Toroidal Coordinates

W. M Elsasser, Phys. Rev. 69, pp. 106-166, 1946 G. Buck, Force-Free Magnetic Fields Solution in Toroidal Coordinates, Ph.D. dissertation, Univ. Michigan, Univ. Microfilms, 1964. 

An alternative would be the toroidal coordinate system, as described by Happel and Brenner (1973), Appendix A-20 (see reference 3, chapter 1). 

Eor fully developed flow field in the curved microchannel, the Eqs. 1, 7, and 8 transformed the toroidal coordinate system (R, Y, 6) to the related coordinate system (X, Y, Z) and can be expressed as... 

Figure 12.15. (a) A cylindrical coordinate system with coordinates (p, z) is used to describe the director field on the central part of the chiral surfaces, while for the outer part a toroidal coordinate system with coordinates is particularly suitable... 

There are two levels of self-assembly in the formation of tetra-, penta-and hexa-nuclear products from the poly-bipyridyls (L) 20 and 21 and iron(II) salts FeCl2, FeBr2 or FeS04 - the products are anion-dependent. The coordination of three bpy units, from different ligand molecules, to the Fe2+ centers produces a helical structure interaction of these helical strands with anions results in further molecular organization to form the final toroidal product. The discussion draws parallels between the helical and toroidal structures here and secondary and tertiary structure in biological systems (482). Thermodynamic and kinetic intermediates have been characterized in the self-assembly of a di-iron triple stranded helicate with bis(2,2/-bipyridyl) ligands (483). 

This theorem could have been used to obtain the drag for fluid and solid spheres in Chapter 3. Explicit analytic solutions are available for bodies whose boundaries are easily described in relatively simple coordinate systems. Results for spheroids and disks (Ol, P3, SI) are discussed below. Solutions are also available for lenses and hemispheres (P3), hollow spherical caps (D3, C3, P3), toroids (P4), long spindles or needles (P5), and pairs of identical spheres (S7). 

Figure 5.24 illustrates an elbow section in a cylindrical channel where the radius of curvature of the section R is comparable to the channel radius r,-. Analysis of the flow field in this section may be facilitated by the development of a specialized orthogonal curvilinear coordinate system, (r, 6, a). The unit vectors are illustrated in the figure. Referenced to the cartesian system, the angle 6 is measured from the x axis in the x-y plane. The angle a is measured from and is normal to the x-y plane. The distance r is measured radially outward from the center of the toroidal channel. 

Discuss the behavior of the continuity equation (and by analogy the other governing equations) as the toroid radius R becomes large compared to the channel radius r,. Under these circumstances, show how the curvilinear equation becomes closer and closer to the regular cylindrical-coordinate equations. 

Numerous metallomacrocycles self-assemble from their components giving species possessing various structures, for instance triangular shape [9.30,9.31] containing a cavity [9.32], which may include a guest molecule [9.33] square [9.34,9.35] or star like [9.36-9.39] shapes wheel-shaped or toroidal hexameric [9.40], octameric [9.41] or decameric [9.42,9.43] structures square [9.44,9.45] rectangular [9.46,9.47] or bent [9.48] boxes into which substrate molecules may bind [9.49,9.50] adaman-tanoid shape [9.51,9.52] with cation inclusion [9.51c], formally related to that of the spheroidal macrotricycle 21 catenane type [9.53]. Coordination species of dendri-mer or arborol nature have been constructed [7.61, 8.27, 9.54]. 

This approach mimics familiar biological self-assembly phenomena such as protein folding [ 192], protein aggregation [ 192] and nucleotide pairing [ 188]. It incorporates features described in each of the above strategies (i.e., I—III), to give specialized nanoscopic structures, that can be precisely designed, usually with excellent control over CMDPs. Recent examples include so called structure directed synthesis by Stoddart [3a] (see Chapter 1 of this book) to produce toroidal bis-bipyridinium cyclophanes that are reminiscent of a molecular abacus , melamine-cyanuric acid lattices by Whitesides [193] and unique helical structures based on coordination of bipyridyl units to copper (II) ions by Lehn [194],... 

The thermal boundary-layer equation, (9-257), also apphes for axisymmetric bodies. One example that we have already considered is a sphere. However, we can consider the thermal boundary layer on any body of revolution. A number of orthogonal coordinate systems have been developed that have the surface of a body of revolution as a coordinate surface. Among these are prolate spheroidal (for a prolate ellipsoid of revolution), oblate spheroidal (for an oblate ellipsoid of revolution), bipolar, toroidal, paraboloidal, and others.22 These are all characterized by having h2 = h2(qx, q2), and either h2/hx = 1 or h2/hx = 1 + 0(Pe 1/3) (assuming that the surface of the body corresponds to q2 = 1). Hence the thermal boundary-layer equation takes the form... 

Figure 15. A Canesian system applied to Figure 14 in a way that simplifies representation of the toroidal surface. The unit of length is the distance between the centers of two hexagons that share a common edge. The choices of a 120° (rather than 60°) angle between axes, and of a left-to-right vector as positive are adopted as standard conventions. In this example the coordinate positions of repeating A" hexagons at (a.c) = (9,0), and (b,d) = (4,1), abbreviated to "(9-4-1) suffice to define the parallelogram, and hence the toroidal surface.

Figure 11. (a) Toroidal trigonal planar, square planar, and pentagonal planar coordination derived from ip d manifolds (b) cylindrical linear coordination derived from an spi manifold (c) idealised examples of d(S- pa and dn- pit bonding found in coinage metal derivatives with toroidal and cylindrical valence orbital manifolds. 

Bearing this in mind, we designed and synthesized a number of P-CD derivatives [27-34] which could i) bind copper(II) forming a multisite recognition system ii) show thermodynamic stereoselectivity in copper(II) ternary complexes iii) perform chiral separation of unmodified amino acid enantiomers. Among the monofunctionalized P-CD derivatives, only those functionalized in position 6 with diamines show chiral molecular recognition [29,32,35-37]. On the contrary, the P-CDs both functionalized in position 3 and those where a triamine was attached to the narrower rim of the toroid do not act as chiral receptors. 2-(aminomethyl)pyridine, histamine and NH3 molecules were used to obtain the three isomers of P-CDs (Figure 3), but only the A,BCD-NH2 molecule, coordinated to the copper(II) ion, is seen to have enatioselective effects on aromatic amino acids [38]. 

---