Dependence on the thermodynamic state of self-diffusion of pseudo-hard-sphere and Lennard-Jones potentials

Self-diffusion $D$ in a system of particles that interact with a pseudo-hard-sphere or a Lennard-Jones potential is analyzed. Coupling with a solvent is represented by a Langevin thermostat, characterized by the damping time $t_d$. The hypotheses that $D=D_0\phi$ is proposed, where $D_0$ is the small concentration diffusivity and $\phi$ is a thermodynamic function that represents the effects of interactions as concentration is increased. Molecular dynamics simulations show that different values of the noise intensity modify $D_0$, but do not have an effect on $\phi$. This result is consistent with the assumption that $\phi$ is a thermodynamic function since the thermodynamic state is not altered by the presence of damping and noise.

Figure 1

Comparison between Lennard-Jones (dashed), hard-core (dash-dotted), and WCA (${\lambda }_r=50,{\lambda }_a=49$) (solid) potentials.

Figure 2

Compressibility factor $Z$ against packing fraction $\eta$ for pseudo-hard-spheres ($T^{*}=1.5$). The symbols represent different values of the damping time $t_d^{*}$: 0.01, 0.1, 1, and 10 (dots for each $t_d^{*}$ overlap almost exactly). The curve corresponds to the Carnahan and Starling EOS, given by Eq. (8).

Figure 3

Reduced diffusivity at small concentration for pseudo-hard-spheres, $D_0^{*}$, against damping time $t_d^{*}$ for $\eta =0.0052$. Crosses are numerical results and the curve corresponds to Eq. (23). Parameters of the simulations: Number of samples = 10, number of time steps (after thermalization) = 100, time step = 0.001, and number of particles per sample = $500000$.

Figure 4

Diffusivity for pseudo-hard-spheres, $D^{*}$, against packing fraction $\eta$ for different values of the damping time $t_d^{*}$: red for 0.1, blue for 1, and green for 10. See Fig. 5 for the details of the simulation.

Figure 5

$D/D_0$ against packing fraction $\eta$ for different values of the damping time $t_d^{*}$: red for 0.1, blue for 1, yellow for 5, and green for 10. The superimposed curve corresponds to the numerical results of Pieprzyk et al. [17] for hard spheres. Parameters of the simulations: Number of samples = 10, number of time steps (after thermalization) = 100, time step = 0.001, and number of particles per sample = $500000$. For $\eta$ greater than 0.35, the time step was changed to 0.0001.

Figure 6

Reduced diffusivity at small concentration for the Lennard-Jones potential, $D_0^{*}$, against damping time $t_d^{*}$ for $\eta =0.0052$ and temperature $T^{*}=4$. Crosses are numerical results and the curve corresponds to Eq. (29). Parameters of the simulations: Number of samples = 100, number of time steps (after thermalization) = 200, time step = 0.001, and number of particles per sample = $108000$.

Figure 7

$D/D_0$ against packing fraction $\eta$ for different values of the damping time $t_d^{*}$: red for 0.1, blue for 1, and green for 10. The solid line corresponds to numerical simulations from Meier [30] for the Lennard-Jones potential, all with temperature $T^{*}=4$ and cutoff radius $r_c=3$. Parameters of the simulations: Number of samples = 100, number of time steps (after thermalization) = 200, time step = 0.001, and number of particles per sample = $108000$. For $\eta$ greater than 0.35, the time step was changed to 0.0001, and for $\eta$ greater than 0.5, the number of steps was changed to 500.