Analytical solution Levich equation

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

The theory describing the mass transport to a rotating-disk electrode is due to Levich [8], who first presented an analytical solution for the flux to the electrode under laminar flow conditions (Re < 2.7x10 ). For the reaction of a minor ionic species B, the limiting current is given by equation (4.105), know as the Levich equation ... 

The equation of the current/potential curve closely resembles that of the analogous, though not identical pattern, observed at an electrode with periodical renewal of the diffusion layer. For a reversible uncomplicated charge transfer the ciurent intensity is proportional to the concentration of the electroactive species at any potential values. Independently of the nature of the charge transfer, in correspondence to the limiting (plateau) value, it is given by the Levich equation, which furnishes the linear relationship of analytical significance between current intensity density and concentration in solution ... 

We will now combine the Levich and Butler-Volmer approaches. The Levich relationship (equation (7.1)) is written in terms of the limiting current / jm, where limiting here means proportional to Canaiyte - in other words, the electrode reaction is so fast that the magnitude of the current is controlled only by the flux of analyte to the electrode solution interface, i.e. /um is mass>transport controlled. 

Amperometric detectors can operate over a range of conversion efficiencies from nearly 0% to nearly 100%. From a mathematical point of view, a classical amperometric determination (conversion of analyte is negligible) is one where the current output is dependent on the cube root of the linear velocity across the electrode surface as described by Levich s hydrodynamic equations for laminar flow. Conversely, the current response for a cell with 100% conversion is directly proportional to the velocity of the flowing solution. While the mathematics describing intermediate cases is quite interesting, it is beyond the scope of this chapter. 

RDEs provide well-controlled diffusion conditions (Figure 1.27). The flow of fhe electiolyfe gives access of fresh solution to the surface, whereas the electrochemical reaction at the electrode surface changes the electrolyte composition. Both effects compensate each other, leading to a constant thickness of fhe Nemst diffusion layer all over fhe disc with a laminar flow of the solution in vicinity of ifs surface. The analytical treatment of this convection and diffusion problem leads to Levich s equation ... 

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