The Boltzmann distribution describes the amount of work that is required to bring an ion from infinity (i.e. the bulk solution, where ψ=0) to a given location having a potential The expression for the Boltzmann distribution is as follows:
( ) (4.1),
where is the ionic number concentration when Ψ=0 and is the number concentration of the species. The relationship between the free charge density and the number
concentration can be expressed as:
∑ (4.2).
The total potential within a nanocapillary under an externally applied field is given as:
( ) ( ) ( ) (4.3),
where ( ) is the potential from the double layer which relates the state of equilibrium where no fluid or externally applied field is present. The second term of Equation 4.3 is the local potential at any axial location within the nanopore. Equation 4 is the Poison equation, which describes the potential distribution within the nanopore:
Substituting the free charge density within the Poison equation gives the Poison-Boltzman (PB) equation, shown in equation 5:
∑ ( ) (4.5),
The PB equation can be simplified by assuming a symmetric (z:z) electrolyte and low surface potential. These two assumptions lead to the Debye-Huckel (DH) approximation. For a cylindrical nanopore, the DB equation is given as
(
) (4.6),
where κ is defined as the inverse of the Debye length. The Debye length is defined as
(
) (4.7).
The DH equation can be analytically solved and expressed as:
( )( ) (4.8),
where is the zeroth-order Bessel function.
The Debye length is a measure of the thickness of the electric double layer. The double layer thickness is usually defined as the distance from the charged pore wall to the point where the potential decays to approximately 33% of the surface potential. As can be seen by Figure 4.1, there are three different parts of the EDL. The inner layer contains non- solvated ions, which are specifically adsorbed to the surface by chemisorption. This layer is defined as the Stern layer or compact layer. The locus of centers of specifically
adsorbed ions is called the Inner Helmholtz Plane (IHP). Within this layer, the ions are assumed to be immobile and adhere to the charged walls. The Outer Helmholtz Plane (OHP) is a region in which counterions are electrostatically attracted to the surface. These ions are nonspecifically adsorbed and are solvated molecules. There is a finite distance at which the proximity of ions may approach the wall. Because of thermal agitation, these ions are distributed in a three dimensional region called the diffuse layer. The OHP is the location in which a shear plane is defined for purposes of defining a no-slip boundary condition, which solves the velocity field for electrokinetic flow. The last layer is the bulk solution. In this regime, the solution is considered to be electro-neutral.
Figure 4.1. Schematic of Electric Double Layer
The governing equations for the velocity fields, pressure, and distribution of ions is given by the following systems of equations:
(4.9),
( ) (4.10),
(4.11),
Equations 4.9 and 4.10 are the conservations of mass and momentum for an incompressible (Newtonian) fluid with constant thermodynamic properties where ρ is the fluid density; u is the fluid velocity vector; p the pressure; E the electric field vector; and, µ is the fluid dynamic viscosity. The last term in the momentum equation is the contribution to the electrical body force. The transport equation of (of the ith
species) within a dilute electrolyte solution within the fluid domain is given as:
( ) (4.12),
where is the diffusion coefficient, F is Faraday constant, is the mobility, and is the valency of the ith species.
By assuming fully-developed laminar flow conditions, the velocity field can be solved for analytically yielding:
Electrokinetic phenomena result from an unbalanced-net charge within the diffuse layer. When there is an electric field applied across a nanopore or nanochannel, the first- term Equation 4.13 cancels, resulting in a fluid flow entirely due to electroosmotic flow. If no pressure gradient is present and the thickness of the electric double layer is much smaller than the diameter of the nanopore, then ( ), eqn 13 reduces to:
(4.14).
This classical result is known as the Helmoltz-Smoluchowski (HS) equation. The fluid moves as a plug flow within the nanopore. Figure 4.2 is a plot of the electroosmotic equation for different normalized radius values. The HS approximation is valid for 1. For , the fluid moves as plug flow:
Figure 4.2. Electroosmotic velocity for a zero pressure gradient plotted as a function of
dimensionless radius. Reprinted with permission from Journal of Separation Science, 30, 1398-1419, Holtzel, A., Tallerek, U., Review, Ionic conductance of nanopores in microscale analysis systems: Where microfluidics meets nanofluidics, Copyright 2007, John Wiley and Sons.
For overlapping double layers ( ) with no pressure gradient, the electroosmotic equation simplifies to:
( ) [ ( ) ] (4.15).
As the radius approaches the size of the Debye length, the effective EMF velocity is reduced to a finite value. As the electroosmotic velocity (ux) reduces to zero. The
term ( ) is similar to the term found in Poiseuillie flow for fully developed flow under an applied pressure across the nanopore. The similar flow characteristics are a result of a liquid body force attributed to the net free charge density across the nanopore. The net liquid body force is similar to the body force found from a pressure gradient across a nanopore.