Curvature Direction

Figure 5.48 Geometry discussed in Section 6.3 for tubule formation based on chiral elastic properties. Here, r is tubule radius, l is tubule length, n is molecular director, m is projection of n into local tangent plane (normalized to unit magnitude), <(> is angle in tangent plane between m and curvature direction (equator running around cylinder), and N is local normal vector. Adapted from Ref. 132 with permission of the author. Copyright 1996 by the American Physical Society.

For the study of border profile Lalchev et al. [35] and Khristov et al. [36] have suggested direct measurements of the radius of border curvature either by a microscope or by a cathetometer. If the condition r/a 1 is not fulfilled and the border shape at each point is chracterised by two variable radii of curvature, direct measurement of the radii of border curvatures with a microscope gives the most reliable results. 

X. Kou et al., Growth of gold bipyramids with improved yield and their curvature-directed oxidation. Small, 3(12), 2103-2113 (2007). 

The subscripts indicate the corresponding partial derivatives. The two eigenvalues in the two main curvature directions, e and g, are defined by the eigenvalue equation T-Cv z Av. Hence, on the one hand, we have... 

If the dimension of the configuration space exceeds n=2, a possible classification scheme of GEs will become inpractical, because there are (n-1) main curvature directions perpendicular to the gradient which influence the classification. Thus, we will get mixed characters of the corresponding GE curves. Nevertheless, we can assume that pure valleys and pure cirques are the right objects for a MEP of a chemical reaction. 

Theorem 1 is simply a group-theoretical reformulation of the Murrell-Laidler theorem, it implies that a PES cannot have more than one negative curvature direction at the transition state point. 

Other cases with a stiff interface curvature directed towards oil-forming oil-in-water droplets with a stiff interface and lamellar... 

Steinhart M, Senz S, Wehrspohn RB, Gdsele U, Wendorff JH. Curvature-directed crystallization of poly(vinylidene difluoride) in nanotube walls. Macromolecules 2003 36 3646-3651. 

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