chapter  2
16 Pages


Figure 2.2 Simple homopolymer and block copolymer architectures. Very few pairs of homopolymers are miscible in the melt; indeed, the tendency of a blend of polymers to macrophase separate is much greater than that of an equivalent blend of the unconnected monomers. The thermodynamics of polymer melts is governed by

the competing influence of energetic and entropic terms in the free energy of mixing. The energy of mixing (ΔUmix) is proportional to the number of monomers present, while the entropic contribution (ΔSmix), which favors homogenous mixing, is proportional to the number of polymer chains. The entropy of mixing per monomer is, therefore, decreased by a factor of the degree of polymerization (N) for a polymer blend compared with a blend of the unconnected monomers. The Flory-Huggins free energy of mixing per monomer (ΔFmix) for a blend of polymers A and B at temperature T, with degree of polymerization N and volume fractions fA and fB, respectively, is given by

ΔFmix = ΔUmix − TΔSmix (2.1)(2.2)where χ is the dimensionless Flory interaction parameter describing the energetic cost per monomer of contacts between A and B monomers ΔFmix kBT

fA N

fB N= lnfA+ lnfB + χ fA fB,

(2.3)where Z is the number of nearest-neighbor contacts in a lattice model of the polymer, and εAB is the interaction energy per monomer between A and B monomers. Polymer mixing is, therefore, controlled by the product χN rather than χ as is the case for a simple monomer blend. A critical value of χN = 2 separates the case in which mixtures of all compositions are stable (χN < 2) from the case in which mixtures at some compositions will phase separate (χN ≥ 2). Since N can be extremely large, only a very small positive value of χ (i.e., very weak repulsive monomer interactions) is required for the free energy of mixing to favor phase separation over a single-phase mixture of polymer chains.In block copolymer melts, repulsive monomer interactions are again a strong driving force for phase separation into A and B rich domains. However, since each block is covalently tethered to its reluctant partner, the scale of phase separation is limited to macromolecular (mesoscopic) dimensions. The result is spontaneous formation of a spectrum of periodic, ordered structures on the 5-100 nm length scale, known as microphase separation. At sufficiently elevated temperatures, as with homopolymer blends, entropic terms overwhelm the energetic interaction terms in the free energy and result in a single disordered phase. The transition from a homogeneous melt phase of copolymer chains to chemically heterogeneous microdomains occurs on cooling at a critical value of χN known as the order-disorder transition (ODT). The morphology of the microphase and the temperature of the ODT depend on the composition of the copolymer, i.e., the volume fraction of one block fA (with fB = 1 − fA). The microphase morphology adopted must balance minimizing unfavorable A-B interactions with the entropic penalty of stretching the two blocks away from each other as the chain adopts an extended configuration. For example, a symmetric diblock (fA = fB) will microphase separate into an alternating lamellar morphology with a flat interface, as illustrated in Fig. 2.3a. A simplified but illuminating treatment of this situation was given by Bates and Fredrickson [2]. At low temperatures (large χ), the lamellar microdomains are nearly pure in A or B components,


12χ = χAB= (εAB− (εAA+εBB)),

separated by interfaces much narrower than the lamellar period λ — so-called “strong segregation.” Assuming that the chains are uniformly stretched, an approximate expression for the free energy per chain for the lamellar phase (FLAM) is given by: (2.4)


Figure 2.3 Microphase separation in diblock copolymers. The shaded regions represent domains rich in A or B monomers. ( ) fA = fB = 1/2, lamellar phase separation with flat domain interfaces. (b) fA > 1/2, spontaneous curvature of domains toward the minority phase in gyroidal, cylindrical, or spherical morphologies. The characteristic domain spacing λ is typically in the range 10-100 nm.