User Guide#
The Ionic Conductivity step calculates the conductivity of an electrolyte from equilibrium molecular dynamics, two ways:
Einstein-Helfand, from the collective displacement of the ions’ charges,
\[\sigma = \lim_{t \to \infty} \frac{\langle \lVert \mathbf{P}(t) - \mathbf{P}(0) \rVert^2 \rangle}{6 V k_B t}, \qquad \mathbf{P} = \sum_{\text{ions}} q_i \mathbf{r}_i\]which includes the correlations between the ions (ion pairing lowers it);
Nernst-Einstein, from the self-diffusion coefficient Dₛ of each ion, σ_NE = e² / (V k_B T) Σₛ Nₛ zₛ² Dₛ, which treats the ions as independent;
and their ratio σ/σ_NE, the ionicity.
The ions are the molecules with a non-zero integer charge, monatomic like Na⁺ or molecular like BF₄⁻; each ion’s self-diffusion is that of its centre of mass. The charges are the forcefield’s: a machine-learned forcefield has no fixed charges, so the step stops with a message for such MD.
Where the MD comes from#
As for the other analysis steps: the step runs its own MD a number of times, or
analyzes the MD earlier in the flowchart or the md.h5md files of earlier runs. The
MD must sample the atoms’ positions at least every picosecond (“Longest interval
between frames”); they are unwrapped, so the displacements are continuous.
Fits and errors#
The mean square displacements are fitted to straight lines between “Fit from” and “Fit
to”, fractions of their length (by default 10 % to 50 %): the early part is not yet
diffusive, and the late part has few time origins. The collective displacement is a
single, noisy quantity per run – this is why conductivities from MD need long runs,
or several. The error is the standard error over the runs, or over blocks of a single
run. The graph ionic_conductivity.graph shows the collective MSD with its fit and
the MSD of each ion.
Finite size. The self-diffusion coefficients, and so σ_NE, depend on the size of the cell. With “Yeh-Hummer correction” the step corrects them with the viscosity from a Viscosity step earlier in the flowchart, as the Diffusivity step does, and says so; without one, the correction is skipped with a note.
Results#
σ (S/m) and its standard error, σ_NE, the ionicity (which can be stored as properties of the configuration), the self-diffusion coefficient of each ion (and corrected), the mean temperature and volume, and the numbers of runs and frames.