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

AN

BN

2

0

0 lnln ABB

BA

A

A

NNkTn

F

where

BBAAABkT

211

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.

BBAAAB

BA

.1 BA

The phase diagram which follows from this expression for the free energy has the form:

The critical point has the coordinates:

BA

Bc

BA

BAc NN

N

NN

NN

;

2

2

For the symmetric case ( ) we have

NNN BA

12

,2

1

Ncc

c 1

c

Binodal

Spinodal

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.

T

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.

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.

A B

AN

BN

Possible resulting morphologies:

Spherical B-micellesin the A-surrounding

Spherical A-micellesin the B-surrounding

Cylindrical B-micellesin the A-surrounding

Cylindrical A-micellesin the B-surrounding

Alternating A- an B- lamellae

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.

Resulting phase diagram for symmetric diblock copolymers (same Kuhn seg-ment length and monomer unit volumes for A- and B- chains).

Nc 10

c

21

c

a b c d e

Af1

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 im-possible and the spontaneous orientation of rods occurs. The resulting phase is called anematic liquid-crystalline phase.

The probability that consecutive “squares” in the row are empty is .The transition occurs when this probability becomes significantly smaller than unity, i.e.

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.

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

1/~ Ldc

2~ cLddL

1cdL

dL /1

For long rods nematic ordering occurs at lowpolymer concentration in the solution.

d

d

L/d squares

Whether concentration of nematic ordering corresponds to a dilute or semidilute range?

Overlap takes place at 3*3* 1~1~ LcLc

22** ~~ LdLdc

0 Ф 2Ld Ld 1

Dilute-semidilutecrossover

Liquid-crystallineordering

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

1~ ldc 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 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.

Statistical Physics of Polyelectrolyte Systems

Polyelectrolytes = macromolecules containing charged monomer units.

Dissociation:

Counter ions are always present in polyelectrolyte system

Schematic pictureof polyelectrolytemacromolecule

Number ofcounter ions

Number of chargedmonomer units=

Neutralmonomerunits

Charged monomerunits

Counterion+

Typical monomer units

for polyelectrolytes:

(b) sodium

methacrylate

(a) sodiumacrylate

(c) diallyldimethylammonium

chloride

(f) methacrylic

acid

(e) acrylic

acid

(d) acrylamide

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 ).

D

ij

ijij r

r

re

rV exp2

Strongly charged( large fraction of links charged )Coulomb interactionsdominate

Weakly charged( small fraction of linkscharged )Coulomb interactions interplay with Van-der-Waals interactions of uncharged links

21

24

ne

kTrD

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.

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.

Counter Ion Condensation

21 rr

e

a

r1r2

counter ion

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.

1

2

1

21 ln~ln~~

rr

kTVV

kTSkTF

1

22

1

22 ln~ln~~

rr

ae

rr

eeF

12

akTe

u 21 FF , then

12

akTe

u 12 FF , then

21ln rr

ae

21 rr

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.

1kT

eu effeff

1kTe

ekT

effThe dependence of the effective chargeon the line as a functionof its initial charge