Fundamental thermodynamic considerations

In document Sacrificial interlayers for all-solid-state batteries (Page 46-50)

ii) The formation of lithium alloys

2.4 Thermodynamic and kinetic aspects of batteries

2.4.1 Fundamental thermodynamic considerations

As shown in Appendix B, a battery consists of an anode, a cathode, and an electrolyte that separates each electrode from the other. For the application as an anode several different materials are considered: metal electrodes, intercalation electrodes or conversion electrodes.

As the aim of this work is to discuss various concepts for the realization of Li–metal electrodes, only this case is considered. The thermodynamic aspects are not limited to the Li anode and can be transferred to other electrode materials in a similar fashion. Basic differences are pointed out briefly in the following paragraphs.

Mainly metal oxides are used as cathode materials, in which lithium can be inserted. In this work they are only discussed as far as necessary to explain the fundamentals of battery cycling.

The basic considerations on stability criteria as outlined in the following are not limited to the anode side and can be transferred to the cathode side and must be applied in a similar way if protection layers for the cathode side are required.

For reasons of simplification, only a linear one–dimensional model is described though it is also applicable for a two–dimensional interface. To derive the basic thermodynamic quantities, the case of a stable electrolyte under open circuit conditions is discussed first.

Lithium ions are the mobile species in a lithium–ion battery. In each component of the battery an electrochemical potential of the lithium ions µ̃Li+ can be defined. In the thermodynamic equilibrium the electrochemical potential of the lithium ions in all components must be equal or a particle flux and current flux will appear:

(µ̃Li+)Anode = (µ̃Li+)Electrolyte = (µ̃Li+)Cathode. (4) The electrochemical potential of the lithium ions consists of the chemical potential µLi+ and the electrical (Galvani)potential 𝜑:

µ̃Li+ = µLi++ z ∙ e 0 ∙ φ, (5) where e 0 is the electric charge and z the charge number. In the case of electrons and lithium ions, z is ± 1 and thus omitted in the following discussion.

The chemical potential depends on the activities ai of the involved species, namely the activity of the lithium ions in the electrodes and the electrolyte:


µLi+ = µLi0+ + RT lnaLi+, (6) with µLi0+ = standard chemical potential of Li+.

R = universal gas constant (8.31447 J/K mol—1) T = absolute temperature [K]

The chemical potential is inaccessible so the values shown in Figure 5 are only relative values.

As the activity of lithium ions in lithium metal is unity, lithium ions have the highest chemical potential in lithium metal.

The electric potential describes the potential energy of a species with charge q in an electric field E. The inner electrical potential (or Galvani potential) consists of two terms: the surface potential 𝜒 and the outer potential or Volta potential ψ. The surface potential depends on the crystal orientation, as different lattice planes have different surface charges depending on the kind of ion that exists at the surface and the number of mirror charges that are formed when an ion is brought in contact with the surface.

The absolute Galvani potential of a charge carrier is not accessible experimentally, as only differences of the Galvani potential can be measured. The potential needs to be lower on the anode side than on the cathode side, as no spontaneous charge transfer would occur otherwise.

The open–circuit voltage or electromotive force (EMF) is a measure for the maximum non–

volumetric work done by the cell. It is derived from the difference in the chemical potentials of the neutral lithium compound in both electrodes when no current is flowing.

EMF = −Δz F rG (7)



p,T,nj≠nLi= µLi (8)

The chemical potential µLi is the measure of the change of the free enthalpy 𝐺 of a homogeneous phase that is required or released for adding or by removing a lithium atom to or from an electrode when pressure p, temperature T and the amount of all other substances nj remain constant. During the charge/discharge process in a battery, the free Enthalpy changes by the value of


ΔµLi = (µLi) Cathode – (µLi) Anode. (9) Thus, the potential difference can be calculated from the EMF. For the chemical potential absolute values cannot be given. To enable calculations, a virtual zero line can be added to the energy scale, which is often set to the theoretical Galvani potential of the anode, i. e. the redox couple Li+/Li.

As a lithium atom can be described as the sum of the positively charged lithium ion and the negatively charged electron, the chemical potential of the neutral lithium µLi is composed of the electrochemical potential of the lithium ions µ̃Li+ and electrons µ̃e. By knowing µLi and µ̃Li+, the potential of the electrons can be calculated:

µLi = µ̃Li+ + µ̃e, (10) whereas the electrochemical potential of the electrons is equal to the Fermi level of the electrons in the electrodes.

The calculation of the electrochemical potential of the electrons is analogous to equation 5, except for the prefix:

µ̃e = µe — z ∙ e 0 ∙ φ. (11) In thermodynamic equilibrium equation 4 is valid. In that case, inserting equation 10 in equation 9 shows that the difference of the chemical potential of lithium in both electrodes is determined solely by the difference of the electrochemical potential of the electrons in both electrodes and results in the open circuit battery voltage U.

ΔµLi = (µ̃e)Cathode − (µ̃e)Anode = U (12)

As the discussion of the cathode side is excluded in this work, it is considered that there are local equilibria and that the potentials in this phase are constant. In the case of the lithium anode, ideal behavior is assumed. Contacting anode, electrolyte and cathode with one another generates a voltage, resulting from the different lithium potentials in the electrodes.

Between the electrodes the potential needs to decline from µLi, Anode to µLi, Cathode and a change of the Galvani potential must occur at the interface. Simultaneously, a change of the chemical potential of lithium across the electrolyte needs to develop. A hint what the slope of the chemical potential of lithium may look like has been given by Nakamura et al. [94].


They developed a model to calculate potential profiles by using the transport properties of the electrolyte and an adjacent layer, e.g. a second electrolyte or a protective layer. The hypothesis of their work was evaluated using oxygen ion conductors, as one requirement for the estimations is the knowledge of the activity dependence of the transport properties. These quantities are known for oxygen ion conductors but currently unknown for typical battery materials. Nakamura et al. showed that the potential drop has a sigmoidal shape. Its width depends on the electronic partial conductivity of the two phases. A smaller electronic Figure 5: Schematic depiction of the potentials in a lithium ion battery. Between anode and cathode, a decline of the lithium potential µLi needs to occur. The electrochemical potential µ̃ is the sum of the chemical potential µ and the Galvani potential z ∙ e 0 ∙ φ. As the electrochemical potential of the ions is constant in all phases (otherwise, a particle flux must occur), the difference of the chemical potential is due to different Galvani potentials in the three phases. The virtual zero line is depicted as dashed line and fixed to the Galvani potential of the anode. Thus, on the anode side the chemical and the electrochemical potential of the electrons are identical and the chemical and electrochemical potential of the Li ions are identical, too. The decay of µLi across the electrolyte is solely determined by the change of the electrochemical potential of the electrons µ̃e. As in an ideal electrolyte µ̃e cannot be formulated, the virtual change of the two values is depicted as dashed lines.


conductivity leads to a stronger potential drop across a phase and the thickness of the phase influences the width of the potential drop. A thin phase causes a narrow drop; a thick phase causes a wide drop.

This potential decline is difficult to determine, and it is indicated as a dashed line. For simplification, a linear drop of the potential in Figure 5 is assigned, but since the potential is a function of µ̃e(the Fermi level) in the electrolyte, which is not constant, a non–linear decay can be expected. As the potential of the lithium ions µ̃Li+ is the same in all battery components, the entire potential drop across the electrolyte must be caused by the change of the electrochemical potential of the electrons (equation 10). This behavior is only valid, if the electrolyte has an (although possibly negligible) electronic partial conductivity. Only in this case can an electrochemical potential of the electrons be defined. As this is not the case in an ideal solid electrolyte, this curve is only indicated as a green dotted line.

Details about the steps of the Galvani potential at the interfaces are unknown, as they cannot be measured. The steps in Figure 5 are only schematic sketches. Their values as well as directions can be different. As at the interface the counter ions of the electrolyte are in contact with the electrode (due to surface charges), a linear potential decay analogous to the Helmholtz layer in aqueous systems should occur. A logarithmic decay is expected only in diffuse interlayers. The width of the steps should be in the range of the Debye length.

For all these considerations it is assumed that no space charge layers exist in the system. In real system they generally cannot be excluded generally [95].

In document Sacrificial interlayers for all-solid-state batteries (Page 46-50)