Commensurate domains

Fig. 28. Soliton lattice solution of the FvdM model with regularly spaced, distance 1, domain walls. The dashed line corresponds to the incommensurate phase with negligible potential modulation (K = 0). The plateaus represent the commensurate domains.

Knowledge of the pressure-induced commensurability led to a series of beautiful experiments searching for evidence for a collective electron-phonon or CDW contribution to the low field conductivity in TTF-TCNQ above Tp. Clear evidence was indeed found for a substantial fall in ah between about 150 and 80 K in the narrow commensurability domain, as shown in Fig. 14 [85]. No such dip was found for the transverse conductivity [86], and the dips in ah were also shown to be suppressed by only a 2 x 10 3 molecular fraction of irradiation induced defects [87]. All of this leads to a consistent picture in favor of a collective electron-phonon CDW contribution to ah above Tp of TTF-TCNQ, as discussed in Ref. 2. However, the extra CDW conductivity is not more than 6000 (fl-cm)-1 at 80 K, that is, about one-half of the ambient pressure conductivity of (TMTSF)2PF6 at the same temperature (Fig. 1) and the latter is usually considered to be a single-particle contribution. So until the mechanism... 

The sides of rectangles, which constitute the domain G are assumed to be commensurable. All this enables us to place in the plane a grid with steps /ij and /ij so that the boundary of the grid domain lies on the boundary of the domain G. One trick we have encountered is to complete the domain G to the rectangle and then denote it by G (see Fig. 14). After that, we construct in G a difference grid and extend it to G. The notation will be used for the grid in the domain G. 

Unlike the bulk morphology, block copolymer thin films are often characterized by thickness-dependent highly oriented domains, as a result of surface and interfacial energy minimization [115,116]. For example, in the simplest composition-symmetric (ID lamellae) coil-coil thin films, the overall trend when t>Lo is for the lamellae to be oriented parallel to the plane of the film [115]. Under symmetric boundary conditions, frustration cannot be avoided if t is not commensurate with L0 in a confined film and the lamellar period deviates from the bulk value by compressing the chain conformation [117]. Under asymmetric boundary conditions, an incomplete top layer composed of islands and holes of height Lo forms as in the incommensurate case [118]. However, it has also been observed that microdomains can reorient such that they are perpendicular to the surface [ 119], or they can take mixed orientations to relieve the constraint [66]. 

From LEED measurements of H monolayers adsorbed on Fe(110) Imbihl et al. proposed a phase diagram as shown in Fig. IS. In addition to lattice gas and lattice fluid phases, two commensurate ordered phases were identifled, denoted as (2 x 1) and (3 x 1) in the figure (cf. Fig. 16). The shaded regions are interpreted as incommensurate phases or as phases composed of antiphase domains their signature is that the LEED spot does not occur at the Bragg position but rather the peak is splitted and satellites appear (Fig. 17). 

Fig. 29. Schematic diagrams showing (a) hexagonal and (b) striped domain wall arrangements (only superlight walls are drawn on a triangular lattice (e.g. the (111) face of f.c.c. metals)). In incommensurate layers, where the monolayer is compressed with respect to the commensurate lattice, domain walls or either heavy or superheavy (c).

A < 0, i.e. attractive walls. A hexagonal network of domain walls (HI) will be formed at the commensurate (C-I) transition, because the number of wall crossings tends to be as large as possible. This C-HI transition should be first order. 

X cos 30° A is the inter-row distance of the commensurate Xe structure in the same direction. From the measured polar and azimuthal peak widths in Fig. 32 we can also estimate average domain sizes of the incommensurate layer. For the FK direction, i.e. parallel to the walls, we obtain 350 A and for the perpendicular TM direction 50 A. 

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