Transcript Заголовок слайда отсутствует

Concentrated Polymer Solutions
Up to now we were dealing mainly with the
dilute polymer solutions, i.e. with single
chain properties (except for the chapter on
the viscosity of entangled polymer
systems). 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 * 

 N a
3
12
3
 1
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
1
  3 12
 N
*
~ 0.2
Concentrated
solution
Polymer
melt
1

The existence of the regime of the semidilute 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
 *  1 2 for -solvents (ideal coils)
N
1
1 for good solvents
*
  3 12  45
 N
N (swollen coil)
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
gradually
c  c * , the coil swelling
diminishes and finally it
vanishes in the melt (i.e. coils are ideal in
the melt -Flory theorem) .
Let us give a qualitative illustration of the
concept of screening of excluded volume
in concentrated systems.
Screening
means
appearance
attraction which neutralizes repulsion.
of
Two sites with excluded
volume repel each other
In the liquid of dimers
two sites normally
exclude 8 possible
dimers positions
If these sites are nearest
neighbors, they exclude
7 dimer positions 
additional attraction
In the liquid of polymers (multimers)
this effect becomes even larger and
leads to the complete screening of
excluded volume.
Polymer coil dimensions in semidilute
solutions: example of scaling arguments
Thus, for the solution of flexible polymer
chains far above the -point: R  aN 3 5
at   * and R  aN 1 2 at   1.
What is the value of R in the intermediate
range *    1 (semidilute polymer
solution)? This problem is easily solved
by scaling method.
Scaling considerations are widely used in
polymer science. We will illustrate this
type of considerations for the problem of
concentration dependence of R in the
semidilute polymer solutions. Scaling
arguments
normally
following steps.
include
the
Step 1.
It is assumed that  * is the only characteristic polymer volume fraction in the range
0    1. Thus
R  aN 3 5 f ( * )
where f(x) is some function (not yet defined)
Step 2.
The asymptotic forms of the function f(x)
are assumed to be the following:
f ( x ) | x1  1 (since for dilute3solu5
tions R  aN );
and
f ( x ) | x 1  const  x n
- power law asymptotic with the exponent n
(not yet defined). Thus, at   *
R  const  aN 3 5 ( * ) n  const  aN 3 5 (N 4 5 ) n
since, *  1 N 4 5 in the good solvent.
Step 3.
The exponent n is chosen from additional
physical arguments. In our case we know: at
  1, R ~ aN 1 2 (Flory theorem). Thus
3 4
1
 n ,
5 5
2
1
n
8
Therefore, for semidilute solutions, i.e. in
the range *    1 we get the following
relation
R  aN 3 5 (  * ) 1 8  aN (N 4 5 ) 1 8 
35
R  aN 1 2 1 8
I.e. size of polymer coil drops with the
increase of  in the semidilute solution range;
at   1 all the swelling vanishes.
This type of scaling arguments has been
successfully used for a number of polymer
problems. This approach allows to obtain
correct answers without complicated
calculations.
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 (1941-1942) for the lattice
model of polymer solution.
Polymer chains are represented as random
walks on the lattice without self-intersections
and with the energy   corresponding to
each close contact of two non-neighboring
along the chain units. In the Flory-Huggins
theory the number of conformations is
counted and the entropy is derived as a
logarithm of this number. The energy is
calculated from the average number of close
contacting monomer units (  Nn ), where n
is the total number of chains and N is the
number of units in each chain.
Flory and Huggins obtained:
F

 ln   (1  ) ln(1  )   2
n0 kT N
where n0 is the total number of lattice sites
and    2kT is the so-called Flory parameter;   0 corresponds to   0 (only
excluded volume; 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
Term responsible for the
attraction of monomer units
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 2
1 3
 ln    (1  2  )    
n0 kT N
2
6

ln 
N
Ideal gas term
1 2
 (1  2  )
2
Binary interactions, second
virial coefficient B
1 3

6
Ternary interactions, third
virial coefficient C
At T   B  0   - point corresponds to   1 2.
  1 2 - good solvent region
  1 2 - poor solvent region
Macroscopic Phase Separation
The typical dependence of the FloryHuggins free energy on the polymer
volume fraction in the solution  :
F
1
2
Spinodal
points
1
Ф
Binodal
points
This dependence contains both convex
and concave parts.
F
Convex part of the function
F(Ф): no macroscopic phase
separation.
F
Free energy of the solution
F
separated into two phases with
  
Ф
  1 and    2
Free energy of homogeneous solution at   
ps
h
1
2
F
Concave part of the dependence F(Ф): macroscopic
phase separation into two
phases.
Fh
F ps
1 
2
Ф
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 (  2 F  2  0 ) spinodal curve.
Spinodal at 1
1

 2  0
N 1  
or
1 1
1 
 


2  N 1   
This dependence is shown in the figure:

c
12
c
Ф
1
1
c  
;
2
N
1
c 
N
Phase diagram with binodal and spinodal

Spinodal
Single
globules
Binodal
c
12
c
Ф
Conclusions:
• Macroscopic phase separation takes
place at the quality of solvent only slightly
poorer than the  - solvent.
1
1
c  
2
N
• The critical point for macroscopic phase
separation corresponds to the dilute
enough solution.
1
c 
N
• The region of isolated globules in solution corresponds to very low polymer
concentrations, especially at the values of 
significantly larger than 1 2.
• 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
12
c
Ф
With the increase of N the critical temperature becomes closer to the  - point,
and the critical concentration becomes
lower.
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.
• What is the connection of the FloryHuggins 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.
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   .
Such situation is called
lower critical solution
temperature (LCST)critical point is “on the

bottom” of the phase
c separation region.
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
polymer- water contacts is more important
at high temperatures).
T
• 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.