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.

The phase diagram which follows from this expression for the free energy has the form: The critical point has the coordinates: For the symmetric case ( ) we have 1 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.

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

Possible resulting morphologies: Spherical B-micelles in the A-surrounding Spherical A-micelles in the B-surrounding Cylindrical B-micelles in the A-surrounding Cylindrical A-micelles in 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). abcde 1

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 a nematic 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 For long rods nematic ordering occurs at low polymer 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 0 Ф 1 Dilute-semidilute crossover Liquid-crystalline ordering 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 >> 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 picture of polyelectrolyte macromolecule Number of counter ions Number of charged monomer units = Neutral monomer units Charged monomer units Counter ion +

Typical monomer units for polyelectrolytes: (b) sodium methacrylate (a) sodium acrylate (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, r D 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

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 r 1 ; in the final state it is confined within the cylinder of radius r 2. Counter Ion Condensation e a r1r1 r2r2 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 (  F 1 and  F 2 ) 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., then

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.   eff The dependence of the effective charge on the line as a function of its initial charge