EDL Potential

What remains is to relate the surface potential to activation potentials for the adsorption/desorption reaction steps. Defining the activation potentials as iji, ijjf for the activation required to overcome the EDL potential for the adsorption, desorption steps, respectively, allows the intrinsic rate constants to be directly related to the rate constants k, k (4), i.e.,... 

Increasing the bulk ion concentration in the liquid results in an increase in /cor a decrease in the EDL thickness 1/k. Correspondingly, the EDL potential field falls off to zero more rapidly with distance, i.e., the region influenced by the EDL is smaller. The ionic concentration effect on the velocity or the flow rate can be understood as follows. Since ionic concentration influences the zeta potential, as the ionic concentration is increased, the zeta potential decreases in value. As the zeta potential decreases, so does the electroosmotic flow velocity (Eq.(7)) and the volumetric flow rate. 

According to the theory of electrostatics, the net charge density in the EDL of channel, pe, is related to the EDL potential that is described by the Poisson equation, which in dimensionless form is expressed as [1]... 

In microfluidic systems, due to the very thin EDL compared to the microchannel dimension, the electroosmotic velocity distribution and EDL potential profile inside the EDL region become insignificant. Thus we do not need to solve the Poisson equation together with the Boltzmann distribution. Instead the electroosmotic flow velocity at the edge of the EDL (i.e., from the diffuse layer to the bulk phase) is given by the Smoluchowski equation [1], expressed as... 

Governing equation Boundary conditions Surface charge balance equation Charge on particle surface EDL potential... 

It is important to reiterate here that in most of the discussions presented in this entry, the celebrated Poisson-Boitzmann equation has been utilized to describe the EDL potential distribution in simple analytical forms. However, as mentioned earlier, the Poisson-Boitzmann equation has its own limited applicability, especially when the surfaces under consideration constitute too narrow a fluid passage within which distinctive intermolecular and surface interactions take place. In brief, for such small separation distances, the following effects are likely to occur, which are not considered, while deriving the Poisson-Boitzmann equation [10] ... 

The pertinent expressions for EDL potential energy and the interaction force are as follows ... 

The second method is to neglect the convection term in Eq. 9 considering that the flow velocity is small for both EOF flow and electrophoretic flow, which decouples the flow field and the EDL potential field. In the second approach, the assumption of a two-species buffer and the application of the Boltzmann distribution are commonly made in order to solve the potential field easily, which yields the well-known Poisson-Boltzmann equation ... 

By solving Eq. 11, the EDL potential and net charge density can be determined, which can be used to determine the electrical driving force in the modified N-S equation for the flow field. 

Because of nonlinearity, the Poisson-Boltzmann equation (i.e., Eq. 10) can be solved numerically. Using the Debye-Hiickel approximation, an analytical solution of Eq. 10 for the EDL potential can be obtained as... 

The EDL potential distribution is governed by the Poisson-Boltzmann equation ... 

Clearly, the EDL potential is required to derive an explicit solution of fluid velocity in Eq. 3. In the present analysis, a symmetric binary electrolyte and a small zeta potential are assumed. Then the net charge density can be linearly related to the EDL potential via Pe = — After... 

The solutions of the EDL potential for the parallel-plate and cylindrical microchannels can be written, respectively, as [2]... 

Prior to solving Eq. 1, one needs to know the net charged density in the EDL, which can be evaluated from the EDL potential, from the Poisson equation... 

Electrical Double Layer General Formulation The EDL potential and the net charge density are related via the Poisson equation ... 

Zeta potential is defined as the electric doublelayer (EDL) potential located at the shear plane between the Stem layer and the diffuse layer of the EDL that is formed in the neighborhood of a charged solid-liquid interface. Zeta potential is an experimentally measurable electrical potential that characterizes the strength and polarity of the EDL of the charged solid-liquid interface. Depending on the solid surface and the solution, zeta potentials values are within a range of —100 mV to +100 mV for most solid-liquid interfaces in aqueous solutions. 

The EDL potential i/r(r) can be determined by the solving the well-known Poisson-Boltzmann equation for cylindrical capillary ... 

Once the EDL potential ij/ r) is known, the local net charge density pe(0 is determined according to Eq. 3. Finally, the equation of motion, Eq. 15a, can be solved numerically to find the velocity u(r). The average velocity can then be determined as follows ... 

When a solid surface is in contact with an electrolyte, the surface usually becomes charged. Due to electrostatic interactions, both anions and cations are preferentially redistributed near the charged surface, leading to the formation of an electric double layer (EDL). According to the theory of electrostatics, the EDL potential distribution, /, is governed by the Poisson equation, written as [9]... 

An approximate expression for the EDL potential energy (Eedl) versus the surface to surface separation distance (D) between two spherical particles of diameter (x) with the same surface charge is (Israelachvili,1992) ... 

The appropriate boundary conditions for solving these governing equations are specified as follows. The liquid velocity tangential and normal to all the channel walls must obey the no-slip and impermeable conditions. The EDL potential at the solid-liquid interface is specified as the zeta potential, which depends on channel material and electrolyte solution chemistry such as ionic strength, pH, etc. The externally applied electrical potential is generally specified by prescribed potentials imposed on inlet and outlet boundaries, and the electrical insulation condition is usually implemented to the channel walls. The prescribed global pressure values (if any) are provided on both inlet and outlet boundaries. 

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