Theories for ordered block copolymer solutions

The position of the structure factor peak, q, for semidilute block copolymer solutions is predicted to scale as (Fredrickson and Leibler 1989 Olvera de la Cruz 1989) 

Mayes et ai. (1994) tested this prediction using SANS on PSPMMA and PMMA-dPMMA diblocks in contrast-matched toluene/d-toluene mixtures in the disordered phase. They did not obtain the scaling (eqn 4.14), instead a best fit to the data yielded q = (p°05 (Fig. 4.34). This slower than expected scaling is presently unexplained. However, the expected concentration dependence of the blob size (Mayes et at. 1994) 

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. 

Gast and co-workers (Gast 1996 McConnell et al. 1994) used SCFT to study interactions between spherical diblock copolymer micelles in solution.The theory was used to calculate intermicellar pair potentials and combined with liquid-state 

Berret, J.-F., Molino, F., Porte, G., Dial, O. and Lindner, P. (1996). Journal of Physics, Condensed Matter, 8, 9513. 

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This chapter is concerned with experiments and theory for semidilute and concentrated block copolymer solutions.The focus is on the thermodynamics, i.e. the phase behaviour of both micellar solutions and non-micellar (e.g. swollen lamellar) phases. The chapter is organized very simply Section 4.2 contains a general account of gelation in block copolymer solutions. Section 4.3 is concerned with the solution phase behaviour of poly(oxyethylene)-containing diblocks and tri-blocks. The phase behaviour of styrenic block copolymers in selective solvents is discussed in Section 4.4. Section 4.5 is then concerned with theories for ordered block copolymer solutions, including both non-micellar phases in semidilute solutions and micellar gels. There has been little work on the dynamics of semidilute and concentrated block copolymer solutions, and this is reflected by the limited discussion of this subject in this chapter. 

The lattice model, as put forth by Flory [84, 85], has been proved successful in the treatments of the liquid crystallinity in polymeric systems, despite its artificiality. In our series of work, the lattice model has been extended to the treatment of biopolypeptide systems. The relationship between the polypeptide ordering nature and the LC phase structure is well established. Recently, by taking advantage of the lattice model, we formulated a lattice theory of polypeptide-based diblock copolymer in solution [86]. The polypeptide-based diblock copolymer exhibits lyotropic phases with lamellar, cylindrical, and spherical structures when the copolymer concentration is above a critical value. The tendency of the rodlike block (polypeptide block) to form orientational order plays an important role in the formation of lyotropic phases. This theory is applicable for examining the ordering nature of polypeptide blocks in polypeptide block copolymer solutions. More work on polypeptide ordering and microstructure based on the Flory lattice model is expected. 

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. 

Rod—coil block copolymers have both rigid rod and block copolymer characteristics. The formation of liquid crystalline nematic phase is characteristic of rigid rod, and the formation of various nanosized structures is a block copolymer characteristic. A theory for the nematic ordering of rigid rods in a solution has been initiated by Onsager and Flory,28-29 and the fundamentals of liquid crystals have been reviewed in books.30 31 The theoretical study of coil-coil block copolymer was initiated by Meier,32 and the various geometries of microdomains and micro phase transitions are now fully understood. A phase diagram for a structurally symmetric coil—coil block copolymer has been theoretically predicted as a... 

In Table 1 we present the Zp values determined in THF and two different THE/DME mixtures. These values, on the order of 1 rm, are comparable to those reported by Discher and coworkers [38,70] and by Bates and coworkers [71,72] for PEO-PI cylindrical micelles with a core diameter of 20 nm in water. Here PEG denotes poly(ethylene oxide). Bates and coworkers deduced their values of Zp from small-angle neutron scattering experiments, whereas Discher and coworkers determined the Zp values using fluorescence microscopy. The fact that the Zp values that we determined from viscometry are comparable to those of the PEO-PI cyUndrical micelles with similar core diameters again suggests the validity of the YFY theory in treating the nanofiber viscosity data. This study demonstrates that block copolymer nanofibers have dilute solution properties similar to those of semi-flexible polymer chains. 

Concentrated solutions of block copolymers have also been used to investigate the kinetics of ordering by using SANS. A Landau-Ginzburg theory for the kinetic process has been suggested by Hashimoto [46], and the expression for the change in scattered intensity with time following a sudden quench from one equilibrium state to a different state is ... 

An-Chang Shi is a professor of physics at the McMaster Univereity, Hamilton, Ontario, Canada. He received his BSc in physics from Fudan Univereity in 1982 and obtained his PhD in physics from Univereity of Illinois at Urbana-Champaign in 1988. From 1988 to 1892 he was a Post-Doctoral Fellow and Research Associate at McMaster University. He joined Xerox Research Centre of Canada as a Member of Research Staff in 1992, and moved to the Department of Physics and Astronomy at McMaster Univereity in 1999. His main research area concerns the theory of phases and phase transitions in block copolymers. Over the last years, An-Chang Shi has worked on a number of problems in theoretical polymer physics, including the development of statistical mechanics theory for polymeric s) tem the investigation of phase diagrams of block copolymer melts, blends and solutions as well as block copolymers tmder confinement and the study of kinetic pathways of order-order transitions of block copolymer phases. 

The solution properties of blocks and grafts are complicated since the copolymer components A and B behave differently in different solvents. In order to simplify the analysis, one usually starts with a solvent in which both A and B are soluble. In this case, the solution properties approach those of a homopolymer, for which accurate theories exist, e.g. in the thermodynamic treatment of Flory and Huggins (1, 2). The latter considers the free energy of mixing of pure polymer with pure solvent, AGmix in terms of two contributions, i.e. the enthalpy of mixing, A//mix, and the entropy of mixing, A mix, as follows ... 

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