Morad Bali
email: morad.bali@iepg.fr

collapse. This example is not the only case of its kind. The same issue exists in [START_REF] Nicoll | Negative outlook for Russian economy as sanctions bite[END_REF] 1 , [START_REF] Shirov | Estimating potential effect of sanctions on economic development in Russia and EU[END_REF] (even if this paper is a great statistical study), [START_REF] Arnold | The true costs of financial sanctions[END_REF], et cetera.

There is, unfortunately, a large number of papers which don't integrate oil price effects in their analysis. This fact is real, even if the importance of oil price has been put forward in [START_REF] Dreger | Between the hammer and the anvil: The impact of economic sanctions and oil prices on Russia's ruble[END_REF] where authors demonstrate that the Russian rouble depreciation is mostly due to Brent prices.

Methodological Biases in Economic Analysis. Do economic coercive measures have an effect on economies targeted? A simple way to answer this question is to check how GDPs of involved economies react to sanctions. Authors tried to do so, but result differs depending on sources, on studied period, or on selected methods (Table A). Thus, it seems that while coercive measures would have cost to the Russian economy from 0.2% to 9% of its GDP per year; the impact on the European Union appears to be much lower. Nevertheless, this conclusion arising from the literature possesses a weak point. One must remember that we cannot compare apples and pears. For example, in [START_REF] Kholodilin | Crimea and punishment: the impact of sanctions on Russian economy and economies of the euro area[END_REF] and [START_REF] Kholodilin | Crimea and punishment: The impact of sanctions on Russian and European economies[END_REF] authors compare sanctions effects on the sum of all Euro Area GDPs to the lone Russian GDP. The problem of doing so is that

Editor and Assistant Editor names.

and the Leningrad Region (Petrostat)

In order to ease the reading of our European colleagues, the formatting of this document differs from the original one (published in the conference collection according to Russian standards).

if it is true that some European economies suffered from Russian embargo -mostly countries of the Baltic Sea and other small economies, see [START_REF] Oja | No milk for the bear: the impact on the Baltic states of Russia's counter-sanctions[END_REF], [START_REF] Havlik | Economic consequences of the Ukraine conflict[END_REF], or [START_REF] Haraldsson | The Economic Impact of the Russian Counter-Sanctions on Trade Between Iceland and the Russian Federation. Nónó EHF[END_REF], some others are less exposed. They are less sensitive to Russian counter measures since Russia represent a small share of their trade (export to Russia ), and because this small share itself is a small share of their GDP (Malta, Spain, Croatia, Portugal, United Kingdom, and so on). Ineluctably, it means that some economies might counterbalance others, which has the ability to create statistical and econometric limitations. 3

In order to ease the reading of our European colleagues, the formatting of this document differs from the original one (published in the conference collection according to Russian standards).

strategies have been set up in order to avoid coercive measures (shrimp or Camembert from "Belarus", for example). It seems complicated (not to say impossible) to study all these nuances without product-by-product analysis. Small economies are more exposed to trade loss.

Davis (2016)

Russian counter-sanctions have negligible effects and Ashford (2016)

Russian embargo created shortgages and increased food prices in Russia; Germany has 400 000 jobs at stake.

Conclusion.

The first identified issue is related to the absence of oil prices in econometric models, when studying effects of sanctions on the Russian economy. It is an essential control variable and must be introduced in any model pretending to study sanctions effects. Secondly, when comparing effects of coercive measures between countries, it is essential to compare what can be compared. Comparing the GDP of one country to the sum of several other countries makes no sense and induces biased results. Thirdly, when focusing on changes in trade, one must be certain to isolate goods depending on whether they are targeted by coercive economic measures or not. The main interest of doing so is to avoid counter-balancing phenomenon between those targeted by sanctions and others. Finally, it seems rational and logical to use a reliable Sanction Index 6 in order to simulate sanctions in econometric models, as for any other economic variable.


Maud Frot

François Mauguière

DUAL REPRESENTATION OF PAIN IN THE OPERCULO-INSULAR CORTEX IN HUMANS

Keywords: Pain, Insula, SII area, Human brain, Intra-cerebral recordings, Evoked potentials

Dual representation of pain in the operculo-insular cortex in humans

Maud Frot, François

Introduction

Based on numerous anatomical and microelectrode studies in monkeys [START_REF] Burton | Second somatosensory cortex and related areas[END_REF][START_REF] Burton | Cortical areas within the lateral sulcus connected to cutaneous representation in areas 3b and 1: a revised interpretation of the second somatosensory area in macaque monkeys[END_REF], Krubitzer et al. 1995), it is now well accepted that the second somatosensory area (SII) and the insular cortex play a major role in the processing of painful and non-painful inputs (Robinson et Burton 1980a,b,c, Mesulam and Mufson 1982, Friedman and Murray 1986, Cusik et al. 1989[START_REF] Burton | Second somatosensory cortical area in macaque monkeys. I. Neuronal responses to controlled, punctate indentations of glabrous skin on the hand[END_REF], 1991[START_REF] Schneider | A modality specific somatosensory area within the insula of the rhesus monkey[END_REF][START_REF] Zhang | Monkey insular cortex neurons respond to baroreceptive and somatosensory convergent inputs[END_REF]. Several clinical observations confirmed these data. Some observations of ictal pain sensation have been indeed reported in patients with epileptic seizures originating in SII [START_REF] Young | Painful epileptic seizures involving the second sensory area[END_REF]). The team of Greenspan (Greenspan andWinfield 1992, Greenspan et al. 1999) has observed reversible loss or decrease of pain and tactile sensation in patients with focal lesions involving the posterior and parietal operculum, confirming the role of these cortices in normal pain and tactile perception. Moreover, recent studies from our department reported painful somatic sensations during direct electrical stimulations of the posterior insular cortex in epileptic patients with intra-cerebral electrodes. Pain evoked by insular stimulation was lateralized to the opposite half of the body performed in epileptic patients with intracerebral electrodes [START_REF] Ostrowsky | Functional mapping of the insular cortex : clinical implication in temporal lobe epilepsy[END_REF][START_REF] Ostrowsky | Representation of pain and somatic sensation in the human insula : a study of responses to direct electrical cortical stimulation[END_REF]. In most cases these painful sensations were elicited by stimulating the non-dominant hemisphere for language. During the past 20 years, many functional imaging studies were in accordance with all these anatomical and clinical data and converged to the conclusion that the operculo-insular cortex is involved in the processing of pain in humans. Positron emission tomography (PET) and functional magnetic resonance imaging (fMRI) studies showed a bilateral pain-related activation in a broad region comprising the depth of the sylvian fissure and the parieto-frontal opercular cortex (for a review see [START_REF] Peyron | Functional imaging of brain responses to pain. A review and meta-analysis[END_REF]. These studies highlighted two distinct sites of activations in this region : i) an antero-inferior activation, in the vicinity of the anterior insular cortex and ii) a posterior activation, at the boundary between the second somatosensory (SII) cortex and the posterior insula [START_REF] Casey | Positron emission tomographic analysis of cerebral structures activated specifically by repetitive noxious heat stimuli[END_REF][START_REF] Xu | Functional localization of pain perception in the human brain studied by PET[END_REF][START_REF] Svensson | Cerebral processing of acute skin and muscle pain in humans[END_REF][START_REF] Paulson | Gender differences in pain perception and patterns of cerebral activation during noxious heat stimulation in humans[END_REF][START_REF] Davis | Functional MRI study of thalamic and cortical activations evoked by cutaneous heat, cold, and tactile stimuli[END_REF][START_REF] Becerra | Human brain activation under controlled thermal stimulation and habituation to noxious heat : an fMRI study[END_REF][START_REF] Baron | Brain processing of capsaicin-induced secondary hyperalgesia. A functional MRI study[END_REF][START_REF] Sawamoto | Expectation of pain enhances responses to nonpainful somatosensory stimulation in the anterior cingulate cortex and parietal operculum/posterior insula : an eventrelated functional magnetic imaging study[END_REF]. Moreover, some electrophysiological studies in humans have localized in the SII/insular cortex the sources of activities peaking on the scalp between 160 and 200 ms after a painful CO 2 laser skin stimulus [START_REF] Tarkka | Equivalent electrical source analysis of pain-related somatosensory evoked potentials elicited by a CO2 laser[END_REF][START_REF] Bromm | Brain electrical sources of laser evoked potentials in response to painful trigeminal nerve stimulation[END_REF], Kakigi et al.1995, 1996[START_REF] Valeriani | Scalp topography and dipolar source modeling of potentials evoked by CO2 laser stimulation of the hand[END_REF][START_REF] Valeriani | Sources of cortical responses to painful CO2 laser skin stimulation of the hand and foot in the human brain[END_REF]. These studies combined the scalp recordings of electrical or magnetic fields evoked by a painful stimulation and the estimation, by dipole source modeling, of the localization of the dipolar intracerebral sources of these activities. [START_REF] Lenz | Painful stimuli evoke potentials recorded from the parasylvian cortex in humans[END_REF] have recorded by means of a sub-dural grid of electrodes CO 2 laser evoked potentials (LEPs) peaking between 162 and 340 ms. The spatial distribution of this response over the cortical surface of the peri-sylvian cortex was considered as compatible with generators located in the parietal operculum and/or in the insular cortices.

Using depth intra-cerebral recordings in humans we have been able to demonstrate the existence of sources of pain evoked potentials in the SII cortex [START_REF] Frot | Intracerebral recordings of responses from the parietofrontal opercular area SII after electrical and CO 2 Laser stimuli in humans. 14 th International Congress of EEG and Clinical Neurophysiology. Florence (It)[END_REF](Frot et al. , 1999[START_REF] Frot | Responses of the supra-sylvian (SII) cortex in humans to painful and innocuous stimuli. A study using intra-cerebral recordings[END_REF].

It remains, however, that the anatomical boundary between SII and insular cortex is not easy to draw in functional neuro-imaging studies, especially along the depth axis from lateral to medial cortical structures (x axis in the Talairach's system of stereotactic coordinates).

Moreover, functional neuro-imaging lacks the temporal resolution required to separate in time the activities of the suprasylvian opercular (SII) and insular cortex, respectively. Such a separation could allow to differentiate these two cortices by their response characteristics and to know how these areas, which are intimately interconnected (Friedman et al 1986), act in the processing of nociceptive inputs.

In this study, we report on response properties of the electrical potentials evoked in the supra-sylvian opercular and insular cortices by a painful skin stimulation delivered by a CO 2 laser. These potentials were recorded by depth intracerebral multi-contact electrodes with a separation of 1.5 mm between two adjacent contacts (2 mm each) along the lateral medial axis (depth axis), and a millisecond resolution in time. By this way, we were able to separate in time and space the supra-sylvian opercular and insular responses to pain.

Patients and methods

Patients

We recorded CO 2 laser evoked potentials (LEPs) in 13 patients (24-47 years, mean age 35 years, 9 females, 4 males). All patients included in this study presented with refractory temporal lobe epilepsy and were investigated using stereotactically implanted intracerebral electrodes before functional surgery. Among other sites (see Table 1) these patients had electrodes chronically implanted in the operculo-insular cortex for the recording of their seizures and cortical functional mapping using electric stimulation and evoked potentials recordings (see [START_REF] Ostrowsky | Representation of pain and somatic sensation in the human insula : a study of responses to direct electrical cortical stimulation[END_REF] for a description of the stimulation procedure). The decision to explore this area resulted from the observation during scalp video-EEG recordings of ictal manifestations suggesting the possibility of a supra-sylvian and/or insular spreading of seizures such as: lip and face paresthesiae or tonic-clonic movements, laryngeal contraction, gustatory illusions, hypersalivation, (see [START_REF] Isnard | The role of the insular cortex in temporal lobe epilepsy[END_REF] for a complete description of the rationale of electrode implantation). This procedure, as well as the recording of somatosensory evoked potentials, is part of the functional mapping of eloquent cortical areas performed routinely before epilepsy surgery in patients implanted with depth electrodes.

According to the French regulations concerning invasive investigations with a direct individual benefit, patients were fully informed of the electrode implantation, stereotactic EEG and EPs recordings (SEEG), and cortical stimulation procedures used to localize the epileptogenic and eloquent brain areas and gave their consent. The CO 2 laser stimulation paradigm has been approved by local ethics committee.

At the time of LEP recordings, patients were under monotherapy with one of the following major anti-epileptic drugs (carbamazepine, phenytoin, valproate, lamotrigine or topiramate). LEPs recordings were performed at the end of the SEEG monitoring, once pertinent seizures had been recorded, so that anti-epileptic drugs daily dosages were at, or slightly under, the minimum of their therapeutic range. Handedness was determined using the Edinburgh Handedness Inventory [START_REF] Oldfield | The assessment and analysis of handedness: the Edinburgh inventory[END_REF], ten patients were right-handed, three were left-handed. The hemisphere dominant for speech was determined using intra-carotid amobarbital test in eleven patients and by cortical stimulation in two. In the ten right-handed patients the left hemisphere was dominant for language, two left-handed patients had a bilateral hemispheric representation of language, in the third left-handed patient language was found to be represented in the right hemisphere.

A complete description of cortical structures explored by depth electrodes, and of the cortical areas found to be involved in seizures propagation is given in Table 1. In all patients but one (case 12) several spontaneous seizures could be recorded, all of which originated in the mesial structures of the temporal lobe. In these 12 patients ictal discharges were propagating outside the mesio-temporal cortex. The areas most frequently involved in seizure propagation were the temporal pole, the temporal neo-cortex, the cingulate gyrus, and the orbito-frontal cortex. In patient 12 (Table 1) no spontaneous seizure occurred during the SEEG monitoring; however seizures could be triggered by stimulating the left hippocampus which spread to the orbito-frontal cortex and spared the suprasylvian operculo-insular cortex.

(Table 1 near here) In seven patients the supra-sylvian operculum showed a rhythmic spike-wave activity during the spread of the discharges; in four of these patients this type of activity was also observed in the insular cortex. The possibility remains that, in these seven patients, the suprasylvian opercular and insular cortices could show some degree of interictal hyper-excitability which might modify their responsiveness to somatosensory or pain inputs. However this possibility seems unlikely for the following reasons; i) none of the patients included in this study showed ictal discharges onset in the operculo-insular cortex and no low voltage fast activity was recorded in this cortex during spontaneous seizures; ii) no sustained after discharge was elicited by electrical stimulation of the operculo-insular cortex in any of our patients; iii) in terms of latency and amplitude somatosensory and pain EPs recorded in the operculo-insular cortex of the patients showing ictal spike-wave activity were not different from those recorded in patients whose seizures were not propagating to these cortical areas.

Electrode implantation

Intracerebral electrodes were orthogonally implanted using Talairach's stereotactic frame [START_REF] Talairach | Co-planar stereotaxic atlas of the Human brain. 3-Dimensional Proportional System : An approach to Cerebral Imaging[END_REF]. The cortical targets were identified on the patient's magnetic resonance (MR) image, previously enlarged at scale one, before the surgery. The implanted procedure has been described elsewhere in details (Frot and Mauguière 1999, Frot et al. 1999, 2001). Each electrode had a diameter of 0.8 mm and 10-15 contacts, each 2 mm long, separated by 1.5 mm; they could be left in place chronically up to 15 days. Each of the contacts could be localized in the Talairach space using its stereotactic coordinates: x for the lateral medial axis, with x=0 being the coordinate of the sagittal inter-hemispheric plane; y for the rostro-caudal axis, y=0 being the coordinate of VAC (Vertical Anterior Commissure) plane and z for the inferior-superior axis, z=0 being the coordinate of the horizontal AC-PC (Anterior Commissure-Posterior Commissure) plane.

In the opercular region, electrodes were implanted caudal and rostral to the VAC plane (y=0). The deepest contacts of the electrodes implanted in the supra-sylvian opercular cortex explored the insula proper. As shown in Table 1 five patients were implanted by a single opercular electrode exploring either the pre-(1 case) or the post-rolandic (4 cases) suprasylvian cortex. In the eight other patients both the frontal operculum and the parietal operculum were each implanted by one electrode exploring the insular cortex. Thus our data were collected using a total number of 21 electrodes. Forty-two contacts explored the insular cortex, distributed along the rostro-caudal axis, 14 mm rostral and 25 mm caudal to the VAC plane (y coordinates).

Apart from the operculo-insular cortex, intra-cerebral LEPs from other regions were recorded in these patients, like the hippocampus, the amygdala, the anterior cingulate gyrus, the temporal pole, the orbito-frontal cortex and in a few cases, the supplementary motor area (see Frot et al. 1999).

CO 2 laser stimulations

The LEPs recordings were performed about 10 days after electrodes implantation.

During the recordings, the patients lay relaxed on a bed in a semidarkened room. Cutaneous heat stimuli were delivered by a CO 2 laser (10.6 µm wavelength, 10 W output power). The power output being fixed, the amount of thermal energy delivered depended on the duration of the pulse. Thresholds for innocuous and painful sensations were first determined in each patient. The intensity of pain perception (pinprick) was rated by the patients between 4 and 7 on a 0-10 Visual Analog Scale (VAS). The energy density of the Laser beam varied between 6.3 and 10.9 mJ/mm 2 delivered on a skin surface of 16 mm 2 . The stimuli were applied to the dorsum of the hand, in the radial nerve territory. A red helium-neon (He-Ne) laser was combined with the CO 2 laser to visualize the stimulated skin area. Two runs of 30 stimuli were performed. For each run of stimuli, the continuous EEG was cut in 30 epochs (each epoch of EEG began 12 ms before the stimulus and ended 500 ms after). All these epochs were averaged to remove the noise and to bring out the evoked potentials. Then, the two runs were averaged after having checked that the averaged waveforms were reproducible. The interstimulus interval varied randomly between 4500 and 5500 ms. The laser beam was slightly moved between two successive stimuli to avoid habituation and especially to avoid peripheral nociceptor fatigue [START_REF] Schwarz | Inactivation and tachyphylaxis of heat-evoked inward currents in nociceptive primary sensory neurones of rats[END_REF]. The analysis time was 512 ms; the signal was bandpass filtered between 1 and 250 Hz and sampled at 500 Hz. The reference electrode was at the earlobe ipsilateral to the stimulated hand and the ground was a circular wrapped electrode at forearm ipsilateral to stimulation.

Responses were labeled according to the polarity-latency nomenclature in which the letters N and P, referring to the polarity of the potential in the contacts close to scalp surface, is followed by the mean latency in milliseconds. In all figures, negative potentials at the intracortical recording site are represented upward. In text and tables mean voltages, latencies and time intervals are given ± 1 standard deviation (SD).

Results

Polarity, latency and voltage of operculo-insular responses.

3.1.a. Responses from the supra-sylvian opercular cortex.

The CO 2 laser stimulus consistently evoked in the supra-sylvian cortex a N140 negative response followed by a P170 positivity (Figure 1). Latencies and voltages of these potentials are given in table 2. These responses, which have been described elsewhere in details (Frot et al. 1999[START_REF] Frot | Responses of the supra-sylvian (SII) cortex in humans to painful and innocuous stimuli. A study using intra-cerebral recordings[END_REF], were picked up by all of the 21 electrodes implanted in the supra-sylvian opercular cortex and were not recorded in the other explored areas including amygdala, hippocampus, anterior cingulate gyrus, temporal pole, orbito-frontal cortex and supplementary motor area (see Frot et al. 1999). Similar responses were equally recorded on the electrodes implanted in the homologous cortex, ipsilateral to the painful stimulus peaking 17 ± 7.5 ms (p= 0.003) and 16 ± 12.9 ms (p= 0.03) later than contralateral N140 and P170 respectively (Table 2). This latency difference between ipsi-and contralateral responses was not different for N140 and P170 potentials (p=0,75).

(Figure 1 near here)

3.1.b. Responses from the insular cortex.

Two potentials contralateral to stimulation were recorded in the insular cortex, consisting in a N180 negative response followed by a P230 positivity (Figure 1). The N180 and P230 responses, of which latencies and voltages are given in table 2, were recorded 49.5 ± 16.3 and 55.5 ± 13.5 ms later than the opercular N140 and P170 potentials respectively. The figure 2 illustrates these opercular and insular responses on several recordings of different patients.

(Figure 2 near here) Negative-positive LEPs were also recorded ipsilateral to stimulus by all of the contacts located in the insular cortex, they had a similar waveform as those recorded contralateral to stimulation but peaked 15 ± 6.6 ms and 17 ± 9.5 ms later than contralateral N180 and P230 respectively. This difference between contra-and ipsilateral responses was statistically significant for each of the two components (p = 0,02 for N180 and p = 0,03 for P230, Student's t test) but there was no difference in this inter-hemispheric transit times between N180 and P230 potentials (p = 0,78).

(Table 2 near here)

Stereotactic localization of the operculo-insular responses.

The maximal amplitude of the N/P deflection was taken to determine the contact likely to be the closer to the source.

3.2.a. Responses from the supra-sylvian opercular cortex.

The N140-P170 responses were recorded along the trajectory of all electrodes penetrating the suprasylvian opercular cortex between vertical planes at 14 mm rostral and 20 mm caudal (y coordinates) to the anterior commissure vertical plane (VAC), and between horizontal planes at 3 mm and 21 mm above (z coordinates) the horizontal anterior commissure -posterior commissure plane (AC-PC). These responses were picked up by the majority of these electrodes contacts, between 38 and 63.5 mm from the mid-sagittal vertical plane (x coordinates) (Table 3).

The cortical volume where opercular LEPs were distributed was assessed using the mean and SD values of x, y and z coordinates of the N140-P170. This volume was of 0.25 cm 3 using a confidence interval of ± 1 SD and of 1.98 cm 3 with a confidence interval of ± 2 SD (Figure 3).

3.2.b. Responses from the insular cortex.

The N180-P230 responses were picked up by the depth contacts of the electrodes penetrating the opercular cortex between vertical planes at 14 mm rostral and 25 mm caudal (y coordinates) to the anterior commissure vertical plane (VAC), and between horizontal planes at 1 mm below and 21 mm above (z coordinates) the horizontal anterior commissureposterior commissure plane (AC-PC). The contacts recording these responses were distributed between 29 and 45 mm from the median line (x coordinates). The average coordinates of these contacts are given in table 3. The cortical volume where insular LEPs were distributed was of 0.35 cm 3 using a confidence interval of ± 1 SD and of 2.8 cm 3 with a confidence interval of ± 2 SD (Figure 3).

(Figure 3 near here)

In the seven patients where the insular cortex was explored by several electrodes (two or three) along the rostro-caudal axis (y) we did not observe any intra-individual differences of the N180-P230 responses between contacts in terms of latency and amplitude. No significant correlation was observed between the localization of the implanted electrodes along the rostro-caudal axis (y) and the latencies of the N180 (r = 0.17, p = 0.5) and P230 (r = 0.003 , p = 0.99) responses (Figure 4). Moreover there was no significant correlation between the localization of the electrodes along the rostro-caudal axis (y) and the amplitudes of the N180 (r = 0.22, p = 0.39) and P230 (r = 0.22, p = 0.39) responses (Figure 4). Thus, we could not define distinct populations of nociceptive neurons along this axis only on the basis of the responses recorded in this cortex.

(Figure 4 near here)

As illustrated in figures 1 and 2 the first insular negativity peaked clearly later than the SII P170 potential (p=0,03, Student's t test). However in two cases (patients C and D in figure 2) this first insular negativity was preceded by a positive potential, the latency of which was the same as that of the SII N140 potential. In these two patients the polarity reversal was observed at the contact located at the border between the insular contact and the deepest SII contact so that it was difficult to decide on which side of the circular sulcus the N140 source was located. This difficulty had already been encountered in our previous studies (Frot et Mauguière 1999, Frot et al. 1999, 2001). Moreover, in one patient (patient E in figure 2) the insular N180 was preceded by a positivity peaking later than the SII N140 and a few milliseconds earlier than the SII P170 (7 ms for the frontal operculum and 14 ms for the parietal operculum). Due to the low spatial electrode sampling in the insular cortex we cannot formulate any firm hypothesis concerning this individual variation of insular responses.

However neither the N180 nor the P230 insular potentials showed any polarity reversal at more superficial contacts located in the SII area. Moreover, when the N180 was spreading to the SII contacts, such as in patient E (figure 2), its amplitude was regularly decreasing from the deepest insular contact to the more superficial SII sites. Therefore we considered that these two potentials were the only consistent components that could be considered as originating in the insula proper.

Discussion

In this study, we were able to separate two cortical areas in the operculo-insular cortex responsive to a painful laser stimulation (Figure 3): a supra-sylvian opercular area where we recorded a response peaking at 140-170 ms after a painful stimulation and a deeper insular area where we recorded a response with a similar pattern but peaking 180-230 ms after the stimulus. The latency range of the operculo-insular responses recorded directly in the cortex is coherent with that of the earliest CO 2 laser potentials recorded on the scalp. Several studies of scalp or subdural LEPs converged on the conclusion that the earliest cortical response consists of a negativity peaking between 150 and 170 ms and maximal in the scalp centro-temporal region contralateral to stimulation, compatible with a generator located in SII/insular areas [START_REF] Treede | Late somatosensory evoked cerebral potentials in response to cutaneous heat stimuli[END_REF][START_REF] Tarkka | Equivalent electrical source analysis of pain-related somatosensory evoked potentials elicited by a CO2 laser[END_REF][START_REF] Miyazaki | Generator mechanism of pain-related evoked potentials following CO2 laser stimulation of the hand: scalp topography and effect of predictive warning signal[END_REF][START_REF] Xu | Pain-related somatosensory evoked potentials following CO2 laser stimulation of foot in man[END_REF][START_REF] Valeriani | Scalp topography and dipolar source modeling of potentials evoked by CO2 laser stimulation of the hand[END_REF][START_REF] Lenz | Painful stimuli evoke potentials recorded from the parasylvian cortex in humans[END_REF]. This negativity was found to be associated with a mid-frontal positivity peaking around 200 ms in mapping studies [START_REF] Treede | Late somatosensory evoked cerebral potentials in response to cutaneous heat stimuli[END_REF][START_REF] Tarkka | Equivalent electrical source analysis of pain-related somatosensory evoked potentials elicited by a CO2 laser[END_REF][START_REF] Valeriani | Scalp topography and dipolar source modeling of potentials evoked by CO2 laser stimulation of the hand[END_REF], suggesting a dipolar source close to the sylvian fissure and tangent to the scalp surface. According to these studies, this dipolar source remains active at the latency of a later response recorded on the scalp at 250 ms after the painful stimulus, which is in the range of our insular positivity latency.

Is the N180-P230 actually generated in the insular cortex?

The question whether the insular responses might reflect the diffusion of the suprasylvian LEPs with a polarity reversal across the sylvian fissure deserved our attention.

Several arguments support the interpretation that the N140-P170 and the N180-P230 are independent responses. In most cases (8 patients) the suprasylvian P170 potential was clearly peaking before the first insular N180 negativity and its latency was significantly shorter by 27 ms (p= 0.0002) (26.8 ± 11.1 ms in average for these patients). In one patient we even recorded a polarity reversal of the supra-sylvian opercular P170 peaking at 162 ms into a negativity peaking with the same latency at the deepest insular contact, where the first insular negativity was peaking at 214 ms (Figure 2: Patient D). Moreover, the insular N180 negativity often spread to the supra-sylvian contacts (Figure 2: Patients B and E), and was then superimposed to the P170 potential (see Results section).

Since only a few contacts per electrode track were exploring the insula proper we were not in a position to show a polarity reversal of the N180-P230 between the surface and the depth of the insular cortex. However, when at least 2 contacts explored the insular cortex, we always recorded on these contacts the N180-P230 responses with a gradient of amplitude along the x axis (Figures 1 and2). Thus, in spite of inter-individual variations in waveforms, the suprasylvian and insular responses are unlikely to be generated by a single source in the upper bank of the suprasylvian fissure.

Location of LEPs activities in the operculo-insular cortex.

Our results are coherent with data of numerous imaging studies showing a double representation of pain in the operculo-insular cortex [START_REF] Coghill | Distributed processing of pain and vibration by the human brain[END_REF][START_REF] Casey | Comparison of human cerebral activation patterns during cutaneous warmth, heat and deep cold pain[END_REF][START_REF] Craig | Functional imaging of an illusion of pain[END_REF][START_REF] Andersson | Somatotopic organization along the central sulcus, for pain localization in humans, as revealed by positron emission tomography[END_REF][START_REF] Rainville | Pain affect encoded in human anterior cingulate but not somatosensory cortex[END_REF][START_REF] Svensson | Cerebral processing of acute skin and muscle pain in humans[END_REF][START_REF] Xu | Functional localization of pain perception in the human brain studied by PET[END_REF][START_REF] Svensson | Cerebral blood-flow changes evoked by two levels of painful heat stimulation : a positron emission tomography study in humans[END_REF][START_REF] Gelnar | A comparative fMRI study of cortical representations for thermal painful, vibrotactile, and motor performance tasks[END_REF]. As shown in table 4 the mean coordinates of the N140-P170 responses contralateral to stimulation are very similar those of the maximal blood-flow responses observed during different painful stimulations, even if our localization is often more anterior.

(Table 4 near here) The restricted spatial sampling along the rostro-caudal (y) and vertical (z) axis, as determined by the number of SEEG electrodes in each patient, could explain this discrepancy.

Nevertheless, the N140-P170 responses were distributed along the y axis between 14 mm rostral and 20 mm caudal to the anterior commissure vertical plane (VAC), thus including the majority of the sites with maximal blood-flow responses (see table 4). In the same way, the mean coordinates of the N180-P230 responses recorded in the insular cortex were very similar to those of the maximal blood-flow responses observed in PET and fMRI studies (see table 4). Moreover, the depth (x) coordinates of operculum and insula reported in PET studies are often on the wrong side of the circular sulcus (Table 4), whereas in this study great care was taken to have precise knowledge of which contacts were in the insula and which ones were in the operculum.

We could not define distinct functional regions in the insula along the y axis on the basis of latencies and amplitudes of pain EPs recorded in this cortex (see results). This observation is in line with data from two PET studies, using a CO 2 laser stimulation, showing a maximal blood-flow response either in the anterior [START_REF] Svensson | Cerebral processing of acute skin and muscle pain in humans[END_REF] or in the posterior insula [START_REF] Xu | Functional localization of pain perception in the human brain studied by PET[END_REF]) (See table 4). This suggests that a stimulus like CO 2 laser, which does not evoke any emotional or affective reaction, is able to activate a largely distributed functional region of the insular cortex.

Transcallosal inputs transmission

This study showed that a painful stimulation elicits responses in the contra and ipsilateral supra-sylvian and insular cortices separated by a delay of 11-18 ms. This delay is compatible with callosal transmission times estimated by numerous studies (e.g. 15 ms between primary visual areas [START_REF] Swanson | Lateral asymetries revealed by simple reaction time[END_REF])); it is in the same range as that measured between ipsi-and contralateral SII magnetic fields evoked by electrical stimulation of the median nerve (20 ms in [START_REF] Hari | Functional organization of the human first and second somatosensory cortices: a neuromagnetic study[END_REF], 10 ms in Mauguière et al. 1997), the shortest callosal transmission time between the two SI areas being estimated at 6-7 ms [START_REF] Noachtar | Ipsilateral median somatosensory evoked potentials recorded from human somatosensory cortex[END_REF]).

However, a callosal transfer from contralateral to ipsilateral supra-sylvian and insular cortices cannot be concluded only on the basis of this time difference. The possibility remains that responses ipsilateral to the stimulus could be triggered via ipsilateral thalamic fibers with slower conduction velocity. Only intracortical recordings of supra-sylvian or insular cortices evoked potentials to ipsilateral stimuli in patients with a lesion of the homologous areas in the opposite hemisphere could address directly this question.

Activation timing of the LEPs supra-sylvian and insular sources

If we consider that the N140-P170 and the N180-P230 are independent responses generated by different cortices, it remains to explain the delay of ~ 50 ms observed between supra-sylvian and insular responses. This delay could reflect the transmission time from the SII area to the insula, a hypothesis which is supported by the observation in monkeys that most of inputs reaching the insular cortex come from the SII area (Friedman and Murray 1986). However, this delay of 50 ms seems quite long for a monosynaptic transmission time between two close cortical areas known to be interconnected by direct projections (Friedman et al. 1986). Alternatively, knowing that both SII and the insula receive direct projections from the thalamus [START_REF] Friedman | Representation pattern in the second somatic sensory area of the monkey cerebral cortex[END_REF]Murray 1986, see also for a review [START_REF] Augustine | The insular lobe in primates including humans[END_REF][START_REF] Augustine | Circuitry and functional aspects of the insular lobe in primates including humans[END_REF], the explanation for the delay between supra-sylvian and insular responses could be that the latter are triggered via thalamo-cortical fibers with a slower conduction than that of thalamic projections to the SII area. To our knowledge, however, no electrophysiological demonstration of this hypothesis is hitherto available. A third hypothesis could be that the supra-sylvian cortex and the insula are activated by inputs conveyed by peripheral fibers with different conduction velocities. The CO 2 laser beam used in this study is known to stimulate the endings of small diameters fibers and mostly those of A fibers (Bromm and Treede, 1981[START_REF] Bromm | Nerve fibers discharges cerebral potentials and sensations induced by CO2 laser stimulation[END_REF], 1991). Some studies have estimated the A conduction velocity in a large range of 7 to 20 m/s (Adriaensen et al. 1983, Naka and[START_REF] Naka | Simple and novel method for measuring conduction velocity of A delta fibers in humans[END_REF] suggesting the existence of different A fibers sub-populations, with different conduction velocities. One can hypothesize that these different sub-populations of A fibers have selective connections with the spino-thalamic neurons in layers I, II and V of the spinal cord dorsal horn (for a review see Wolf 1994). If such an organization was maintained along the spinothalamic and thalamo-cortical tracts, one could assume that these different subpopulations of peripheral fibers could project in distinct cortical regions. However, to our knowledge, no electrophysiological study has been devoted to the identification of separate sub-populations of fibers with different conduction velocities in the spinothalamic tract or thalamocortical projections. It remains that, in spite of conjectures about their parallel or sequential activation by pain inputs, the SII and insular areas are those showing the earliest responses to pain and are both receiving projections from the posterior thalamus. responses. The data have been normalized according to the maximal latency or amplitude of each component recorded in each patient. For example, the patient A had three electrodes implanted in the insular cortex (each of them with a different y coordinate, i.e. -6 mm, 2 mm and 6 mm), on which different latencies of the N180 response were recorded; i.e. 177 ms, 183 ms and 180 ms respectively. For this patient, we normalized the data according to the maximal amplitude of the response, which is 183 ms. Thus we had a latency of '0.96' for y= -6mm, a latency of '1' for y=2mm and a latency of '0.98' for y=6mm.

Figures:Figure 1 :

 1 Figures:

Figure 2 :

 2 Figure 2: Contralateral LEPs recorded in the operculo-insular cortex of five patients (earlobe

Figure 3 :

 3 Figure 3: Mean distribution of the contralateral N140-P170 and N180-P230 responses,

Figure 4 :

 4 Figure 4: Correlation between the localization of the implanted electrodes along the rostro-

Acknowledgements:

Grateful thanks are due to Dr Marc Guénot (Dept of Functional Neurosurgery: Pr M.

Sindou) for stereotactic electrode implantation.


L García-Fernández

P Fruton

Henri Bataller

J M Ortiz De Zárate

Fabrizio Croccolo

Coupled non-equilibrium fluctuations in a polymeric ternary mixture

Keywords: Multicomponent mixtures, Polymer, Thermodiffusion, Shadowgraphy, Nonequilibrium, Fluctuations

come

Introduction

The majority of the fluids encountered in nature and industry are multicomponent mixtures.

The knowledge of their transport properties is crucial for many different applications from both scientific and technological points of view. The efficiency of exploitation of the crude oil wells, as well as the environmental alternative focused on the CO2 storage in deep brine aquifers, are some recent examples showing the interest of studying the transport properties of multicomponent mixtures subjected to non-equilibrium conditions [1,2]. Complex fluids in non-equilibrium conditions exhibit giant fluctuations of the thermodynamic variables, the socalled non-equilibrium fluctuations (NEFs) [START_REF] Ortiz De Zarate | Hydrodynamic Fluctuations in Fluids and Fluid Mixtures[END_REF][START_REF] Croccolo | [END_REF][5]. By analysing NEFs through light scattering techniques, the transport properties of the fluid can be determined both at atmospheric [6] and at high pressure [7].

The non-equilibrium condition can be induced, for example, by applying a temperature gradient to a multicomponent fluid mixture, thus inducing a composition gradient within the fluid by means of the thermodiffusion, or Soret effect [8][9][10]. This transport process can be investigated by optical techniques and particularly by light scattering thanks to its ability to visualize NEFs without altering the intrinsic properties of the fluid. In this work, dynamic shadowgraphy has been adopted [11][12][13] to study the fluctuations of the refractive index as generated by the NEFs of the thermodynamic variables, like temperature and concentrations. By shadowgraphy, a large range of fluctuations sizes 𝜆 or, conversely, wave numbers 𝑞 = 2𝜋 𝜆 ⁄ , can be investigated at the same time. This approach, combined with statistical analysis, represents a powerful characterization technique able to provide simultaneous reliable measurements of different transport properties, like mass diffusion coefficient or thermal diffusivity as well as Soret coefficient [6,14,15].

A suitable characterization of transport processes in complex mixtures requires a deep understanding of simpler fluids. So far, only binary mixtures [10,16] have been extensively

characterized. The extension of theories and experiments from binary to ternary mixtures requires further development due to the intrinsic and significant increase of difficulty with the number of components of the mixture. Currently, a great effort is devoted to investigate the transport phenomena in ternary mixtures, as performed in the present study. Thermodiffusion experiments in ternary mixtures are performed on ground, by using different optical techniques [10,14,17,18], or in microgravity in order to avoid both convection and sedimentation. The thermodynamic characterization of ternary mixtures is one of the objectives of the following ESA projects: Diffusion Coefficient Measurements in ternary mIXtures (DCMIX) [18][19][20][21][22][23],

Soret Coefficients in Crude Oil (SCCO) [1,[24][25][26][27] and Giant Fluctuations [28], as further elaborated in a recent colloquium paper [29].

As a benchmark ternary mixture, the tetraline, isobutylbenzene and n-dodecane sample has been studied by digital interferometry on ground and in microgravity within the DCMIX experiments [18,19,[30][31][32][33][34][35][36][37][38]. In addition, Bataller et al. [17] investigated this mixture on ground by onewavelength shadowgraphy. Since no other alternative was available at that time, Bataller et al.

[17] compared their experimental results with the theory of concentration NEFs developed for ternary mixtures in the absence of gravity [39]. The two concentration modes related to the presence of the two independent components could not be distinguished experimentally, and only an average of the two eigenvalues of the mass diffusion matrix could be provided. This result is expected, as the mixture is composed of similar-size molecules, and consequently the two eigenvalues of the diffusion coefficient matrix are very similar to each other. The two eigenvalues 𝐷 ̂1 and 𝐷 ̂2 of the diffusion matrix 𝐃 need to be different enough to distinguish the two concentration modes. An equivalent way to set this statement is to define a dimensionless number 𝐷𝑟 = 𝐷 ̂2 𝐷 ̂1 ⁄ (𝐷𝑟 for Diffusion eigenvalue Ratio and 𝐷 ̂1 representing the slowest diffusion mode) and require it to satisfy the relation 𝐷𝑟 ≫ 1. A ternary mixture containing a polymer dissolved in a binary mixed solvent is a good candidate for such measurement, due to the very different molecular size of its components and, correspondingly, a value of 𝐷𝑟 ≈ 10.

This was experimentally proved for the diluted polymeric ternary mixture of polystyrenetoluene-n-hexane, as reported in [14], where a layer of 𝐿 = 2 𝑚𝑚 thick was investigated by using one-wavelength shadowgraphy at a single image acquisition frequency of 28 Hz. By adopting the theoretical model of ternary mixtures in the absence of gravity, the dynamics of the NEFs could be analysed for large wave numbers [14]. From such analysis, it was only possible to determine the two eigenvalues of the mass diffusion matrix and the thermal diffusivity. Afterwards, the diffusion matrix was simplified by neglecting the off-diagonal elements and the result was utilised for obtaining an analytical relationship between the two Soret coefficients, without an independent measurement of them. More recently [40], we have further developed the theory of NEFs to include the effect of gravity, but not confinement so that a direct comparison with the data of Ref. [14] is impossible since fluid layer thickness of only 2 mm makes confinement effects important, as evidenced in related literature for binary mixtures [41,42].

In the present study, we perform a new shadowgraph experiment on a similar ternary mixture, but in this case a thicker sample with vertical extension of 𝐿 = 5 𝑚𝑚 is analysed in order to avoid the confinement effect, and be able to compare the experimental results with the theory of NEFs including gravity [40]. The present measurements are also carried out with a faster camera, allowing a more accurate characterization of thermal fluctuations and, thus, a better separation of them from concentration ones. Additionally, a new procedure of image acquisition combined with the concatenation of the structure functions allows us to investigate a wider range of correlation times, as required for a complete analysis of this mixture that has a wide range of decay times. The experimental results are thus compared for the first time with the theoretical model developed for ternary mixtures in the presence of gravity [40], as well as with the recently developed theory that includes the coupling of velocity, thermal and solutal NEFs in the case of binary mixtures [43]. These comparisons suggest that a further development of the theory is required, in particular to include the coupling between the different NEFs in the case of the ternary mixture. As part of the contents of this paper we present such a development and compare with the experimental data.

Experimental procedure

Polymeric solution preparation and characterization

The sample is prepared by dissolving 2 wt% of polystyrene (PS, 𝑀 𝜔 = 4730 𝑔 𝑚𝑜𝑙 ⁄ , DIN-Poly(styrene) 4730 DA -PDI 1.03 by PSS-polymer) in 49 wt% toluene (Sigma-Aldrich, 24,451-1, 99.8%) and 49 wt% n-hexane (Sigma-Aldrich, 13,938-6, >99%) at ambient temperature by using a magnetic stirrer. The thermophysical properties of the mixture, such as density, viscosity and both thermal and mass expansion coefficients are determined.

The kinematic viscosity  is measured at 25 o C by a capillary viscometer (Ubbelohde SCHOTT). The thermal and mass expansion coefficients, 𝛼 and 𝛽 𝑖 , respectively, are defined as follows:

𝛼 = - 1 𝜌 ( 𝜕𝜌 𝜕𝑇 ) 𝜔,𝑝 , (1)
𝛽 𝑖 = 1 𝜌 ( 𝜕𝜌 𝜕𝜔 𝑖 ) 𝑇,𝑝 , ( 2
)
where 𝜌 is the mixture density, 𝑇 is the temperature and 𝜔 𝑖 is the concentration of component 𝑖 in mass fraction. Following a common notation, components are ordered according to their (pure) density, hence, in this study 1 corresponds to PS, 2 corresponds to toluene and 3 corresponds to n-hexane.

Thermodiffusion experiment

The experimental set-up is quite similar to the one used in previous studies and a more detailed description can be found elsewhere [6,7]. Before filling our thermodiffusion cell, its interior is evacuated by using a vacuum pump for more than four hours in order to evaporate all the residual fluids. Subsequently, the mixture is injected into the cell. The sample is sandwiched between two square sapphire windows of 8 mm thick, mechanically separated to provide a vertical thickness of the fluid layer of 𝐿 = (5.0 ± 0.1) 𝑚𝑚. Two thermoelectric (TEC) Peltier devices are used to apply the temperature gradient to the fluid mixture. Both TECs are independently connected to two proportional-integral-derivative (PID) temperature controllers (Wavelength Electronics, MODEL LFI -3751), being able to set the temperatures of the two sides of the fluid slab with an absolute accuracy of 0.01 K and a relative RMS stability of 1 mK over 24 hours. The temperature gradient is applied in the 𝑧 direction, and the sample is always heated from above in order to avoid convection, since both Soret coefficients are expected to be positive [44][45][46]. When applying a temperature gradient, the two TECs actually act as heat pumps in the same direction, thus a heat exchanger flushed with water coming from a thermostat (HUBER, ministate 125) removes the excess heat from the external side of the Peltier elements.

The thermodiffusion experiments are performed by imposing temperature differences of ∆𝑇 = Dynamic Algorithm (DDA) briefly described in the following section [48].

Dynamic shadowgraphy

The DDA analysis consists of the following steps: images are recorded at different times as bidimensional intensity maps 𝐼(𝑥 ⃗, 𝑡) and are 2D-Fourier transformed and normalized to get The concatenation procedure has been already applied in our precedent work [43], but here it is described in more detail. We point out that this scheme becomes useful whenever it is necessary to investigate NEFs with quite different decay times and that it is similar to the multi-tau approach utilised in dynamic light scattering [50].

Figure 1: Structure function of the thermodiffusion experiment carried out at temperature difference of 20 K for a series of images of 2048 × 2048 pixels at 100, 10 and 1 Hz, as a function of the wave number (a, c, e, respectively) and as a function of correlation time (b, d, f, respectively). Concatenated structure function as a function of the wave number (g) and as a function of time (h). Purple squares, cyan circles and orange triangles correspond

to the data acquired at 100, 10 and 1 Hz, respectively.

Analysis of experimental signals

After the DDA and the concatenation process one gets the c-SF; it represents the raw experimental signals, obtained only from the acquired image series. For the analysis of these concatenated 〈|∆𝑖(𝑞, 𝑑𝑡)| 2 〉, the physical optics theory of shadowgraph proposes the following expression [12,48]:

〈|∆𝑖(𝑞, 𝑑𝑡)| 2 〉 = 2{𝑇(𝑞)𝑆(𝑞)[1 -𝐼 𝑆𝐹 (𝑞, 𝑑𝑡)] + 𝐵(𝑞)}, (3)
where 𝑇(𝑞) is the shadowgraph optical transfer function [11], 𝑆(𝑞) is the static power spectrum, A simple observation of Fig. 1h, without any calculation, shows that, depending on the wave number range, up to three different decay times can be distinguished in the c-SF. In some cases, these decay times differ by more than one order of magnitude. In addition, and also depending on the wave number range, one can observe in Fig. 1h damped oscillations in 〈|∆𝑖(𝑞, 𝑑𝑡)| 2 〉.

𝐵
This oscillating (propagating) phenomenon means that at least two of the relaxation times 𝜏 𝑖 (𝑞)

present in the 𝐼 𝑆𝐹 (𝑞, 𝑑𝑡) of Eq. 3 (and their associated amplitudes) form a pair of complex conjugate numbers. We conclude that a complex scenario of NEFs with different decay times is present in the case of a ternary mixture with large 𝐿𝑒 and 𝐷𝑟 numbers, and a key question is the selection of the number of modes in the 𝐼 𝑆𝐹 (𝑞, 𝑑𝑡) of Eq. 3 and whether they are purely diffusive (monotonically decaying) or propagating (oscillating). A rational choice needs to be guided by the available theoretical models for NEFs in ternary mixtures, which we review next.

Theoretical framework

Thermodiffusion in ternary mixtures

In a ternary mixture, the mass concentrations of the three components are related by: 𝜔 1 + 𝜔 2 + 𝜔 3 = 1, which means that only two concentrations, e.g. 𝜔 1 and 𝜔 2 , are independent. The corresponding diffusion fluxes 𝐽 1 and 𝐽 2 in the centre-of-mass frame of reference have thermodiffusion and Fickean components, as expressed by [51]:

( 𝐽 1 𝐽 2 ) = -𝜌 [( 𝐷 11 𝐷 12 𝐷 21 𝐷 22 ) ( ∇𝜔 1 ∇𝜔 2 ) + ( 𝜔 1 (1 -𝜔 1 ) -𝜔 1 𝜔 2 -𝜔 1 𝜔 2 𝜔 2 (1 -𝜔 2 ) ) ( D 𝑇1 D 𝑇2 ) ∇𝑇], (4)
where 𝜌 is the density of the mixture, 𝐷 𝑖𝑗 are the components of the centre-of-mass diffusion matrix, ∇𝜔 𝑖 are the concentration gradients, 𝐷 𝑇 𝑖 are the invariant thermodiffusion coefficients and ∇𝑇 is the applied temperature gradient.

At the stationary state, the diffusion fluxes become zero and, following Ortiz de Zárate [51],

the Soret coefficients 𝑆 𝑇 𝑖 can be expressed by:

( 𝜔 1 (1 -𝜔 1 ) -𝜔 1 𝜔 2 -𝜔 1 𝜔 2 𝜔 2 (1 -𝜔 2 ) ) ( S 𝑇1 S 𝑇2 ) ∇𝑇 = -( ∇𝜔 1 ∇𝜔 2 ), (5)
The introduction of the matrix concentration pre-factors in Eqs. 4 and 5 forces the thermodiffusion coefficients 𝐷 𝑇 𝑖 and the Soret coefficients 𝑆 𝑇 𝑖 to display the same frameinvariance properties of the single 𝐷 𝑇 or 𝑆 𝑇 of a binary mixture, that is, they are numerically the same irrespective of whether mass fractions or mole fractions are used to express the composition [51]. Here, for simplicity, we display the equations only in the mass fraction framework.

Fluctuating Hydrodynamics (FHD) has been recently extended to ternary mixtures first considering equilibrium fluctuations [52] and then non-equilibrium fluctuations [39,40]. The first NE theory [39] neglected the effects of gravity and confinement. In that case, there is no mixing between the decay rates of the four hydrodynamic modes (fluctuations of velocity, temperature and two concentrations) which continue to decay with their well-known equilibrium values given by kinematic viscosity, thermal diffusivity and the two eigenvalues 𝐷 ̂𝑖 of the diffusion matrix 𝐃. These 𝐷 ̂𝑖, are always real and positive, and can be related to the four components 𝐷 𝑖𝑗 of Eq. 4 by [39,40]:

𝐷 ̂1,2 = 1 2 [𝐷 11 + 𝐷 22 ∓ √(𝐷 11 -𝐷 22 ) 2 -4𝐷 12 𝐷 21 ]. (6)
In accordance with the aforementioned common notation (i.e. 1 for PS, 2 for toluene and 3 for n-hexane), 𝐷 ̂1 < 𝐷 ̂2 since 𝐷 ̂1 represents the slowest diffusion mode essentially related to the diffusion of the polystyrene in the binary solvent.

Since refractive index does not depend on fluid velocity, only three of the four hydrodynamic modes are directly observable by optical techniques. Hence, within the approximations of Ref.

[39] (no gravity and no confinement) we conclude that for a thermodiffusion experiment in a ternary mixture, a suitable expression for the 𝐼 𝑆𝐹 (𝑞, 𝑑𝑡) can be provided by the sum of three exponential decays given by two concentration modes, plus one mode for temperature fluctuations:

𝐼 𝑆𝐹 (𝑞, 𝑑𝑡) = 𝑎 1 exp[-𝑑𝑡 𝜏 1 (𝑞) ⁄ ] + 𝑎 2 exp[-𝑑𝑡 𝜏 2 (𝑞) ⁄ ] + (1 -𝑎 1 -𝑎 2 )exp[-𝑑𝑡 𝜏 3 (𝑞) ⁄ ], (7)
where 𝑎 𝑖 are the normalised amplitudes of the different modes and 𝜏 𝑖 (𝑞) the corresponding relaxation times.

Concentration NEFs in ternary mixtures including gravity effects

For NEFs, first the theory was developed in the absence of gravity force [39]; while gravity effects were incorporated by Martínez Pancorbo et al. [40]. Both of these papers adopted the so-called large Lewis number approximation that uncouples temperature fluctuations from concentration fluctuations. Next, we briefly summarize the main results of Ref. [40] and recall the expressions useful for the experimental data analysis.

The presence of gravity causes a mix between the two concentrations NEFs. The corresponding decay times can be described as 𝜏 𝑖 (𝑞) = 1 [𝛾 𝑖 (𝑞)𝑞 2 ] ⁄ , where the diffusivities 𝛾 𝑖 (𝑞) in dimensionless form, are given by the following equation [40]:

𝛾 𝑖 (𝑞 ̃) = 1 2 (𝐷 ̂1 + 𝐷 ̂2 - 𝑅𝑎 𝑖 𝐷 ̂𝑖 𝑞 ̃4 ) × [1 ∓ √1 - 4𝐷 ̂1𝐷 ̂2 (𝐷 ̂1+𝐷 ̂2) 2 [1- 𝑅𝑎 𝑆,𝑡𝑒𝑟 𝑞 ̃4 ] [1- 𝑅𝑎 𝑖 𝐷 ̂𝑖 𝑞 ̃4(𝐷 ̂1+𝐷 ̂2) ] 2 ], (8)
where 𝑞 ̃= 𝑞𝐿 is the dimensionless wave number, and 𝑅𝑎 𝑖 are the two solutal Rayleigh numbers defined for ternary mixtures by:

𝑅𝑎 𝑖 = 𝑔𝐿 4 𝜈𝐷 ̂𝑖 𝛽′ 𝑖 ∇𝜔′ 𝑖 , (9)
𝑅𝑎 𝑖 𝐷 ̂𝑖 = 𝑅𝑎 1 𝐷 ̂1 + 𝑅𝑎 2 𝐷 ̂2, (10)
𝑅𝑎 𝑆,𝑡𝑒𝑟 = 𝑅𝑎 1 + 𝑅𝑎 2 , (11)
In Eq. 9, the primes indicate the concentrations (mass fractions) in which the diffusion matrix is diagonal, for a generic diffusion matrix [53]:

( 𝜔′ 1 𝜔′ 2 ) = ( 1 𝐷 22 -𝐷 ̂2 𝐷 21 𝐷 11 -𝐷 ̂1 𝐷 12 1 ) ( 𝜔 1 𝜔 2 ). (12)
The use of "diagonal" concentrations simplifies the theory but, as we shall see, somewhat complicates the interpretation of experimental results. In general, they differ from the measurable mass fractions.

In summary, in this model [40] the two concentration modes couple, while temperature fluctuations remain uncoupled, keeping their equilibrium decay rate unaltered. Since for typical values of 𝑅𝑎 𝑖 the two decay times of Eq. 8 are real numbers, this theory [40] suggests to still use Eq. 7 for the 𝐼 𝑆𝐹 (𝑞, 𝑑𝑡). We note that the notation of Eqs. 9-11 implies that, for a ternary mixture, two Lewis numbers (𝐿𝑒 1 and 𝐿𝑒 2 ) can be defined as 𝐿𝑒 𝑖 = 𝑎 𝑇 𝐷 ̂𝑖 ⁄ .

Coupling of concentration and temperature NEFs in binary mixtures

In the simplest case of binary mixtures, most of the available literature on concentration and temperature NEFs makes use of the simplification that the two modes are always decoupled. This is essentially true whenever the large Lewis number approximation is valid, that is the case of ordinary liquid mixtures whose mass diffusion coefficient is much smaller than the thermal diffusivity. However, the decoupling assumption fails to be valid for very large fluctuations, i.e. for small wave numbers, as recently demonstrated both experimentally and theoretically [43]. In particular, only the full coupling of velocity, temperature and concentration NEFs in the bulk fluid can explain the appearance of propagating modes giving rise to a 𝐼 𝑆𝐹 (𝑞, 𝑑𝑡) that includes oscillations at very small wave numbers. This mechanism also induces a slowing-down of the non-propagating concentration NEFs similar to that caused by the confinement effect [41,42].

This theory of full coupling between NE velocity, temperature and concentrations fluctuations has been published only for binary mixtures. We briefly recall here the main steps of the theoretical framework, while for details we address the reader to the literature [43].

In order to take into account a more complete description of the behaviour of NEFs within a thermodiffusion experiment, one should write the evolution equations of the fluctuations of velocity 𝑣 𝑧 , temperature and concentration of the single independent component:

where the inverse linear response matrix in its dimensionless form can be written as:

G(𝑞 ̃) = [ 𝑃𝑟𝐿𝑒𝑞 ̃2 -1 1 -𝑅𝑎 𝑠 𝑃𝑟𝐿𝑒 𝜓 ⁄ 𝐿𝑒𝑞 ̃2 0 𝑅𝑎 𝑠 𝑃𝑟𝐿𝑒 𝜓𝑞 ̃2 𝑞 ̃2], (14)
In the previous equation the well-known dimensionless numbers appear, namely the Lewis number 𝐿𝑒 = 𝑎 𝑇 𝐷 ⁄ , where 𝑎 𝑇 is the thermal diffusivity; the Prandtl number 𝑃𝑟 = 𝜈 𝑎 𝑇 ⁄ ; the separation ratio 𝜓 = 𝑐 0 (1 -𝑐 0 )𝑆 𝑇 𝛽 𝛼 ⁄ ; and the solutal Rayleigh number 𝑅𝑎 𝑆 =

[𝛽𝑔∇𝜔 (𝜈𝐷) ⁄ ]𝐿 4 .

The solution of Eqs. 13 and 14 requires the computation of the eigenvalues and eigenvectors of the matrix G(𝑞 ̃):

det[𝐆(𝑞 ̃) -𝜆𝟏] = [Γ 𝜈 𝑧 (𝑞 ̃) -𝜆][Γ 𝑇 (𝑞 ̃) -𝜆][Γ 𝜔 (𝑞 ̃) -𝜆], (15)
with NE fluctuations dimensionless decay times given by 𝜏 𝑖 (𝑞 ̃) = 1 [Γ 𝑖 (𝑞 ̃)] ⁄ . Although Eq. 15 can be solved analytically, the resulting expressions of the roots as a function of q are so complicated that, in practice, it is more useful to have a numerical solution for fixed values of the parameters 𝑃𝑟, 𝐿𝑒 , 𝑅𝑎 𝑆 and 𝜓, in the whole range of 𝑞.

Depending on the parameter values and on the wave number 𝑞, Eq. 15 may have a pair of complex conjugate solutions, which typically happens at small 𝑞 values. The resulting 𝐼 𝑆𝐹 (𝑞, 𝑑𝑡) function in those cases shows a sinusoidal term:

𝐼 𝑆𝐹 (𝑞, 𝑑𝑡) = 𝑎 1 exp[-𝑑𝑡 𝜏 1 (𝑞) ⁄ ] + { 1 -𝑎 1 cos[𝜙(𝑞)] } cos[Ω(𝑞)𝑑𝑡 + 𝜙(𝑞)] exp[-𝑑𝑡 𝜏 2 (𝑞) ⁄ ] , (16)
where Ω(𝑞) is the oscillation frequency and 𝜙(𝑞) a phase term.

Coupling of concentration and temperature NEFs in ternary mixtures

The next step in the development of FHD is to include all the couplings among the fluctuating modes for a ternary mixture. Of course, in that case one needs to consider the evolution of fluctuations in velocity 𝛿𝑣 𝑧 , temperature and concentration of the two independent components:

where the inverse linear response matrix for "diagonal" concentrations can be written in dimensionless form as:

𝐆 𝒕 (𝑞 ̃) = [ 𝑃𝑟𝐿𝑒 𝑡 𝑞 ̃2 -1 1 1 -𝑃𝑟𝐿𝑒 𝑡 2 𝑅𝑎 𝐿𝑒 𝑡 𝑞 ̃2 0 0 𝑃𝑟𝐿𝑒 𝑡 2 𝑅𝑎𝜓 1 ′ 𝜓 1 ′ 𝑞 ̃2 𝑞 ̃2 0 𝑃𝑟𝐿𝑒 𝑡 2 𝑅𝑎𝜓 2 ′ 𝐷𝑟𝜓 2 ′ 𝑞 ̃2 0 𝐷𝑟𝑞 ̃2] . ( 18
)
To make dimensionless Eqs. 17 and 18, L is used as unit of length and 𝐿 2 coefficients as defined in Ref. [18]. We point out that this approach is different from the one used in our previous publication [40] and reported in section 3.2, where two Lewis numbers were introduced.

As in the case of binary mixtures, the solution of Eq. 17 requires to numerically evaluate the dimensionless decay rates Γ 𝑖 (𝑞 ̃) as the four eigenvalues of the matrix 𝐆 𝒕 (𝑞 ̃), namely:

det[𝐆 𝒕 (𝑞 ̃) -𝜆𝟏] = [Γ 𝜈 𝑧 (𝑞 ̃) -𝜆][Γ 𝑇 (𝑞 ̃) -𝜆][Γ 𝜔 1 (𝑞 ̃) -𝜆][Γ 𝜔 2 (𝑞 ̃) -𝜆]. (19)
A more detailed explanation of the full-coupling theory for ternary mixtures is outside the scope of the present publication and will be the subject of a future paper. We simply mention that, depending on 𝑞, complex solutions of Eq. 19 may exist. As was the case with binaries, for parameter values typical of ternary liquid mixtures, these propagating modes do appear as a mixing between the temperature and the viscous (velocity) modes. The consequence is an oscillatory 𝐼 𝑆𝐹 (𝑞, 𝑑𝑡) such as the one expressed in Eq. 16, but with an additional exponential term to represent the extra concentration mode.

Results

Intermediate scattering function analysis

As mentioned in section 2.5, for most wave numbers the 𝐼 𝑆𝐹 (𝑞, 𝑑𝑡) of Eq. 3 is usually defined as a sum of exponential decays. However, a preliminary analysis of the experimental SF is required to check the suitability of this model. As it can be seen in the c-SF shown in Fig. 1h for 𝑞 = 94/𝑐𝑚, three exponential decays can be detected, corresponding (from left to right) to the relaxation of the temperature and concentration (faster and slower mass modes) NEFs.

Then, the 𝐼 𝑆𝐹 (𝑞, 𝑑𝑡) can be modelled for most wave numbers by a sum of three exponential decays, as defined in section 3.1 by Eq. 7 for a system including one thermal mode plus two solutal, well separated, modes. In the studied system, the corresponding three exponential decays represent the following three modes: long decay times for the two concentration modes, slower for mass diffusion of the denser component (PS in the binary solvent), and faster for mass diffusion of the two components of the solvent mixture (toluene and n-hexane); and the shortest decay time corresponds to the thermal diffusivity of the overall mixture.

Nevertheless, for small wave numbers such as 𝑞 = 28/𝑐𝑚, oscillations are detected in the c-SF (see Fig. 1h), indicating the presence of propagating modes [43]. Fig. 2 shows the oscillations in the experimental c-SF for several wave numbers in the range 19/𝑐𝑚 < 𝑞 < 47/𝑐𝑚. These need to be taken into account by using a proper 𝐼 𝑆𝐹 (𝑞, 𝑑𝑡) function, such as the one described in section 3.3 by Eq. 16. The latter equation is defined for a binary mixture, and includes one solutal mode, i.e. the exponential decay, and a coupled thermo-viscous mode, i.e.

the damped oscillation. For analysing ternary mixtures, one should supplement Eq. 16 with an additional exponential term, representing the second solutal mode. However, after carrying out a careful comparison of the corresponding fitting output, we have found that the addition of this term gives more scattered results due to the increase in the number of fitting parameters. For that reason, Eq. 16 has been used to fit the raw data in the wave number range where oscillations are detected in the c-SF. The slower concentration mode is, thus, unobservable in this wave number range. Generally, the quality of these fittings is good, as demonstrated in Fig. 2, with an average R 2 of about 0.99, which translates into a 3% mean relative uncertainty for the decay times determined by these fittings. However, both R 2 and the relative uncertainty of the decay times depend on the wave number. The best accuracy is obtained for intermediate wave numbers where the signal-to-noise ratio of the c-SF is the largest (see Fig. 1g). Decay times with a relative uncertainty larger than 10% have been disregarded for the further analysis.

Decay times of the NEFs

Figure 3 shows the decay times of the different fluctuating modes as resulting from the modelling of the experimental c-SF by using Eqs. 3 and Eq. 7 or Eq. 16 depending on the analysed q. Three decay times are easily distinguished (from top to bottom): the slowest mode (black symbols) describes the concentration mode of the polymer into the binary solvent; the intermediate one (blue symbols) the concentration mode of the binary solvent mixture; and the fastest one (red symbols) the temperature mode. These decay times cover a span of about three orders of magnitude, which is quite impressive for an imaging technique. This justifies the necessity of performing measurements at different acquisition frequencies and of carrying out the concatenation procedure adopted here. Besides decay times, when the Eq. 16 is used, the fitting procedure also gives the oscillation frequency Ω(𝑞) of the propagating modes appearing in Fig. 2 for wave numbers 𝑞 < 47/𝑐𝑚.

Figure 3: Decay times of NEFs on the ternary mixture of PS-toluene-n-hexane in a thermodiffusion experiment performed at temperature difference of 20 K, as a function of the wave number q. Lines represent fitting functions through Eqs. 20-21 over the diffusive wave number range, as indicated in the text.

The experimental decay times shown in Fig. 3 are to be discussed first in the diffusive regime only (for large enough wave numbers), where the three decay times are well-separated. As elucidated at the end of section 3.1, in this region of large wave numbers the effect of gravity is negligible for both concentration modes, and the decay times of the two concentration modes are given by the eigenvalues of the diffusion matrix: 2. These values in Table 2 are calculated by averaging the fitting results

𝜏 𝑖 (𝑞) = 1 (𝐷 ̂𝑖𝑞 2 ) ⁄ . ( 20
of the experiments performed at three temperature differences, 20, 15 and 10 K, and the uncertainties are the corresponding standard deviations. These values are comparable with those obtained in our previous study [14] for a similar polymeric mixture prepared with the same components of this study but a slightly different polymer molecular weight, and with those reported in [43,45] for the diluted binary mixture of PS-toluene and in [44,46] for the binary solvent mixture of toluene-n-hexane. Furthermore, the value of the 𝐷𝑟 number can be determined as the ratio between the two eigenvalues, resulting in 𝐷𝑟 = 8.4 (see Table 2). This value totally justifies the selection of the polymeric mixture as a suitable sample for experimentally studying well-separated concentration modes.

In a similar way, in order to get information about the thermal diffusivity 𝑎 𝑇 of the ternary mixture, one can try to fit the data points of the thermal mode with the theoretical predictions for thermal NEFs only. In the diffusive regime, the decay time of the thermal mode can be expressed as [54]:

𝜏 𝑡ℎ (𝑞) = 1 (𝑎 𝑇 𝑞 2 ) ⁄ , (21)
and the fitting of temperature NEFs for 𝑞 > 90/𝑐𝑚 in Fig. 3 provides the thermal diffusivity value reported in Table 2. This result is again comparable with the one obtained in previous studies for a similar polymeric mixture [14] as well as with that reported in [46] for the equimolar toluene-n-hexane binary solvent mixture.

From the two diffusion eigenvalues and the thermal diffusivity, the 𝐿𝑒 𝑖 numbers can be determined, which are also listed in Table 2.

Discussion

Comparison between experimental and theoretical decay times

As described in section 3.2, Martínez-Pancorbo et al. [40] presents the analysis of the fluctuations of two independent concentrations together with the effect of buoyancy, but considering neither the effect of confinement nor the presence of temperature and velocity fluctuations. Consequently, the coupling between modes which is known to be important at small wave numbers, as demonstrated for binary mixtures [43], is not taken into account. In order to be insensitive to confinement effects the experiments reported here have been performed in a thick cell with vertical dimension of 𝐿 = 5 𝑚𝑚. In these conditions, the effect of confinement should not be visible in the investigated wave number range. In fact, confinement effects appear only at dimensionless wave numbers 𝑞 ̃= 𝑞𝐿 < 5, corresponding here to dimensional wave numbers 𝑞 < 10/𝑐𝑚, at the very bottom of the investigated range of q [41,42].

Regarding coupling effects, it is mandatory to consider them whenever oscillations are visible in the experimental c-SF at small 𝑞 (see Fig. 2). The complete FHD theory, including the presence of velocity and temperature fluctuations and the various couplings among all modes, has been previously published by us [43] for the case of binary mixtures, as summarised in section 3.3. The results of [43] can be very useful for discussing the impact of the velocitytemperature coupling on the measured decay times.

In the following, we compare the experimental values of decay times at different wave numbers (obtained by fitting the c-SF through Eqs. 3 and 7 or 16), with the two published models [40,43]. This allows to understand the impact of the different effects on the measured c-SF, and to evaluate if a more complete theoretical framework is needed or if on the contrary, the available theories can be simply applied in different wave number ranges in order to extract the transport properties of the investigated ternary mixture. As described in the previous section, at least the diffusion coefficients can be safely determined by fitting decay times at large wave numbers,

where NEFs are dominated only by the diffusion phenomenon. The measurement of the Soret coefficient is more complicated as it requires the analysis of the effect of gravity [6], or other suitable analysis [55]. The simultaneous and independent characterization of the two Soret coefficients of a ternary mixture requires a complete knowledge of the diffusion matrix including the cross-diffusivities. This has not been achieved yet by Shadowgraphy and is the ultimate goal of our current research. Although we estimate that this can be done with onewavelength Shadowgraphy, a more precise information can be obtained with a two-wavelength version of the technique. The two-wavelength Shadowgraphy is currently being developed in our lab, also in relation to the future space experiments of the NEUF-DIX project [28].

Figure 4 shows the experimental decay times and the oscillation frequency obtained from the c-SF (data points are the same as in Fig. 3), together with theoretical predictions provided either by Eq. 8 (dashed yellow curves) or by numerically solving Eqs.14-15 (solid green curves). The experimental data are fitted to Eq. 8 by fixing the diffusion matrix eigenvalues (𝐷 ̂1 and 𝐷 ̂2)

to the corresponding values presented in Table 2. The parameters obtained from this fitting to Eq. 8 are: 𝑅𝑎 𝑖 𝐷 ̂𝑖 = (260 ± 40) 𝑐𝑚 2 𝑠 ⁄ , and 𝑅𝑎 𝑆,𝑡𝑒𝑟 = (4.6 ± 1.0) × 10 7 . As visible in Fig. 4a, the experimental results suggest that the available theory for the ternary mixture reported by Martínez Pancorbo et al. [40] cannot describe the behaviour of c-NEFs in the present experiment for the whole range of wave numbers. The theory provides suitable predictions for intermediate and especially for large wave numbers, but fails in the limit of small 𝑞 (in particular for the fastest concentration mode) where the effect of the coupling with temperature fluctuations, and the consequent appearance of propagating modes in the experimental structure function become significant (see section 3.3) [43]. Hence, investigation of the coupling with temperature fluctuations is required for a full understanding of the present experimental results.

We proceed into two steps: first we compare with the published theory [43] for the binary mixture case (one concentration mode) as reviewed in section 3.3; second we compare with the theory for ternary mixtures (two concentration modes) as briefly presented here in section 3.4.

This allows appreciating the features that are genuinely related to the ternary mixture case.

As already mentioned, Fig. 4 shows, superposed on the experimental data, the theoretical predictions (green solid lines) of the theory [43] with one concentration mode. gravitational [6]. The resulting values for the free fitting parameters are the following: 𝑃𝑟 = 9.0 and 𝜓 = 0.9. As it can be observed in Fig. 4a, the decay times provided by the numerical solution agree very well with the experimental decay times of the temperature NEFs in the whole range of wave numbers. However, for the diffusion mass mode, at the smallest wave number range, the agreement is not as good as for the temperature, and a slowing-down of the experimental decay times is visible in comparison to the corresponding numerical solution (green solid line). The theoretical results of the oscillation frequency are likewise similar to the experimental data (see Fig. 4b). This last comparison between the experimental and theoretical decay times shows the importance of including temperature fluctuations in the theoretical analysis.

Next, we finally take into account the more complete theory of NEFs, which includes the coupling of thermal and viscous fluctuations to the two independent concentration modes present in a ternary mixture, as briefly described in section 3.4. The comparison between experimental and theoretical decay times is shown in Fig. 5. The effect of considering the two concentration modes and the associated couplings is evident by comparing Fig. 4 and5. The theoretical curves shown in Fig. 5 2, and the applied temperature gradient. The resulting values for the free fitting parameters are the following: 𝑃𝑟 = 10, 𝜓 1 ′ = 0.7 and 𝜓 2 ′ = 0.4. The theoretical results show that for large wave numbers (𝑞 ≥ 47𝑐𝑚 -1 ), four real solutions are obtained, demonstrating that all the modes are decoupled in this wave number range. The velocity mode cannot be observed experimentally, as it does not give rise to refractive index fluctuations. For small wave numbers (𝑞 < 47𝑐𝑚 -1 ), two real solutions and a pair of complex conjugate ones are obtained numerically, indicating that velocity and temperature fluctuations are mixed at small q. This mixing induces the appearance of propagating modes, observed in the experimental c-SF of Fig. 2 as oscillations at small q.

Applying the full theory of section 3.4, we found a reasonable agreement between the experimental and the theoretical data for the three modes, especially for large and intermediate wave numbers. However, we would like to point out that the numerical values of 𝑃𝑟, 𝜓 1 ′ and 𝜓 2 ′ must be taken into account with caution as the fitting procedure is not robust. The focus here is mainly on the qualitative understanding of the behaviour of the different modes in a ternary mixture when the coupling among them is included.

One relevant point is the slowing-down of the two concentration modes and their consequent diffusive-like behaviours at very small wave numbers (see Fig. 5a). In addition, it is worth noting the good agreement obtained for the frequency of the propagating mode, as can be seen in Fig. 5b. It clearly improves the results of the binary model shown in Fig. 4b.

Of course, the effect of confinement should be also included in the theory if one wants to describe layers of ternary mixtures of any vertical thickness. Unfortunately, this additional development requires more effort and no analytical solution can be obtained.

Conclusions

The concentration and temperature NEFs in a thermodiffusion experiment of a ternary mixture composed of polystyrene, toluene and n-hexane are characterized by dynamic shadowgraphy.

[ 11 ,

 11 13,47]. In general, velocity fluctuations are not supposed to induce fluctuations of the density or of the refractive index.In our optical setup, a super-luminous diode (Superlum, SLD-MS-261-MP2-SM) with a wavelength of 𝜆 = (675 ± 13) nm acts as the light source, illuminating the sample parallel to the temperature gradient. The light is provided out of a monomode optical fibre and the beam is collimated by an achromatic doublet lens with focal length 𝑓 = 150 𝑚𝑚 and placed at its focal distance from the fibre output. After the collimating lens, a linear polariser is positioned before the sample and a second linear polariser (analyser) after it, providing a control of the light intensity independent of the integration time of the detector. Finally, the light beam impinges onto a scientific-CMOS camera (Hamamatsu Digital Camera C13440, ORCA -Flash 4.0) whose detector size is s = 1.33 cm, placed in the near field, i.e. at a distance 𝑧 = (200 ± 5) 𝑚𝑚 from the sample midplane. The s-CMOS camera allows a 'fast' image acquisition frequency up to 100Hz at full frame (2048 × 2048 pixels), thus enabling a fruitful characterization of the temperature fluctuations in the observable wave number range. The shadowgraph signal is the result of the interference of the transmitted beam with the ones scattered by the sample refractive index fluctuations. In our experiment series of 𝑁 = 2500 images of 2048 × 2048 pixels, corresponding to 𝑞 𝑚𝑖𝑛 = 2𝜋 𝑠 ⁄ = 4.72 𝑐𝑚 ⁄ , are recorded for three different frequencies of 𝑓 = 100, 10, 1 𝐻𝑧, corresponding to minimum correlation times of 𝑑𝑡 𝑚𝑖𝑛 = 0.01, 0.1, 1 𝑠. The recorded images are eventually analysed by the Differential

  𝑖(𝑞 ⃗, 𝑡) = 𝐼(𝑞 ⃗, 𝑡) 𝐼(0, 𝑡) ⁄ . Then, differences between pairs of normalised Fast Fourier Transforms (FFTs) are calculated and their square moduli are determined |∆𝑖(𝑞 ⃗, 𝑡, 𝑑𝑡)| 2 = |𝑖(𝑞 ⃗, 𝑡) -𝑖(𝑞 ⃗, 𝑡 + 𝑑𝑡)| 2 . These quantities are computed for all possible delay times 𝑑𝑡, multiples of the 𝑑𝑡 𝑚𝑖𝑛 set by the acquisition frequency of the detector. Finally, the results are averaged first over time 〈|∆𝑖(𝑞 ⃗, 𝑑𝑡)| 2 〉 = 〈|∆𝑖(𝑞 ⃗, 𝑡, 𝑑𝑡)| 2 〉 𝑡 and second over the modulus of the wave vector 𝑞 ⃗, 〈|∆𝑖(𝑞, 𝑑𝑡)| 2 〉 = 〈|∆𝑖(𝑞 ⃗, 𝑑𝑡)| 2 〉 |𝑞 ⃗⃗| . The resultant function is the so-called Structure Function (SF) 〈|∆𝑖(𝑞, 𝑑𝑡)| 2 〉. All these calculations are performed by a custom software running on a graphic card and taking advantage of the massive parallelization of the Graphic Processing Unit (GPU), thus requiring about one hour of computational time for each experiment, i.e. the three series of 2500 images of 2048 × 2048 pixels [49]. 2.4 Concatenation of the experimental structure function As mentioned above, in the experiments reported in this study, sets of images of 2048 × 2048 pixels have been recorded at three different acquisition frequencies at the steady state of the thermodiffusion process. Then, the experimental SF is determined by analysing each stack of images acquired at a specific frequency. Examples of 〈|∆𝑖(𝑞, 𝑑𝑡)| 2 〉 are shown in Fig. 1, as a function of the wave number (Fig. 1a, 1c, 1e) and as a function of the correlation time 𝑑𝑡 for different wave numbers (Fig. 1b, 1d, 1f). The SFs obtained at three different frequencies are then merged in order to form a single concatenated SF, or c-SF (Fig. 1g), that covers a large range of correlation times of about four orders of magnitude, as shown in Fig. 1h. As stated above, the c-SF is the result of merging the SFs determined at three different frequencies (from left to right in Fig. 1h): the first points correspond to the data acquired at 100 Hz (from Fig. 1b, dark purple squares at short dt), the second ones to 10 Hz (from Fig. 1d, light cyan circles at intermediate dt) and the third ones to 1 Hz (from Fig. 1f, dark orange triangles at large dt). The concatenation procedure is made by simply selecting the relevant part of the SFs as shown in Fig. 1. About the 10% of the delay times of each data set are used for the concatenation, disregarding those that are already included in the slower frequency slot. This means that from the fastest acquisition set only delay times in the range 0.01-2.5 s are taken, from the intermediate one the selected range is 2.5-25 s and from the slowest one 25-250 s.

Figure 2 :

 2 Figure 2: Concatenated structure function of the thermodiffusion experiment carried out at

)

  By fitting the experimental decay times in the diffusive regime (𝑞 > 290/𝑐𝑚 for the concentration NEFs, c-NEFs, of component 1 and 𝑞 > 190/𝑐𝑚 for the c-NEFs of component 2) to Eq. 20, the eigenvalues of the diffusion matrix can be determined. The diffusion coefficient 𝐷 ̂1 obtained from the slowest concentration mode represents the mass diffusion eigenvalue of the polymer in the binary solvent mixture. 𝐷 ̂2 calculated from the intermediate decay times (i.e. faster concentration mode) represents the mass diffusion eigenvalue of toluene in n-hexane (i.e. interdiffusion of the solvents in the solvent mixture). The resulting two eigenvalues 𝐷 ̂1 and 𝐷 ̂2 are reported in Table

Figure 4 :

 4 Figure 4: a) Decay times and b) oscillation frequency of NEFs on the ternary mixture of PS-

Figure 5 :

 5 Figure 5: a) Decay times and b) oscillation frequency of NEFs on the ternary mixture of PS-

  The decay times of the NEFs are determined and three modes are clearly distinguished. The fastest decay times are associated to the temperature NEFs, the intermediate ones to the concentration NEFs of the molecular solvent and the slowest ones to the concentration NEFs of the polymer. To encompass the very different decay times (i.e. large range of correlation times), images are acquired at various frequencies, and the corresponding SFs are merged into the so-called c-SF by following the described concatenation procedure.In order to exploit the obtained decay times, first, they are fitted with a simple diffusive model at large wave numbers to obtain the diffusion eigenvalues 𝐷 ̂1 and 𝐷 ̂2 and the thermal diffusivity 𝑎 𝑇 . Secondly, the theory developed for the two concentration modes of a ternary mixture in the presence of gravity is compared with the experimental results for the first time. The experimental decay times of the concentration modes at intermediate and large wave numbers appear to be in good agreement with this theoretical model. However, for smaller wave numbers, a slowing-down of the experimental decay times of NEFs is detected with respect to the predictions of that theory. Due to the large thickness of the thermodiffusion cell, confinement effects cannot be considered as the origin of such discrepancy. In addition, evidence of propagating modes on the structure function suggests that the aforementioned slowing-down could be originated by the coupling of the temperature and viscous modes to the concentration ones. Comparison with the decay times from the theory developed to describe such coupling in a binary mixture shows a better agreement for the thermal and the faster concentration modes at intermediate wave numbers. This positive outcome lead us to further develop the theory of NEFs in the case of ternary mixtures, by including the coupling of velocity, temperature and the two concentration modes under strong gravitational stabilization and for large size fluctuations (small wave numbers). A better qualitative agreement is found between the new theoretical development and the experimental data points.

Table 1 : Thermophysical properties of the polystyrene-toluene-n-hexane mixture

 1 The coefficients 𝛼 and 𝛽 𝑖 are indirectly determined from measurements of the mixture density performed at different conditions through a Density Meter (ANTON PAAR, DMA 5000). For the thermal expansion coefficient, the density is measured at different temperatures (from 23 to 27 o C), while keeping the concentrations constant at 𝜔 1 = 0.02 and 𝜔 2 = 0.49. The mass expansion coefficients are obtained after measuring the density at different concentrations while keeping the temperature constant at 25 o C. To obtain the mass expansion coefficient of the polymer in the mixed solvent, 𝛽 1 , the concentration of the polystyrene is changed from 𝜔 1 = 0.000 to 𝜔 1 = 0.035, keeping 𝜔 2 = 𝜔 3 . For the mass expansion coefficient of the mixed solvent, 𝛽 2 , the composition of the mixed solvent is modified from 𝜔 2 = 0.47 to 𝜔 2 = 0.51, keeping 𝜔 1 = 0.02 constant. Measured values of 𝜌, 𝜈, 𝛼 and 𝛽 𝑖 are summarized in Table1. The values of 𝜌, 𝜈, 𝛼 and 𝛽 2 are consistent with those reported in[44], determined for the equi-

	massic toluene-n-hexane binary mixture at 𝑇 𝑚𝑒𝑎𝑛 = 25 °𝐶. The measurements of viscosity and
	density are repeated at least three times for each sample. The values reported in Table 1 are
	calculated as the average of the three measurements and the corresponding uncertainties are the
	standard deviations. The parameter 𝛼 is determined through Eq. 1, i.e. dividing the slope of the
	density vs. temperature by the density of the sample at the nominal temperature. Similarly, 𝛽 𝑖
	are calculated through Eq. 2, i.e. dividing the slope of the density vs. concentration by the
	density of the sample at the nominal concentration.
	Parameter	Value
	𝜌	(750.0  0.4) × 10 -3 g/cm 3
	𝜈	(5.24  0.01) × 10 -3 cm 2 /s

Table 2 : Diffusion coefficient eigenvalues, Lei and Dr numbers of the polymeric ternary mixture

 2

	Parameter	Value
	𝐷 ̂1	(3.8  0.1) × 10 -6 cm 2 /s
	𝐷 ̂2	(3.1  0.1) × 10 -5 cm 2 /s
	𝑎 𝑇	(8.5  0.2) × 10 -4 cm 2 /s
	𝐿𝑒 1	227  12
	𝐿𝑒 2	27  2
	𝐷𝑟	8.4  0.6

  The theoretical curves have been obtained numerically (solving Eqs. 14 and 15) by keeping constant the Lewis number 𝐿𝑒 = 27 to the same value of 𝐿𝑒 2 shown in Table 2, and varying the values of 𝑃𝑟 and

	𝜓 to best fit the data. The solutal Rayleigh number 𝑅𝑎 𝑆,2 = -(𝑞 𝑆,2 * 𝐿)	4 is also kept constant at
	-5.8 × 10

6

, after evaluating the value of the solutal rolloff wave number 𝑞 𝑆,2 * . This value represents the wave number where the fluctuation 𝛿𝜔 2 behaviour changes from diffusive to

  have been obtained numerically (solving Eqs. 18 and 19) by keeping constant 𝐿𝑒 𝑡 = 227 and Dr = 8.4 (the 𝐿𝑒 1 and Dr values shown in Table 2), and varying the values of 𝑃𝑟, 𝜓 1 ′ and 𝜓 2 ′ to best fit the data. Additionally, the term 𝑃𝑟𝐿𝑒 𝑡 2 𝑅𝑎 = 𝛼𝑔𝐿 4 ∇𝑇/𝐷 ̂12 appearing in the matrix 𝐺 𝑡 (𝑞 ̃) of Eq. 18 is also kept fix at 1.6 × 10 11 , considering for this number the 𝛼 value of Table 1, the 𝐷 ̂1 value of Table

Acknowledgements

The authors of the University of Pau kindly thank the financial support from the Centre National d'Etudes Spatiales (CNES) and from the funding partners of the Industrial Chair CO2ES: E2S-UPPA, TOTAL, CNES and BRGM. L. García-Fernández gratefully acknowledges the CNES for the post-doctoral research grant. The research at the Complutense University was supported by grant ESP2017-83544-C3-2-P of the Spanish Agencia Estatal de Investigación.

Authors contribution statement

The experimental measurements and data analysis were performed by L. García-Fernández, H.