Interfacial symmetry

The two surfaces that comprise a contact can be oriented in any number of specific ways however, for crystalline surfaces, interfacial symmetries correspond to either of two broad classifications. The first type of orientation is called the commensurate case and is found when two identical surfaces are perfectly aligned. The term incommensurate corresponds to the case in which two crystalline surfaces are misoriented or have different periodicities. An example of a commensurate systems is given as structure A in Figure 7, whereas structures B through D are incommensurate. Interestingly, the orientation of the surfaces within a contact has a tremendous influence on... 

Surface and Double-layer Properties Valette [19] has analyzed earlier experimental data on the inner-layer capacity at PZC for Ag(lll), Ag(lOO), and Ag(llO) surfaces in order to estimate the surface area and capacitance contributions of superficial defects for real electrodes, as compared to ideal faces. Considering the application of surface spectroscopy techniques to single-crystal Ag electrodes, one should note that anisotropy of the SHG response for metal electrode allows one to analyze and correlate its pattern with interfacial symmetries and its variations by changing nonlinear susceptibility and the surface structure. Early studies on Ag(lll) single-crystal electrodes have... 

An ordering phase transition is characterized by a loss of symmetry the ordered phase has less symmetry than the disordered one. Hence, an ordering process leads to the coexistence of different domains of the same ordered phase. An interface forms whenever two such domains contact. The thermodynamic behavior of this interface is governed by different forces. The presence of the underlying lattice and the stability of the ordered domains tend to localize the interface and to reduce its width. On the other hand, thermal fluctuations favor an interfacial wandering and an increase of the interface width. The result of this competition depends strongly on the order of the bulk phase transition. 

The question of how to terminate the box is fundamental to all the calculations of interfacial energy in compounds, including the calculation of surface energies. It has been addressed previously for particular cases by Chetty and Martin [11,12]. These authors pointed out that a suitable termination is one which is on a symmetry plane of the crystal, or which follows symmetry planes if it is not parallel to the boundary. However, it may not always be possible to find a symmetry plane. I offer a solution here which is more general. It reconciles the atomistic picture with the thermodynamic limit. 

It has been mentioned in Sect. 3.1 that the molecular conformation can be altered strongly compared to that in solution if the side chains of a brush molecule are specifically adsorbed on the substrate or tend to spread on the surface to minimize the interfacial energy (Fig. 27). Moreover, the substrate changes the dimensionality of the system and breaks its symmetry [169,170]. Depending on the interfacial interactions and distribution of the side chains we can discuss a number of distinct conformations (a) stretched brush, (b) two-dimensional helix [170], and (c) globule state. 

A central assertion of homogeneous nucleation theory is that interfacial free energy costs induce a spherical symmetry in the phase embryo. However, these simulation studies indicate that inter molecular interactions may not permit the development of spherical symmetry when these interactions are strong and highly asymmetric. 

O Keeffe (1991Z)) has used bond valences to model the coherent interface that occurs between the semiconductors Si and MSi2 with M = Ni or Co (27139). Although these systems contain Si-Si bonds and therefore do not obey the assumptions of the bond valence model (condition 3.2), the mathematical formalism of the model still works because of the high symmetry. As both Si-Si and Si-Ni bonds are found in NiSi2, the cubic structure is strained (cf. BaTiOs in Section 13.3.2) and this strain affects the structure of the interface. Of the six possible interfacial structures examined, the two with the lowest BSI eqn (12.1) are those that are believed to occur in NiSi2 and CoSi2 respectively, and in both cases the strain introduced at the interface is correctly predicted. 

There is therefore one essential conclusion from the comparison of electrodic e-i junctions and semiconductor n-p junctions The symmetry factor P originates in the atomic movements that are a necessary condition for the charge-transfer reactions at electrode/electrolyte interfaces. Interfacial charge-transfer processes that do not involve such movements do not involve this factor. By understanding this, ideas on P become a tad less underinformed. Chapter 9 contains more on this subject. 

Figure 19.20 shows a cross section through the center of a critical nucleus that has cylindrical symmetry around the vertical axis EF. AB and CD are the traces of flat facets that possess the interfacial energy (per unit area) 7, and AC and BD are the traces of the spherical portion of the interface that possesses the corresponding energy 7. 

The nucleus has cylindrical symmetry around an axis normal to the boundary and mirror symmetry across the grain-boundary plane. Figure 19.27 shows a cross section of the nucleus centered in a patch of boundary of constant circular area, Ac. The area of the nucleus projected on the boundary is indicated by A. The total interfacial energy of this configuration is then... 

Symmetry is the easiest to apply. It is based on the correct selection of the coordinate system for a given problem. For example, a temperature field with circular symmetry can be described using just the coordinates (r, z), instead of (x, y, z). In addition, symmetry can help to get rid of special variables that are not required by the conservation equations and interfacial conditions. For example, the velocity field in a tube, according to the Navier-Stokes and continuity equations, can have the functional form uz(r). 

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