Helmholtz-Smoluchowski model

This equation, known as the Helmholtz-Smoluchowski equation, relates the potential at a planar bound surface region to an induced electro-osmosis fluid velocity 6. Recall that in the previous section surface charge was related to a potential in solution. In the following section surface charge will be related to the chemistry of the surface. A model for the development of surface charge in terms of acid-base dissociation of ionizable surface groups is introduced. 

Expression (V.25), referred to as the Helmholtz-Smoluchowski equation, relates the rate of relative phase displacement to some potential difference, Acp, within the electrical double layer. In order to understand the nature of this quantity, let us examine in detail the mutual phase displacement due to the external electric field acting parallel to the surface, taking into account the electrical double layer structure. Let us assume that the solid phase surface is stationary. Figure V-7 shows the distributions of the potential, cp(x ) (line 1), the rate of displacement of the liquid layers relative to the surface in the Helmholtz model, u(x) (line 1/), and the true distribution of the potential in the double layer (curve 2). 

In the small nanochaimels (from a few to about 100 nm), the electric double layer (EDL) thickness becomes larger or at least comparable with the nanochaimels lateral dimensions. It affects the balance of bulk ionic concentrations of co-ions and counterions in the nanochannels. Thus, many conventional approaches such as the Poisson—Boltzmann equation and the Helmholtz-Smoluchowski slip velocity, which are based on the thin EDL assumption and equal number of co-ions and counterions, lose their credibility and cannot be utilized to model the electrokinetic effects through these nanoscale channels. The Poisson equation, the Navier-Stokes equations, and the Nemst-Planck equation should be solved directly to model the electrokinetic effects and find the electric... 

Extension to 2D and 3D Systems In the majority of microfluidic cases where 1/k is much smaller than the channel height, the Helmholtz-Smoluchowski equation provides a reasonable estimate of the flow velocity at the edge of the double layer field. As such when modeling two- and three-dimensional flow systems, it is common to apply this equation as a slip boundary condition on the bulk flow field. Since beyond the double layer by definition... 

Electroosmosis produces an effective slip of the liquid outside the double layer past to the solid surface. In the classical continuum model of the diffuse layer, the slip velocity is given by the Helmholtz-Smoluchowski formula ... 

Particle trajectories are determined by a combination of fluid-flow, electrophoresis and DEP. The fluid-flow is driven by electroosmosis for the case of interest here. For the thin Debye layer approximation, electroosmotic flow may be simply modeled with a slip velocity adjacent to the channel walls that is proportional to the tangential component of the local electric field, as shown by the Helmholtz-Smoluchowski equation 12). Here, the proportionality constant between the velocity and field is called the electroosmotic mobility, tigo- Fluid-flow in microchannels becomes even simpler for ideal flow conditions where the zeta potential, and hence /ieo, is uniform over all walls and where there are no pressure gradients. For these conditions, it can be shown that the fluid velocity at all points in the fluid domain is given by the product of the local electric field and// o(/i). 

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