Shear viscosity and Stokes-Einstein violation in supercooled light and heavy water

We report shear viscosity of heavy water supercooled $33\mathrm{K}$ below its melting point, revealing a 15-fold increase compared to room temperature. We also confirm our previous data for the viscosity of supercooled light water and reach a better accuracy. Our measurements, based on the spontaneous Brownian motion of $350\mathrm{nm}$ spheres, disagree at the lowest temperature with the only other available data, based on Poiseuille flow in a narrow capillary, which may have been biased by electro-osmotic effects. Here we provide a detailed description of the experiment and its analysis. We review the literature data about dynamic properties of water (viscosity, self-diffusion coefficient, and rotational correlation time), discuss their temperature dependence, and compare their decoupling in the two isotopes.

Figure 1

Decorrelation time $\tau$ as a function of wave vector $q$ in heavy water (mass fraction ${\mathrm{D}}_2\mathrm{O}97\%$) at $-{31}^{\circ }\mathrm{C}$. The line is a fit with $\tau =1/\left(D{q}^2\right)$, which yields $D=5.03{10}^{-2}{\mu \mathrm{m}}^2{\mathrm{s}}^{-1}$.

Figure 2

Side view of the sample placed in the temperature-controlled stage of the microscope.

Figure 3

$\mathrm{\Delta }T=T_{m,\mathrm{act}}-T_{m,\mathrm{Linkam}}$ as a function of $T_{m,\mathrm{Linkam}}$ for the four chemicals listed in Table 1. The line is a linear least-${\chi }_2$ fit. Inset: Determination of the melting point of dodecene during heating in the thermal bath: $T_{m,\mathrm{act}}$ is given by the intersection of the two straight lines.

Figure 4

Shear viscosity $\eta$ as a function of ${\mathrm{D}}_2\mathrm{O}$ molar fraction $x$ for various temperatures ($\pm 0.05\mathrm{K}$). The lines are least-${\chi }_2$ fit with Eq. (5).

Figure 5

Bottom panel: raw viscosity data for pure ${\mathrm{D}}_2\mathrm{O}$. The red curve is a least-square fit with the Speedy-Angell power-law [Eq. (3)]. Top panel: Relative deviation between data and fit. The $1-\sigma$ relative uncertainty corresponds to a $68\%$ confidence interval.

Figure 6

Lower panel: viscosity of ${\mathrm{H}}_2\mathrm{O}$ as a function of temperature. The red curve is ${\eta }_{\mathrm{smoothed},{\mathrm{H}}_2\mathrm{O}}\left(T\right)$. Upper panel: reduced residuals $[\eta \left(T\right)-{\eta }_{\mathrm{smoothed},{\mathrm{H}}_2\mathrm{O}}\left(T\right)]/\sigma \left(T\right)$.

Figure 7

Relative deviation of literature data [9, 12, 13, 25, 26, 27] from our results ${\eta }_{\mathrm{smoothed},{\mathrm{H}}_2\mathrm{O}}$ (see Appendix). The curves show the uncertainty band ($\pm 1$ standard deviation) for our previous data [9] (dashed orange) and the present ones (solid red).

Figure 8

Relative deviation of literature data [13, 24, 29, 30] from our results ${\eta }_{\mathrm{smoothed},{\mathrm{D}}_2\mathrm{O}}$ (see Appendix). The red curves show the uncertainty band ($\pm 1$ standard deviation) of our measurements.

Figure 9

Viscosity of light water in different representations. The three panels include the 8 datasets listed in Table 3 and our data (pink circles), together with a best-fit with various models (lines, see below). The lowest temperature pink circle is the one that was previously published in Ref. [9]. (Left) Arrhenius plot, showing an apparent activation energy increasing from 1200 to 6000 K upon cooling (solid lines). (Center) VTF representation, with best-fit parameters in Table 12. (Right) Speedy-Angell representation, with best-fit parameters in Table 4. (Top) The reduced residuals $({\eta }_{\exp }-{\eta }_{\mathrm{fit}})/{\sigma }_{\exp }$, where ${\eta }_{\exp }$ and ${\eta }_{\mathrm{fit}}$ are the experimental and fitted viscosity, respectively, and ${\sigma }_{\exp }$ is the experimental uncertainty (1 SD). Note the different vertical scale in the top right panel.

Figure 10

Viscosity of heavy water in different representations. The three panels include the 7 datasets listed in Table 6 and our data (pink circles), together with a best-fit with various models (lines, see below). (Left) Arrhenius plot, showing an apparent activation energy increasing from 1300 to 7000 K upon cooling (solid lines). (Center) VTF representation, with best-fit parameters in Table 12). (Right) Speedy-Angell representation, with best-fit parameters in Table 4). (Top) The reduced residuals $({\eta }_{\exp }-{\eta }_{\mathrm{fit}})/{\sigma }_{\exp }$, where ${\eta }_{\exp }$ and ${\eta }_{\mathrm{fit}}$ are the experimental and fitted viscosity, respectively, and ${\sigma }_{\exp }$ is the experimental uncertainty (1 SD). Note the different vertical scale in the top right panel.

Figure 11

Fit of the self-diffusion coefficient of light water by a Speedy-Angell law, with parameters given in Table 4.

Figure 12

Fit of the self-diffusion coefficient of heavy water by a Speedy-Angell law, with parameters given in Table 4.

Figure 13

Fit of the rotational correlation time of light water by a Speedy-Angell law, with parameters given in Table 4.

Figure 14

Fit of the rotational correlation time of heavy water by a Speedy-Angell law, with parameters given in Table 4.

Figure 15

Test of the predictions of the mode coupling theory on the dynamic quantities of light water: $D_s\eta$ (top) and $\eta /{\tau }_{\theta }$ (bottom) should be constant according to the mode coupling theory.

Figure 16

Test of the predictions of the mode coupling theory on the dynamic quantities of heavy water: $D_s\eta$ (top) and $\eta /{\tau }_{\theta }$ (bottom) should be constant according to the mode coupling theory.

Figure 17

Stokes-Einstein (top) and Stokes-Einstein-Debye (bottom) ratios as a function of temperature in light water.

Figure 18

Stokes-Einstein (top) and Stokes-Einstein-Debye (bottom) ratios as a function of temperature in heavy water.

Figure 19

Apparent hydrodynamic radius $R_h$ as a function of temperature calculated from Eq. (8) for ${\mathrm{H}}_2\mathrm{O}$ (blue circles) and ${\mathrm{D}}_2\mathrm{O}$ (red squares); the colored areas indicate the $1-\sigma$ uncertainty. The solid curves show $R_{\mathrm{rcp}}$ from Eq. (9) for ${\mathrm{H}}_2\mathrm{O}$ (blue) and ${\mathrm{D}}_2\mathrm{O}$ (red).

Figure 20

Fractional Stokes-Einstein (top) and Stokes-Einstein-Debye (bottom) ratios as a function of viscosity in light water. Straight lines: Stokes-Einstein and Stokes-Einstein-Debye relations. Dashed line: $D_s\propto {(\eta /T)}^{-0.8}$ (top); ${\tau }_{\theta }\propto {(\eta /T)}^{0.9704}$ (bottom).

Figure 21

Fractional Stokes-Einstein (top) and Stokes-Einstein-Debye (bottom) ratios as a function of viscosity in heavy water. Straight lines: Stokes-Einstein and Stokes-Einstein-Debye relations. Dashed line: $D_s\propto {(\eta /T)}^{-0.7}$ (top); ${\tau }_{\theta }\propto {(\eta /T)}^{0.9567}$ (bottom).