Looking Inside the Widom Region: Non-Equilibrium Stratification in Supercritical CO2

Abstract

The supercritical state of matter is usually described as a continuous phase without sharp boundaries between liquid and gas regions. However, under non-equilibrium conditions, this view breaks down. Here we report an experimental investigation of non-equilibrium fluctuations in supercritical carbon dioxide (CO2) subjected to a stabilising temperature gradient. Using shadowgraphy, we reveal spontaneous stratification of the fluid into different layers, separated by transition regions, where thermodynamic properties vary dramatically. These signatures are particularly evident when the system crosses the Widom lines, loci of the extrema of the response function in the supercritical domain. The analysis of the intermediate scattering function of temperature fluctuations highlights the presence of Brunt-Vaisala oscillations within the fluid at multiple frequencies. These oscillations arise from the coupling of thermal and viscous modes under gravity and are a clear signature of the layered structure of the fluid. Our approach enables systematic exploration of a wide range of thermodynamic conditions in a single experiment. These findings suggest that the Widom region cannot be described as a homogeneous phase, but rather as a dynamic assembly of phase-like behaviours, challenging the applicability of classical thermodynamics in non-equilibrium supercritical regimes.

Figures (5)

• Figure 1: Density map of pure CO2 in the $p$-$T$ diagram. Indications of the Widom region limits are provided by the minimum sound speed line (dashed gray line) and the isenthalpy line (dashed and dotted blue line), while the isochore is displayed as well (dashed dark line). The thermal gradients corresponding to the experiments analysed and discussed here are displayed with solid horizontal segments. Inset: close-up of the $p$-$T$ diagram around the critical point.
• Figure 2: Differential Dynamic Algorithm analysis of shadowgraph images. Sample data at $p$ = 7.5 MPa, $T_{\mathrm{mean}}$ = 31.1 °C, $\Delta T$ = 4 K. (a) Shadowgraph raw image, (b) image difference between two images acquired with a time delay $dt=5$ s, (c) normalised 2D-structure function of the density fluctuations for $dt=5$ s, normalised 1D-structure functions (d) for different $dt$ from 0.005 to 0.13 s, and (e) for three different wave numbers $q$ from 160 to 689 $\mathrm{cm^{-1}}$.
• Figure 3: Experimental results for case 1 under thermodynamic conditions far from the Widom region. Data at $p$ = 15 MPa, $T_{\mathrm{mean}}$ = 31.1 °C, $\Delta T$ = 8 K, blue gradient in Fig.\ref{['fig:densMap']}. Top line: (a) thermodynamic properties of the system: (a.i) density, (a.ii) thermal expansion coefficient, (a.iii) thermal diffusivity, and (a.iv) kinematic viscosity. All the properties are plotted vs. the temperature or the height, assuming a linear relationship between the two. The temperature and height are on the vertical axis for all graphs. The vertical dotted lines show the measured data from the fitting of the SFs, as explained in the text. Bottom line: (b) 1D-Structure functions for 5 wave numbers (33, 52, 90, 142, and 189 $\mathrm{cm^{-1}}$), (c) time decay $\tau(q)$ of the density fluctuations and (d) Brunt-Vaisälä frequency $\Omega(q)$ as obtained by fitting the structure functions shown in (b) with the model of equation (\ref{['sf_with_1_oscillation']}).
• Figure 4: Experimental results for case 2 under thermodynamic conditions crossing the Widom region. Data at $p$ = 15 MPa, $T_{\mathrm{mean}}$ = 40 °C, $\Delta T$ = 44 K, green gradient in Fig.\ref{['fig:densMap']}. Top line: (a) thermodynamic properties of the system inside the cell: (a.i) density, (a.ii) thermal expansion coefficient, (a.iii) thermal diffusivity, and (a.iv) kinematic viscosity. All the properties are plotted vs. the temperature or the height assuming a linear relationship between the two. The temperature and height are on the vertical axis for all graphs. The vertical dotted lines show the measured data from the fitting of the SFs, as explained in the text. Bottom line: (b) 1D-Structure functions for 5 wave numbers (33, 52, 90, 142, and 189 $\mathrm{cm^{-1}}$), (c) time decays $\tau(q)$ of the density fluctuations and (d) Brunt-Vaisälä frequencies $\Omega(q)$ as obtained by fitting the structure functions shown in (b) with the model of equation (\ref{['eq:ISF_2exp']}).
• Figure 5: Experimental results for case 3 under thermodynamic conditions crossing the Widom region, close to the critical point. Data at $p$ = 7.7 MPa, $T_{\mathrm{mean}}$ = 32.86 °C, $\Delta T$ = 4 K, red gradient in Fig.\ref{['fig:densMap']}. Top line: (a) thermodynamic properties of the system inside the cell: (a.i) density, (a.ii) thermal expansion coefficient, (a.iii) thermal diffusivity, and (a.iv) kinematic viscosity. All the properties are plotted vs. the temperature or the height supposing a linear relationship between the two. The temperature and height are on the vertical axis for all graphs. Bottom line: (b) 1D-Structure functions for 5 wave numbers (66, 104, 179, 283, and 378 $\mathrm{cm^{-1}}$), (c) time decay $\tau$ of the density fluctuations and (d) Brunt-Vaisälä frequency $\Omega$ as obtained by fitting the structure functions shown in (b) with the model of equation (\ref{['sf_with_1_oscillation']}).