Diffusion in dense supercritical methane from quasi-elastic neutron scattering measurements

Methane, the principal component of natural gas, is an important energy source and raw material for chemical reactions. It also plays a significant role in planetary physics, being one of the major constituents of giant planets. Here, we report measurements of the molecular self-diffusion coefficient of dense supercritical CH4 reaching the freezing pressure. We find that the high-pressure behaviour of the self-diffusion coefficient measured by quasi-elastic neutron scattering at 300 K departs from that expected for a dense fluid of hard spheres and suggests a density-dependent molecular diameter. Breakdown of the Stokes–Einstein–Sutherland relation is observed and the experimental results suggest the existence of another scaling between self-diffusion coefficient D and shear viscosity η, in such a way that Dη/ρ=constant at constant temperature, with ρ the density. These findings underpin the lack of a simple model for dense fluids including the pressure dependence of their transport properties.

The literature measurements of supercritical high-pressure methane at room temperature by QENS 2,4 were analysed with a roto-translational model, which assumed that the two types of motion are uncoupled. Rotational diffusion coefficients D R ranging from 40 ps -1 at 0.05 GPa to 6 ps -1 at 0.4 GPa were reported, and their accuracy was not better than 25% 2 . 40 ps -1 would be too fast to be analysed by our experiment. 6 ps -1 corresponds to a rotational timescale of 0.0833 ps (=1/2D R ), which is just within the accessible timescales in our experiment, suggesting that the rotational motion (overall rotation of the molecule around the centre of mass) should be accessible at pressures of 0.4 GPa and above, but only on the anti-Stokes side of the spectra and at high Q.
Moreover, modelling of the rotational motion can be challenging and we decided to ignore it in the present study. To limit the effect that the rotational contribution might have on our results, we excluded the points at Q≥1.4Å -1 from the fits of Γ(Q), as the intensity of the quasi-elastic terms of a rotational scattering function must tend to zero for Q→0. For example, in the model employed in ref. 2 , the area of the translational component dominates over that of the rotational component at small wave vector transfers (Q≤0.8Å -1 ), while the opposite is true for Q≥2.0Å -1 . In addition, we used a narrow fitting range close to zero energy transfer (always within -4.5 to +2 meV). The previous estimation 2 of D R up to 0.4 GPa indicates that the quasi-elastic rotational contribution is so broad that it must be essentially seen as a flat background over this narrow region of the spectra.
At high pressure, molecules penetrate deeper into the repulsive force fields of their neighbours and are likely to experience strong orientational correlations, as they also do in liquid methane at ambient pressure 5 . Orientational correlations introduce further correlations ("coupling") between the translational and rotational motions. The authors of refs. 2,4 pointed out that their model did not perfectly reproduce the experimental spectra and that the disagreement was probably due to the neglected coupling effects between the translations and the rotations.

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" " ,  Figure 2. Lennard-Jones model. Logarithm of the dimensionless reduced selfdiffusion coefficient D + =D m/ σ 2 as a function of dimensionless reduced pressure P + =P σ 3 / from our data (the employed values of and σ are indicated in the legend) and from the results of the molecular dynamics simulations of the Lennard-Jones fluid reported in ref. 16 for It can be seen that a linear dependence describes our data at high pressure fairly well, but the slope is very different from that predicted for the Lennard-Jones fluid.

Supplementary Note 3: Reduced self-diffusion coefficient D *
The dimensionless reduced self-diffusion coefficient D * defined by Dymond 17 is given by where (nD) 0 is the value of nD for the dilute gas, V is the molar volume and V 0 the volume of close packing: For hard spheres, (nD) 0 is given by the gas kinetic Chapman-Enskog theory: Using Eqs. (2) and (3), Eq. (1) can be written as Finally, Eq. (4) is equivalent to if the unit of D is m 2 s -1 , the unit of V is m 3 mol −1 , the unit of M is kg mol −1 , and the unit of T is K. We calculated D * for our experimental values of D, V =M /ρ, M =16.04 g mol −1 and T =300 K. The result is plotted in Supplementary Fig. 3 as a function of reduced number density nσ 3 , where n=ρN A /M and σ was chosen to be 3.6Å. Similarly, we calculated D * and nσ 3 for literature data 11,14 along six other isotherms reaching a maximum pressure of 0.22 GPa, using the values of σ indicated in the legend of Supplementary Fig. 3. This set of six values was found in ref. 14 so that the reduced diffusivity isotherms fall on a common curve when plotted against reduced number density. It can be seen from Supplementary Fig. 3 that the six literature isotherms do fall on a common curve while our isotherm deviates significantly. Supplementary Figure 3 also reports D * as a function of nσ 3 for the smooth hard-sphere fluid as obtained computationally in ref. 18 . Supplementary Figure 3 is another way of visualizing the finding presented in Fig. 4a of the main text: There is no constant sphere diameter σ for which our experimental results can be matched with the expectation for a smooth hard-sphere fluid. It is also interesting to note that the wide pressure range explored in the current study allows us to see a considerably larger deviation from the smooth hard-sphere behaviour than that observed in liquid methane for nσ 3 >0.86 in ref. 14 . Supplementary Figure 3 also shows that there is a clear deviation of the literature results 11,14 from the smooth hard-sphere behaviour for nσ 3 below ∼0.6.
The reduced number density corresponding to freezing of the hard sphere fluid is 0.9392 18 but ref. 18  therefore n=0.02289Å −3 and one finds σ=3.449Å. This is in fairly good agreement with the value mentioned above (3.43Å) or the (identical) one that could be deduced from Fig. 4a, showing the essential correctness of our experimental results. It must be stressed that the density value at 1.38 GPa used here had to be extrapolated from the literature data 6 existing up to 1 GPa. In figure 5 of ref. 14 , Harris and Trappeniers showed that σ values from i) the comparison of the experimental diffusion coefficient with that of the smooth hard-sphere fluid at moderate pressures and ii) the experimental density of fluid methane at freezing coincide at low temperature. However, the two approaches provide progressively diverging diameters above ∼170 K 14 . One can then infer that the equivalent hard-sphere diameter of methane must be reduced upon compression along isotherms for T >170 K, but entire isotherms have to be measured for this effect to be visible (pressures comparable to the freezing points have to be achieved).
! "   ! " Supplementary Figure 5. Test of the Stokes-Einstein-Sutherland relation at low temperature. Pressure dependence of the product between the self-diffusion coefficient D and the viscos-  ,    11,12,14 and η values are predicted values from ref. 19 .