Collective dynamics and molecular interactions in liquid ${\text{CO}}_2$ by inelastic neutron scattering and computer simulations

We report an inelastic neutron scattering determination of the dynamic structure factor of liquid carbon dioxide in the wave-vector range $3<Q<17{\text{nm}}^{-1}$ and molecular dynamics simulations performed using several site-site intermolecular interaction models. The comparison of neutron and simulation dynamical spectra allows effective model-potential selection. The good performance of some of the ${\text{CO}}_2$ anisotropic interaction models enables a thorough investigation of the collective properties, taking full advantage of the direct access that simulations provide to the purely translational modes of a molecular liquid. Center-of-mass collective excitations of liquid ${\text{CO}}_2$ can be fully described through a viscoelastic modeling of the second-order memory function if a free $Q$ dependence of the parameters is allowed. The system behaves in a hydrodynamiclike way up to about $Q\simeq 5{\text{nm}}^{-1}$, where the thermal relaxation has a dominant role. A rather different situation is found at higher $Q$, ruled by both structural and dynamical effects, namely, the rapid growth of the static structure peak, the increased damping of the Brillouin lines, and the shortening of the relaxation time. Acoustic excitations propagate up to $Q\simeq 14{\text{nm}}^{-1}$, while beyond this value a transition to overdamped modes takes place.

Figure 1

Single scattering intensity of the vanadium empty can at $Q=7{\text{nm}}^{-1}$ (full circles with error bars) and Gaussian fit to the data (solid curve).

Figure 2

$I\left(Q,E\right)$ of liquid ${\text{CO}}_2$ at various experimental $Q$ values (full circles with error bars).

Figure 3

Central peak of the experimental ${\tilde{S}}^{\text{expt}}\left(Q,\omega \right)$ of liquid ${\text{CO}}_2$ (circles with error bars) at various experimental $Q$ values compared with the MD spectra obtained with the EPM2 (full dots), BBV (solid), and TT (dashed) potentials.

Figure 4

$\tilde{S}\left(Q\right)$ obtained by frequency integration of the inelastic neutron data (circles with error bars connected by a solid line) and of the simulated spectra: ZD (diamonds), EPM2 (full dots), TraPPE (stars), TT (empty squares), and BBV (full squares). The full diamond at $Q=0$ is the thermodynamic limit (see text). Experimental error bars, smaller than the size of the symbols at $Q$ values below $11{\text{nm}}^{-1}$, take into account the uncertainties on both the normalization and spectral integration procedures.

Figure 5

Simulated $S_{\text{CC}}\left(Q,\omega \right)$ from the EPM2 potential (dots with error bars) and best-fit curves (solid lines) obtained with viscoelastic model (9) at three $Q$ values. At $Q=3{\text{nm}}^{-1}$, the spectrum calculated from memory function (11) is also shown as a dotted line. The calculated spectrum has been convoluted with the MD resolution function and normalized so as to have a frequency integral equal to the fitted $S_{\text{CC}}\left(Q\right)$.

Figure 6

Center-of-mass static structure factor $S_{\text{CC}}\left(Q\right)$ of liquid ${\text{CO}}_2$ from the VE fits (full circles) and from frequency integration (solid line) of the EPM2-based simulated spectra. The dotted line shows the level of the thermodynamic $S_{\text{CC}}\left(0\right)$ limit (full diamond at $Q=0$) as derived from PVT data (Ref. 43).

Figure 7

(a) Fitted ${\omega }_L^2$ (full circles) as a function of $Q$ compared with the parabola (dotted line) $c_L^2{Q}^2$ with $c_L=2.27\text{nm}/\text{ps}$. (b) Fitted normalized second frequency moment $⟨{\omega }_Q^2⟩$ (open circles). The solid line linking full dots is the theoretical prescription $k_{B}T{Q}^2/m{S}_{\text{CC}}\left(Q\right)$ where the fitted $S_{\text{CC}}\left(Q\right)$ is used. The dotted curve is the $Q\to 0$ limit $c_s^2{Q}^2/{\gamma }_0$.

Figure 8

Relevant quantities in VE memory function (7) calculated from the best-fit parameters of the MD spectra (EPM2 model). (a) Generalized specific-heat ratio $\gamma \left(Q\right)$; the $Q=0$ value ${\gamma }_0$ is shown as a full diamond. (b) Inverse time constants of the two exponential terms of Eq. (7): $1/\tau \left(Q\right)$ (full circles), estimated (see text) $1/{\tau }_0$ value (full diamond at $Q=0$), ${\Gamma }_T$ (open circles), and its limiting $Q\to 0$ trend (dotted line). Both latter curves are multiplied by a factor of 4. (c) Amplitudes of the two exponential terms in memory function (7): ${\Delta }_L^2\left(Q\right)$ (full circles) and $\left[\gamma \left(Q\right)-1\right]⟨{\omega }_Q^2⟩$ (open circles) are compared with their $Q\to 0$ behaviors (solid and dotted lines, respectively). The curves for $\left[\gamma \left(Q\right)-1\right]⟨{\omega }_Q^2⟩$ are multiplied by a factor of 16. (d) Time integrals of the two terms in memory function (7) $M_{1,\text{int}}\left(Q\right)$ (full circles), $M_{2,\text{int}}\left(Q\right)$ (open circles), and their sum $M_{\text{int}}\left(Q\right)$ (squares). The solid and dotted lines are the low-$Q$ parabolic behavior of $M_{1,\text{int}}\left(Q\right)$ and $M_{\text{int}}\left(Q\right)$. The diamond is the $Q=0$ value of $M_{2,\text{int}}\left(Q\right)$.

Figure 9

(a) Amplitudes and (b) half widths at half maximum of the two central Lorentzians from VE fitting [Eq. (9)] of MD spectra (symbols), compared with their respective low-$Q$ trends (lines) or $Q=0$ values (diamonds). In (a), $I_1$ (full circles and diamond) and $I_2$ (open circles and dotted line). In (b), $z_1$ (full circles and solid line) and $z_2$ (open circles and diamond).

Figure 10

(a) $Q$ dependence of the undamped frequency $\Omega \left(Q\right)$ (full circles) of the VE equivalent harmonic oscillator (see text) and of the damping $z_s$ (open circles) of collective excitations. The line shows the low-$Q$ quadratic behavior of $z_s$. (b) Dispersion curve of collective excitations. The frequency ${\omega }_s$ (full circles) and the low-$Q$ straight line $c_{s}Q$ (dashes) are shown.