Curvature Terms Links

The existence of the two lengthscales in the intermediate regime has a simple physical interpretation and can be explained in terms of the LCA theory, which links the velocity of the interface with its local curvature. Namely, one can show [223] that the quantity p(t) = (LH(t)/LK(t))2 estimates the average number of necks piercing the surface of a sphere of radius LH(t) [LH(t) is the characteristic... 

The topology of the electronic charge density ip r)), as pointed out by Bader [55], is an accurate mapping of the chemical concepts of atom, bond and structure. The main topological properties are summarized in terms of their critical points (CP) [55,56]. The nuclear positions behave topologically as local maxima in p(r). A bond critical point (BCP) is found between each pair of nuclei, which are considered to be linked by a chemical bond, with two negative (Ai and ki) and one positive (A3) curvature (denoted as (3,-1) CP). The ellipticity (e) of a bond is defined by means of the two negative curvatures in a BCP as ... 

Fletcher (46) showed experimentally that the interfacial partition coefficients of different solutes depend on both the external salinity and on the interfacial curvature. Leodidis and Hatton (37b) extended this work by illustrating a curvature dependence of the interfacial partition coefficient Kx- Figures 9.12 (37a) and 9.13 (37b) demonstrate this curvature effect. The term 1/Vko is directly related to the curvature 1// , if / w is the radius of the droplet. When the curvature is increasing, the partition coefficient is decreasing. This is attributed to an increase of the rigidity of the interface, which induces a squeezing-out effect, as in the lamellar phase. Thus, a local equilibrium of solute adsorption at the interface can be directly linked to the mean curvature of the surfactant film. 

It is no accident that both wave motion and the fundamental theory of chemistry are best described in terms of natural numbers. However, conventional wave mechanics in three dimensions offers only a partial elucidation of the periodic table of the elements. On the other hand, a detailed reconstruction, also of the more general periodicity of stable nuclides, derives directly from elementary number theory. It shows, in addition, how the periodic function responds to the state of space-time curvature and identifies the golden ratio as a possible parameter that links perceptions in tangent space to the simation in curved space-time. 

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