Radius, analytical

They argue that the Hertzian load (Ph) is not signifieantly affected by the finite size effeets, therefore the JKR expression relating the load to the contact radius and adhesion energy (Eq. 11) should still be valid. Using a combined analytical and computational approach, Hui et al. [36] found that a correction given by Shull et al. for the eompression of such thin lenses was accurate for moderately large eontact radius... 

Due to difficulties and uncertainties in the experimental separation of the porous media [93], and the inevitability of approximations in the analytical treatment [87,89], the nature of the chain movement in a random environment is still far from being well understood, and theoretical predictions are controversial [87,89]. Thus, on the ground of replica calculations within a variational approach, one predicts three regimes [87] in which the chain gyration radius Rg scales with the number of repeatable units N as rI (X for low, R x N for medium, and R x for high... 

It has been discovered recently that the spectrum of solutions for growth in a channel is much richer than had previously been supposed. Parity-broken solutions were found [110] and studied numerically in detail [94,111]. A similar solution exists also in an unrestricted space which was called doublon for obvious reasons [94]. It consists of two fingers with a liquid channel along the axis of symmetry between them. It has a parabolic envelope with radius pt and in the center a liquid channel of thickness h. The Peclet number, P = vp /2D, depends on A according to the Ivantsov relation (82). The analytical solution of the selection problem for doublons [112] shows that this solution exists for isotropic systems (e = 0) even at arbitrary small undercooling A and obeys the following selection conditions ... 

The simplest shape for the cavity is a sphere or possibly an ellipsoid. This has the advantage that the electrostatic interaction between M and the dielectric medium may be calculated analytically. More realistic models employ moleculai shaped cavities, generated for example by interlocking spheres located on each nuclei. Taking the atomic radius as a suitable factor (typical value is 1.2) times a van der Waals radius defines a van der Waals surface. Such a surface may have small pockets where no solvent molecules can enter, and a more appropriate descriptor may be defined as the surface traced out by a spherical particle of a given radius rolling on the van der Waals surface. This is denoted the Solvent Accessible Surface (SAS) and illustrated in Figm e 16.7. 

In the preceding section, we have established the importance of the power series q x) r(x), 5(x), t x) in combinatorics. Here we examine their analytical properties radius of convergence, singularities on the circle of convergence, analytic continuation. We derive these characteristics from the functional equations whose solutions these series present. I start with a summary of the equations and some notations. 

A platinum on silica gel catalyst was prepared by impregnation of silica gel (BDH, for chromatographic adsorption) by a solution containing 0.5% (wt.) of sodium hydroxide and 0.5% (wt.) of chloroplatinic acid (both of analytical grade). The dried catalyst contained 1% (wt.) of platinum and a corresponding amount of the alkaline component. The BET surface area of the catalyst was 40 m2/g, the mean pore radius 150 A. The catalyst was always reduced directly in the reactor in a stream of hydrogen at 200°C for 2 hr. 

Something that is not obvious from the simple sag equation is how the local radii of curvature change with radial distance, p. Using analytical geometry one can show that the local radius of curvature in the radial direction goes as... 

Rodbard, D Chrambach, A, Estimation of Molecular Radius, Free Mobility, and Valence Using Polyacrylamide Gel Electrophoresis, Analytical Biochemistry 40, 95, 1971. 

The series (A. 4) has here the radius of convergence he = 2ir, but it can be continued analytically beyond its radius of convergence. 

For analytical solutions, it is more convenient to work with nondimensional forms of the diffusion equations. We choose the following nondimensional substitutions. The time coordinate / is replaced by the nondimensional parameter /, and a is the root radius ... 

The theory has been verified by voltammetric measurements using different hole diameters and by electrochemical simulations [13,15]. The plot of the half-wave potential versus log[(4d/7rr)-I-1] yielded a straight line with a slope of 60 mV (Fig. 3), but the experimental points deviated from the theory for small radii. Equations (3) to (5) show that the half-wave potential depends on the hole radius, the film thickness, the interface position within the hole, and the diffusion coefficient values. When d is rather large or the diffusion coefficient in the organic phase is very low, steady-state diffusion in the organic phase cannot be achieved because of the linear diffusion field within the microcylinder [Fig. 2(c)]. Although no analytical solution has been reported for non-steady-state IT across the microhole, the simulations reported in Ref. 13 showed that the diffusion field is asymmetrical, and concentration profiles are similar to those in micropipettes (see... 

The ability to selectively excite a particular ion (or group of ions) by irradiating the cell with the appropriate radiofrequencies provides a level of flexibility unparalleled in any other mass spectrometer. The amplitude and duration of the applied RF pulse determine the ultimate radius of the ion trajectories. Thus, by simply turning on the appropriate radiofrequency, ions of a single m/z may be ejected from the cyclotron. In this way, a gas-phase separation of analyte from matrix is achieved. At a fixed radius of the ion trajectories the signal is proportional to the number of orbiting ions. Quantitation therefore requires precise RF control. 

Figure 3. The phase lag produced by the Gouy phase, calculated using the analytic model described in the text for 0)3 + 3t0i excitation, with two additional coi photons in one of the channels. The calculations are performed for various ratios of the molecular beam radius d to the Rayleigh range zr.

To develop analytical models for processes employing porous catalysts it is necessary to make certain assumptions about the geometry of the catalyst pores. A variety of assumptions are possible, and Thomas and Thomas (15) have discussed some of these. The simplest model assumes that the pores are cylindrical and are not interconnected. Develop expressions for the average pore radius (r), the average pore length (L), and the number of pores per particle (np) in terms of parameters that can be measured in the laboratory [i.e., the apparent particle dimensions, the void volume per gram (Vg), and the surface area per gram (Sg). ... 

---