3. Theoretical modeling and simulation
The transport of the charged reactive species, referred to as
electromigration in the present study, was modeled based on the
Nernst-Plank equation.27-29 The molar flux of speciesi , Ji , is expressed in Equation (1),
describing the diffusion by concentration difference and migration by
electrostatic force. The convection term is not included due to the fact
that the experiments were conducted in a quiescent hydrodynamic
environment.
\(J_{i}=-D_{i}\nabla c_{i}-\frac{D_{i}z_{i}F}{\text{RT}}c_{i}\nabla\Phi\)(1)
where Di is the diffusion coefficient
(m2/s), ci is the ion
concentration (mol/m3), zi is
the valence number, F is the Faraday constant, R is the
gas constant, T is the absolute temperature (K), and Φ is
the electric potential (V). The Wilke-Chang equation was applied to
estimate the mutual diffusion coefficient if
necessary.30
The electric field in the domain is described by the Poisson
differential equation30:
\(\nabla\bullet(\varepsilon_{0}\varepsilon_{r}\nabla\Phi)={-\rho}_{v}\)(2)
where ε0 is the dielectric constant of the free
space, εr is the relative permittivity of
materials, ρv is the electric space charge
density.
The equations were solved using finite element method in COMSOL
Multiphysics 5.4. Simulation was conducted to understand the mass
transfer of reactive species in the proposed reaction system, assuming
no reaction nor flux at/across the interface. Concentration profiles of
Ru catalyst and formate at the interface were calculated under various
external electric fields.