Double-Gyroid

All IMDSs can be approximated by constant mean curvature (CMC) surfaces that minimize interfacial area subject to a volume (or volume fraction) constraint. This fact is not surprising, since the microdomain structure is the result of the balance between chain-stretching and the interfacial energies, where the latter term seeks to minimize surface area and thus favors CMC surfaces. 

Since analytical expressions for only a few continuous triply periodic CMC surfaces are known (e.g. the Enneper-Weierstrass parameterization of the single-gyroid minimal surface with H = 0 and a volume fraction of 50 % [9]), these surfaces are typically modeled with the help of level surfaces. 

The single-gyroid (SG) IMDS with 74i32 (No. 214) symmetry was first discovered in 1967 by Luzzati et al. as a cubic phase occurring in strontium soap surfactants and in pure lipid-water systems [12, 13]. In 1970, Schoen identified the minimal 

The possible values for t and the resulting structures are discussed below  

The resulting level surface closely approximates the minimal gyroid or gyroid minimal surface discovered by Schoen, see Fig. 2.2a. Minimal surfaces have the 

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The best-known and simplest class of block copolymers are linear diblock copolymers (AB). Being composed of two immiscible blocks, A and B, they can adopt the following equilibrium microphase morphologies, basically as a function of composition spheres (S), cylinders (C or Hex), double gyroid (G or Gyr), lamellae (L or Lam), cf. Fig. 1 and the inverse structures. With the exception of the double gyroid, all morphologies are ideally characterized by a constant mean curvature of the interface between the different microdomains. 

A systematic comparative study of triblock terpolymers in the bulk and thin-film state was carried out on polystyrene-fo-poly(2-vinyl pyridine)-b-poly(ferf-bulyl methacrylate), PS-fr-P2VP-fr-PfBMA. A diblock precursor with a minority of PS leading to a double gyroid structure was used. Upon increase of PfBMA content this morphology changed from lamellae with... 

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...

Fig. 38 Milner s diagram for morphologies of A-b-B copolymers as reported in literature [111]. Observed morphologies of linear and miktoarm-star copolymers of BS-b-P2MP system are represented. Lamellar structure, cylinders of P2MP in PS matrix spheres of P2MP in PS matrix, biphasic structure of lamellar/double gyroid microdomains. From [113]. Copyright 2003 American Chemical Society...

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... 

Fig. 28 SEM images of about 60 nm thick films of SVT block terpolymers along with expected structural elements of the thin-film structure, (a) Core-shell cylinders (b) helices wound around a cylindrical core (c) (112) plane of an ideal double gyroid structure. Copyright (2002) Wiley. Used with permission from [18]...

Fig. 15.11. SEM micrograph showing the bicontinuous nanochannels in a PS matrix (a) and a computer graphics 3D view of the double gyroid network (b). (Reprinted with permission from Langmuir, 1997, 73, 6869. 1997 American Chemical Society [47].)...

Fig.20 Top row single unit-cell models of core-shell double gyroid (Q °), orthorhombic (O °), and alternating gyroid (Q ) cross-sectioned to reveal interfacial configuration. Bottom row. direct projections of cross-sectioned interfaces. Sketches of PI-fi-PS-fi-PEO chains show how each morphology is assembled. Projections appear to scale that is, the core-shell double gyroid unit cell is roughly twice the thickness of the other two. From [75]. Copyright 2004 American Chemical Society...

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