Infinite velocity method

In an attempt to broaden the applicability of the RSF method to snbstrates for which reference materials were not available, the matrix-tranrferable RSF approach was snggested. This is covered in Section A.9.1. Shortly thereafter, another approach that also did not reqnire matrix-matched reference materials was suggested. This method, termed the Infinite Velocity method, is discnssed in Section A.9.2. These, however, suffered in that they could not provide the precision levels required for high sensitivity/detection limits commonly expected of SIMS. 

Infinite velocity method A suggested SIMS quantification method... 

A recent proposal for quantification of SIMS data from first principles is the infinite velocity (IV) method of van der Heide et al. [93], The basis for this method is the quantum mechanical argument derived by Norskov and Lundquist [94] the SIMS matrix effect is removed if secondary ions are emitted from the sample surface with infinite velocity (i.e., the secondary ion yield for all elements is the same at infinite emission velocity). Under this condition, the relative concentration of an element, /, can be calculated from... 

The Infinite Velocity (IV) method represents an alternative approach in that RSFs are not derived/required in defining the concentration/s of the element/s of interest. As the name suggests, the IV method describes the extrapolation of data to infinite emission velocity. This is made possible by collecting the intensities of secondary ions of the elements of interest at several different emission energies and then extrapolating to infinite emission velocity. This method was first introduced in 1994 (van der Heide et al. 1994) and refined thereafter (van der Heide et al. 1998). 

Coefficient Equations.—To determine the coefficients of the expansion, the distribution function, Eq. (1-72), is used in the Boltzmann equation the equation is then multiplied by any one of the polynomials, and integrated over velocity. This gives rise to an infinite set of coupled equations for the coefficients. Only a few of the coefficients appear on the left of each equation in general, however, all coefficients (and products) appear on the right side due to the nonlinearity of the collision integral. Methods of solving these equations approximately will be discussed in later sections. 

Elutriation differs from sedimentation in that fluid moves vertically upwards and thereby carries with it all particles whose settling velocity by gravity is less than the fluid velocity. In practice, complications are introduced by such factors as the non-uniformity of the fluid velocity across a section of an elutriating tube, the influence of the walls of the tube, and the effect of eddies in the flow. In consequence, any assumption that the separated particle size corresponds to the mean velocity of fluid flow is only approximately true it also requires an infinite time to effect complete separation. This method is predicated on the assumption that Stokes law relating the free-falling velocity of a spherical particle to its density and diameter, and to the density and viscosity of the medium is valid... 

The various physical methods in use at present involve measurements, respectively, of osmotic pressure, light scattering, sedimentation equilibrium, sedimentation velocity in conjunction with diffusion, or solution viscosity. All except the last mentioned are absolute methods. Each requires extrapolation to infinite dilution for rigorous fulfillment of the requirements of theory. These various physical methods depend basically on evaluation of the thermodynamic properties of the solution (i.e., the change in free energy due to the presence of polymer molecules) or of the kinetic behavior (i.e., frictional coefficient or viscosity increment), or of a combination of the two. Polymer solutions usually exhibit deviations from their limiting infinite dilution behavior at remarkably low concentrations. Hence one is obliged not only to conduct the experiments at low concentrations but also to extrapolate to infinite dilution from measurements made at the lowest experimentally feasible concentrations. 

The procedure for determining APs that will be presented here is that of Molerus (1993). The basis of the method is a consideration of the extra energy dissipated in the flow as a result of the fluid-particle interaction. This is characterized by the particle terminal settling velocity in an infinite fluid in terms of the drag coefficient, Cd ... 

A number of authors from Ladenburg (LI) to Happel and Byrne (H4) have derived such correction factors for the movement of a fluid past a rigid sphere held on the axis of symmetry of the cylindrical container. In a recent article, Brenner (B8) has generalized the usual method of reflections. The Navier-Stokes equations of motion around a rigid sphere, with use of an added reflection flow, gives an approximate solution for the ratio of sphere velocity in an infinite space to that in a tower of diameter Dr ... 

The mathematical method of Aris assumes a doubly infinite pipe (as does Taylor), with both the velocity distribution and the diffusion coefficients constant in the direction of flow. Hence in any real pipe, the length would have to be long enough so that the buildup of the velocity profile at the entrance would not invalidate the doubly infinite pipe assumption. Thus there are some practical restrictions on the method used by Aris. 

The generalized graphical correlation presented in Fig. 2.5 gives one method of estimating terminal velocities of drops and bubbles in infinite liquid media. For more accurate predictions, it is useful to have terminal velocities correlated explicitly in terms of system variables. To obtain such a correlation is especially difficult for the ellipsoidal regime where surface-active contaminants are important and where secondary motion can be marked. 

Stefan gave an exact solution for the constant-velocity melting of a semi-infinite slab initially at the fusion temperature. This was extended by Pekeris and Slichter (P2) to freezing on a cylinder of arbitrary surface temperature and Kreith and Romie (K6) to constant-velocity melting of cylinders and spheres by a perturbation method, in which the temperature is assumed to be expressible in terms of a convergent series of unknown functions. To make the method clear, consider the freezing of an infinite cylinder of liquid, of radius r0, at constant surface heat flux. For this geometry the heat equation is... 

The friction velocity at the minimum transport condition may be related to the system configuration and the operating condition by a two-step correlation method, i.e., first, to obtain the velocity at the minimum transport condition under infinite dilutions and, second, to correct for the concentration dependence [Thomas, 1962]. The functional dependence of Uf to the solids concentration is given by... 

The methods used to inhibit and effectively stop corrosion can be divided into two large groups depending on the corroding situation. If the body to be protected is in an infinitely large solution, typically the sea, then the methods have to rely upon what can be done electrochemically to the metal itself. Usually, as in the widely practiced cathodic protection, a circuit is fixed up in which the metal of the object to be protected is moved away from the corrosion potential in the cathodic direction (or electronation), thus reducing the anodic (or deelectronation) dissolution velocity, and hence the corrosion. 

When a pile is driven into unconsolidated material which is infinite in extent, a relation may be developed giving the resistance encountered by methods of elementary mechanics. The common velocity v between... 

O Brien s method was extended to study the electrophoresis of a nonuniformly charged sphere with thin but polarized ion cloud in a symmetric electrolyte [32]. The electrophoretic mobility depends on the charge distribution at the particle surface. It is found that the polarization effect of the ion could leads to different electrophoretic mobilities for particles with different zeta potential distributions but having an identical velocity for the limit of infinite Ka. This intriguing result is due to the fact that the theory for undistorted ion cloud is linear in the distribution of zeta potential, whereas the polarization effects are nonlinear. 

Keh and Chen [33] employed O Brien s method [7] to examine the polarization effect on the electrophoresis of an infinitely long circular cylinder. They found that neglecting the end effect, the transverse electrophoretic velocity is identical to that for a spherical particle with the same radius. The polarization effects were also investigated for a spheroidal particle [34,35] and an infinitely long elliptical cylinder [36]. An interesting feature discovered from these studies is that the electrophoretic velocity decreases with the reduction of the maximum length of the particle in the direction of the migration. 

Boundary effects on the electrophoretic migration of a particle with ion cloud of arbitrary thickness were also investigated by Zydney [46] for the case of a spherical particle of radius a in a concentric spherical cavity of radius d. Based on Henry s [19] method, a semi-analytic solution has been developed for the particle mobility, which is valid for all double layer thicknesses and all particle/pore sizes. Two integrals in the mobility expression must be evaluated numerically to obtain the particle velocity except for the case of infinite Ka. The first-order correction to the electrophoretic mobility is 0(A3) for thin double layer, whereas it becomes 0(A) for thick double layer. Here the parameter A is the ratio of the particle-to-cavity radii. The boundary effect becomes more significant because the fluid velocity decays as r l when the double layer spans the entire cavity. The stronger A dependence of the first order correction for thick double layer than that obtained by Ennis and Andersion [45] results from the fact that the double layers overlap in... 

An analytical study using the method of reflections was conducted by Chen and Keh [9] to investigate the electrophoretic motion of two freely suspended nonconducting spherical particles with infinitely thin double layer. The particles may differ in size and zeta potential, and they are oriented arbitrarily with respect to the imposed electric field. The resulting translational and angular electrophoretic velocities are given by... 

Howells (1974) restricted his attention to fixed particles, extending the method of Childress (1972) by considering a given number of particles chosen from an infinite set. This partly self-consistent scheme furnishes terms valid in the small-solids concentration limit. In a very readable paper, Hinch (1977) combined some of the above procedures in formulating an averaged-equation approach to particle interactions, providing expressions for the bulk stress, average sedimentation velocity, and effective permeability in suspensions and fixed beds. 

In contrast to fhe static methods discussed in the previous section, molecular dynamics (MD) includes thermal energies exphcitly. The method is conceptually simple an ensemble of particles represents fhe system simulated and periodic boundary conditions (PBC) are normally apphed to generate an infinite system. The particles are given positions and velocities, fhe latter being assigned in accordance with a... 

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