Matsen-Schick

Fig. 2.44 Phase diagram for a conformationally-symmetric diblock copolymer, calculated using self-consistent mean field theory. Regions of stability of disordered, lamellar, gyroid, hexagonal, BCC and close-packed spherical (CPS), phases are indicated (Matsen and Schick 1994 ). All phase transitions are first order, except for the critical point which is marked by a dot.

Matsen, M. W. and Schick, M. (1994a). Physical Review Letters, 72, 2660. 

Self-consistent field theory (SCFT, see Sections 2.3.3 and 3.4,2) has recently been applied to the phase behaviour of ordered micellar solutions. Noolandi et al. (1996) compared continuum SCFT to the lattice version of this theory for triblock copolymers such as the Pluronics in aqueous solution. From a different viewpoint, this work represents an extension of the SCFT employed by Hong and Noolandi (1981, 1983) and Matsen and Schick (1994) for the phase behaviour of block copolymer melts to block copolymers in solution. The approximations introduced by the adoption of a lattice model are found to lead to some significant differences in the solution phase behaviour compared with the continuum theory, as illustrated by Fig. 4.44. For example, the continuum theory predicts ordered phases for Pluronic L64 (PE013PP03oPEO 3), whereas the lattice theory (neglecting polydispersity) predicts none. 

As mentioned in Section 2.3,3, because SC IT is the most general theory for the ordering of block copolymers to date, a brief outline is given here. The simplest case of a diblock copolymer melt is considered, following Matsen and Schick (1994). The extension to other melts of other architectures, solutions, blends or semicrystalline copolymers is discussed in the appropriate chapter. 

A system of n AB diblock copolymers each with a degree of polymerization N and A-monomer fraction, /, is considered. The A and B monomers occupy a fixed volume, l/g0, and the system is incompressible with a total volume, V, equal to hN/qq. A variable s is used as a parameter than increases continuously along the length of a polymer. At the A monomer end, s = 0, at the junction point, s = f, and at the other end, s = 1. The functions r (s) define the space curve occupied by the copolymer a (Matsen and Schick 1994). 

Matsen MW, Schick M (1996) Curr Opin Colloid Interface Sci 1 329... 

The book by Hamley [65] is a good general resource for self-consistent mean field theory. This formalism is based on the assumptions that (a) every chain in the system obeys Gaussian statistics, (b) the fluid is incompressible, and (c) the interactions between different structural units are local so that they depend only on the chemical nature but not on the positions of the units along their respective chains. As a result, the equations describe an ensemble of ideal chains in an external field which, in turn, is determined self-consistently from the structural unit probability distributions. As illustrated by Matsen and Schick [66], solving the exact equations requires a significant amount of computational effort to determine the equilibrium... 

Matsen M W and Schick M 1994 Stable and unstable phases of a diblook oopolymer melt Phys. Rev. Lett. 72 2660... 

Helfand E, Wasserman ZR (1982) In Goodman 1 (ed) Developments in block copolymers. Applied Science, London, p 99 Matsen MW, Schick M (1994) Phys Rev Lett 72 2660 Matsen MW, Schick M (1996) Curr Opin Colloid Interface Sci 1 329 Matsen MW, Bates FS (1996) Macromolecules 29 1091 Matsen MW (2001) J Phys Condens Matter 14 R21 Drolet F, Fredrickson GH (1999) Phys Rev Lett 83 4317 Fredrickson GH (2002) J Chem Phys 117 6810 Schweizer KS, Curro JG (1994) Adv Polym Sci 116 319 Schweizer KS, Curro JG (1997) Adv Chem Phys 98 1 David EF, Schweizer KS (1994) J Chem Phys 100 7767 David EF, Schweizer KS (1994) J Chem Phys 100 7784 Chandler D, Andersen HC (1972) J Chem Phys 57 1930 Chandler D, McCoy JD, Singer SJ (1986) J Chem Phys 85 5977 Gutin AM, Sfatos CD, Shakhnovich El (1994) J Phys A 27 7957 Angerman H, ten Brinke G, Erukhimovich lY (1996) Macromolecules 29 3255 Angerman H, ten Brinke G, Ernkhimovich lY (1996) Macromol Symp 112 199 Potemkin 11, Panyukov SV (1998) Phys Rev E 57 6902 Semenov AN (1997) J Phys II (France) 7 1489 Semenov AN, Likhtman AE (1998) Macromolecnles 31 9058... 

FIGURE 11.9 Phase diagram for conformationally symmetric diblock-copolymer melts showing regions of stability for the disordered (D), lamellar (L), gyroid (G), hexagonal (H), and cubic (C) phases. Dashed lines denote extrapolated phase boundaries and the dot marks the critical point. (Adapted from Matsen, M.W. and Schick, M., Phys. Rev. Lett., 72, 2660, 1994. With permission from the American Physical Society.)... 

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