Sphere oscillation

Stokes GG (1851) On the effect of the internal friction of fluids on the motion of pendulums. Section II Solution of the equations in the case of a sphere oscillating in a mass of fluid either unlimited, or confined by a spherical envelope concentric with the sphere in its position of equilibrium. Trans Cambridge Phil Soc 9 (pt II) 8-106... 

However, at the larger values (15.0-120) the sphere oscillated across the tube axis one or more times before the migration ceased—the frequency of the oscillation increasing with increasing Rep and flow rate that is, the radial motion was such that while remaining in the same vertical meridian plane the particle crossed the axis, continued to the other side of the tube (but not all the way to the wall), turned, and moved back towards the centerline, etc. The two distinct modes of motion observed in the different Rep ranges are termed overdamped and damped by Denson. 

These results were confirmed on a number of other solvent systems, including benzene, cyclohexane and carbon tetrachloride, which all behave as fairly rigid spheres. Oscillations of attraction and repulsion were observed for up to ten molecular layers. With more flexible molecules, such as -octane and 2,2,4-trimethyl pentane, the oscillations died faster, after about four molecular layers. Polar molecules, e.g. propylene carbonate or methanol, gave oscillations plus double-layer repulsions. But water was the most interesting solvent, which must be considered in more detail. 

Hough and Ou-Yang used optical tweezers to drive a 1.58 p,m silica microsphere through solutions of 85 kDa end-capped polyethylene oxide in water(97). Drive frequencies were as large as 40000 rad/s. Measurements of the ampUtude and phase (relative to the driving force) for sphere oscillations were inverted, treating the sphere as a forced damped harmonic oscillator, to obtain G (co) and G"( )). The dynamic moduli were. .. quite different from those obtained by a macroscopic rheometer, and are sensitive to surface treatment of the bead. 

Figure B3.2.4. A schematic illustration of an energy-independent augmented plane wave basis fimction used in the LAPW method. The black sine fimction represents the plane wave, the localized oscillations represent the augmentation of the fimction inside the atomic spheres used for the solution of the Sclirodinger equation. The nuclei are represented by filled black circles. In the lower part of the picture, the crystal potential is sketched.

I 1 11 Schrodinger equation can be solved exactly for only a few problems, such as the particle in a box, the harmonic oscillator, the particle on a ring, the particle on a sphere and the hydrogen atom, all of which are dealt with in introductory textbooks. A common feature of these problems is that it is necessary to impose certain requirements (often called boundary... 

It is well known, that in aqueous solutions the water molecules, which are in the inner coordination sphere of the complex, quench the lanthanide (Ln) luminescence in result of vibrations of the OH-groups (OH-oscillators). The use of D O instead of H O, the freezing of solution as well as the introduction of a second ligand to obtain a mixed-ligand complex leads to either partial or complete elimination of the H O influence. The same effect may be achieved by water molecules replacement from the inner and outer coordination sphere at the addition of organic solvents or when the molecule of Ln complex is introduced into the micelle of the surfactant. 

Fig. 10(b)). One of the reasons for the differences between both theories is a different form of a hard sphere part of the free energy functional. Segura et al. have used the expression resulting from the Carnahan-Starhng equation of state, whereas the Meister-Kroll-Groot approach requires the application of the PY compressibility equation of state, which produces higher oscillations. 

As in previous theoretical studies of the bulk dispersions of hard spheres we observe in Fig. 1(a) that the PMF exhibits oscillations that develop with increasing solvent density. The phase of the oscillations shifts to smaller intercolloidal separations with augmenting solvent density. Depletion-type attraction is observed close to the contact of two colloids. The structural barrier in the PMF for solvent-separated colloids, at the solvent densities in question, is not at cr /2 but at a larger distance between colloids. These general trends are well known in the theory of colloidal systems and do not require additional comments. 

To give a simple classical model for frequency-dependent polarizabilities, let me return to Figure 17.1 and now consider the positive charge as a point nucleus and the negative sphere as an electron cloud. In the static case, the restoring force on the displaced nucleus is d)/ AtteQO ) which corresponds to a simple harmonic oscillator with force constant... 

FIGURE A.l A molecular representation of the three states of matter. In each case, the spheres represent particles that may be atoms, molecules, or ions, (a) In a solid, the particles are packed tightly together, but continue to oscillate, (b) In a liquid, the particles are in contact, but have enough energy to move past one another, (c) In a gas, the particles are far apart, move almost completely freely, and are in ceaseless random motion. 

The change in the inner-sphere structure of the reacting partners usually leads to a decrease in the transition probability. If the intramolecular degrees of freedom behave classically, their reorganization results in an increase in the activation barrier. In the simplest case where the intramolecular vibrations are described as harmonic oscillators with unchanged frequencies, this leads to an increase in the reorganization energy ... 

In order to accurately describe such oscillations, which have been the center of attention of modern liquid state theory, two major requirements need be fulfilled. The first has already been discussed above, i.e., the need to accurately resolve the nonlocal interactions, in particular the repulsive interactions. The second is the need to accurately resolve the mechanisms of the equation of state of the bulk fluid. Thus we need a mechanistically accurate bulk equation of state in order to create a free energy functional which can accurately resolve nonuniform fluid phenomena related to the nonlocality of interactions. So far we have only discussed the original van der Waals form of equation of state and its slight modification by choosing a high-density estimate for the excluded volume, vq = for a fluid with effective hard sphere diameter a, instead of the low-density estimate vq = suggested by van der Waals. These two estimates really suggest... 

Noordsij and Rotte (N10) studied mass transfer at a vibrating sphere. The oscillating motion was achieved by means of an attached rod, held by a spring against an eccentric wheel. The results, showing considerable scatter, are represented approximately by a Pe1/2 dependence, with an additive constant term accounting for pure diffusion. 

Another noteworthy example is x-ray absorption fine structure (EXAFS). EXAFS data contain information on such parameters as coordination number, bond distances, and mean-square displacements for atoms that comprise the first few coordination spheres surrounding an absorbing element of interest. This information is extracted from the EXAFS oscillations, previously isolated from the background and atomic portion of the absorption, using nonlinear least-square fit procedures. It is important in such analyses to compare metrical parameters obtained from experiments on model or reference compounds to those for samples of unknown structure, in order to avoid ambiguity in the interpretation of results and to establish error limits. 

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