Calculating Conductivity from Molecular Dynamics Simulation

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Methods to calculate the conductivity of a species from molecular dynamics simulation

Green-Kubo relations

The velocity autocorrelation function, CV(t), is

CV(t)=1NNi=1(vi(0)vi(t))=<vi(0)vi(t)>

where N is the total number of particles (atoms/molecules in the selection) and vi is a vector storing the three components of the velocity (vx, vy, and vz) for the i-th particle.

vi(0)=vi(t=t0) and vi(t)=vi(t=t0+nΔt), where n is the timestep and Δt is the timestep size

Given this function decays to zero at long time, the diffusion constant D may be found from the integral of CV(T) as

D=13t=t=0<vi(0)vi(t)>dt

Electrical conductivity, σ, is calculated using the normalized autocorrelation function of the total current J(t) as

J(t)=<(ivi+(t)jvj(t))>×<(ivi+(0)jvj(0))>

where vi+ and vj are the velocity vectors for the cations and anions in the system, respectively

Conductivity is then calculated as

σ=e23VkBT0J(t)dt

Nernst-Einstein

Compute the mean-square displacement (MSD) of atoms or molecules from a set of initial positions

MSD≡<(x(t)x0)2>=1NNi=1|x(i)(t)x(i)(0)|2

where N is the total number of particles (atoms/molecules) in the selection, vectors xi(t) and x(i)(0) are the position of the i-th particle at time t and the reference position of the i-th particle.

The command will calculate a diffusion constant, D, from MSD according to the Einstein relation

MSD=2Dt

where t is the simulation time used to calculate the MSD

Calculating D for cations and anions in the system (D+ and D, respectively), the ionic conductivity can be calculated by the Nernst-Einstein equation

σ=e2VkBT(N+z+D++NzD)

where e and kB are the unit charge and Boltzmann constant; V, T, N±, and z± are the volume, temperature, number of each ionic species, and charge of the ionic species in the system.

References

The Velocity Autocorrelation Function

Statistical mechanics of dense ionized matter. IV. Density and charge fluctuations in a simple molten salt