Multiscale empirical mode decomposition of density fluctuation images very near above and below the critical point of S
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 (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 ). 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 S 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 S 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 clearly shows that it saturates for a delay time below one second (see Fig. 4A). The time-dependent structure functions
Discussion
Should there be any separate spatial scale for IMF2 above ? 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): where 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 S critical density fluctuations data extremely close to [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 S. 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.
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