Current local variations

By putting these equations to zero, we can calculate the equilibrium lateral displacements. In other words, we can calculate the deformations of the elastic layers in the x and y directions, from the current local variations in film height. Furthermore, by taking the functional derivative of the stretching free energy with respect to the height of the film, we also obtain the pressure contribution, which acts on the fluid layer. 

Fig. 6.3 Optical micrographs of edges of cleaved Si wafers showing different crystal planes anodized at 100 mA cm"2 in ethanoic HF (1 1). (a) The growth rate of meso PS formed on a highly doped n-type substrate (2xl018 crrT3, 2 min) shows a clear dependence on crystal orientation, (b) An orientation dependence is not observed for micro PS formed on moderately doped p-type samples (1.5 xlO16 cm-3, 4 min) but the PS thickness becomes inhomogeneous because of local variations in the current density caused by the edge geometry.

A local variation in porosity can be produced by an inhomogeneous illumination intensity. However, any image projected on the backside of the wafer generates a smoothed-out current density distribution on the frontside, because of random diffusion of the charge carriers in the bulk. This problem can be reduced if thin wafers or illumination from the frontside is used. However, sharp lateral changes in porosity cannot be achieved. 

On small-scale profiles (such as in the case of printed circuitry), CE varies not only with current density but also with local variations of effective thickness of the diffusion layer. In cyanide-type baths these opposite influences tend to compensate each other. 

Circulation within the bank depends upon the position of the currents and these are not necessarily constant across the width of a calender. Examination of a bank may reveal sluggish areas, some rotating in opposite directions, or what appears to be spiralling (the last, because of local variation in the temperature of the stock, often will result in flow marks in the film). Defects of this nature in the flow are influenced by the composition, and it may be necessary to adjust the amount of lubricant however, if a mix is long-established the indication probably is that the conditions of calendering have drifted away from the optimum—and appropriate corrections must be made. The size of a bank can be altered by adjusting the previous nip or by changing the speed ratio of the rolls, and often this will rectify a fault. 

Precisely symmetrical models of aqueous electrode interfaces are often shown for educational purposes. Polar water molecules and positive ions are assembled in a repetitive array. In reality the array would be interrupted by impurities and irregularities, all thermally agitated. The precise achievement of a potential difference V , corresponding to Nernst s equation (4.1), would require microscopically uniform conditions at the interface between electrode and electrolyte, so as to avoid local variations of potential difference and circulating currents in the electrode. 

Uncertainties in the above values come from the dependence of the number and sizes of the inclusions in the matrix and non-linearities in the dependence of the Li ion yield with concentration. The main contributor to the large disparity between the Li concentrations found for samples DNA 1 and 2 is regarded as the local variation in the number and size of the inclusions. To reduce the uncertainty in the results a larger number of inclusions over a wider area would need to be sampled but even then the natural variation in the microstructure may still give a difference between the samples for practicable numbers of measurements. The calibration was for Li on a Fe substrate whereas for actual samples the Li was associated with A1 inclusions. This may result in a different RSF than used in the quantification of the Li to matrix ion intensity and so a variation from the reported Li concentration. The combined effect of the sources of uncertainty are compatible with the ten-fold difference between the measured results so that the present data are currently regarded as contributing to a best estimate of magnitude. 

An alternative is to apply a constant current to drive the reaction. This usually results in more even film growth although local variations in current density will produce a heterogeneous polymer. The rate of polymerization is dictated by the current density applied. Again, if the rate is too low, oxidation without deposition may occur. However, if the rate is too high, the potential may stray into the region where overoxidation of the polymer occurs. 

The classical model for porous or mixed-phase electrodes is therefore formulated in terms of equations that describe the local variation of electric current and potential in each phase in the layer of thickness L by the following equations ... 

VjUq = V/(q2 -2Vju = -2Vju oc i/cr the local electronic conductivity is determined by the total current divided by the local change of the electrons electrochemical potential and simultaneously by the local variation in stoichiometry (oc = 1/2V//q ). If the conductivity is approximately constant, the electronic conductivity is obtained from the ratio of current density and voltage, which is direcdy obvious from the equivalent circuit. More accurately, the partial conductivity is derived from the slope of the current-voltage relationship. 

Fundamentals. Localized very small variations of the electrode potential that are caused by current flow across the metal/solution interface over the surface of an electrochemically active material (e.g. a corroding metal) can be measured with a scanning reference electrode [178]. The local variations are picked up by a pair of very fine tips about 10 pm above the surface. The response of a twin platinum electrode has been modelled and results could be matched satisfactorily with real... 

Potential and Current Distribution at Electrodes with Local Variations of Polarization Parameter . Electrochem. Acta, Vol. 12, pp. 131-136, 1967. 

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