Theory for the Electrophoretic Transport Interface
The Electrophoretic Transport Interface models transport of charged and uncharged species, and in addition sets up a charge balance equation for the electrolyte potential.
The species concentrations are denoted, ci (SI unit: mol/m3), and the potential, ϕl (SI unit: V).
The species are transported by diffusion, migration, and (optionally) convection according the Nernst–Planck set of equations. The total flux of species i is denoted Ni (SI unit: mol/(m2·s)) according to
(3-117)
where Di (SI unit: m2/s) is the diffusion coefficient, zi (1) the corresponding charge, um,i (SI unit: s·mol/kg) the mobility and u (SI unit: m/s) the velocity vector. Ji denotes the molar flux relative to the convective transport (SI unit: mol/(m2·s)). For a detailed description of the theory of these equations and the different boundary conditions, see The Transport of Diluted Species Interface.
Charge Balance Equation
The current vector, il (A/m2), is defined as
(3-118)
where the summation is made over all species in the electrolyte.
The governing equation for the electrolyte potential is
(3-119)
where Ql (SI unit: A/m3) is the electrolyte current source stemming from, for example, porous electrode reactions. For nonporous electrode domains this source term is usually zero.
Proton and Hydroxide Concentrations and Electroneutrality
Assuming the total number of species to be N + 2, the assumption of electroneutrality is
(3-120)
In water-based systems the species H+ and OH- are always present. The auto ionization reaction for water is
(3-121)
This reaction is fast, following the equilibrium relation
(3-122)
where Kw ≈ 10−14.
Now, the electroneutrality condition, including the two additional species H+ and OH-, reads
(3-123)
Combining these two equations results in the following algebraic expressions for the concentrations of H+ and OH-.
(3-124)
and
(3-125)
where
(3-126)
Mass Balance Equations for Dissociation Species
Assume a set of species Si describing k dissociation steps from
(3-127)
to
(3-128)
where z0 is the charge (valence) of species S0 (which has no dissociable protons) and Ka,j is the acid (equilibrium) constant of the jth dissociation reaction. The brackets “[ ]” here represent the species activity. The charge of each species is always deductible from the index i according to z0+i and will be dropped from now on.
If the proton activity is known, any species Sm may be expressed using any other species Sl according to
(3-129)
if m > l and
(3-130)
if l > m.
Setting m = i and denoting the flux of species i by Ni using equation Equation 3-117, the mass balance equation for the concentration ci of each subspecies i in the dissociation chain is
(3-131)
where Req,i,j is the reaction source stemming from the jth dissociation step (with Req,i,k+1 = 0), and Ri any additional reaction sources.
The reaction source contributions from the dissociation steps are generally not known, but may be canceled by taking the sum of all mass balance equations, resulting in
(3-132)
Average Ionization Formulation for Large Molecules
Large protein molecules are typically subject to a large number of dissociation steps.
As an alternative way to a long chain of dissociation steps for describing protein transport, you can instead formulate the protein transport based on the average number of protons (1) removed from the molecule. is typically provided as a function of pH, based on experimental data. The average charge of the species then is , so that the flux of the species is written as:
(3-133)
and similarly the addition to the charge neutrality condition is .
When considering the contribution to the current and the charge balance equation you need to take into account that the squared average charge, , is not equal to the “average squared charge”, (Ref. 1).
The addition to the current density vector is
(3-134)
The average number of protons removed from the proton typically depends on the pH. If the average number of removed protons depend only on the pH, the averaged squared number of protons removed can be written as
(3-135)
And from this you can derive the average squared charge according to
(3-136)
Diffusivity–Mobility Relations
The Stokes radius r of a molecule is related to the diffusivity according to
(3-137)
where is the μ (SI unit: Pa·s) is the dynamic viscosity and k the Boltzmann constant.
For small molecules, there is the frequently used Nernst–Einstein relation between the diffusivity and the mobility
(3-138)
For larger molecules, such as proteins, the mobility may instead be calculated based on the Debye–Hückel–Henry expression (Ref. 2 ) according to
(3-139)
where κ (1/m) is the Debye parameter, which depends on the ionic strength of the solution, is defined for ideal solutions as
(3-140)
where ε is the dielectric constant of the fluid and ε0 the permittivity of free space. (should be used if available in the formula above when calculating the ionic strength).
The function f above is based on a sigmoidal function so that it ranges from 1 for κr = 0 to 1.5 for κr = ∞. Note that the Debye–Hückel–Henry expression approaches the Nernst–Einstein mobility as r → 0.
For larger molecules (macro ions), where the distance between the charges is large compared to 1/κ, the Linderstrøm–Lang approximation postulates a smaller contribution of to the ionic strength so that the z-valent ion behaves as a monovalent ion with a z-fold concentration. For an assemble of N − M smaller molecules and M macro ions, the Debye parameter then is defined as
(3-141)
References
1. R.A. Mosher, D.A. Saville, and W. Thormann, The Dynamics of Electrophoresis, VCH Verlagsgesellschaft mbH, Weinheim, Germany, 1992.
2. R.A. Mosher, P. Gebauer, J. Caslavska, and W. Thormann, “Computer Simulation and Experimental Validation of the Electrophoretic Behavior of Proteins. 2. Model Improvement and Application to Isotachophoresis,” Anal. Chem., vol. 64, pp. 2991–2997, 1992.