Transport of ions and electrons in mixed conductors

Diffusion of the electroactive species within the electrode toward or away from the interface with the electrolyte is an irreversible process. The sum of the products of the forces and fluxes corresponds to the entropy production. In order to avoid space charge accumulation, the motion of at least two types of charged species has to be considered for charge compensation. Onsager s equations read in the isothermal case (neglecting energy fluxes) 

On the other hand, from a phenomenological point of view, the flux is given by the product of the concentration and the average velocity of the species. 

The average velocity is the result of the force F = —grad p. The ratio of the velocity over the force is defined as the general mobility b 

This may be separated into two terms, one involving an activity gradient (dilfusion) and the other an electrical field (migration) 

The expression d In a/d In c is known as the thermodynamic factor and is a special case of the Wagner factor (or thermodynamic enhancement factor) which plays an important role for the kinetic properties of electrodes. This term indicates the deviation from ideality of the mobile component. For ideal systems this quantity becomes 1 and comparison with Pick s first law yields 

---