The Edge Effect

The current density distribution in a cell with plane parallel electrodes, with edges not touching the cell wall, is illustrated in Fig. 3.4 [1]. 

It can be assumed that a homogeneous electric field and, consequently, a uniform current density distribution are present over the entire electrode surface up to the very edge of the electrode, where the current density increases abruptly. This problem has been studied in detail in Refs. [2, 3, 5-7], and a relatively rough approximation will be used here. This approximation is based on the following assumptions  

This approximation is more appropriate for the tip of a stationary wire electrode due to the dissipation of current lines occurring through the space [6]. On the other hand, in the case of the edges of plane parallel electrodes, the dissipation of current fines occurs in one plane normal to the electrodes to which the two symmetrically positioned points belong. 

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When an isotropic material is subjected to planar shock compression, it experiences a relatively large compressive strain in the direction of the shock propagation, but zero strain in the two lateral directions. Any real planar shock has a limited lateral extent, of course. Nevertheless, the finite lateral dimensions can affect the uniaxial strain nature of a planar shock only after the edge effects have had time to propagate from a lateral boundary to the point in question. Edge effects travel at the speed of sound in the compressed material. Measurements taken before the arrival of edge effects are the same as if the lateral dimensions were infinite, and such early measurements are crucial to shock-compression science. It is the independence of lateral dimensions which so greatly simplifies the translation of planar shock-wave experimental data into fundamental material property information. 

When using air curtains the edge effects are neglected and the flow is treated as two-dimensional. The different parts of a two-dimensional jet are sketched in Fig. 10.63. 

Figure 4.1.2 is a photograph of a coimterflow burner assembly. The experimental particle paths in this cold, nonreacting, counterflow stagnation flow can be visualized by the illumination of a laser sheet. The flow is seeded by submicron droplets of a silicone fluid (poly-dimethylsiloxane) with a viscosity of 50 centistokes and density of 970 kg/m, produced by a nebulizer. The well-defined stagnation-point flow is quite evident. A direct photograph of the coimterflow, premixed, twin flames established in this burner system is shown in Figure 4.1.3. It can be observed that despite the edge effects.

Application of the sample as a continuous streak is also possible using commercially available simple instruments (sample applicators), which give a sample zone for preparative separation less than 3 to 4 mm wide. It is also advisable to apply the streak across the plate starting 2 cm from both edges to avoid the edge effect, which may cause the motion of the mobile phase to be faster or slower at the edges than across the center of the plate. 

In turbulent flow, the edge effect due to the shape of the support rod is quite significant as shown in Fig. 6. The data obtained with a support rod of equal radius agree with the theoretical prediction of Eq. (52). The point of transition with this geometry occurs at Re = 40000. However, the use of a larger radius support rod arbitrarily introduces an outflowing radial stream at the equator. The radial stream reduces the stability of the boundary layer, and the transition from laminar to turbulent flow occurs earlier at Re = 15000. Thus, the turbulent mass transfer data with the larger radius support rod deviate considerably from the theoretical prediction of Eq. (52) a least square fit of the data results in a 0.092 Re0 67 dependence for... 

The application of RHSE is primarily in the laminar boundary layer flow regime of Re < 15000, where the edge effect is negligible and the mass transfer theory has been confirmed by experimental investigations. An important consideration in the design of a practical RHSE system is to conform to the theoretical requirement that the boundary layer thickness be thin in comparison to the radius of the RHSE (<5 a). 

The UMEs used in bioarrays can be divided into three types disk, ring, and strip electrodes. The theory of the disk, ring, and strip UMEs has been extensively studied [97-100], Due to the edge effect, the profile of the mass diffusion to the ultramicroelectrode surface is three dimensional, and can significantly enhance the mass transportation in comparison to the conventional large electrode with one-dimensional mass transportation. The steady-state measurement at a planar UME can be expressed as... 

Diffusion of electroactive species to the surface of conventional disk (macro-) electrodes is mainly planar. When the electrode diameter is decreased the edge effects of hemi-spherical diffusion become significant. In 1964 Lingane derived the corrective term bearing in mind the edge effects for the Cotrell equation [129, 130], confirmed later on analytically and by numerical calculation [131,132], In the case of ultramicroelectrodes this term becomes dominant, which makes steady-state current proportional to the electrode radius [133-135], Since capacitive and other diffusion-unrelated currents are proportional to the square of electrode radius, the signal-to-noise ratio is increased as the electrode radius is decreased. 

When the plates are placed in the incubator, temperature and evaporation gradients start to form inside the plates which result in positional biases (edge effects) that introduce noise in the final data. The most affected samples are the ones in the periphery of the plate. Filling the spaces between the wells with PBS reduces the edge effect. Stacking the plates inside the incubator will also influence the evaporation gradient inside the plate, especially when the top plate of a stack is compared to the bottom plate of a stack. If the plates must be stacked then the plates that are to be compared direcdy should be in the same position on different stacks. 

Baker [17] derived an elegant description of the edge effect based on the mechanics of the pad and of the wafer. He argued that the peak in the edge exclusion region is proportional to where C is the slope of... 

Even in the absence of the edge effect, several other issues limited the ability of these earners to produce very flat wafers. First, the application of pressure to the center of the wafer, in spite of the rigid structure of the... 

A. R. Baker, The Origin of the Edge Effect in Chemical Mechanical Planarization, presented at the ECS Fall 1996 meeting, San Antonio, TX. 

Baker, A. (1997). The origin of the edge effect in chemical mechanical planarization. Electrochem. Soc. Proc. 96-122, p. 228. 

The unconfined critical diam for propagation of the deton wave varied from about the edge effect value 0.6cm (for ideal expls of very high reaction rate) to very large values,... 

The photograph is included to make two points. First, the particle paths show qualitatively that the flow follows the anticipated streamlines. Even for the relatively small dimensions, the edge effects that could interrupt similarity behavior at the outflow appear to be minor. Second, and more striking, is the fact that the flame zone is extremely flat. Here is a situation that includes a considerable amount of chemistry (methane combustion) and complex heat and mass transfer. The fact that the flame zone shows no radial dependence is is convincing evidence that the fluid mechanical similarity is indeed valid. 

Of the many complex transport geometries, some of the most important are those encountered with inlaid electrodes (Fig. 14) [25— 30], in which the electrode is planar and embedded in an insulator whose surface is a continuation of the electrode plane. If such an electrode is large, it may be appropriate to treat it as a case of planar transport, with a correction for the edge effect [31, 32], For a small inlaid electrode, however, or for experiments of long duration, such an approximation is no longer valid. 

Nakajima and Hamel50 have derived expressions for calculating shear stresses from the Mooney torque values to give viscosities in agreement with those obtained from other instruments, and also an expression to correct for the edge effects.51 Bristow52 derived non-standard Mooney parameters for natural rubbers to improve the distinction between different grades. 

In this derivation it is assumed that the field and displacement are uniform throughout the volume of the condenser, and that the field is zero outside the condenser. This is true for the isotropic dielectric medium with the edge effects neglected. If this were not true, then the work done on differential volumes would have to be considered and the total work would be obtained by integration over the volume of the condenser. 

Cheneau E, Waksman R, Yazdi H, et al. How to fix the edge effect of catheter-based radiation therapy in stented arteries. Circulation 2002 106( 17) 2271 -2277. 

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