Broken translational invariance

This section deals with the dynamics of collective surface vibrational excitations, i.e. with surface phonons. A surface phonon is defined as a localized vibrational excitation of a semi-infinite crystal, with an amplitude which has wavelike characteristics parallel to the surface and decays exponentially into the bulk, perpendicular to the surface. This behavior is directly linked to the broken translational invariance at a surface, the translational symmetry being confined here to the directions parallel to the surface. 

Note that the first two terms on the right hand side tell us that there is a change in the strain energy density that is incurred because the defect has moved and it has dragged its elastic fields with it. However, these terms do not reflect the presence of broken translational invariance. By exploiting equality of mixed partials, and by rearranging terms via a simple application of the product rule for differentiation, this expression may be rewritten as... 

Figure 4.19. Various stages of the calculation of an effective medium from a single cell. A) The cell is immersed in a non-self-consistent medium. (B) The mean scattering on the cell A allows one to obtain a superlattice of mean cells. (C) The translational invariance, broken in stage B, is restored by averaging, at the cost of introducing new interactions between neigbors.

For a bulk block copolymer melt, there is translational invariance and hence the phase ((> could be omitted in Eq. (45) by a suitable choice of the coordinate origin. In a thin film, the translational invariance in z-direction is broken, and hence it is indispensable to allow for a non-zero phase < > in Eq. (53). In fact, Eq. (56) then yields [61]... 

Figure 6 Segment of a typical polydiacetylene PDA) backbone. The side chain groups are substituted by hydrogen atoms and the translationally invariant unit cell is surrounded by broken lines. The carbon-carbon bond distances vary for different models as defined in Table 4...

A difference between the perturbation considered here and that in the section on LRT considered earlier is the term involving qo, the position at which the drift velocity (i.e., the velocity contribution from the external field) of the fluid is zero. This term was chosen to be zero in the treatment for bulk fluids for simplicity it must be used here because confinement has broken the translational invariance of the system. The perturbation generates a planar Couette flow in the fluid between two surfaces ... 

In a situation compatible with the lubrication approximation, perturbations due to the proximity of a solid surface are weak. In this case, the translational invariance of an unbounded two-phase system is weakly broken, and both the shift of the equilibrium chemical potential due to interactions with the solid surface and the deviation from the zero-order density profile are small. Since molecular interactions have a power decay with a nanoscopic characteristic length, this should be certainly true in layers exceeding several molecular diameters. A necessary condition for the perturbation to remain weak even as the liquid-vapor and liquid-solid interfaces are drawn together still closer, as it should happen in the vicinity of a contact line, is smallness of the dimensionless Hamaker constant % = asps/p — 1- Even under these conditions, the perturbation, however, ceases to be weak when the density in the layer adjacent to the solid deviates considerably from p+. This means that low densities near the solid surface are strongly discouraged thermodynamically, and a... 

Let the director of the nematic phase is perpendicular to a flat interface. Then we can anticipate two effects. First, a polar surface layer should appear due to the break of the cylindrical symmetry, n — n. Second, due to some positional correlation of the centers of molecules in several layers adjacent to the surface, the nematic translational invariance can be broken. It means that the surface induces the short-range smectic A order. In the framework of the Landau theory, the smectic order decays with distance from the interface according to the exponential law... 

For low grafting densities, isolated side chains collapse onto the backbone of the bottle brush. For high grafting densities, a homogeneous cylindrical brush collapses onto itself. For intermediate grafting densities, however, the translation invariance along the backbone is broken upon side-chain coUapse, and a microphase-separated pearl-necklace stmcture is formed. 

Certain two-dimensional 5 = quantum antiferromagnets can imdergo a direct continuous quantum phase transition between two ordered phases, an antiferromagnetic Neel phase and the so-called valence-bond ordered phase (where translational invariance is broken). This is in contradiction to Landau theory, which predicts phase coexistence, an intermediate phase, or a first-order transition, if any. The continuous transition is the result of topological defects that become spatially deconfined at the critical point and are not contained in an LGW description. Recently, there has been a great interest in the resulting deconfined quantum critical points. ... 

Most of the microfluidic devices consist of microchannel of different dimensions interconnected with each other. In the case of two straight channels of different dimensions connected to form one long channel, the translation invariance, that is, fully developed flow assumption, is broken. Hence, the expressions for the ideal Poiseuille flow no longer apply. However, it is expected that the ideal description is approximately correct if the Reynolds number (Re) of the flow is sufficiently small. This is because a very small value of Re corresponds to a vanishing small contribution from the nonlinear term (V - V)F in the Navier-Stokes equation (N-S equation), a term that is strictly zero in ideal Poiseuille flows due to translation invariance. 

It is not yet known how T (ro) behaves in the vicinity of a solid boundary, when translational invariance is broken. The compact support of the weighting function A suggests that is a local correction and therefore largely independent of macroscopic boundary conditions. Numerical simulations with periodic boundary conditions (Sect 4.6) show that g is independent of system size and fluid viscosity. The weak system-size dependence reported in [140] is entirely accounted for by the difference between the periodic Green s function [165] and the Oseen tensor. Thus, in a periodic unit cell of length L, (292) requires a correction of order 1/L [19],... 

The crystallization of a liquid is a change of phase in which symmetry is broken a spatially periodic state arises from one that was invariant under translations. A full description of such a phase change would involve three components (1) an understanding of the equilibrium aspects of the transition (at what temperature and pressure it takes place, what changes in thermodynamic properties such as volume and entropy accompany it), (2) a micro-... 

As we have seen, either is a. (pure) fluid phase, is E -invariant, extr. R -inv, uniform clustering, or its intrinsic symmetry E. Within the second alternative, only three cases can occur which can be characterized equivalently by (1) how much the translation symmetry is broken, i.e., a classification according to (2) spectrum properties, and (3) clustering properties. 

Liquid crystals are classified by symmetry. As it is well known, isotropic liquids with spherically symmetric molecules are invariant under rotational, 0(3), and translational, T(3), transformations. Thus, the group of symmetries of an isotropic liquid is 0(3)xT(3). However, by decreasing the temperature of these liquids, the translational symmetry T(3) is usually broken corresponding to the isotropic liquid-solid transition. In contrast, for a liquid formed by anisotropic molecules, by diminishing the temperature the rotational symmetry is broken 0(3) instead, which leads to the ap>p)earance of a liquid crystal. The mesophase for which only the rotational invariance has been broken is called nematic. The centers of mass of the molecules of a nematic have arbitrary positions whereas the principal axes of their molecules are spontaneously oriented along a preferred direction n, as shown in Fig. 1. If the temperature decreases even more, the symmetry T(3) is also partially broken. The mesophases exhibiting the translational symmetry T(2) are called smectics (see Fig. 1), and those having the symmetry T(l) are called columnar phases (not shown). 

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