Transverse dynamics of water across the melting point: A parallel neutron and x-ray inelastic scattering study

Joint inelastic neutron and x-ray scattering measurements have been performed on heavy water across the melting point. The spectra bear clear evidence of low- and high-frequency inelastic shoulders related to transverse and longitudinal modes, respectively. Upon increasing the momentum transfer, the spectral shape evolves from a viscoelastic regime, where the low-frequency mode is clearly over-damped, toward an elastic one where its propagation becomes instead allowed. The crossover between the two regimes occurs whenever both the characteristic frequency and the linewidth of the low-frequency mode match the inverse of the structural relaxation time. Furthermore, we observe that the frequency of the transverse mode undergoes a discontinuity across the melting, whose extent reduces upon increasing the exchanged momentum.

Figure 1

In the left side of each panel, selected $S(Q,\omega )/S\left(Q\right)$ line-shapes are reported as measured by IXS ($•$) and INS ($\circ$) after background subtraction and normalization. In the left bottom panel, we also report the energy resolution functions of IXS and INS experiments (solid and dotted lines, respectively). The corresponding current spectra ${(\omega /Q)}^{2}S(Q,\omega )/S\left(Q\right)$ are shown in the right side of each panel. The black solid line and the thick grey one indicate the IXS and INS profiles, respectively, while vertical arrows show the characteristic energy of low- and high-frequency excitations of IXS curve. Data were collected at $T=293$ $\mathrm{K}$ (INS spectra) and $T=295$ $\mathrm{K}$ (IXS spectra) and at the indicated $Q$ values. In the $Q$ $=$ 16 ${\text{nm}}^{-1}$ plot, the black line across the curve is obtained as a polynomial interpolation and is intended as a guide to the eye.

Figure 2

Selected INS spectra ($\circ$) are reported with the corresponding best fit line-shape (solid grey line) along with the DHO function (solid black line) accounting for the low-frequency transverse mode [see Eq. (5)]. Data were collected at $T=293$ $\mathrm{K}$ and at the $Q$ values indicated in the individual panels.

Figure 3

The $Q$-dependence of best fit parameters ${\Gamma }_t$ ($\square$) and ${\Omega }_t$ ($•$) are reported along with the ones of $1/\tau$ ($-☆-$) and the longitudinal frequency, ${\Omega }_l^{\mathrm{IXS}}$, extracted from spectral current maxima ($-▴-$). Dashed curve represent the boundary of the dynamic region allowed for the INS experiment.

Figure 4

The left panel compares the dispersion curves of water (${\text{H}}_2\text{O}$ and D${}_2$O) as determined by (i) INS measurements [Ref. 1 ($\square$), Ref. 2 ($▿$) and Refs. 3 and 4 ($◊$)]; (ii) IXS measurements [Ref. 8 ($\times$)] and Refs. 5, 6, and 7 ($★$); (iii) MD simulations [Ref. 10 ($⊠$) and Ref. 11 (lines)] compared with the findings of the present work: ${\Omega }_t$ ($•$), the peak position of current spectra (see text) ${\Omega }_t^{\mathrm{INS}}$ ($⧫$) and ${\Omega }_l^{\mathrm{IXS}}$ ($▴$). All quantities measured in this work are averaged over the temperature. The right plot displays the density of state of water, $Z\left(E\right)$, as measured by incoherent neutron scattering [Ref. 53, ($\circ$)].

Figure 5

Temperature dependence of $\tau$ at $Q=20$ ($-\blacksquare -$), $Q$ $=$ 18 ($-\square -$), 16 ($-•-$), 14 ($-\circ -$), 12 ($-⧫-$), 10 ($-◊-$), and 8 ($-▴-$) ${\mathrm{nm}}^{-1}$. These curves are compared with those derived from the unconstrained best fit of IXS spectra (see text) measured at three lowest $Q$'s: $Q$ $=$ 2 ($-$;$-$), 4 ($-▵-$), and 6 ($-\nabla -$) ${\mathrm{nm}}^{-1}$. The $\tau$ values derived from US measurements (Ref. 54) (dotted line) and the arbitrarily rescaled shear viscosity (Ref. 43) are also reported for comparison (dash-dotted line).

Figure 6

Best fit values of parameters ${\omega }_l$ ($◊$), ${\Omega }_l^{\mathrm{IXS}}$ ($-▴-$) and $1/\tau$ ($-☆-$) are reported along with ${\omega }_0$ ($\circ$), as calculated through Eq. (7), and the low-$Q$ hydrodynamic linear dispersion (dashed lines), as derived from sound velocity data (Ref. 43). Dashed straight lines represent the hydrodynamic dispersions as evaluate from the NIST database.[43]

Figure 7

The $Q$-dependence of $\tau$ at $T=295$ K, as obtained from the totally unconstrained best fit of the IXS data ($\circ$) and from the iterative fitting routine of both the IXS and the INS spectra ($•$).

Figure 8

Representative INS spectra of solid D${}_2$O ($T=268$ K), collected at the indicated $Q$ values, are reported against the respective best fit line-shape (solid line). The additional, longitudinal, peak is also reported for comparison (dashed line).

Figure 9

(Left) Parameters ${\Gamma }_t$ ($\square$) and ${\Omega }_t$ ($•$) measured in the liquid phase ($T=295$ K) are reported along with the corresponding quantities measured in the solid phase (dashed and solid lines, respectively). We also report for comparison the dispersion curves of single-crystalline D${}_2$O ($\circ$) and H${}_2$O ($◊$), as well as those of polycrystalline H${}_2$O (▵). The $-⧫-$ data represent the measurement of the longitudinal branch of Ih measured by Criado et al. (Ref. 56). (Right) Water density of states, $Z\left(E\right)$ as evaluated by incoherent neutron scattering in Ref. 58.