Polymer Mixtures

1
Polymer Mixtures
Suppose that now instead of the solution (polymer
A dissolved in low-molecular liquid B), we have
the mixture of two polymers (po-lymer A, number
of links in the chain and polymer B, number
of links in the chain ). 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
where
and , and are the energies
asso-ciated with the contact of corresponding
monomer units and are the volume
fractions for A- and B- components and
First two terms are connected with the entropy
of mixing, while the third term is energetic.
2
The phase diagram which follows from this
expression for the free energy has the form
Spinodal
Binodal
1
The critical point has the coordinates
For the symmetric case ( ) we
have
3
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 phase
diagrams with upper and low critical mixing
temperatures are possible.
4
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 diblock-copolymer molecule
consists of monomer units of type A and
monomer units of type B.
B
A
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 B-rich microdomains.
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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
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Resulting phase diagram for symmetric diblock
copolymers (same Kuhn seg-ment length and monomer
unit volumes for A- and B- chains).
1
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
. The type of resulting morphology is
controlled by the composition of the
diblock. Microphase separation is an example of
self-assembly phenomena in polymer systems with
partial ordering.
7
Liquid-Crystalline Ordering in Polymer Solutions
Stiff polymer chains l gtgt d. If the chain is so
stiff that l gtgt L gtgt d macromolecules can be
considered as rigid rods. Examples short
fragments of DNA ( Llt50 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 im-possible
and the spontaneous orientation of rods occurs.
The resulting phase is called a nematic
liquid-crystalline phase.
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Let us estimate the critical concentration c for
the emergence of the critical liquid-crystalline
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
The probability that consecutive
squares in the row are empty is
. The transition occurs when this probability
becomes significantly smaller than unity, i.e.
For long rods nematic ordering occurs at
low polymer concentration in the solution.
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Whether concentration of nematic ordering
corresponds to a dilute or semidilute range?
Overlap takes place at
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
ap-roximately rectilinear), and the above
consideration for the rigid rods of length l and
diameter d can be applied. Then
if l gtgt d , i.e for stiff chains
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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 right-left asymmetry) then the
so-called choleste-ric 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.
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Statistical Physics of Polyelectrolyte Systems
Polyelectrolytes macromolecules containing
charged monomer units.
Dissociation ? Counter ions are always
present in polyelectrolyte system
12
Typical monomer units for polyelectrolytes

13
Polyelectrolytes Coulomb interactions
in the Debye-Huckel approximation
where ? is the dielectric
constant of the solvent, rD is the so-called
Debye-Huckel radius, n is the total concentration
of low-molecular ions in the solution ( counter
ions ions of added low-molecular salt ).
Strongly charged ( large fraction of links
charged ) Coulomb interactions dominate
Weakly charged ( small fraction of links charged
) Coulomb interactions interplay with
Van-der-Waals interactions of uncharged links
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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. I
n 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.
r2
r1
counter ion
a
e
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The gain in the entropy of translational
motion Decrease in the average energy of
attraction of counter ion to the charged line
( - linear charge
density) One can see that both contributions (
?F1 and ?F2 ) are proportional to . Therefore
the net result depends on the coefficient before
the logarithm. If and this means
that the gain in entropy is more important the
counter ion goes to infinity. On the other hand,
if and counter ion should
approach the charge line and condense on it.
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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 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
that the condensation of counter
ions stops. All the remaining counter ions are
floating in the solution. 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.