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 over- lapping coils.
It is to be reminded that the overlap concentration of monomer units is
3 2 1 3 3
2 3 3 3
* 1
3
4 N a N a
N R
c N
The corresponding volume fraction 1
* 3 12 3
*
c N a
Since , 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) . Such solutions are called semidilute.
1
*
1
*
0 0.2 1
2 1 3
* 1
N
Dilute
solution Semidilute
solution Concentrated
solution Polymer melt
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
2 1
* 1
N
for -solvents (ideal coils)
5 4 2
1 3
* 1 1
N N
for good solvents
(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 , the coil swelling gradually 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 of attraction which neutralizes repulsion.
c*
c
In the liquid of polymers (multimers) this effect becomes even larger and leads to the complete screening of excluded volume.
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
Thus, for the solution of flexible polymer chains far above the -point:
at and at .
What is the value of R in the intermediate range (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 include the following steps.
5
aN3
R
2
aN1
R
*
1
1
*
Polymer coil dimensions in semidilute solutions: example of scaling arguments
Step 1.
Step 2.
It is assumed that is the only characte- ristic polymer volume fraction in the range
. Thus 1
0
)
( *
5
3
aN f R
where f(x) is some function (not yet defined)
*
The asymptotic forms of the function f(x) are assumed to be the following:
1
| )
(x x1
f (since for dilute solu-
tions );R aN35
and
n x const x x
f ( ) | 1
- power law asymptotic with the exponent n (not yet defined). Thus, at *
n n const aN N
aN const
R 35( *) 35( 45)
since, in the good solvent. * 1 N 4 5
Step 3.
The exponent n is chosen from additional physical arguments. In our case we know: at
(Flory theorem). Thus
2
~ 1
,
1 R aN
8 1
2, 1 5
4 5
3 n n
Therefore, for semidilute solutions, i.e. in the range we get the following relation
1
*
aN3 5( *)18 aN 3 5( N 4 5)18
R
8 1 2
1
aN R
I.e. size of polymer coil drops with the increase of in the semidilute solution range;
at 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.
1
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).
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.
Supernatant phase Precipitant
phase
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 ( ), where n is the total number of chains and N is the number of units in each chain.
Nn
Flory and Huggins obtained:
2 0
) 1
ln(
) 1
(
ln
N kT
n F
where is the total number of lattice sites and is the so-called Flory para- meter; corresponds to (only excluded volume; very good solvent).
n0
kT
2
0
0
N ln
) 1
ln(
) 1
(
2
This term describes translatio- nal entropy of coils (free
energy of ideal gas of coils)
Term responsible for exclu- ded volume interaction
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 :
2 3
0 6
) 1 2 1 2 (
ln 1
N kT
n F
N ln
) 2 1 2 (
1 2
3
1 6
Ideal gas term
Binary interactions, second virial coefficient B
Ternary interactions, third virial coefficient C
At - point corres- ponds to .
B 0 T
2
1
2
1
2
1
- good solvent region - poor solvent region
Macroscopic Phase Separation
The typical dependence of the Flory- Huggins free energy on the polymer volume fraction in the solution :
This dependence contains both convex and concave parts.
Convex part of the function F(Ф): no macroscopic phase separation.
Free energy of the solution separated into two phases with
and1
2
Concave part of the depen- dence F(Ф): macroscopic phase separation into two phases.
Free energy of homogeneous solution at
F
Ф F
Ф
1 2 Fps
Fh
Fps
Fh
F
Ф
Binodal points
Spinodal points
2
1 1
Conditions for the phase separation (minimum possible free energy) are determined from common tangent straight line - binodal curve. Condi- tions for the absolute stability of homo- geneous phase at a given concentration are determined from the positions of inflexion points ( ) - spinodal curve.
Spinodal at
0
2
2
F
0 1 2
1
1
N
or
1 1 1
2 1
N
This dependence is shown in the figure:
N N
c c
1 1 ; 2
1
Ф
c
c
2 1
Conclusions:
• Macroscopic phase separation takes place at the quality of solvent only slightly poorer than the - solvent.
c N
1 1 2
• The critical point for macroscopic phase separation corresponds to the dilute enough solution.
c N
1
• The region of isolated globules in solu- tion corresponds to very low polymer
concentrations, especially at the values of
significantly larger than .1 2
c Ф
c
2 1
Binodal Spinodal
Single globules
Phase diagram with binodal and spinodal
• The precipitant phase close enough to the - point is very diluted.
• For different values of N the binodal curves (boundaries of the phase separa- tion region) have the form:
With the increase of N the critical tempe- rature becomes closer to the - point, and the critical concentration becomes lower.
c Ф
c
2 1 N1
N2
N3
N1 N2 N3
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 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.
c
• What is the connection of the Flory- Huggins parameter and the temperatu- re T ? Within the framework of the
lattice model in the
experi-mental variables T, c the phase diagram has the form shown in the figure, i.e. the poor solvent region corresponds to
kT
T
Such situation is called upper critical solution temperature (UCST) - critical point is “on the top” of the phase separa- tion region.
Examples: poly(styrene) in cyclohexane (around ), poly(isobutylene) in benzene, acetylcellulose in chlorophorm.C
35
T
c
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
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).
Such situation is called lower critical solution temperature (LCST)- critical point is “on the bottom” of the phase separation region.
T
c
• 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:
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 chromatogra-phic columns.
T
1 c
Tg
