E6 Ionic Transport

Topic

The point defect established in another model within this set—the one describing the crystal's ionic point defects—is what enables, in the current model, the movement of an ion through the crystal lattice: adjacent to a vacancy or an interstitial site, a neighboring ion finds an available spot to jump into, and this jump transforms the ion into a mobile one. Because the point defect facilitates this jump, the ion remains trapped at its own lattice site—unable to move—in the absence of such defects.

The mobility of this mobile ion—the ease with which it moves under an applied electric field—is directly related, via the Nernst-Einstein equation, to its diffusion coefficient—the ease with which the same ion disperses due to thermal agitation, without the need for an external field. Fundamentally, both quantities describe the same frequency of ion jumps between lattice sites, with one process driven by the electric field and the other by thermal agitation. Consequently, higher ionic mobility results in a proportionally higher diffusion coefficient, while lower mobility results in a proportionally lower diffusion coefficient.

This same diffusion coefficient governs the spatial and temporal evolution of the ionic concentration within the crystal, in accordance with Fick's second law; this law describes how an ionic concentration gradient disperses—or relaxes—over time. A higher diffusion coefficient causes the ionic concentration to disperse more rapidly through the crystal, whereas a lower coefficient leads to slower dispersion. To describe this evolution, the model employs Fick's second law—the same law used for the same purpose in the equivalent model of diffusion in metals. The movement of all these mobile ions through the crystal lattice is, collectively, what gives rise to the ceramic's ionic conductivity; this conductivity, in turn, depends on temperature according to an Arrhenius-type relationship: higher temperatures cause ionic conductivity to increase rapidly, as more mobile ions possess the thermal energy required to hop between lattice sites, whereas lower temperatures cause that conductivity to drop—also rapidly—due to the lack of such thermal energy.

Thus, in this model—defined by the point defect and mobile ion introduced at the outset—the ionic conductivity resulting from movement through the crystal lattice is precisely the quantity experimentally measured by another model in this set: the one describing electrical characterization via impedance spectroscopy. Ultimately, what that model derives from the electrical circuit representing the ceramic is the same ionic transport that this model explains in terms of the point defect, ionic mobility, and temperature dependence.

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