Theory of Shape Deformations

In early applications of the jellium model, the shape of metal clusters was assumed in all instances to be spherical [33, 34], but soon it became apparent that the spherical symmetry was too restrictive [35, 36]. Indeed clusters with open electronic shells (between the magic numbers = 2, 8, 20, 40, 58, 92, etc.) are subjected to Jahn-Teller distortions [37]. By now it has been well established that a quantitative description of the underlying shell effects and of fragmentation phenomena (as well as of other less complicated phenomena such as ionization and vertical electron detachment) requires a proper description of the deformed shapes of both parent and daughter clusters (of both precursor and final ionic or neutral product in the case of ionization and vertical electron detachment). 

A most successful method for describing both deformation and shell effects in simple metal clusters (i.e. those that can be described by the Jellium background model) is the SCM, originally developed in the field of nuclear physics [38, 2]. In a series of recent publications [25, 26, 28, 39-45], the SCM was further developed, adapted and applied in the realm of finite-size, condensed-matter nanostructures (i.e. metal clusters [25, 26, 39-43], but also multiply charged fullerenes [28], He clusters [44], and metallic nanowires and nanoconstrictions [45]). Additionally, Refs [46-49] have used 

The SCM derives its justification from the local-density-approximation (LDA) functional theory and has been developed as a two-level method. 

At the microscopic level, referred to as the LDA-SCM, the method has been shown to be a nonselfconsistent approximation to the Kohn-Sham (KS)-LDA approach [50]. Apart from computational efficiency, an important physical insight provided by the LDA-SCM is that the total KS-LDA energy totai(A ) (or in another notation Eks( )) of a finite system of interacting delocalized electrons (or more generally of other fermions, like nucleons or - He atoms) can be divided into two contributions, i.e. 

Starting from the fundamental microscopic separation in Eq. (4), various semiempirical implementations (referred to as SE-SCM, see Section 4.2.2) of such a division consist of different approximate choices and methods for evaluating the two terms contributing to this separation. 

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PLASTIC DEFORMATION. When a metal or other solid is plastically deformed it suffers a permanent change of shape. The theory of plastic deformation in crystalline solids such as metals is complicated but well advanced. Metals are unique among solids in their ability to undergo severe plastic deformation. The observed yield stresses of single crystals are often 10 4 times smaller than the theoretical strengths of perfect crystals. The fact that actual metal crystals are so easily deformed has been attributed to the presence of lattice defects inside the crystals. The most important type of defect is the dislocation. See also Creep (Metals) Crystal and Hot Working. 

The principles of the modem physical theories of the deformation of rubber xerogels have already been dealt with in Chapter IV, p. 123. If a strip of raw rubber is rapidly extended, the molecules, which initially assumed randomly kinked forms, are stretched too. The more extended shape of the chains is a statistically less probable one corresponding to a lower entropy. en the piece of rubber is rapidly released again, it reassumes its original form in that the chains return to their, most probable configurations The entropy-character of rubber elasticity has been proven in that it exhibits a positive temperature coefficient. 

The size and shape of polymer chains joined in a crosslinked matrix can be measured in a small angle neutron scattering (SANS) experiment. This is a-chieved by labelling a small fraction of the prepolymer with deuterium to contrast strongly with the ordinary hydrogenous substance. The deformation of the polymer chains upon swelling or stretching of the network can also be determined and the results compared with predictions from the theory of rubber elasticity. 

The exact Eq. (4.2.17) takes into account the effect of the reservoir (the condensed phase) on the spectral line shape through the parameter 77. Consideration of a concrete microscopic model of the valence-deformation vibrations makes it possible to estimate the basic parameters y and 77 of the theory and to introduce the exchange mode anharmonicity caused by a reorientation barrier of the deformation vibrations thereby, one can fully take advantage of the GF representation in the form (4.2.11) which allows summation over a finite number of states. 

In a laminar flow or in an external potential field a polymer molecule is subjected to forces that can both make it rotate as a whole and cause a relative shift of its parts leading to a deformation, i.e. changing its conformation. Which of these two mechanisms of motion predominates depends on the ratio of times required for the deformation and rotation of the molecule. If the time of the rotation of the molecule as a whole, tq, is shorter than the time required for its deformation, tkinetically rigid. In the opposite case, when tq > r<, the deformation mechanism of motion will predominate and the molecule will be kinetically flexible. To characterize quantitatively the kinetic rigidity of chain molecules Kuhn has introduced the concept of internal viscosity - a quantity describing the resistance of the molecule to a rapid charge in its shape. Later, the theory of internal viscosity has been developed by Cerf ... 

The theory of fuzzy sets [382-385] has numerous contemporary applications in various fields of engineering [386-388], in the description of quantum mechanical uncertainty [389-393], in the study of molecular identity preserving deformations [106,251], in new approaches to the description of approximate symmetry [252,394,395], as well as in both static and dynamic shape characterization and dynamic shape similarity analysis of molecules [55,396]. 

As we have emphasized aheady, the study of plasticity is one of the centerpieces of the mechanics of materials. A wide array of technologies depend upon the ability to deform materials into particular desirable shapes, while from a scientific perspective I personally find the subject of great interest because it is an intrinsically dissipative process featuring history dependence and is a strong function of the material s microstructure. In addition, from the effective theory perspective, the study of plasticity is built around the motion and entanglement of dislocations which requires the construction of theories of lines and their interaction thus ushering in a certain nonlocality to the phenomenon right from the outset. 

Chen, Y.H. and Lagoudas, D.C. (2008) A constitutive theory for shape memlarge deformations. Journal of the Mechanics and Physics of Solids, 56, 1752-1765. 

Finally, a few words may be added with regard to the viscosity at high rates of shear. From the hydrodynamic theory of rod-shaped particles it can be inferred that the viscosity contribution of the rods is decreased as the rate of shear increases This is due to the fact that the particles are more and more oriented in the direction of flow. With flexible long-chain molecules this effect does not exist. It is true that here also the orientation is more pronounced at high rates of shear. This would result in a lower viscosity. At the same time, however, the elastic forces generated by the deformation of the molecules are increased. As the result of these two effects the viscosity is independent of the rate of shear. 

The theoretical explanation of these effects can be derived from the energy theory of fracture. Consider a crack as it just penetrates into a stiffer region of material, shown in Fig. 16.8. When the crack is at the interface, it exhibits the shape of a long bent beam under load. However, when the crack tries to penetrate into the stiffer material, there is more elastic resistance to bending deformation... 

The properties of strongly deformed nuclei can be well described assuming axially symmetric shape. However, in the transitional regions it is necessary to consider also triaxial (i.e., not axially symmetric) shapes. The departure from axial symmetry may be either static (stable deformation) or dynamic (y surface oscillation). The theory of y-soft nuclei was first developed by Wilets and Jean (1956). 

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