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.