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Polymer Mixtures
Suppose that now instead of the solution
(polymer A dissolved in low-molecular liquid
B), we have the mixture of two polymers (polymer A, number of links in the chain N A and
polymer B, number of links in the chain N B).
Flory-Huggins method of calculation of the
free energy can be applied for this case as
well. The result for the free energy of a
polymer mixture is
F0


 A ln  A  B ln  B   2A
n0kT N A
NB
where

1 
1








 AB
AA
BB 
kT 
2

and  AB,  AA and  BB are the energies associated with the contact of corresponding
monomer units;  A and  B are the volume
fractions for A- and B- components; and
 A   B  1.First two terms are connected
with the entropy of mixing, while the third
term is energetic.
The phase diagram which follows from this
expression for the free energy has the form:

Spinodal
Binodal
c
1
c

The critical point has the coordinates:
c


NA  NB 
NB
; c 
2N A N B
NA  NB
2
For the symmetric case ( N A  N B  N )
we have
1
c  ,
2
2
 c   1
N

For polymer melt it is enough to have a
very slight energetic unfavorability of the
A-B
contact
to
induce
the
phase
separation.
Reason: when long chains are segregating
the energy is gained, while the entropy is
lost but the entropy is very low (polymer
systems are poor in entropy).
Thus, there is a very small number of
polymer pairs which mix with each other;
normally polymer components segregate
in the melt.
Note:
for
polymer
mixtures
the
T   phase diagrams with upper and low
critical mixing temperatures are possible.
Microphase Separation in Block-Copolymers
Suppose that we prepare a melt of A-B
diblock copolymers, and the blocks A and B
are not mixing with each other. Each diblockcopolymer molecule consists of N A monomer
units of type A and N B monomer units of
type B.
A
B
The A- and B- would like to segregate, but
the full-scale macroscopic phase separation
is impossible because of the presence of a
covalent link between them. The result of
this conflict is the so-called microphase
separation with the formation of A- and Brich microdomains.
Possible resulting morphologies:
Spherical B-micelles
in the A-surrounding
Cylindrical B-micelles
in the A-surrounding
Alternating A- an B- lamellae
Spherical A-micelles
in the B-surrounding
Cylindrical A-micelles
in the B-surrounding
Resulting phase diagram for symmetric
diblock copolymers (same Kuhn segment length and monomer unit volumes
for A- and B- chains).

a b
c
d e
c
12
1
fA
 c  10 N  to induce microphase separation
one needs a somewhat stronger repulsion of
components than for disconnected blocks. Near
the critical point the boundaries between the
microdomains are smooth, while they are
becoming very narrow at    c.
The type of resulting morphology is controlled
by the composition of the diblock.
Microphase separation is an example of selfassembly phenomena in polymer systems with
partial ordering.
Liquid-Crystalline Ordering in Polymer Solutions
Stiff polymer chains: l >> d. If the chain is so
stiff that l >> L >> d macromolecules can be
considered as rigid rods.
Examples:
short fragments of DNA ( L<50 nm ), some
aromatic polyamides, -helical polypeptides,
etc.
Let us consider the solution of rigid rods, and
let us increase the concentration.
Starting from a certain concentration the
isotropic orientation of rods becomes impossible and the spontaneous orientation of
rods occurs. The resulting phase is called a
nematic liquid-crystalline phase.
Let us estimate the critical concentration c
for the emergence of the critical liquidcrystalline phase. Let us adopt the lattice
model of the solution.
d
d
L/d squares
The liquid-crystalline ordering will occur
when the rods begin to interfer with each
other. This means that it is impossible to put
L/d “squares” of the rod in the row without
intersection with some other rod.
Volume fraction of rods is  ~ cLd 2
The probability that L d
consecutive
“squares” in the row are empty is  1   L /.d
The transition occurs when this probability
becomes significantly smaller than unity,
i.e. L d  c  1
 c ~ d / L  1
For long rods nematic ordering occurs at low
polymer concentration in the solution.
Whether concentration of nematic ordering
corresponds to a dilute or semidilute range?
Overlap takes place at c*L3 ~ 1  c* ~ 1 L3 
* ~ c*Ld 2 ~ d L2
Liquid-crystalline
ordering
Dilute-semidilute
crossover
0
d L2
d L
1
Ф
Liquid-crystaline ordering for rigid rods
occurs in the semidilute range.
Real stiff polymers always have some
flexibility. Then the chain can be divided
into segments of length l (which are aproximately rectilinear), and the above
consideration for the rigid rods of length l
and diameter d can be applied. Then
 c ~ d l  1 if l >> d , i.e for stiff chains
Examples of stiff-chain macromolecules
which form liquid-crystalline nematic phase:
DNA, -helical polypeptides, aromatic poly-
amides, stiff-chain cellulose derivatives.
Nematic phase is not the only possibility for
liquid-crystalline ordering. If the ordering
objects (e.g. rods) are chiral (i.e. have rightleft asymmetry) then the so-called cholesteric phase is formed: the orientational axis
turns in space in a helical manner. E.g.
liquid-crystalline ordering in DNA solutions
leads
to
cholesteric
phase.
Another
possibility is the smectic phase, when the
molecules are spontaneously organized in
layers.
Statistical Physics
of Polyelectrolyte Systems
Polyelectrolytes = macromolecules containing
charged monomer units.
Schematic picture
of polyelectrolyte
macromolecule
Dissociation:
Charged
Counter
monomer + ion
units
Neutral
monomer
units
 Counter ions are always present in
polyelectrolyte system
Number of
counter ions
=
Number of charged
monomer units
Typical monomer units
for polyelectrolytes:
(a)
sodium
acrylate
(d)
acrylamide
(b)
sodium
methacrylate
(e)
acrylic
acid
(c)
diallyldimethylammonium
chloride
(f)
methacrylic
acid
Polyelectrolytes
Strongly charged
( large fraction of links
charged )
Coulomb interactions
dominate
Weakly charged
( small fraction of links
charged )
Coulomb interactions
interplay with Vander-Waals interactions
of uncharged links
Coulomb interactions in the Debye-Huckel
approximation
 rij 
e
V rij  
exp  
rij
 rD 
2
kT 

rD  
2
 4ne 
1
2
where  is the dielectric constant of the
solvent, rD is the so-called Debye-Huckel
radius, n is the total concentration of lowmolecular ions in the solution ( counter ions +
ions of added low-molecular salt ).
Counter Ion Condensation
The main assumption used in the derivation of
Debye-Huckel potential is the relative weakness
of the Coulomb interactions. This is generally
not the case, especially for strongly charged
polyelectrolytes. The most important new effect
emerging as a result of this fact is the
phenomenon of counter ion condensation.
r2 r1
a
e
counter ion
In the initial state the
counter ion was confined
in the cylinder of radius r1;
in the final state it is
confined within the cylinder
of radius r2.
r1  r2
r1  r2
The gain in the entropy of translational
motion
V2
r2
F1 ~ kTS ~ kT ln ~ kT ln
V1
r1
Decrease in the average energy of
attraction of counter ion to the charged line
 r2
e 2 r2
F2 ~ e ~ e ln ~  ln
 r1
a r1
(   e a - linear charge density)
One can see that both contributions ( F1
lnr.1 r2 
and F2 ) are proportional to
Therefore the net result depends on the
coefficient before the logarithm. If
e2
u
 1, then F1  F2
akT
and this means that the gain in entropy is
more important; the counter ion goes to
infinity. On the other hand, if
e2
u
 1, then F2  F1
akT
and counter ion should approach the charge
line and “condense” on it.
Now we take the second, third etc. counter
ions and repeat for them the above
considerations. As soon as the linear charge  on
the line satisfies the inequality e kT  1 the
counter ions will condense on the charged line.
When the number of condensed counter ions
neutralizes the charge of the line to such extent
 eff e
that
ueff 
1
kT
the condensation of counter ions stops. All the
remaining counter ions are floating in the
solution.
eff

 
kT
e
The dependence of
the effective charge
on the line as a function
of its initial charge

One can see that in the presence of counter
ions there is a threshold * such that it is
impossible to have a charged line with linear
charge density above this threshold.