Electrical conductivity and ionic conductivity

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Chapter II. Characterization of fluoride solid electrolytes

2. Electrical conductivity and ionic conductivity

The electrical conductivity 𝜎 (S.cm-1) quantifies how strongly a given material allows the flow of electric current. It is characterized by the ratio between the current density 𝑗 (A.cm-2) and the applied electric field 𝑬 (V.cm-1), in accordance with Ohm’s law.

𝜎 = 𝑗

𝑬 37.

The conductivity 𝜎 is the inverse of the resistivity 𝜌 (Ξ©.cm) and can be determined by measuring the resistance 𝑅 (Ξ©) that the material opposes to the electric current, providing that its length 𝐿 (cm) and cross sectional area 𝐴 (cmΒ²) are uniform and measurable.

𝜎 =𝑅1𝐴𝐿 =𝜌1 38.

Moreover, 𝜎 is the sum of the electronic πœŽπ‘’ and ionic πœŽπ‘–conductivity contributions.

𝜎 = πœŽπ‘’ + πœŽπ‘– 39.

The ionic (or electronic) transference number 𝑑𝑖 (or 𝑑𝑒) characterizes the contribution of the ionic (or electronic) conductivity to the total conductivity.

𝑑𝑖 ≑ πœŽπœŽπ‘–

π‘‘π‘œπ‘‘ 40.

𝑑𝑒 ≑ πœŽπœŽπ‘’

π‘‘π‘œπ‘‘ 41.

From a practical point of view, the resistance of electrolyte can be determined experimentally by performing electrochemical impedance spectroscopy (EIS) measurements. An introduction to EIS is available in appendix 2.

Now that the fundamental notions have been introduced, we can take a look at what lies behind the conductivity of ionic solids.

2.2. Conductivity in ionic solids 2.2.1.Generalities

All ionic solids conduct electricity due to unavoidable defects present in their crystalline

Characterization of fluoride solid electrolytes

ionic solids exhibit poor conductivity and are generally considered as insulators in standard conditions.

Only a few solids can offer conductivities that are sometimes high enough to reach the values of liquid electrolytes. These ionic solids exhibit specific structural features, allowing for high conductivity (10-1 - 10-5 S.cm-1)[35] well below their melting point. They are sometimes referred as fast ion conductors or superionic conductors, but are more commonly named solid electrolytes.

Only a few solid fluorides exhibit significant electrical conductivity. The electrical conductivity of fluorides usually rises from their ionic conductivity, since that fluorides are generally electronic insulators due to their large band gap, as predicted by the band theory.

However, they can still exhibit significant residual electronic conductivity by motion of electrons and/or holes[36], [37]. This is of major importance as in electrochemical systems, it is crucial to determine both contributions to the total conductivity. Indeed, to avoid spontaneous discharge of the FIB and to avoid faradaic loss, the electronic conductivity of the electrolyte must be several orders of magnitude lower than the ionic conductivity. For the practical use of a solid-electrolyte in an electrochemical cell, an ionic transference number 𝑑𝑖 as close as possible to 1 is desired. If 𝑑𝑖 becomes significantly lower than 1, electronic leakage will take place in the cell and the faradaic yield will be accordingly degraded.

2.2.2. Mechanism of ionic conductivity

The mechanism relies on the hopping of the ionic species (the fluoride ion in our case) from site to site, along channels that are present in the rigid crystal framework of the material.

Increasing hopping rate, and multiplicity of diffusion channels, result in improved conductivity of the electrolyte. In a phenomenological approach, πœŽπ‘– can be expressed as the product of concentration 𝑐𝑖, charge π‘ž, and mobility 𝑒𝑖 of its ionic species.

πœŽπ‘– = π‘π‘–π‘žπ‘’π‘– 42.

Two conduction mechanism for the mobility of ions can be distinguished. They both rely on point defects (defects at, or around, a single lattice point, involving only a few atoms) that are inherently present in the crystal sub lattice: vacancies and interstitials.

In a vacancy type conduction, the ion hops from its site to another vacant site with equivalent energy. In an interstitial type conduction, the ion hops from its site to an interstitial position

Characterization of fluoride solid electrolytes

where no ions should be present. The interstitial position generally has a higher energy value.

The presence of conduction pathways (e.g. diffusion channels) where these mechanisms are energetically favored results in an increased mobility of the ionic species and thus a better conductivity.

The conductivity of a material is temperature dependent. In an ionic solid, for an ion to hop, it must overcome the energetic barrier that separates it from the next site. The displacement of ions along conduction channels can be seen as a series of potential wells and barriers as outlined by Figure II.1. The drawing illustrates how an ion can hop from one site to another when it has enough energy to overcome the barrier that separates it from the next lattice point.

Figure II.1. Illustration of the energy profile for an ionic species migrating in a solid electrolyte. The interstitial site has a higher potential energy than the vacancy sites.

This energetic barrier, called activation energy, πΈπ‘Ž, is the major factor controlling the mobility 𝑒𝑖. The activation energy can somewhat be seen as the equivalent of the band gap 𝐸𝑔 in the band theory. It is linked to the conductivity and temperature by an Arrhenius type law:

πœŽπ‘– =𝐴𝑇

𝑇 𝑒π‘₯𝑝 (βˆ’πΈπ‘Ž

π‘˜π΅π‘‡) 43.

Where 𝐴𝑇 is called the pre-exponential factor, or prefactor, which contains many terms like the number of mobile ions[38], T is the absolute temperature (K), and π‘˜π΅ is the Boltzmann constant (π‘˜π΅ β‰ˆ 8.6173324. 105 𝑒𝑉. πΎβˆ’1) [39].

To continue the discussion and illustrate the Arrhenius-type law of conductivity, let us take LaF as example. Figure II.2 represents (a) the ionic conductivity of a LaF single crystal in a

Characterization of fluoride solid electrolytes

respective representations of LaF3 crystal structure. Arrhenius plots are commonly presented by plotting the log of the conductivity in function of the inverse temperature in order to visualize and compare πΈπ‘Ž, which is proportional to the slope of the curve. A larger πΈπ‘Ž will lead to a greater increase of the conductivity with increasing temperature (steeper slope). The ionic conductivity of the LaF3 single crystal was measured perpendicular and parallel to its c axis.

We can observe that the increase of σ𝑖 with increasing temperature is greater when the conductivity of LaF3 is measured perpendicular to c axis. Consequently, πΈπ‘Ž is higher for the direction perpendicular to c axis.

2,2 2,4 2,6 2,8 3,0 3,2 3,4

10-5 10-4 10-3 10-2 10-1 100 101

LaF

3

- Perpendicular to c axis LaF

3

- Parallel to c axis

T (S.K.cm-1 )

1000/T (K-1) Anisotropic

(a)

Isotropic

180 160 140 120 Temperature (Β°C)100 80 60 40

Figure II.2. (a) Arrhenius plot of the bulk ionic conductivity of a LaF3 single crystal perpendicular and parallel to its c axis. Experimental data collected by Schoonman et al.[40]

(b) Crystal structure of LaF3 perpendicular (left) and parallel (right) to c axis. The blue balls

3+

-Characterization of fluoride solid electrolytes

This observation illustrates the anisotropic nature of σ𝑖 and the presence of preferable conduction pathways due to preferable motion of fluoride ions in certain directions of the crystal structure. The Arrhenius plot shows that at RT, F- move faster along c axis due to preferable hopping energies in this direction, as illustrated by Figure II.2.b, where it is possible to see some F- ions perfectly aligned along the c axis. As temperature increases, conduction pathways perpendicular to c get thermally activated faster than the ones parallel to c. Between 140 and 180 Β°C, conductivity becomes isotropic βŠ₯ and βˆ₯ to c.

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