
T

Tg
1
c
Upon the temperature jump to the region of macroscopic phase separation,
the separation begins, but it cannot be completed, because of the formation
of the glassy nuclei which “freeze” the system. As a result, microporous
system is formed, and this is one of the methods of preparation of
microporous chromatographic columns.
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The Vitrification Effect
The effect of the solvent at high polymer volume fraction is to plasticize the polymer.
However, if the polymer is below its glass transition temperature, the concentrated
polymer solution may vitrify, or become glassy
Berghmans’ point
Vitrification
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Polymer Blends
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POLYMER–POLYMER Miscibility
O
P
Y
P
O
R
L
L
M
Y
E
E
M
R
PHASE SEPARATION
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Why Phase Separation?
BECAUSE OF
REDUCED
COMBINATORIAL
ENTROPY
OF MIXING
Endotherm Enthalpy
change Plus small Entropy
Change
Not necessary NEGATIVE
FREE ENERGY of MIXING
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Temperature Effect
T
Phase Separation
Hmix=TSmix
Lower Critical Solution Temp.
LCST
One Phase
1
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PHASE DIAGRAMS
Spinodal Lines
Binodal Line
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Triple Mixtures
Solvent
MII
Enters Phase
Separation
Region
PI
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PII
PII
PI
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HOWEVER, It is possible to have:
TYPE I
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TYPE II
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TYPE III
One
TYPE IV
One
Phase
Phase
Two
Phase
TYPE V
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TYPE VI
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Kinetics of Phase Separation
Major mechanisms by which two components of a mutual
solution can phase-separate: nucleation and growth, and
spinodal decomposition
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Nucleation and Growth
Nucleation and growth are associated with metastability, implying the
existence of an energy barrier and the occurrence of large composition
fluctuations.
Domains of a minimum size, the so-called critical nuclei, are a necessary
condition.
Nucleation and growth (NG) are the usual mechanisms of phase
separation of salts from supersaturated aqueous solutions, for
example.
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Spinodal Mechanism:
Spinodal decomposition (SD), on the other hand, refers to phase
separation under conditions in which the energy barrier is negligible, so
even small fluctuations in composition grow.
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Comparison of the two mechanisms:
Nucleation and growth could be seen as tiny spheres, while spinodal
decomposition looked like tiny overlapping worms.
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Investigation under
Microscope
Which kind of phase
separation does the
picture show?
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Phase separation by spinodal decomposition () and nucleation and growth (•),
as observed under a microscope
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Thermodynamics of Mixing
The Flory–Huggins theory provides an expression for the
free energy density of mixing of two homopolymers labeled
A and B
Entropy Origin
Enthalpy Origin
Ni
is the number of monomers in chain i, and i is the volume of each
monomer on chain i, A is the volume fraction of component A in the mixture, v
is an arbitrary reference volume,  is the Flory–Huggins interaction parameter,
Gm is the free energy change on mixing per unit volume, k is the Boltzmann
constant, and T is the absolute temperature.
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If we consider:
Then:
Note that Ni depends weakly on temperature (because i depends on temperature)
while ^ Ni does not.
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At the critical point:
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In the case of (NA=NB=N)
The spinodal curve:
The binodal curve:
and critical point is located at:
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Temperature Dependence of 
The  parameters obtained from polymer blends are often
linear functions of 1/T.
However, in some cases a distinct nonlinearity is observed when  is plotted
versus 1/T. In such cases the data can be fit to a quadratic function in 1/T.
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Numerical Values of A, B and C,
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Types of Phase Diagram
• Type I.  is Positive and Increases Linearly with
1/T (B > 0, C = 0)
Example:SPB(88)/dSPB(78)
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UCST= 105 C
in our example
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• Type II.  is Negative and Decreases Linearly
with 1/T (B < 0, C = 0)
• PIB/dHHPP blend as an example
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Lower Critical Solution Temperature
(LCST)

Transition from single phase to
two phase occurs at 170 5 C regardless of
composition.
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Type III.  is Positive and Increases Nonlinearly with
1/T(C  0, d/dT  0)
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Type III (cont.)
An example of such behavior is PEB/dSPI(7)
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One
Phase
One
Phase
Two
Phase
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Type IV.  is Positive and Nonmonotonic
with Temperature, and C > 0
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Type IV (cont).
Example system is
HHPP/dSPI(7)
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Type V.  is Positive and Nonmonotonic
with Temperature, and C < 0
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Type V (cont.)
The SPI(7)/dPP blend exhibits such
behavior
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Type VI. Athermal Mixing,  is Independent
of T (B = 0 and C = 0)
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Type VI (cont.)
SPI(50)/SPB(78) blends are the examples
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Thermodynamics of phase separation
according to Sperling
where V is the volume of the sample (usually taken as 1 cm3), Vr is the volume
of one cell , z is the lattice coordination number (z is usually between 6 and 12), and Nc is
the number of molecules in 1 cm3. V/Vr is a count of the number of cells in 1 cm3.
Compare it to our previously studied one:
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Thermodynamics of phase separation
according to Sperling (Cont.)
•An Example Calculation: Molecular Weight Miscibility Limit
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Solution
•An Example Calculation: Molecular Weight Miscibility Limit
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Equation of State Theories:
Deficiency of the proposed Eq.
A serious deficiency in the classical theory, is the
assumption of incompressibility.
This deficiency can easily be remedied by the addition of free volume in the form of
“holes” to the system. These holes will be about the size of a mer and occupy one lattice
site. In materials science and engineering, “holes” are frequently called “vacancies.”
Imagine that a multicomponent mixture is mixed with N0 holes of volume fraction v0.
Then the entropy of mixing is:
This allows for compressibility, since the number of holes may be varied.
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Reduced Quantities:
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Thus, an equation of state requires
three reducing quantities,
Reduced Temperature:
Reduced Pressure
Reduced Volume
Reduced Density
The starred quantities represent characteristic values for particular
polymers, often referred to as the “hard core” or “close-packed” values,
that is, with no free volume.
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Hartmann equation of state
(An example)
the classical Doolittle equation
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~
Properties of Reduced Density, 
When all the sites are occupied,
~ 1

Fractional free volume (vacant sites):
*
*
V
V

V
 0  1  ~  1 

V
V
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The Entropy of Mixing, Now!
The entropy of mixing vacant sites with the molecules in equation of
state terminology is given by
When all the sites are occupied,
~

= 1, and the right hand side is zero.
where the quantity * is a van der Waals type of energy of interaction.
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Note that ˜ = ˜ (P,T); ˜  1 as T  0; ˜  1
as P  .
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~0
By taking GM / 
theory is obtained
, the equation of state via the lattice fluid
where r is the number of sites (mers), in the chains, and
For high polymers, r goes substantially to infinity, yielding a general equation
of state for both homopolymers and miscible polymer blends,
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Criteria for Miscibility:
Basically, T* values must be similar for miscibility.
Polymers
must have similar coefficients of
expansion for miscibility; that is,
the ratio of the densities must
remain similar as the temperature
is changed.
If T*1 > T*2, then it is desirable to have P*1 > P*2
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Equation of state parameters for
some common polymers
From such information, it may be found that the system poly(2,6-dimethyl
phenylene oxide)–blend–polystyrene is miscible but that poly(dimethyl
siloxane), which is so different from the others, should be immiscible with
all of them.
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Very Important Conclusions:
• For the Flory–Huggins theory of incompressible polymer
mixtures, the small entropy of mixing is dominated by the
heat of mixing, leading to the conclusion that the heat of
mixing must be zero or negative to induce miscibility.
• When two compressible polymers are mixed together,
negative heats of mixing cause a negative volume change.
• Since reducing the volume of the system reduces the
number of holes, the entropy of mixing change is negative.
• At high enough temperatures, the unfavorable entropy
change associated with the densification of the mixture
becomes prohibitive, that is, TS > H, and the mixture
phase-separates.
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Very Important Conclusions:
• The two-phase system has a larger volume as the
holes are reintroduced, and hence a larger
positive contribution to the entropy.
• Significantly differing coefficients of expansion
contribute to phase separation.
• In virtually all polymer–polymer systems
exhibiting critical phenomena, both H and TS
are relatively small quantities. Thus relatively
modest changes in either the enthalpy or the
entropy alter the phase diagram significantly.
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Let’s try softwares
Applet by
Maye’s group
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Radius of Gyration of a Polymer Coil
The radius of gyration Rg is defined as the RMS distance
of the collection of atoms from their common centre
of gravity.
R
For a solid sphere of radius R;
2
Rg 
R  0.632R
5
For a polymer coil with rms end-to-end distance R ;
1 2
Rg  R
6
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1/ 2
a 1/ 2
 N
6
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The excluded volume effect
• Steric hindrance on short distances limits the
number of conformations
• At longer distances we have a topological constraint
– the self avoiding walk – or the excluded volume
effect:
Instead of <R2>1/2=aN1/2 we will have
<R2>1/2=aNν
where v>0.5
Experiments tells us that in general: v~0.6
Why?
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Excluded volume according to Flory
Consider a cube containing N segments of a polymer
V=r3 where r is the radius of gyration.
The concentration of segments is c~N/r3
Each segment with volume ‫“ ע‬stuffed” into the cube reduces the
entropy with –kb‫ע‬N/V = -kb‫ע‬N/r3
(for small x; ln(1-x)~lnx)
The result is a positive contribution to F; Frep= kb‫ע‬TN/r3 (expansion of the
coil)
From before; Coiling reduces the entropy; Fel=kbT3R2/2Na
The total free energy F is the sum of the two contributions!
Search for equilibrium!
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Scaling Law Theories
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Equilibrium and stability
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Lever Rule
Metastable
Stable
fraction fa of the volume of the material having composition Фα
(and fraction fβ = 1 —fα having composition Фβ,
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• According to thermodynamic principles, the condition for
equilibrium between two phases requires that the partial molar
free energy of each component be equal in each phase.
• This condition requires that the first and second derivatives of
in equation :

( 1  10 )  RT ln(1  v2 )  v2 (1  1 )  v22
x

with respect to 2 be zero.
• The critical concentration at which phase separation occurs may
be written:
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The Critical Values
n= number of segments per polymer
chain

For large n, 2c =1/n0.5.
 For n = 104,2c = 0.01, a very dilute solution.
 The critical value of the Flory–Huggins polymer–solvent interaction
parameter, 1, is given by
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as n approaches infinity, 1c approaches 1/2
111
The Critical Temp.
Where 1 is a
constant.
Plot of 1/Tc versus
(1/n0.5 + 1/2n) should
yield the Flory
-temperature at
n = infinity
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What do you understand from the
following picture?
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Critical Concentration
114
Now we consider more systematically the equilibrium properties of
concentrated polymer solutions of overlapping coils.
It is to be reminded that the overlap concentration of monomer units
is
N
N
1
c 
 3 32 3  3 12 3 
3
4 3 R
 N a
 N a
*
The corresponding volume fraction
  c * 
*
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
 N
3
12
 is molar volume
a
3
1
N
 1
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Since   1, the overlap occurs already at very low polymer concentration.
There is a wide concentration region where 
(i) coils are overlapping and strongly entangled; and
(ii)
. 1
Such solutions are called semidilute.
 
*
*
Dilute
solution
0
Semidilute
solution
* 
Concentrated
solution
~ 0.2
1
3
 N12
Polymer
melt
1

The existence of the regime of the semi-dilute polymer solutions is a specific
polymer feature, for low-molecular solutions such regime does not exist.
The crossover volume fraction between the two regimes is
* 
1
N12
for
* 
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-solvents (ideal coils)

1
1

 3N1 2
N45
for good solvents
(swollen coil)
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Behavior of Polymer Solutions in Poor Solvents
In poor solvent (below the - point) the attraction between monomer units
prevails. Single chains (or chain in dilute enough solutions) collapse and form a
globule. However, in concentrated solutions the macroscopic phase separation
can take place as well (a kind of intermolecular collapse).
Supernatant
phase
Precipitant
phase
What are the conditions for macroscopic phase separation? To answer this
question it is necessary to write down the free energy of polymer solution.
This problem was first solved independently by Flory and Huggins (19411942) for the lattice model of polymer solution.
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117
Flory and Huggins obtained:
F


ln   (1   ) ln(1   )   2
n0 kT
N
where n0is the total number of lattice sites and    2kT
is the so-called
Flory para-meter;   0corresponds to   0 ( very good solvent).

ln 
N
This term describes translational entropy of coils (free
energy of ideal gas of coils)
(1   ) ln(1   ) Term responsible for excluded volume interaction
 2
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Term responsible for the attraction of monomer
units
118
With the increase of  the quality of solvent becomes poorer. Which

value of
 corresponds to the
- point? The expansion of F in the
power
 of :
F

1
1

ln  
 2 (1  2  ) 
3  
n0 kT
N
2
6

ln 
N
Ideal gas term
1
 2 (1  2  )
2
1
3
6
At T  
Binary interactions, second
virial coefficient B
Ternary interactions, third
virial coefficient C
B  0  -point
 corresponds to
 1 2
.
  1 2 - good solvent region
  1 2 - poor solvent region
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119
The Polymer-Solvent Diagram
Excluded
Volume=v
Semidilute
Marginal
Concentrated
dilute
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
120
The Polymer-Solvent Diagram
Semidilute
Marginal
dilute
Excluded
Volume=v
Ideal
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Concentrated

121
Excluded Volume
• A chain cannot cross itself in space.
• Is a long-range interaction; eliminates
conformations in which two widely separated
segments would occupy the same space.
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123
Excluded Volume
• At longer distances we have a topological
constraint – the self avoiding walk – or the
excluded volume effect:
Instead of <r2>1/2=aN1/2 we will have
<r2>1/2=aNν where v>0.5
Experiments tells us that in general: v~0.6
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