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)=1NN∑i=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=13∫t=∞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)=<(∑i→vi+(t)−∑j→vj−(t))>×<(∑i→vi+(0)−∑j→vj−(0))>
where →vi+ and →vj− are the velocity vectors for the cations and anions in the system, respectively
Conductivity is then calculated as
σ=e23VkBT∫∞0J(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>=1NN∑i=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++N−z−D−)
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.