Undulator planar

A study of the Cu-Sn reaction couple at temperatures between 250 and 325 °C (482 and 617 °F), observed that the e-layer always grew as a somewhat undulated planar layer in phase with the q-phase scallops (Ref 40). This morphology indicated that the e grows by reaction of Cu with... 

Because strings of the same atoms come to be adjacent when these layers are stacked, alternating layers of atoms of one kind each are formed. These layers are planar in AuCu they are inclined relative to the plane of the paper in the unit cell (Fig. 15.1) they are parallel to the base plane. The layers of equal atoms are undulated in the other two structure types. 

We start with the ground state (°), fi(° defined by the simple shear flow y(°), Fig. 17. The principal effect is, as expected, the appearance of a small tilt of the director from the layer normal (flow alignment), predominantly in z direction (Fig. 18). Note that the configuration of layers is also modified by the shear (Figs. 19 and 20), i.e., the cylindrical symmetry is lost. This is analogous to the shear-flow-induced undulation instability of planar layers (wave vector of undulations in the... 

Thermodynamic Equations. We briefly describe the thermodynamic formalism of Ruckenstein.10 Let us consider a liquid crystal of volume V per unit area, in equilibrium, and denote by <5i the average thickness of the water layer and by d2 the average thickness of the oil layer. The Helmholtz free energy of the system can be written as the sum of a free energy F0 of a hypothetical system in which the lamellae are treated as bulk, planar, phases and a free energy Pi, which accounts for the smallness of the lamellae, the interactions between them, and their thermal undulations. One can write... 

The free energy per unit area for a pair of successive lamellae, which includes the effect of thermal undulations, is computed as follows. Let us denote by U(x, X2) the interaction potential between planar water and oil lamellae of thicknesses X and x2, which include the van der Waals, hydration, steric, and electrostatic interactions... 

Figure 1. The distribution of distances between an undulating membrane and a rigid surface interacting via a harmonic potential is the same as the Boltzmann distribution of distances between a rigid surface and independent planar pieces of a suitable chosen area A, interacting via the same potential.

Throughout this section, the energy of the film will be calculated by assuming planar, non-undulating, interfaces. Most of the equations of section IIA—E were taken from our previous paper.5 We include them without detailed derivation for the completeness of the presentation. 

As a demonstration of the use of the perturbation method we refer to the work of Fogden, Mitchell and Ninham [33] who explicitly consider a single undulating charged monolayer in contact with an electrolyte. The surface is presumed to possess a sinusoidal undulation about the planar shape, but is assumed to maintain a state of constant charge ... 

The structures were originally described in terms of undulating chains of MO7 pentagonal bipyramids, with a composition of M Os- The cation coordination was described as a mixture of octahedral and pentagonal bipyramidal. Interstitial metal atoms situated in sites between sheets of pentagonal bipyramids were invoked to achieve the correct composition. It is now clear that the cations are in a hexagonal planar array and that the anion array is continuously modulated, and expands or contracts to fit the anion to cation ratio. In some places, the cation coordination is well represented by a pentagonal bipyramid, and in other places by an octahedron. Elsewhere the coordination polyhedra can be described as distorted variants of these. 

Show that a planar interface, governed only by its surface tension is stable with respect to smaii unduiations of the interface, whiie a cyiindricai interface is unstabie to iong-waveiength undulations. 

The structure and orientation of the reinforcing fibers in the matrix system are essential for the mechanical properties of the work-piece. The choice of a particular architecture is dependent on multiple factors like drapeabiUty of the fabric, geometry/shape of the workpiece, mechanical requirements, and manufacturing process. Compraients made of unidirectional layers (laminates) are showing the best mechanical properties since the fibers are completely stretched (no undulation). Usually the single layers of a laminate are showing different fiber orientations. This causes anisotropic material behavior in the planar direction. The structure of a multiaxial layered laminate is shown in Fig. 2. 

The effect of glycerol on planar La-phase has been discussed in detail in our previous paper. For nonclosed planar structures, the bilayers have a well-defined spacing that is the result of equilibrium between attractive van der Waals forces and repulsive undulation forces. Upon substitution of the solvent water gradually by glycerol-water mixed solvent, the attractive van der Waals forces approach zero. 

Joint Roughness Rough, Planar Undulating Rough, Undulating Smooth, Undulating... 

---