Section 4. Measuring transport properties

Measuring the mean squared displacement of the particles.
We are going to follow these steps in order to estimate \(D\), \(η\), \(C_v\), and \(k\)
• Estimate the Mean Squared Displacement (\(MSD\)).
• Compute the self diffusivity coefficient (\(D\)).
• Calculate the viscosity (\(η\)) of liquid Argon using the value of (\(D\)).
• Calculate the thermal conductivity (\(k\)) from the value of \(η\) obtained.
• Estimate the heat capacity (\(C_v\)) by inducing a change in the system's temperature and computing the change in the system's energy.

4.1 Estimate the Mean Squared Displacement (\(MSD\)).
The Mean Square Displacement in the \(n^{th}\) dimension (\(MSD_n\)) is given by: $$MSD_n = \left.\left\langle\left(P_n\left(t\right)-P_{n,\ 0}\right)^2\right.\right\rangle,$$ where \(P_n\left(t\right)\) is the position of the particle at some time point from Stage 3, \(P_{n,\ 0}\) is the position of the particle at the beginning of Stage 3, and the brackets represent the average over the number of particles. To calculate the \(MSD_n\) in each dimension using Scymol, go into the Statistical section of the Analyze tab and select the Pos.x option (to the left) and the Mean square (MS) to the right. Click on the Check button to get the data points. Repeat the process for Pos.y and Pos.z. The values you obtain for the \(MSD_n\) should be close to (for a 1000 ps simulation): $$MSD_n \approx 400 \ Å^2$$ or $$MSD_n \approx 0.4 \ Å^2/ps$$ The \(MSD\) (in three dimensions) is given by: $$MSD =\sum_{n} MSD_n\ ,$$ The value you obtain for the \(MSD\) should be close to (for a 1000 ps simulation) $$MSD \approx 1200 \ Å^2$$ or $$MSD \approx 1.2 \ Å^2/ps$$

4.2 Estimate the self-diffusivity coefficient \(D\). $$D_n=\frac{1}{2}\lim_{t\rightarrow\infty}{\frac{d}{dt}\ MSD}\ ,\ $$ where \(t\) is the simulation time elapsed. The time used to measure the \(MSD\) is long enough to accurately measure \(D\), meaning that the previous expression is reduced to the following: $$D \approx \frac{MSD}{6t} ,\ $$ The self diffusivity coefficient obtained should be Self‐Diffusion in Liquid Argon. https://doi.org/10.1063/1.1742938 $$D\approx0.2 Å^2/ps,\ $$ The value of \(D\) obtained compares well with the literature values reported for the self-diffusivity coefficient (\(D_{theoretical}\)) of liquid Argon, where \(D_{theoretical}\approx \ 2.05 Å^2/ps \). To improve the accuracy of the value obtained using Scymol, try increasing the number of atoms and/or the simulation times used for both the Equilibration and Production stages.

4.3 Estimate the dynamic viscosity (\(η\)).
The dynamic viscosity of liquid argon can be approximated using Li and Chang’s Li and Chang’s modification to the Sutherland formula, that correlates the self-diffusivity constant (\(D\)) with dynamic viscosity \(η\) in homogenous liquids. https://doi.org/10.1063/1.1742022 modification to the Sutherland formula, that correlates the self-diffusivity constant (\(D\)) with dynamic viscosity \(η\) in homogenous liquids (like pure liquid Argon) organized in a perfect cubic cell. The approximation is given by: $$\eta=\frac{K_bT}{2\pi D}\left(\frac{N_{av}}{V}\right)^\frac{1}{3},$$ where \(k_b\) is the Boltzman constant, \(T\) is the system's temperature, \(D\) is the self-diffusivity coefficient, \(N_{av}\) is Avogadro's number, and \(V\) is the volume of the system. The computed value for \(η\) is $$η \approx 2.8 \times 10^{-4}$$ which compares well with the reported value of \(η_{theoretical}=2.7 \times 10^{-4}\).

4.4 Estimate the isochoric heat capacity (\(C_v\)).
The isochoric heat capacity (\(C_v\)) can be approximated by applying the fluctuation dissipation theorem to measure the instantaneous thermal and energy fluctuations that occur in the simulation of Stage 3 (i.e., a constant N, V, and E simulation) at equilibrium. Therefore, we do not need to run another simulation to induce a temperature change to estimate the heat capacity of our system. The fluctuation dissipation theorem to measure heat capacity in an NVE system at equilibrium is given by $$\left \langle K_e^2 \right \rangle - \left \langle K_e \right \rangle^2 = \frac{3Nk_b^2T^2}{2}\left(1-\frac{3Nk_b}{2C_v}\right) ,$$ where \(K_e\) is the kinetic energy, \(N\) is the number of particles, \(k_b\) is the Boltzmann constant, and \(T\) is the average kinetic temperature. Go into the Analyze tab, enter a range that includes only steps from Stage 3, and select the NVE - Isochoric Heat Capacity (in the Physical Quantities option). The computed \(C_v\) should be close to: $$C_v \approx 1.6 E/K/atom$$ If you need more accurate values for \(C_v\), you would need to either increase the number of particles, make sure the system is properly equilibrated at 85 K, or run a longer simulation.

4.5 Estimate the thermal conductivity (\(k\)).
The thermal conductivity \(k\) is estimated using the Andrade's Andrade's relationship between the \(η\) and \(k\) in monoatomic saturated liquids. https://doi.org/10.1038/170794b0 relationship between the \(η\) and \(k\) for monoatomic saturated liquids: $$k=\beta\times\eta\times C_v,$$ where k is the thermal conductivity, \(\beta\) is an input parameter that depends on the nature of the system, \(η\) is the dynamic viscosity, and \(C_v\) is the isochoric heat capacity computed above.

The computed value should be close to: $$k \approx 1.32 E/K/atom $$ which compares well to the reported values of thermal conductivity of liquid Argon.