Spherical coordinates Subject

In an electrolyte solution the ions are interspersed by water molecules, moreover they are subject to thermal motion. Debye and Huckel simplified the description of this problem to a mathematically manageable one by considering one isolated ion in a hypothetical, uniformly smeared-out sea of charge, the ionic cloud, with the total charge just opposing that of the ion considered. For this case the Poisson equation in terms of spherical coordinates is given by... 

There are three possibilities corresponding to the dimension of the distribution. The first is a ID concentration distribution (d = 1), in which the diffusing species spreads evenly in the z directions from an initial line pulse at z = 0 on the xz plane. In this case, the variable r in (6-37) is the Cartesian variable z. The second case is a circularly symmetric distribution for c (d = 2), which evolves by diffusion on a plane from an initial compact planar pulse. In this case, r in (6 37) is the radial component of a polar (or cylindrical) coordinate system that lies in the diffusion plane. The third case is a spherically symmetric distribution corresponding to d = 3, which evolves at long times from a compact 3D pulse that diffuses outward into the frill 3D space. In this case, r is the radial variable of a spherical coordinate system. To obtain the long-time form of the distribution we must solve (6-37), but subject to the integral constraint that the total amount of the diffusing species is constant, independent of time ... 

A simple example for which problems (9-16) and (9 17) can be solved easily is the case of a heated sphere. In this case, we may choose the sphere radius as the characteristic length scale for nondimensionalization, and the problem is to solve (9 16) subject to the boundary condition 0 = 1 at r = 1. This can be done easily. We may first note that a general solution of Laplace s equation in spherical coordinates is... 

There are now two independent angular coordinates, 6 and cp (besides the fixed distance R to the centre of the sphere) and wavefunctions that can be expressed as products of 0(0) and (p). The wave equation can be split into two, one for each variable 6 and cp. Each one of the functions 0(0) and (0) is subject to boundary conditions, in a similar way to the motion on a circular ring. Accordingly, two quantum numbers arise. The complete solutions Q 6) (p) are known as spherical harmonic functions and the allowed energies are given by an expression that resembles Eq. (2.72) ... 

Consider convection diffusion toward a spherical particle of radius R, which undergoes translational motion with constant velocity U in a binary infinite diluted solution [3], Assume the particle is small enough so that the Reynolds number is Re = UR/v 1. Then the flow in the vicinity of the particle will be Stoke-sean and there will be no viscous boundary layer at the particle surface. The Peclet diffusion number is equal to Peo = Re Sc. Since for infinite diluted solutions, Sc 10 and the flow can be described as Stokesian for the Re up to Re 0.5, it is perfectly safe to assume Pec 1. Thus, a thin diffusion boundary layer exists at the surface. Assume that a fast heterogeneous reaction happens at the particle surface, i.e. the particle is dissolving in the liquid. The equation of convective diffusion in the boundary diffusion layer, in a spherical system of coordinates r, 6, (p, subject to the condition that concentration does not depend on the azimuthal angle [Pg.128]

Modified poly(methacrylic acid) microparticles complexed with gadolinium(lll) (Gd ) ions were prepared at 100 nm by Michinobu [81]. The emulsion tcrpolymerization of methacryhc acid, ethyl acrylate, and aUyl methacrylate and the following complexation with Gd ions yielded the polymer particles with different Gd ion contents. Potentiometric titration of the complexation of the particle with Gd + ions indicated the formation of a very stable tris-carboxylate coordinate complex with the Gd + ion. The microparticles dispersed on a mica substrate were subjected to AFM, followed by MFM. AFM showed 100-nm-sized and monodispersed spherical images. The following MFM... 

It is a trivial step to expand rotation of a particle or rigid rotor to three dimensions. The radius of a particle from a center is still fixed, so three-dimensional rotation describes motion on the surface of a sphere, as shown in Figure 11.11. However, in order to be able to describe the complete sphere, the coordinate system is expanded to include a second angle 6. Together, the three coordinates (r, 6, ) define spherical polar coordinates. The definitions of these coordinates are shown in Figure 11.12. In order to treat the subject at hand more efficiently, several statements regarding spherical polar coordinates are presented without proof (although they can be proven without much effort, if desired). 

---