Gyroid phase

Another phase which has attracted recent interest is the gyroid phase, a bicontinuous ordered phase with cubic symmetry (space group Ia3d, cf. Fig. 2 (d) [10]). It consists of two interwoven but unconnected bicontinuous networks. The amphiphile sheets have a mean curvature which is close to constant and intermediate between that of the usually neighboring lamellar and hexagonal phases. The gyroid phase was first identified in lipid/ water mixtures [11], and has been found in many related systems since then, among other, in copolymer blends [12]. 

Figure 1. Phase diagrams of the systems M0-H20 (Adapted from ref. 1) and MO-DOPC-H20 (at 28 C, Adapted from ref. 2), respectively. G (gyroid) and D (diamond) are cubic phases with similar structures to the type shown schematically in the figure (Adapted from ref. 18).

Fig. 1 Morphologies of diblock copolymers cubic packed spheres (S), hexagonal packed cylinders (C or Hex), double gyroid (G or Gyr), and lamellae (L or Lam). Inverse phases not shown. From [8], Copyright 2000 Wiley...

Fig. 18 Phase space of PI-fc-PS-fc-PEO in vicinity of ODT. Filled and open circles-. ordered and disordered states, respectively, within experimental temperature range 100 < T/° C< 225. Outlined areas compositions with two- and three-domain lamellae (identified by sketches) shaded regions three network phases, core-shell double gyroid (Q230), orthorhombic (O70), and alternating gyroid (Q214). Overlap of latter two phase boundaries indicates high- and low-temperature occurrence, respectively, of each phase. Dashed line condition tfin = 0peo associated with symmetric PI-fc-PS-fc-PEO molecules. From [75]. Copyright 2004 American Chemical Society...

The appearance and persistence of core-shell structures as well as the occurrence of phase separation are attributed to a small asymmetry in the X -parameters (xPS-pi = 0.06, xpi-pdms = 0.09 and xps-pdms = 0.20). Hence, a PDMS core surrounded by a PI shell embedded in a PS matrix results in a smaller inner diameter interfacial area, relative to that for the PS-PI case. In a blend of a PS-fo-PI-fc-PDMS triblock with PS and PDMS homopolymers, more PS homopolymer is expected to be found in the corona of the PS block than PDMS homopolymer in the corona of the PDMS block because the penalty for contact between the PI block and PDMS homopolymer is larger. In consequence, the distribution of homopolymers favours an expanded PS-PI interface, making the core-shell morphologies, gyroid and cylinder, more prevalent. 

In binary mixtures of water, surfactants, or lipids the most common structure is the gyroid one, G, existing usually on the phase diagram between the hexagonal and lamellar mesophases. This structure has been observed in a very large number of surfactant systems [13-16,24—27] and in the computer simulations of surfactant systems [28], The G phase is found at rather high surfactant concentrations, usually much above 50% by weight. 

An A-B diblock copolymer is a polymer consisting of a sequence of A-type monomers chemically joined to a sequence of B-type monomers. Even a small amount of incompatibility (difference in interactions) between monomers A and monomers B can induce phase transitions. However, A-homopolymer and B-homopolymer are chemically joined in a diblock therefore a system of diblocks cannot undergo a macroscopic phase separation. Instead a number of order-disorder phase transitions take place in the system between the isotropic phase and spatially ordered phases in which A-rich and B-rich domains, of the size of a diblock copolymer, are periodically arranged in lamellar, hexagonal, body-centered cubic (bcc), and the double gyroid structures. The covalent bond joining the blocks rests at the interface between A-rich and B-rich domains. 

Figure 14. The phase diagram of the gradient copolymer melt with the distribution functions g(x) = l — tanh(ciit(x —fo)) shown in the insert of this figure for ci = 3,/o = 0.5 (solid line), and/o — 0.3 (dashed line), x gives the position of ith monomer from the end of the chain in the units of the linear chain length. % is the Flory-Huggins interaction parameter, N is a polymerization index, and/ is the composition (/ = J0 g(x) dx). The Euler characteristic of the isotropic phase (I) is zero, and that of the hexagonal phase (H) is zero. For the bcc phase (B), XEuier = 4 per unit cell for the double gyroid phase (G), XEuier = -16 per unit cell and for the lamellar phases (LAM), XEuier = 0.

The Euler characteristic of the hexagonal phase [Fig. 13(c)] is 0, since the Euler characteristic of each cylinder is zero. The double gyroid phase (Fig. 5) within the two-shell approximation is represented as... 

At very asymmetric compositions, the free-energy difference between the double-gyroid and hexagonally perforated lamellar (HPL) phases becomes very... 

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