Description of defect transport processes in MIECs and EIS modeling

elevated temperatures under uniaxial mechanical tensile loads. By fitting the model predictions with experimental data, the physical properties of polycrystalline SDC are extracted. The effects of mechanical stress on the conductivities of grains and grain boundaries are elucidated.

6.2 DESCRIPTION OF DEFECT TRANSPORT PROCESSES IN MIECS AND EIS MODELING

A mixed ionic and electronic conductor (MIEC) solid solution is capable of conducting multiple charged species. There are two fundamental equations governing such transport processes. One is the continuity of charged species, driven by the electrochemical potential. Another one is the Poisson’s equation relating the electrical potential field with the charges in the system. It is worth noting that the MIECs are usually polycrystalline in practical devices. Accordingly the Poisson’s equation within grain boundaries is modified using the grain boundary core-space-charge layer model. In the following sections, the governing equations are described in details.

6.2.1 Charge Transport in MIEC solid solution

The electrochemical potential of a defect species in an ideal solid solution is represented by [77]:

(6-1)

Where is the electrochemical potential of species j; R the gas constant; T the temperature; xj the molar fraction of species j; zj the effective charge of species j; F the

The driving force for charge transport in a solid solution is the gradient of electrochemical potential. According to non-equilibrium thermodynamics, the current density of a charged species in a solid solution driven by an electrochemical potential can be expressed as,

(6-2)

Where is the diffusion coefficient of the species j; is the mobility

of the species j; cj is the concentration of diffusion component, e.g., oxygen vacancy,

electron or hole.

Substitution of Equation (6-1) into (6-2) gives,

( ) ( ) (6-3)

Clearly, the diffusion of mobile defects in a solid solution is driven by the gradients of mobile species concentration and the electrical potential field.

Substituting the Equation (6-3) into the equation of species conservation,

, we have,

( ( )

( )

) (6-4)

Equation (6-4) is used to describe the transport process of charged species in a solid solution, e.g., oxygen vacancy and electron or hole. When the steady-state condition is considered, the first term at the left side of Equation (6-4) disappears.

6.2.2 Poisson’s equation

The overall charge density distribution is closely related to the electrical potential distribution within a grain domain and is described by the Poisson’s equation [181]

,

where is the vacuum permittivity and is the dielectric constant or relative permittivity of the material.

As mentioned above, the doped ceria electrolyte is generally polycrystalline in practical SOFCs, consisted of grains and grain boundaries. The Poisson’s equation can mathematically describe the relations between the electrical potential distribution and charge density distribution in a grain. However, it cannot be directly used for grain boundaries due to the presence of space charge layer.

A grain boundary is the crystallographic mismatch zone between two grains. The defect structure of grain boundaries is different from that of grains. Taking the Sm0.15Ce0.85O1.925-δ (SDC) material as an example, to achieve the thermodynamic

equilibrium, Sm segregation takes place at the SDC grain boundaries. The segregation of Sm at the grain boundary leads to the depletion of oxygen vacancies and the accumulation of electrons, forming a space-charge layer. The space-charge theory is then suggested based on the experimental observations[182]. As shown in Figure 6.1, the grain boundary consists of a grain boundary core and two adjacent space charge layers. It is generally realized that compared to the concentrations of dopants, the concentrations of

𝑥

Potential/log concentration Grain boundary core

𝜙

[𝑒 ]

[𝑉_{𝑂 ]}

[𝑆𝑚_𝐶𝑒]

Grain interior Space charge

Figure 6.1 Schematic potential and concentration profiles in grain boundary

ions and electrons are negligible in the space charge region, i.e., [183]. Therefore the Poisson’s equation for grain boundary becomes,

( ) (6-6)

6.2.3 Perturbation analysis for impedance calculation

In experiment, impedance is obtained by linear perturbation method. A periodic current perturbation with a small magnitude is applied to a steady state system. The corresponding voltage perturbation is measured. By varying the frequency of current perturbations from very low to extremely high, a series of voltage perturbations are obtained. The impedance then can be calculated using the ratio between voltage perturbations and corresponding current perturbations.

When a current perturbation is applied to an equilibrium system, all parameter variables involved in the system will have perturbations, such as electrical potential, species concentration, and charged species flux. Mathematically, we have

{

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )

where ( ), ( ), and ( ) represent electrical potential, species concentration, and charge flux in steady state respectively while ( ), ( ), and ( ) represent the corresponding perturbations.

Substituting (6-7) into Equations (6-4) and (6-5), subtracting the corresponding steady state equations while neglecting the second-order terms, yield,

( )

( ( )) (6-8)

Here ( ) ( ) ( ( )) ( ) ( ) ( ) ( ) ( )

( ( )).

Since Equation (6-8) holds for each kind of charge species, summing up all kinds of species, we have,

(∑ ( )) (∑ ( )) (6-10)

Combining Equations (6-10) and (6-9), we obtain,

(∑ ( )) ( (

( )

))=0 (6-11)

Equation (6-11) indicates that the total charge flux ( ) is composed of every charge flux and a displacement flux ( ) induced by time-varying electrical field, i.e.,

( ) ∑ ( ) ( ), with ( ( ) ). (6-12)

To simplify the problem into a manageable form, we consider the case where the majority of mobile defects are vacancies and electronic species only. The Laplace transform of Equations (6-8) and (6-9) yields,

( ) ( ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) ( ( ))) (6-13) ( ) ( ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) ( ( ))) (6-14) ( ) ( ) ( ) ( ) (6-15)

In the grain boundary, ( ) ( ) terms on the right of Equation

to notify that there are 6 unknowns: the profiles of ( ), ( ) and ( ) under steady state, as well as their perturbations, ( ), ( ) and ( ).

The impedance spectroscopy then can be obtained using the perturbations of electrical potential and the total charge flux [184]:

( ) ( ) ( ) _{( )} (6-16) { ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) ( ( )) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) ( ( )) ( )