Ionic Conductivity

Ionic conductivity describes how charge is conducted by the carriers in a system. In the case of molecular dynamics simulations on an atomistic level, these carriers consist of the atoms in the system with a charge assigned to them by some level of theory.

Einstein Method

The Einstein approach to calculating the ionic conductivity utilizes the mean square displacement of the translational dipole moment written as

\[\sigma_{E} = \frac{d}{dt}\frac{1}{6 k_{B} T V} \Bigg \langle \Bigg ( \mathbf{M}(t) - \mathbf{M}(0) \Bigg )^{2} \Bigg \rangle\]

where \(\mathbf{M}(t)\) is the translational dipole moment of the system, calculated by

\[\mathbf{M}(t) = q \sum_{i} z_{i}\mathbf{r}_{i}(t)\]

Green-Kubo Method

In keeping with the typical relationship between the Green-Kubo (GK) and the Einstein methods, the GK approach takes the autocorrelation with the respect to the ionic current in the system \(\mathbf{J}\) defined by

\[\mathbf{J}(t) = q \sum_{i} z_{i}\mathbf{v}_{i}(t).\]

The full equation for the conductivity is

\[\sigma_{GK} = \frac{V}{k_{B} T} \int_{0}^{\infty} dt \langle \mathbf{J}(t) \cdot \mathbf{J}(0) \rangle\]

Diffusion Summation

As an alternative to using a direct calculation, a summation of diffusion coefficients may be used to calculate the ionic conductivity as

\[\sigma_{DS} = \rho q^{2} \frac{1}{k_{B} T} \Bigg ( x_{\alpha} z^{2}_{\alpha} D_{\alpha} + x_{\beta} z^{2}_{\beta} D_{\beta} + x^{2}_{\alpha} z^{2}_{\alpha} D_{\alpha \alpha} + x^{2}_{\beta} z_{\beta}^{2} D_{\beta \beta} + 2 x_{\alpha} x_{\beta} z_{\alpha} z_{\beta} D_{\alpha \beta} \Bigg )\]

Nernst-Einstein Equation

In the case of the above equation, we have included diffusion terms associated with correlated ion motion. In the case of a system wherein distinct atoms do not interact, the above equation reduces to what is knowns as the Nernst-Einstein equation of ionic conductivity as

\[\sigma_{DS} = \rho q^{2} \frac{1}{k_{B} T} \Bigg ( x_{\alpha} z^{2}_{\alpha} D_{\alpha} + x_{\beta} z^{2}_{\beta} D_{\beta} \Bigg )\]

The Nernst-Einstein equation can be used to determine how the ions within a system interact with each other, and how this interaction impacts the dynamics of the material.