Applications of a general random-walk theory for confined diffusion

A general random walk theory for diffusion in the presence of nanoscale confinement is developed and applied. The random-walk theory contains two parameters describing confinement: a cage size and a cage-to-cage hopping probability. The theory captures the correct nonlinear dependence of the mean square displacement (MSD) on observation time for intermediate times. Because of its simplicity, the theory also requires modest computational requirements and is thus able to simulate systems with very low diffusivities for sufficiently long time to reach the infinite-time-limit regime where the Einstein relation can be used to extract the self-diffusivity. The theory is applied to three practical cases in which the degree of order in confinement varies. The three systems include diffusion of (i) polyatomic molecules in metal organic frameworks, (ii) water in proton exchange membranes, and (iii) liquid and glassy iron. For all three cases, the comparison between theory and the results of molecular dynamics (MD) simulations indicates that the theory can describe the observed diffusion behavior with a small fraction of the computational expense. The confined-random-walk theory fit to the MSDs of very short MD simulations is capable of accurately reproducing the MSDs of much longer MD simulations. Furthermore, the values of the parameter for cage size correspond to the physical dimensions of the systems and the cage-to-cage hopping probability corresponds to the activation barrier for diffusion, indicating that the two parameters in the theory are not simply fitted values but correspond to real properties of the physical system.

Figure 1

Schematic of unsuccessful and successful cage-to-cage moves. On the left, an unsuccessful move is reflected back into the original cage, maintaining the length of the trajectory. On the right, a successful move puts the particle in a new cage with center located a distance $R_{\mathrm{cage}}$ along the vector of the trajectory.

Figure 2

Structures of RDX (on the left) and IRMOF-1 (on the right). Color legend: blue, N; red, O; gray, C; white, H; maroon, Zn.

Figure 3

Cross section from a snapshot of a molecular dynamics simulation of Nafion (EW$=$1144) at 300 K and a water content of λ$=$6. In this snapshot, all atoms except the sulfur of Nafion have been rendered invisible. Color legend: white, H; red, O of water; green, O of hydronium ion; orange, S.

Figure 4

Impact of cage-to-cage hopping probability $p_{\mathrm{cage}}$. Plot of the mean square displacement as a function of observation time on linear (a) and logarithmic (b) scales for a dimensionless system in which $\langle \Delta r\rangle =2.87\times {10}^{-3},\hspace{0.5em}\Delta t=1,\hspace{0.5em}{R}_{\mathrm{cage}}=1$, and $p_{\mathrm{cage}}$ is varied from 0 to 1.

Figure 5

Impact of cage radius $R_{\mathrm{cage}}$. Plot of the mean square displacement as a function of observation time on linear (a) and logarithmic (b) scales for a dimensionless system in which $\langle \Delta r\rangle =2.87\times {10}^{-3},\hspace{0.5em}\Delta t=1,\hspace{0.5em}{p}_{\mathrm{cage}}=0.001$, and $R_{\mathrm{cage}}$ is varied from 0 to 10.

Figure 6

Comparison of the mean square displacement from MD (symbols) and CRW simulation (lines) for diffusion of RDX in IRMOF-1 at different temperatures. From the bottom to the top: 300, 350, 400, 450, 500, 550, and 600 K.

Figure 7

Arrhenius plot showing the diffusivities (obtained from the CRW model) for the RDX–IRMOF-1 system as a function of temperature.

Figure 8

Comparison of the mean square displacement from MD (symbols) and CRW simulation (lines) for diffusion of water in Nafion 1144 with different levels of hydration. From the bottom to the top λ$=$3, 6, 9, 15, and 22 ${\mathrm{H}}_2\mathrm{O}/{\mathrm{SO}}_3\mathrm{H}$.

Figure 9

Comparison of the mean square displacement from MD (symbols) and CRW simulation (lines) for diffusion of glassy iron at 300 K.