Point Defects at a Surface

The relevant surface energy is again given by Faurf in equation (3.395) by the sum of Fi and F2 in cylindrical polar coordinates where C( ) is the displacement of the liquid crystal surface in the 2 -direction. For instance, the slope of the surface at the peak shown in Fig. 3.25 is clearly — and so, analogous to equation (3.389) 

The Euler-Lagrange equation for Fgurf provides the equihbrium equation 

This is an inhomogeneous Bessel equation and its general solution is [1, p.496] 

Substituting this value for a and setting b to zero in equation (3.425) results in the solution 

This solution can be given an alternative formulation via the identity [278, p.338] 

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The point defect at a surface of an ordered medium can represent either the end of a line that is topologically stable in the bulk or a true surface point defect with no bulk singularity attached [61]. In cholesteric liquid crystals, all points with A = i 1 are the ends of bulk disclinations. Only when k = 2 An rotations of the director field), the point defect might be an isolated surface singularity. However, even in this case one should take care of the requirement of the layers equidistance. For example, the classical boojum configuration cannot be observed in a cholesteric vessel when 1. 

The ability of STM to image at the atomic scale is particularly exemplified by the two other chapters in the book. Thornton and Pang discuss the identification of point defects at Ti02 surfaces, a material that has played an important role in model catalyst studies to date. Point defects have been suggested to be responsible for much of the activity at oxide surfaces and the ability to identify these features and track their reactions with such species as oxygen and water represents a major advance in our ability to explore surface reactions. Meanwhile, Baddeley and Richardson concentrate on the effects of chirality at surfaces, and on the important field of surface chirality and its effects on adsorption, in a chapter that touches on one of the fundamental questions in the whole of science - the origins of life itself ... 

The results presented in the previous sections demonstrate the importance of point defects at the surface of oxide materials in determining the chemical activity of deposited metal atoms or clusters. A single Pd atom in fact is not a good catalyst of the cyclization reaction of acetylene to benzene except when it is deposited on a defect site of the MgO(lOO) surface. A detailed analysis of the reaction mechanism, based on the calculation of the activation barriers for the various steps of the reaction, and of a study of the preferred site for Pd binding, based on the MgO/Pd/CO adsorption properties, has shown that the defects which are most likely involved in the chemical activation of Pd are the oxygen vacancies, or F centers, located at the terraces of the MgO surface and populated by two (neutral F centers) or one (charged paramagnetic F centers) electrons. 

Very recently, based on a combined use of chemical evidence, IR and EPR spectroscopies, as well as of ab initio calculations it has been suggested that oxygen vacancies form preferentially at low coordinated sites (steps, edges, comers) [133] it has also been observed that the nature of the point defects at the surface changes dramatically depending on the pretreatment conditions [135]. 

The interaction of very small clusters with point defects was addressed in a recent BP86 study of Ni species deposited on regular and defective sites of the MgO(OOl) surface, using models with Mg +PC embedding [178]. As defects, neutral F centers and paramagnetic Fs centers with only one electron in the cavity were considered. The stability of two- and three-dimensional Ni clusters adsorbed on MgO was compared, allowing a discussion of the role of point defects at the surface in nucleation and growth of supported metal particles. 

It is instructive to examine boundary disclination lines at a nematic surface or interface by introducing some additional modelling and approximations that incorporate surface tension and the effect due to gravity. Many of these aspects introduced here in this Section are common throughout the literature on liquid crystals and will also form a basis for the discussion on point defects at a free surface of nematic discussed in the next Section. The results presented below are based on those derived by de Gennes [107] and have been further elucidated in physical terms by de Gennes and Prost [110]. 

The impressed current enters the pipe surface at a defect of radius r. The defect can be considered as a point source at a sufficiently large distance compared to the diameter of the pipe. The potential on the surface of the soil is from Table 24-1, line 5 ... 

The nucleation behavior of transition metal particles is determined by the ratio between the thermal energy of the diffusing atoms and the interaction of the metal atoms at the various nucleation sites. To create very small particles or even single atoms, low temperatures and metal exposures have to be used. The metal was deposited as metal atoms impinging on the surface. The metal exposure is given as the thickness (in monolayer ML) of a hypothetical, uniform, close-packed metal layer. The interaction strength of the metals discussed here was found to rise in the series from Pd < Rh < Co ( Ir) < V [17,32]. Whereas Pd and Rh nucleate preferentially at line defects at 300 K and decorate the point defects at 90 K, point defects are the predominant nucleation center for Co and V at 300 K. At 60 K, Rh nucleates at surface sites between point defects [16,33]. 

Fig. 4 Osmium clusters supported on MgO(OOl) a OssC/MgisOs and b OS5C at a surface point Vs defect site [33] these were represented by density functional theory, and the samples were characterized by EXAFS spectroscopy, transmission electron microscopy, and other techniques [15]...

In principle, the optical absorptions in this region could also be associated with point defects on the surface either with or without trapped electrons and holes. However, the properties of the charged defects have been studied extensively, and the trapped charges can be thermally annealed at temperatures far lower than the normal preparation temperatures of these samples. In addition, they are characterized by optical and EPR spectra which are not observed in these samples. Contributions from point defects with no trapped charges cannot easily be eliminated. In fact, such a surface vacancy or divacancy would represent a localized state on a low-index surface associated with 4-coordinated ions. 

Electronic point defects can presumably exist at a surface, provided only that the necessary atomic configuration for trapping can exist there. An analog of the F center, for example, should be stable at the surface. The detailed energy levels and symmetry will naturally be altered by the missing layer of atoms. 

In a recent review, Pacchioni presented a detailed classification of point defects on the surfaces of oxide materials [9]. He described at least four majors kinds of irregularities low-coordinated sites, divacancies, impurity atoms, and surface... 

Surface properties can differ from the bulk stracturally, both as clean surfaces or because of products formed on reactive surfaces (physisorbed or chemisorbed). The former can experience relaxation, that is, surface reconstruction due to the distortion in bonding for surface atoms which are lacking bonds. Impurity segregation at a surface can further alter properties, as can second phases formed on a surface. The activity of heterogeneous catalysts and corrosion is controlled by such surface properties and by the bulk and surface point-defect structures. 

Fig. 8.18 Boodjooms. Structure of the director with two boodjooms in a nematic drop with tangential alignment of molecules at the surfaces (a), linear disclination with a point defect at the boundary of a nematic layer (b), and the same point defect (boodjoom) after aimihilation of the linear disclination (c)...

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