Multiscale empirical mode decomposition of density fluctuation images very near above and below the critical point of SF6

https://doi.org/10.1016/j.physa.2020.125293Get rights and content

Highlights

  • Short-range critical fluctuations separated from long-range phase separation.

  • Correlation times of short- and long-range fluctuations were separated.

  • Diffusion coefficients estimated for multi-scale fluctuations.

Abstract

We use a multiscale approach to investigate the dynamics of fluctuations near the critical point of sulfur hexafluoride (SF6) in microgravity. Rather than increasing the fitting model’s complexity during the critical temperature crossing, we used a different approach to finding the thermal diffusivity coefficient (above critical temperature), which can then be distinguished from an effective diffusion coefficient (below critical temperature). We first separate different spatial scales from the original images using the Bidimensional Empirical Mode Decomposition (BEMD) technique. The spatial scale represented by an Intrinsic Mode Function (IMF) image was analyzed using the Dynamic Differential Method (DDM). The Intermediate Scattering Function (ISF) of each IMF was used for computing the structure factor and the relaxation time of fluctuations. We found that the first IMF returns over 90 % of the spatial and temporal knowledge contained in the original image, providing thus thermal diffusivity coefficient above the critical temperature and effective diffusion coefficients below the critical temperature very close in magnitude. The relaxation time associated with the distinguishable structures observed in the second IMF could be attributed to the fractal nature of fluctuations. and to light scattering at low wavenumber during the stationary behavior and the transient evolution of the critical fluid cell, which are not easy to detect in the original image. The third order IMF presents no noticeable structure, and the associated relaxation time is not physically significant.

Introduction

Dynamic Differential Microscopy (DDM) is an appealing experimental technique to extract the relaxation time of dynamical phenomena [1], [2], [3], [4]. It has recently been applied to critical density fluctuations from light scattering images of systems approaching the liquid–gas critical point of pure fluids from the homogeneous domain [5]. In such experiments, the image processing performed with the dynamic structure factor (DSF) algorithm produces results consistent with the modern theory of critical phenomena. One prediction is that such systems are characterized by only one spatial scale of critical density fluctuations related to their characteristic size, the critical correlation length, and a single critical relaxation time of density fluctuations [6], [7].

The recent extension of the classical theory of fluctuations to nonequilibrium processes [8], [9] showed that the temporal relaxation of fluctuations could be directly obtained from fluctuation images. Such an approach led to experimental advances in measuring thermal diffusivity coefficient and viscosity coefficient. The recent generalization of DDM to investigating the dynamics of nonequilibrium fluctuations was introduced by Cerbino and Vailati [1], [2], [3], [4]. DDM has also been applied to investigating equilibrium fluctuations close to critical conditions in binary mixtures [10] and under nonequilibrium conditions in dense colloids [11]. DDM method allows quantitative investigation of fluctuations in the fluid outside thermodynamic equilibrium, e.g., thermal, concentration, or density gradients. DDM also allowed low wavenumber range investigation where gravity dominates the dynamics of fluctuations, and new propagation modes influenced by viscosity and gravity were observed [7], [12]. The information regarding the evolution and the scaling of the fluctuation relaxation time is contained in the Intermediate Scattering Function (ISF). For a pure fluid in thermal equilibrium, the ISF is a Gaussian with width proportional to the diffusion time. There are cases when ISF contains multiple time scales, and one approach has been the fitting of ISF with multiscale exponentials to capture each characteristic time separately [13], [14]. This approach allowed, for example, the separation of the thermal diffusivity coefficient from the mass diffusivity [13].

However, when the experiments are performed within µK macroscopic finite time, and the finite size of the fluid observations can affect wavenumbers whose time and length characteristics are substantially different. The effect of multiple temporal and spatial scales that governed the energy transfer between probing photons and probed molecular systems is reflected in the existence of multiple decay exponentials in the Intermediate Scattering Function (ISF) [13]. Therefore, the traditional approach to the multiscale analysis of the images is by fitting the ISF with exponential functions with multiple characteristic times [13], [14].

Although there is always a tradeoff between parsimony and the goodness of fit, here our goal was to both achieve low parsimony and good accurate description of experimental data by separating the dominant spatial scale related to critical density fluctuations with the characteristic size of the order of the correlation length from any other significantly different spatial scales. To achieve this goal, we used the Bidimensional Empirical Mode Decomposition (BEMD) algorithm for the multiscale separation of the original image in multiple Intrinsic Mode Functions (IMFs) images. After obtaining the IMF images, we applied the DDM method to each IMF image set, as described in [5].

In this paper, after the selected recalling of the main experimental setup features in Section 2 and a brief description of optical features in Section 3.1, we focus the IMFs results in Section 4. The first IMF for the shortest spatial scale is related directly to the critical density fluctuations above Tc (Section 4.1). The second IMF with coarser structures can be linked (Section 4.2) to the initial stage of cluster formation and phase separation process (below Tc). The relaxation time of fluctuations in the third-order IMF presents no noticeable structure. Section 5 focuses on the possible mechanisms that could explain the multiscale results. The concluding remarks in Section 6 compare the results of this multiscale analysis and the existing theoretical predictions for critical fluctuations. The two subsections of the Appendix briefly recall the main characteristic features of the DDM technique (Appendix A.1) and the BEMD method (Appendix A.2), with related references.

Section snippets

Experiments: setup aspects

Direct imaging of large density fluctuations near the liquid–gas critical point of SF6 in microgravity environment was performed with ALICE 2 facility [15] on-board the MIR space station [16]. A cylindrical sample with an inner diameter of 12 mm and a thickness of 4.34 mm was filled with electronic quality SF6 corresponding to 99.98 % purity (from Alphagaz-Air Liquide). The fluid is sandwiched between two sapphire windows with a 10 mm thickness each. The cell was housed inside a large sample cell

Optical aspects

The following description of the optical characteristics refers to the 30 years old technologies used in the ALICE 2 facility here recalled anticipating a possible application of the BEMD method to the upgraded images which can be recorded from the real-time monitoring of the current and future similar experiments performed with the DECLIC and DECLIC-EVO instruments on-board of the International Space Station (ISS) [18], [19], [20], [21], [22], [23], [24], [25].

ALICE 2 has a modular optical

Main features of the image processing using the DDM method applied to the UP set

The above applications of the DDM method were mainly based on the final determination of the relaxation time versus the scattered light’s wavenumber. We note, however, that the experimental decay of the relaxation time is not precisely matching the expected theoretical one, as shown in Eq. (2).

Our experimental determination of time-dependent structure functions Cm(q,δt) clearly shows that it saturates for a delay time δt below one second (see Fig. 4A). The time-dependent structure functions Cm(q

Discussion

Should there be any separate spatial scale for IMF2 above Tc? We investigated the possibility that the values shown in Fig. 11A as diffusion coefficients for IMF2 (see the solid triangles) could be related to a viscosity mode. Previously studies on nonequilibrium systems under concentration and gravity gradients [7], [12] used a more general relaxation time of fluctuations than Eq. (2): τ=1Dq21+qcq41+2ηkinematicq2,where ηkinematic is the kinematic viscosity. Based on [32], we estimated the

Concluding remarks

Using the MIR space station’s microgravity conditions and the performing thermal and optical environments of the ALICE 2 facility, we have observed the SF6 critical density fluctuations data extremely close to Tc [15], [16]. We previously used the DDM method [5] with a single characteristic time (single exponential) and fitted the ISF of the critical density fluctuations in the homogeneous domain (UP) above the critical point of SF6. The DDM method was extended to the nonhomogeneous domain

CRediT authorship contribution statement

Ana Oprisan: Data analysis, Wrote the manuscript, Made all required revisions, Reviewed the manuscript, Provided comments and feedback. Yves Garrabos: Designed the experiment, Carried out the experimental data collection from ALICE 2, Reviewed the manuscript, Provided comments and feedback. Carole Lecoutre-Chabot: Designed the experiment, Carried out the experimental data collection from ALICE 2, Reviewed the manuscript, Provided comments and feedback. Daniel Beysens: Designed the experiment,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

A.O acknowledges a mini-Research and Education Awards Project grant from NASA South Carolina Space Grant/EPSCoR. Y.G., C.L., and D.B. acknowledge a research grant from Centre National d’Études Spatiales (CNES) and a NASA grants NAG3-1906 and NAG3-2447.

References (76)

  • CerbinoR. et al.

    Differential dynamic microscopy: Probing wave vector dependent dynamics with a microscope

    Phys. Rev. Lett.

    (2008)
  • CroccoloF. et al.

    Non-local fluctuation phenomena in liquids

    Eur. Phys. J. E

    (2016)
  • VailatiA. et al.

    Fractal fronts of diffusion in microgravity

    Nature Commun.

    (2011)
  • OprisanA. et al.

    Dynamic structure factor of density fluctuations from direct imaging very near (both above and below) the critical point of SF6

    Phys. Rev. E

    (2012)
  • BondarchukO. et al.

    Correlation time for step structural fluctuations

    Phys. Rev. B

    (2005)
  • CroccoloF.

    Dynamics of non equilibrium fluctuations in free diffusion

    (2006)
  • VailatiA. et al.

    Nonequilibrium fluctuations in time-dependent diffusion processes

    Phys. Rev. E

    (1998)
  • CroccoloF. et al.

    Shadowgraph analysis of non-equilibrium fluctuations for measuring transport properties in microgravity in the GRADFLEX experiment

    Microgravity Sci. Technol.

    (2016)
  • GiavazziF. et al.

    Equilibrium and non-equilibrium concentration fluctuations in a critical binary mixture

    Eur. Phys. J. E

    (2016)
  • GiavazziF. et al.

    Structure and dynamics of concentration fluctuations in a non-equilibrium dense colloidal suspension

    Soft Matter

    (2016)
  • CroccoloF. et al.

    Use of dynamic schlieren interferometry to study fluctuations during free diffusion

    Appl. Opt.

    (2006)
  • BatallerH. et al.

    Analysis of non-equilibrium fluctuations in a ternary liquid mixture

    Microgravity Sci. Technol.

    (2016)
  • Ortiz de ZarateJ. et al.

    Non-equilibrium fluctuations induced by the soret effect in a ternary mixture

    Eur. Phys. J. E

    (2014)
  • MarcoutR. et al.

    ALICE 2, an advanced facility for the analysis of fluids close to their critical point in microgravity

  • LecoutreC. et al.

    Turbidity data of weightless SF6 near its liquid–gas critical point

    Int. J. Thermophys.

    (2009)
  • OprisanA. et al.

    Universality in early-stage growth of phase-separating domains near the critical point

    Phys. Rev. E

    (2008)
  • DurieuxA. et al.

    Declic: design, integration and testing of a multi configurable instrument using optical diagnostics to study directional solidification and critical fluids

  • GarrabosY. et al.

    Crossover equation of state models applied to the critical behavior of xenon

    J. Stat. Phys.

    (2015)
  • GarrabosY. et al.

    Critical crossover functions for simple fluids: Towards the crossover modelling uniqueness

    J. Stat. Phys.

    (2016)
  • GarrabosY. et al.

    Liquid-vapor rectilinear diameter revisited

    Phys. Rev. E

    (2018)
  • LecoutreC. et al.

    Weightless experiments to probe universality of fluid critical behavior

    Phys. Rev. E.

    (2015)
  • NikolayevV. et al.

    Boiling crisis dynamics: Low gravity experiments at high pressure

    Microgravity Sci. Technol.

    (2015)
  • CroccoloF. et al.

    Nondiffusive decay of gradient-driven fluctuations in a free-diffusion process

    Phys. Rev. E

    (2007)
  • OnukiA.

    Phase transition dynamics

    (2002)
  • OhJ. et al.

    Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bénard convection

    Phys. Rev. E

    (2004)
  • TanakaH. et al.

    Direct determination of the probability distribution function of concentration in polymer mixtures undergoing phase separation

    Phys. Rev. Lett.

    (1987)
  • MoldoverM. et al.

    Gravity effects in fluids near the gas-liquid critical point

    Rev. Modern Phys.

    (1979)
  • WilkinsonR.A. et al.

    Equilibration near the liquid-vapor critical point in microgravity

    Phys. Rev. E

    (1998)
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