Dynamics of Uniaxial Hard Ellipsoids

We study the dynamics of monodisperse hard ellipsoids via a new event-driven molecular dynamics algorithm as a function of volume fraction $\varphi$ and aspect ratio $X_0$. We evaluate the translational $D_{\mathrm{trans}}$ and the rotational $D_{\mathrm{rot}}$ diffusion coefficients and the associated isodiffusivity lines in the $\varphi -X_0$ plane. We observe a decoupling of the translational and rotational dynamics which generates an almost perpendicular crossing of the $D_{\mathrm{trans}}$ and $D_{\mathrm{rot}}$ isodiffusivity lines. While the self-intermediate scattering function exhibits stretched relaxation, i.e., glassy dynamics, only for large $\varphi$ and $X_0\approx 1$, the second order orientational correlator $C_2(t)$ shows stretching only for large and small $X_0$ values. We discuss these findings in the context of a possible prenematic order driven glass transition.

Figure 1

Grid of state points simulated (crosses and red-filled circles) and relevant boundary lines of coexistence regions. Long-dashed curves are coexistence curves of all first order phase transitions in the phase diagram of HE evaluated by Frenkel and Mulder (FM) [9]. Solid lines are coexistence curves for the $I\mathrm{\text{−}}N$ transition of oblate and prolate ellipsoids, obtained analytically by Tijpto-Margo and Evans (TME) [6]. Dotted lines (TME-2) are coexistence curves of prolate ellipsoids for the $I\mathrm{\text{−}}N$ transition, taken from [6].

Figure 2

Isodiffusivity lines. Solid lines are isodiffusivity lines from translational diffusion coefficients $D_{\mathrm{trans}}$ and dashed lines are isodiffusivities lines from rotational diffusion coefficients $D_{\mathrm{rot}}$. Arrows indicate decreasing diffusivities. Left and right arrows refer to rotational diffusion coefficients. Diffusivities along left arrow are 1.5, 0.75, 0.45, 0.3, 0.15. Diffusivities along right arrow are 1.5, 0.75, 0.45, 0.3, 0.15, 0.075, 0.045. Central arrow refers to translational diffusion coefficients, whose values are 0.5, 0.3, 0.2, 0.1, 0.04, 0.02. Thick long-dashed lines are FM and TME coexistence lines from Fig. 1.

Figure 3

Shape of $F_{\mathrm{self}}$ and $C_2$ at $\varphi =0.50$ for different $X_0$. Symbols are data from MD simulations. Solid lines are fits to exponential functions, while long-dashed lines are fits to stretched exponentials ($\beta$ is the stretching parameter). $t^{*}$ is the time at which correlation functions ${\varphi }_{vv}$ and ${\varphi }_{\omega \omega }$, for $F_{\mathrm{self}}$ and $C_2$, respectively, reach $1/e$ of their initial values. Top: Prolate ellipsoids with $X_0=3.2$, $C_2$ shows a significant stretching while $F_{\mathrm{self}}$ decays exponentially. Middle: $X_0=1.0$ for $F_{\mathrm{self}}$ and $X_0=1.1$ for $C_2$, the dashed line is the theoretical decay of a free rotator $C_2^f$ [$C_2^f(t)=1-\frac{3}{2}(t/{\tau }_f)\exp [-t^2/{\tau }_f^2]\tilde{\Phi }(t/{\tau }_f)$, where ${\tau }_f^2=1/{\varphi }_{\omega \omega }(0)$ and $\tilde{\Phi }(t)={\int }_0^t\exp [x^2]dx$]. Bottom: Oblate ellipsoids with $X_0=0.348$.

Figure 4

${\beta }_{C_2}$ and ${\tau }_{C_2}$ are obtained from fits of $C_2$ to a stretched exponential for $\varphi =0.40$, 0.45, and 0.50. Top: ${\tau }_{C_2}$ as a function of $X_0$. Bottom: ${\beta }_{C_2}$ as a function of $X_0$. The time window used for the fits is chosen in such a way as to exclude the microscopic short times ballistic relaxation (see text for details). For $0.588<X_0<1.7$ the orientational relaxation is exponential.