Connection of translational and rotational dynamical heterogeneities with the breakdown of the Stokes-Einstein and Stokes-Einstein-Debye relations in water

We study the Stokes-Einstein (SE) and the Stokes-Einstein-Debye (SED) relations, $D_t=k_{B}T∕6\pi \eta R$ and $D_r=k_{B}T∕8\pi \eta R^3$, where $D_t$ and $D_r$ are the translational and rotational diffusivity, respectively, $T$ is the temperature, $\eta$ the viscosity, $k_B$ the Boltzmann constant, and $R$ the “molecular” radius. Our results are based on molecular dynamics simulations of the extended simple point charge model of water. We find that both the SE and SED relations break down at low temperature. To explore the relationship between these breakdowns and dynamical heterogeneities (DHs), we also calculate the SE and SED relations for subsets of the 7% “fastest” and 7% “slowest” molecules. We find that the SE and SED relations break down in both subsets, and that the breakdowns occur on all scales of mobility. Thus these breakdowns appear to be generalized phenomena, in contrast with a view where only the most mobile molecules are the origin of the breakdown of the SE and SED relations, embedded in an inactive background where these relations hold. At low temperature, the SE and SED relations in both subsets of molecules are replaced with “fractional” SE and SED relations, $D_t\sim {(\tau ∕T)}^{-{\xi }_t}$ and $D_r\sim {(\tau ∕T)}^{-{\xi }_r}$, where ${\xi }_t\approx 0.84(<1)$ and ${\xi }_r\approx 0.75(<1)$. We also find that there is a decoupling between rotational and translational motion, and that this decoupling occurs in both the fastest and slowest subsets of molecules. Further, we find that, the decoupling increases upon cooling, but that the probability of a molecule being classified as both translationally and rotationally fastest also increases. To study the effect of time scale for SE and SED breakdown and decoupling, we introduce a time-dependent version of the SE and SED relations, and a time-dependent function that measures the extent of decoupling. Our results suggest that both the decoupling and SE and SED breakdowns originate at a time scale corresponding to the end of the cage regime, when diffusion starts. This is also the time scale when the DHs are more relevant. Our work also demonstrates that selecting DHs on the basis of translational or rotational motion more strongly biases the calculation of diffusion constants than other dynamical properties such as relaxation times.

Figure 1

(Color online) (a) Stokes-Einstein ratio $D_t\tau ∕T$ for the 7% most translationally mobile molecules (fastest), for the 7% least translationally mobile molecules (slowest), and for the entire system (all). There is a breakdown of the Stokes-Einstein relation (constant Stokes-Einstein ratio) at low temperatures in both the fastest and slowest subsets, as well as for the entire system. (b) Stokes-Einstein-Debye ratio $D_r\tau ∕T$ for the 7% most rotationally mobile molecules, for the 7% least rotationally mobile molecules, and for the entire system (all). As in (a), there is a breakdown of the Stokes-Einstein-Debye relation (constant Stokes-Einstein-Debye ratio). (c) and (d) Normalization of the curves in (a) and (b), respectively, by the corresponding quantities at $T=350\mathrm{K}$. The collapse of these curves demonstrates that the relative deviations from the Stokes-Einstein and Stokes-Einstein-Debye relations are approximately the same for the corresponding mobility subsets.

Figure 2

(Color online) (a) Power-law fits of translational diffusivity $D_t$ as functions of $\tau ∕T$, $D_t\sim {(\tau ∕T)}^{-{\xi }_t}$, for the eight values of temperature $T=210,\dots ,280\mathrm{K}$ (but not for the remaining values $T=290,\dots ,350\mathrm{K}$), for fastest translational heterogeneities (TH), slowest TH, and all molecules. We estimate ${\xi }_t\approx 0.84$. The dot-dashed line represents the normal Stokes-Einstein behavior $({\xi }_t=1)$. Consistently with the results of Fig. 1, the deviation of these three curves from the Stokes-Einstein behavior is almost identical as reflected in the values of these fractional exponents. (b) Power-law fits of rotational diffusivity $D_r$ as functions of $\tau ∕T$, $D_r\sim {(\tau ∕T)}^{-{\xi }_r}$, of simulations in the same temperature range as in (a) for fastest rotational heterogeneities (RH), slowest RH, and all molecules. We estimate ${\xi }_r\approx 0.75$. The dot-dashed line represents the normal Stokes-Einstein-Debye behavior $({\xi }_r=1)$. Also for RH, a fractional law is found with the same exponents for the three families considered, and, noticeably, the deviation from the normal case $({\xi }_r=1)$ is stronger for $D_r$ than for $D_t$, since ${\xi }_r<{\xi }_t$.

Figure 3

(Color online) (a) Ratio of rotational and translational diffusivities $D_r∕D_t$ as a function of temperature. As temperature decreases, this ratio increases, indicating a decoupling between rotation and translational motion. The deviation of $D_r$ is stronger than that of $D_t$. The line is a guide for the eye. (b) Same as (a) where the rotational diffusivity $D_r$ is replaced by the inverse of the rotational relaxation time ${\tau }_{\ell }$ with $\ell =1,2$, as is usually done in experiments. An opposite decoupling is observed in (a) and (b). The lines are guides for the eye.

Figure 4

(Color online) Ratio of rotational and translational diffusivities, $D_r$ and $D_t$ respectively, for the following choices of subsets: $D_r$ for fastest translational heterogeneities (TH) divided by $D_t$ for fastest TH (◇), $D_r$ for slowest TH divided by $D_t$ for slowest TH (▵), $D_r$ for fastest rotational heterogeneities (RH) divided by $D_t$ for fastest RH (엯), $D_r$ for slowest RH divided by $D_t$ for slowest RH (◻). The values were normalized by the $T=350\mathrm{K}$ values for every curve. The fact that for these four cases $D_r∕D_t$ deviates from unity (dashed line) to approximately the same degree indicates that the decoupling occurs across all four cases.

Figure 5

(Color online) (a) Time-dependent extension $b_{\mathrm{TH}}\left(\Delta t\right)$ of the Stokes-Einstein relation for the fastest translational heterogeneities (TH) at different $T$. For the sake of clarity the curve corresponding to $T=290\mathrm{K}$ was removed. (b) Time-dependent extension $b_{\mathrm{RH}}\left(\Delta t\right)$ of the Stokes-Einstein-Debye relation for the fastest rotational heterogeneities (RH) at different $T$. For the sake of clarity the curve corresponding to $T=290\mathrm{K}$ was removed. In both (a) and (b), the maxima occur at the time scales corresponding to the end of the cage regime, when DH are more pronounced. These maxima increase upon cooling, as the DH become more pronounced.

Figure 6

(Color online) Temperature dependence of (a) $t_b$, the time at which the time-dependent extensions of the Stokes-Einstein and Stokes-Einstein-Debye relations, $b_{\mathrm{TH}}$ and $b_{\mathrm{RH}}$, respectively, have maxima, and (b) $t^{*}$, the time at which the non-Gaussian parameter ${\alpha }_2\left(\Delta t\right)$ reaches a maximum. $t^{*}$ indicates the time scale corresponding to the end of the cage regime. We show the results when considering molecules belonging to translational heterogeneities, rotational heterogeneities, and also for the entire system.

Figure 7

(Color online) (a) Temporal behavior of the ratio of the time-dependent rotational diffusivity and translational diffusivity for the fastest translational heterogeneities (TH), normalized by the average over the entire system. We show all the simulated temperatures. (b) Temporal behavior of the ratio of the time-dependent rotational diffusivity and translational diffusivity for the fastest rotational heterogeneities (RH), normalized by the average over the entire system. We show all the simulated temperatures. The figure shows that the decoupling of rotation from translation is increasingly more pronounced as $T$ decreases and is a maximum (a) or minimum (b) on the time scale of the DH.

Figure 8

(Color online) Example of time correlation functions limited to subsets of DH. (a) Mean square displacement (MSD) and (b) rotational mean square displacement (RMSD) at $T=210\mathrm{K}$ for the fastest and slowest translational heterogeneities (TH) and rotational heterogeneities (RH), respectively, as well as for the entire system. Intermediate scattering function $F(q,\Delta t)$ at $T=210\mathrm{K}$ for (c) the fastest and slowest TH, and entire system and (d) the fastest and slowest RH and the entire system.

Figure 9

(Color online) Fraction of molecules belonging simultaneously to both fastest translational heterogeneities (TH) and fastest rotational heterogeneities (RH) versus observation time $\Delta t$, at different temperatures. This overlap of fastest TH and fastest RH is maximum at the end of the cage regime and increases upon cooling. It is almost 45% at the lowest $T$.