A symmetric molecule separates into stripes (lamellae) of red and blue, while a molecule with more blue than red separates into cylinders of red embedded in a matrix of blue (micelles), or spheres or, depending on how asymmetric the molecule is, any number (so it seems!) of exotic bicontinuous phases, such as the Gyroid phase pictured here (the observant reader will notice that this is not the actual gyroid phase: if you want to know what's wrong with this picture email me). For a good page devoted to minimal surfaces like this Gyroid phases, see Robert Holyst's Research Group. There is a wealth of interesting physics here, including the crystallography of these phases, molecular and collective fluctuation effects near the order-disorder transition and elsewhere, and the behavior under flow.

PD Olmsted and IW Hamley,

We consider micro- and macro- phase separation in blends of AB and BC diblock copolymers. We show that, depending on architecture, a number of phase diagram topologies are possible. Microphase separation or macrophase separation can occur, and there are a variety of possible Lifshitz points. Because of the rich parameter space, Lifshitz points of multiple order are possible. We demonstrate Lifshitz points of first and second order, and argue that, in principle, up to 5th order Lifshitz points are possible.

** Strong-Segregation Theory of Bicontinuous Phases in Block Copolymers**

PD Olmsted and ST Milner,
*Macromolecules* **31** (1998) 4011-4022
(abstract ,
reprint in .ps
or .pdf format).

We compute phase diagrams for diblock star copolymers in the strong-segregation regime as a function of volume fraction $\phi$, including bicontinuous phases related to minimal surfaces (G, D, and P surfaces) as candidate structures. We present the details of a general method to compute free energies in the strong segregation limit, and demonstrate that the gyroid G phase is the most nearly stable among the bicontinuous phases considered. We explore some effects of conformational asymmetry on the topology of the phase diagram.

**Analytic weak-segregation theory of bicontinuous phases in diblock
copolymers** S. T. Milner and P. D. Olmsted, *Journal de Physique II
***7** (1997) 249-255. (at
Journal de Physique)

We compute phase diagrams for diblock copolymers in the mean-field weak-segregation regime as a function of the fraction f of A-monomers and the repulsive interaction \chi M. We include the ordered bicontinuous double-diamond (OBDD) phase [space group Pn3m] and the gyroid phase [space group Ia3d] as well as lamellae, hexagonal cylinders, and BCC spheres. We find a stable region of gyroid phase between cylinders and lamellae just above the mean-field critical point, in agreement with numerical mean-field calculations. The stability of gyroid depends on the presence of the next higher [220] reflections in addition to the [211] fundamental. The gyroid free energy is favored by terms of the form \psi^3[211]\psi[220] and \psi^2[211]\psi[220]; the analogous terms are not permitted for OBDD.

**Fluctuation Corrections to Mean-Field Theory for Homopolymer-Copolymer
Macrophase Separation: Sequence Distribution Effects** P. D. Olmsted
and S. T. Milner, *Macromolecules ***27** (1994) 1964-1967.

Mean-field theories (Flory-Huggins, RPA) predict that the critical repulsive
interaction parameter \chi_c for polymer-polymer phase separation is independent
of the monomer sequence distribution. In this work we generalize the RPA
to account for sequence distribution effects on the spinodal by including
both composition and chemical potential fluctuations. We calculate \chi_c(p)
for a blend of homopolymers and multiblock copolymers with 2p blocks, and
find an architecture-dependent shift of relative order N^{-1/2}. This shift
either increases or decreases as p is increased (corresponding to a more
evenly distributed copolymer), depending roughly on whether the homopolymer
is shorter or longer than the copolymer. Fluctuations also shift the critical
*composition *away from the Flory-Huggins value, even in asymmetric
homopolymer/homopolymer blends.

**Strong-Segregation Theory of Bicontinuous Phases in Block Copolymers**
P. D. Olmsted and S. T. Milner, *Physical Review Letters ***72**
(1994) 936-939; **74** (1995) 829.

We compute phase diagrams for diblock copolymers in the strong-segregation regime as a function of volume fraction \phi. We include the ordered bicontinuous double-diamond (OBDD) phase, which is found to be stable between cylinders and lamellae, consistent with experiment, for \phi (or 1-\phi) between 0.19 and 0.27. Our phase diagram relies on taking into account the hexagonal shape of the unit cell for the cylindrical phase, and the unique geometry of the OBDD phase.

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