Sebastian Peukert

# Michał Kijak

Jakub Ostapko

Jerzy Sepioł

Catherine Le Bris

Anne Zehnacker-Rentien

Michał Gil
email: mgil@ichf.edu.pl

Jacek Waluk

Supersonic Jet Spectroscopy of Parent Hemiporphycene: Structural Assignment and Vibrational Analysis for S0 and S1 Electronic States

Keywords: Tautomerism, Proton Transfer, Laser Induced Fluorescence, Dispersed Fluorescence, Tetrapyrrolic ring, Supersonic Expansion

Hemiporphycene (HPc), a constitutional isomer of porphyrin, is studied under supersonic expansion conditions by means of laser-induced fluorescence (LIF), visible-visible hole-burning experiments, single vibronic level fluorescence (SVLF) techniques, and quantum chemical calculations. Only one trans form of jet-cooled HPc is observed, in contrast with solutions studies that evidence a mixture of two trans tautomeric forms separated in energy by ca. 1 kcal/mol. Reliable structural assignment is provided by simulating absorption and emission patterns at the DFT and TD-DFT levels of theory. The vibronic spectra are nicely reproduced for both electronic ground and lowest excited singlet states for the most stable trans form. In contrast with another porphyrin isomer, porphycene (Pc), no tunneling or photoinduced hydrogen transfer are detected. The lower symmetry of HPc compared with Pc and the concomitant non-equivalent positions of the inner-cavity nitrogen atoms result in a nonsymmetric double minimum potential for tautomerization, larger energy barrier, and a longer tunneling distance, with the average intramolecular hydrogen bonds length larger in HPc than in Pc. HPc readily forms hydrates that show red-shifted absorption relative to the bare molecule.

Introduction

Porphyrin (Pr) is an aromatic system where four pyrrolic subunits are arranged into a macrocycle having a planar, rectangular cavity of four nitrogen atoms connected via two inner hydrogen bonds. Well-defined configuration of this cavity and relatively good protection of the inner H-bonds from the environment makes Pr and its constitutional isomers interesting model systems for studying intramolecular hydrogen transfer. [1][2][3][4][5][6] The rate of hydrogen transfer strongly depends on geometrical parameters of the inner cavity (NH … N distance and linearity). Special attention has been paid to porphycene (Pc), where the transfer of two internal hydrogens has been investigated in various environments by using multitude of stationary and time-resolved spectroscopic and microscopic techniques. 3,4,7,8,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] Tautomerization rates in Pc are larger than in Pr by many orders of magnitude (picoseconds vs. microseconds). In Pc, the reaction takes place in both ground and excited electronic states. It proceeds via concerted movement of two hydrogen atoms along a symmetric double minimum on the potential energy surface, assigned to two isoenergetic trans tautomeric forms (Figure 1). The process has been found to be dominated by tunnelling, not only in isolated molecules at low temperatures, 10,[23][24][25] but also in room temperature solutions, as demonstrated by ultrafast transient absorption anisotropy measurements. 16 Laser-induced fluorescence and dispersed fluorescence spectra recorded under jet-cooled conditions clearly exhibit the band splitting, characteristic of coherent delocalization of two inner hydrogens, 23,24,26,27 in contrast with Pr that exhibits single bands. 28 The splitting is strongly sensitive to deuteration, confirming the tunnelling nature of the process. Moreover, tunnelling splitting in Pc is vibrational-mode selective, due to the multidimensional nature of the reaction coordinate. [24][25][26][27] The sensitivity to structural perturbations, such as substitution 24 or deuteration of the peripheral rim 26 pointed out the importance of weak structural perturbations or symmetry breaking in the process. Supersonic jet studies of tetraalkyl-substituted Pc have shown the distortion of the PES caused by four methyl or n-propyl groups. 24 The energy degeneracy of the tautomeric forms can be lifted by asymmetric chemical substitution of formerly symmetric framework, e.g., a single substitution at one of the meso positions (-CH=CH-bridges between bipyrrole subunits). Such an effect of the PES distortion has been reported for tetra-n-propyl porphycene singly substituted at the meso position with an acetoxy group. 29,30 The studies carried out in condensed media at different temperatures have shown the presence of two almost isoenergetic non-equivalent trans tautomers. While the PES in the ground electronic state is only slightly asymmetric, resulting in comparable populations of both tautomers even at cryogenic temperatures down to ~20 K, that of the excited electronic state is strongly asymmetric. Thus, the emission of only one tautomer has been detected, due to fast exoenergetic excited-state hydrogen transfer in the S1 state. Quite different behavior has been reported for 9-amino-substituted tetraphenylporphycene, although the potential-energy surface is also almost symmetric in the ground state and strongly asymmetric in S1. In this case, dual emission is observed from the non-equivalent trans tautomers. 31,32 Figure 1 Other porphyrin isomers obtained so far include corrphycene, 33,34 hemiporphycene, 35,36 isoporphycene, 37 inverted (confused), 38,39 and neo-confused 40 porphyrins. Except for inverted porphyrin, all these isomers have been obtained so far as substituted derivatives.

Recently, Ostapko et al. achieved the synthesis of hemiporphycene (HPc) in its parent, unsubstituted form. 41 HPc is of lower symmetry (Cs) than Pr (D2h) and Pc (C2h) and therefore, in contrast to Pr and Pc, its trans forms are not equivalent. Figure 1 shows the structure of two trans tautomers in Pc and in HPc, and the schematic plots of their potential energy surfaces along the hydrogen transfer coordinate. Trans1 form of HPc is theoretically predicted as the most stable one in the ground electronic state at the B3LYP/6-311++G(d,p) level, while the energy of trans2 is higher by ~1 kcal/mol. This calculated value is in excellent agreement with electronic absorption and fluorescence measurements. UV/Vis absorption spectra of HPc in dodecane recorded as the function of temperature indicate the presence of two inequivalent forms, assigned to trans1 and trans2 tautomers. 41 Fluorescence from both trans forms is observed upon excitation of the higher energy trans2 tautomer and the rate of ~10 7 s -1 for trans2 → trans1 reaction has been estimated at room temperature. At 293 K, the efficiency of trans2 → trans1 phototautomerization is low (~10 %), but increases to ~50 % at 373 K. The rate at room temperature is four orders of magnitude lower than the rate determined for S1 tautomerization in Pc, but still larger than that in Pr. 15,42 This finding agrees with the results of electronic structure calculations of bare HPc and the crystal structure analysis of HPc 41 and its octaethyl derivative, 36 which confirm the planarity of the hemiporphycene framework and demonstrate that the dimensions of the inner cavity in HPc lie between those of Pc and Pr.

Combining the energy difference between trans1 and trans2 with the 0-0 transition energies of both tautomers leads to the conclusion that the distortion of the potential is larger in the ground than in the excited state, although trans1 remains the lower energy form in both S0 and S1. 41 Thus, the distortion of inner cavity due to a less symmetrical framework affects the potential energy surface differently than substitutions in meso position discussed above. This finding may be important for future synthesis, in designing new derivatives or isomers. Supersonic jet technique combined with laser spectroscopy has proved to be very useful for studying tautomerism in porphycene derivatives, owing to highly selective excitation of a well-defined ground state population of cold and isolated molecules. 23,24,26,27 In the present work, we report and discuss experiments for HPc, applying this technique in combination with laser-induced fluorescence excitation (LIF) and single vibronic level fluorescence (SVLF), as well as visible-visible fluorescence depletion spectroscopy. The ground and electronic excited state geometries and harmonic frequencies are calculated within the frame of the density functional theory (DFT).

The main purpose of this work is to confirm the structure of the lowest energy tautomeric form of HPc under conditions of molecular isolation. The computed difference in energy, 1 kcal/mol, is at the limit of the accuracy of DFT calculations. A question arises whether the influence of the surroundings could not modify this difference. Moreover, if the difference in energy between trans2 and trans1 decreases under jet-cooled conditions, tunnelling splitting should be observed.

Next, even if the ground-state potential well is asymmetric in S0, one can ask the question whether it is possible to observe two tautomeric forms under jet-cooled conditions. Indeed, the relative population of conformers in a supersonic expansion depends not only on their energy difference, but also on the barrier separating the two potential wells. It might happen that kinetic trapping allows stabilization of higher-energy forms, even though the ground state energy difference of ca. 1 kcal/mol seems to exclude at first sight the possibility of populating the higher energy trans2 tautomer in S0. The trans2 form could also be formed in the electronically excited state. Condensed-phase studies have shown that phototautomerization proceeds in the trans2 → trans1 direction only, which indicate that trans 1 is more stable in S1 as well. However, in a jet, the reverse trans1 → trans2 process could take place upon excitation of a long-lived higher-energy S1 vibronic level of trans1.

We therefore undertake the study of HPc in isolated conditions and show that the analysis based on comparison between the measured and simulated LIF and SVLF spectra provides reliable arguments for the determination of the observed tautomeric structure. We also report the effect of hydration on the vibronic spectra of HPc.

Experimental and Theoretical Methods

In Warsaw, experiments in a supersonic expansion were performed using a home-made setup described previously. 43 The samples were heated up to ~520 K under the flow of helium (stagnation pressure of ~3.5 atm) used here as a carrier gas. A cold trap was used to freeze out water traces and the gas supply system has been purged with carrier gas. The gas mixtures were expanded to a vacuum chamber through a home-made pulsed valve (based on IOTA Series 9, General Valve) with a nozzle of 500 μm diameter. The jet was probed by a laser beam at 8 mm downstream from the nozzle. A narrow band (< 0.1 cm -1 ) optical parametric oscillator (OPO, Sunlite Ex, Continuum) pumped by a seeded Powerlite 8000 Nd:YAG laser (at 10 Hz repetition rate) was used with average pulse energy in the range of 0.2 -1 mJ.

The total fluorescence from the sample was collected by a toroidal mirror and focused through a RG645 nm long pass filter onto a cooled Hamamatsu R2949 photomultiplier connected to a Yokogawa DL9140 oscilloscope for the LIF measurements or, to a slit of an Acton Research 0.275 m (1200 lines/mm) spectrograph equipped with a Princeton Research LN-cooled CCD for measuring SVLF spectra. 44 For fluorescence depletion spectroscopy, a homemade dye laser (Rh6G dye in methanol, energy ~ 100 μJ, spectral width ~ 0.2 cm -1 ) pumped by a Nd:YAG laser (Surelite I-10, Continuum) was used as a pump, while the OPO served as a probe with a pump -probe delay of 200 ns. In our procedure, the pump laser was fixed at a wavelength resonant with a vibronic transition of a selected species while scanning the probe laser. The measurement is done in the presence and absence of the pump laser beam by using a rotary shutter at each step of the scan, which results in two LIF excitation spectra (a so-called active baseline procedure).

After a few hours of heating we have observed thermal degradation of HPc. In order to exclude the possibility that thermal decomposition of the sample has an impact on the spectroscopy, LIF and SVLF spectra of HPc were also measured in Orsay using the laser desorption technique. Molecules were desorbed by laser pulses (frequency doubled Nd:YAG; Quantel, Brilliant B) from the surface of a graphite pellet fixed at the nozzle of a pulsed valve generating the supersonic jet. In these experiments argon was used as a carrier gas and expanded under a backing pressure of ~5 atm. LIF spectra were obtained by exciting the jet-cooled molecules by a tunable dye laser (Rh610 and Rh640 dyes, spectral width ~0.2 cm -1 at 603 nm) pumped by the second harmonic of a Nd:YAG laser. The fluorescence signal from the sample was collected by a two-mirror system and a monochromator (Princeton Instruments Acton SP2300) used under broad band conditions and then detected by a photomultiplier tube. The light collected and focused by the first mirror (a toroidal mirror, protected Aluminium, radius of curvature R -478.76, r-422.89, angle of incidence 20°) located at 700 mm from the fluorescent molecules that act as a point source. The light collected by the toroidal mirror was propagated onto a rotating planar mirror (coated BK7) and then reflected onto the entrance slit of a monochromator. The position (distance and rotation angle) of the planar mirror were carefully adjusted so that the emitted light was focused exactly on the entrance slit of the monochromator. For recording SVLF spectra, the fluorescence light was dispersed through this monochromator with a grating of 1200 lines/mm and the light was detected with an intensified and gated CCD (Princeton Instruments PI-MAX 4). 44 Quantum chemical modeling was performed using the GAUSSIAN09 program package. 45 Ground and excited state geometries were optimized and vibrational frequencies were calculated with DFT and TDDFT methods, respectively, applying the B3LYP functional and the 6-311++G(d,p) basis set. Vibrational frequency scaling factor of 0.969 has been used for both the ground and excited states. Vibrationally-resolved absorption and emission spectra were obtained by applying the Franck-Condon principle within the harmonic approximation, as implemented in Gaussian09. The half-widths of the bands, 1 and 15 cm -1 , respectively, have been arbitrarily chosen to match the experimental ones.

Results and Discussion

Lower symmetry of HPc relative to Pc increases the number of possible tautomeric forms obtained by moving the inner hydrogen atoms within the nitrogen cavity. Two trans and four cis forms can be envisaged. Table S1 (Supporting Information) contains the calculated relative energies of these isomers in S0 and S1 states. The calculations predict trans1 to be the most stable form of isolated HPc in both S0 and S1. The second most stable form is trans2, higher in energy by 0.97 (0.89 with ZPVE correction) kcal/mol in S0 and by 0.71 (0.73) kcal/mol in S1. The lower-energy cis form, cis1, is in S0 ~4 -4.5 kcal/mol less stable than trans1. Therefore, all possible cis forms are unlikely to be observed under jet-cooled conditions and hence will not be considered further.

It was established for different symmetrically substituted Pc derivatives that proton transfer rates clearly correlate with the distances between H-bonded N-atoms. 17

(dN1-H -dN2•••H)+(dN3•••H -dN4-H)]/2
) is much larger in HPc than in Pc for both, S0-and S1-optimized structures (being 90.3 and 95.2 pm in trans1 of HPc, and 65.5 and 69.1 pm in trans tautomer of Pc, in S0 and S1 states, respectively). Therefore, we expect that in HPc the tautomerization rate will be larger than in Pr but lower than in Pc.

The LIF spectrum of HPc presented in Figure 2 consists of well-resolved bands in the measured spectral range, indicating sufficient cooling conditions in the jet. The half-width of the bands is ~1 cm -1 and similar for spectra recorded using an oven and helium carrier gas or laser desorption and argon. The origin of the spectrum (0-0 transition) consists of a single band at 16566 cm -1 , and, in comparison to Pc, is shifted to higher energies by 400 cm -1 . No doublet structure characteristic for tunneling splitting is observed, which contrasts with Pc. The absence of "hot" bands was checked by varying the distance between the jet nozzle and the exciting laser beam from 5 to 11 mm. No modification of the spectrum was observed. Besides vibronic bands that could be clearly assigned (see below), the spectrum contains several weak bands in the low frequencies region (0 -100 cm -1 ) above the origin. Neither the LIF spectrum of bare Pc, nor that of Pr has shown transitions in this region. 23,28 The presence of these bands could be an indication of the presence of several species, or be due to a more complex vibronic pattern caused by lower symmetry. Thus, to determine how many species are present in the jet, we conducted visible-visible double-resonance fluorescence depletion experiments. Figure 2 shows that all intense bands respond with similar relative depletion (35 -50%) as the 0-0 band (~40%) of HPc. The same set of spectral dips was obtained while pumping the vibronic transitions at 138, and at 142 cm -1 (not shown). This indicates that only one conformer of bare HPc is present in the S0 state in the jet. This observation agrees with previously-reported temperature dependence of the absorption spectra of HPc in solution, which showed a disappearance of the absorption band assigned to trans2 tautomer upon lowering the temperature. 41 It seems safe to assume that the trans1 form remaining in the condensed media upon lowering the temperature is also the one present in the cold jet. One can notice in the LIF spectrum several bands separated by a few wavenumbers, for example at 138/142, 255/257, or 471/475 cm -1 . In contrast to tunneling doublets observed for Pc, these bands respond as spectral dips to a common probe wavelength in the hole-burning experiment. Therefore, they are due to one form of HPc excited from a common vibrational level of the ground state and are due to excitation of vibronic bands with similar frequencies in S1, as will be seen later.

Figure 2

The identification of the tautomeric form observed in the LIF spectrum can be supported by modeling. First, we compared the frequencies of the experimental transitions with the calculated ones for the fundamental vibrations of the two trans tautomers. These comparisons result in equally good linear fits for both forms with root mean square errors of ~4 cm -1 for trans1 and ~5 cm -1 for trans2. Based only on the calculated frequencies, it is therefore not possible to clearly conclude which tautomer is observed in the experiments. Consequently, we also analyzed the experimental and simulated intensity patterns. Figures 3 and4 show the comparison between the experimental LIF and SVLF spectra of HPc and the theoretically predicted ones. In both spectra of Figure 3 and4, the calculated intensities of the 0-0 transitions are very large in comparison to other bands. Very similar vibrational frequencies are observed in absorption and emission (compared in Table 1), and within the limitation imposed by a different resolution of both spectra, the dispersed fluorescence resembles relatively well a mirror image of the LIF spectrum. In particular, strong transitions due to in-plane fundamental vibrations 1A' -5A' and some of their combinations (e.g. 1A'+5A', 2A'+5A') are observed both in LIF and SVLF. This observation supports lack of excited state tautomerization reaction upon the lowest energy (0-0 transition) excitation; a result expected from energy levels considerations (Figure 1).

Figures 34Table 1 In the experimental LIF spectrum (Figure 3), the prominent vibronic bands at 138 and 142 cm -1 can be assigned to skeletal in-plane vibrations with similar predicted frequencies (1A' and 2A' modes) for both tautomers. They have comparable intensities, which is much better reproduced by calculations for the trans1 tautomer. The experimental bands at 255 and 257 cm - 1 can be assigned to the 3A' in-plane vibration calculated at 256/255 cm -1 for trans1 and trans2 and to a combination of two out-of-plane modes at 257 cm -1 , respectively. In the simulated spectra, only the former transition has significant intensity, comparable for both tautomers. Thus, we extended the analysis by including Herzberg-Teller contributions in the simulation of the spectra (Figure S1, Supporting Information). This attempt did not improve the agreement between the experiment and theory in the region 0-500 cm -1 above the 0-0 transition. In particular, an experimental doublet at 255/257 cm -1 was still not reproduced in LIF simulation regardless of chosen tautomer. A more precise diagnostic can be a triplet structure appearing in the experiment at 276, 280, and 284 cm -1 , with an approximate intensity ratio of 1/2/1. These bands are readily assigned as the overtones of 1A' and 2A' modes and their combination (1A' + 2A'). The spectrum simulated for trans1 reproduces this sequence of harmonics and combination with three bands at 270, 278, and 286 cm -1 , which is not the case for the trans2 tautomer. The bands observed at 293, 333, 355, and 375 cm -1 are assigned to in-plane fundamental vibrations 4A' -7A', respectively. The simulation predicts them to be observed in the spectrum regardless of the chosen tautomer, although their intensity ratio is slightly better reproduced in the case of trans1. Confirmation of the assignment to the trans1 structure is given by the analysis of the vibronic transitions at 471, 475 and 484 cm -1 . These bands are due to the combinations of 1A' + 5A' (138 and 333 cm -1 ), 2A' + 5A' (142 and 333 cm -1 ), and the fundamental 9A' mode. The simulation for trans1 shows bands at 464 (135 + 329), 472 (143 + 329), and 476 cm -1 with relative intensities similar to those observed in the experiment. Again, the higher energy tautomer shows worse agreement in this respect. Thus, the above comparison of the experimental and simulated LIF spectra clearly is in favor of trans1 tautomer as the species observed in the supersonic jet.

Comparison between experimental and simulated SVLF spectra confirms this assignment. Upon excitation at the 0-0 transition (Figure 4), the experimental spectrum shows four fluorescence bands in the region 0 -400 cm -1 , at 140, 258 (weak), 291, and 336 cm -1 . For the band at 140 cm -1 , we assign it to fundamental in-plane vibrations 1A' and 2A', calculated at 135 and 145 cm -1 , but unresolved in the experiment due to insufficient resolution of SVLF spectra. Other three fluorescence bands in this region are assigned to 3A', 4A', and 5A' in-plane modes (Table 1). Comparison between the experiment and the calculations for both trans tautomers shows that trans1 gives better prediction with respect to the relative intensities of these four bands. An even stronger suggestion in favor of trans1 tautomer is provided by the next group of four bands at ~ 712, 749, 788, and 818 cm -1 . They are assigned to fundamental modes 12A', 13A', 15A', and 16A'. These bands have practically no predicted intensity in the simulation for trans2, while they show excellent agreement with the spectrum of the trans1 form. The next hint comes from a medium intensity experimental transition at 1167 cm -1 which in the simulation of the trans1 tautomer is predicted as an intense fundamental in-plane vibrational transition at 1188 cm -1 . According to calculations, this mode strongly engages inplane wagging of the hydrogen atom in the N1-H bond, the one which is simply absent in the structure of trans2 (Figure 1). In the latter, the calculated frequency of the corresponding mode is considerably shifted, to 1231 cm -1 , and the predicted intensity is much lower than that observed in the experiment. Between 1300 -1400 cm -1 , four relatively weak bands are observed at 1313, 1343, 1396, and 1416 cm -1 . While a definite assignment is difficult due to a multitude of skeletal in-plane vibrations predicted for this region, only the simulation for the trans1 form shows significant fluorescence intensity in this region. Finally, a fluorescence band at 1606 cm - 1 is assigned to the in-plane fundamental mode calculated at 1583 cm -1 in the trans1 tautomer.

In contrast, the simulated spectrum for trans2 does not show significant intensity bands above 1500 cm -1 . Visualization of this vibration in trans1 indicates a large displacement vector of the hydrogen atom bonded to N3. This particular bond is absent in the case of trans2 which, in parallel to the explanation presented above for the band observed at 1167 cm -1 , could be a reason for a worse agreement with experiment.

Based on the above analysis, one can safely conclude that it is the trans1 tautomer which is observed in the supersonic jet. Consequently, the calculated frequencies of this form have been used for the assignment of vibronic bands in the experimental LIF and SVLF spectra in Table 1. Moreover, the fact that the emission spectrum is also characteristic of trans1 experimentally confirms that it is also the most stable one in S1.

Previous experiments in solutions have suggested that the endothermic trans1 → trans2 reaction could take place when increasing the excitation energy. This process is however difficult to follow in solution due to an overlap of broad absorption bands of both trans forms. The latter obstacle is not present in the jet, where only the trans1 tautomer is present. In search of any evidence of phototautomerization, we compared in Figure 5 the SVLF spectra obtained upon excitation at the origin (0-0 transition at 16566 cm -1 ) with those recorded using an excess of excitation energy (the respective transitions are marked in Figure 2). All spectra are background-corrected by subtracting the scattered light measured without seeding HPc into the jet stream. Due to uncertainties in this procedure, which in some cases leads even to a negative band in the corrected spectra, the intensities at the exact wavelength of excitation should be taken with care. The upper spectrum in Figure 5 was excited at the 0-0 transition. Due to slightly better s/n ratio than the corresponding part of Figure 4, it reveals two additional weak transitions at 477 and ~610 cm -1 , nicely predicted in the simulation of the trans1 form (Figure 4, middle panel). When using an excess of excitation energy at the (0-0) + 138, 142, 293, and 333 cm -1 , the sequences built on "pseudo 0-0" transitions are the same as upon the lowest energy excitation. This is also true for SVLF spectra recorded upon excitation of other bands (not shown). The SVLF spectra presented in Figure 5 do not show any indication of fluorescence emission from two tautomeric forms. However, due to lower resolution and sensitivity of dispersed fluorescence, the possibility of tautomerization in the S1 state with very small yield cannot be completely excluded.

Figure 5

As mentioned above, the LIF spectrum presented in Figure 2 shows some additional weak bands in the region 0 -100 cm -1 above the origin of bare HPc, which do not respond as spectral dips in the fluorescence depletion spectrum. In turn, their intensity is growing upon addition of H2O vapors to the carrier gas. Thus, they are due to water remaining in the gas supply system, which leads to formation of HPc/H2O complexes. Formation of complexes of porphycene with water has been reported by our group. 23 A spectral signature of such species was observed as a LIF band at +119 cm -1 above the 0-0 transition of bare Pc, with the intensity dependent on the content of water admixed to helium gas. The SVLF spectrum (not published) recorded upon excitation into this band at 16294 cm -1 revealed a disappearance of doublet structures assigned to the tunneling splitting, but otherwise showed vibrational progression very similar to that observed for origin-excited bare Pc. Figure 6 shows significant parts of the LIF spectrum of HPc obtained upon increasing the vapor pressure of water in the jet. The lowestenergy transition is observed at 16233 cm -1 and has been assigned to the origin of the S0-S1 transition of the HPc/(H2O)n complexes. It is shifted down in energy by 333 cm -1 relative to bare HPc, which contrasts with the +119 cm -1 blue shift observed for the hydrate of Pc. The main vibronic transitions are observed at 137, 143, and 292 cm -1 from the origin and these frequencies are practically unchanged in comparison with 1A', 2A' and 4A' vibrations of bare HPc (Table 1). An additional weaker band has been observed in the hydrate at 140 cm -1 , which is reminiscent of a band at 144 cm -1 in bare molecule, tentatively assigned to a combination of out-of-plane modes 2A'' and 3A''. Figure 7 shows the SVLF spectra obtained for excitation into the 0-0 band of the complex as well as into the bands at +143 and +292 cm -1 . The SVLF spectrum resulting from the excitation of the origin (Figure 7A) shows peaks at 142, 258, and 291 cm -1 . They are in clear correspondence to the spectrum of bare HPc. However, no further fluorescence bands were observed above 300 cm -1 , where bare HPc shows several transitions due to in-plane vibrations of the macrocycle (e.g., at 336, 712, 749, or 788 cm -1 ). Interestingly, such weak spectral activity in this region has been predicted by the simulation for trans2 tautomer (Figure 4, bottom). Another noticeable feature of HPc/H2O complexes is the red shift of the 0-0 transition in the LIF spectrum, which is opposite to that observed for the complexes of parent porphycene. Its value of 330 cm -1 is quite substantial, and also comparable to the energy difference between the absorption bands of both trans tautomers of HPc at room temperature (612 and 622 nm, for trans1 and trans2, respectively, i.e. ~260 cm -1 ) ). While it may be a pure coincidence, it may also suggest that hemiporphycene complexed by molecule(s) of water is in a different tautomeric form (trans2) than the "normal", isolated one. Although interesting, this suggestion needs more detailed fluorescence studies, and possibly massselective jet experiments to verify the stoichiometry of observed water complexes. for both trans tautomers. The overall agreement between the experiment and the simulated spectra using trans1 molecular structure is very good, regarding both the vibrational frequencies and relative intensities. The prominent bands in LIF (e.g. 138, 142, 293, 333, and 355 cm -1 ) and SVLF spectra (e.g. 140, 336, 712, 749, and 788 cm -1 ) can be assigned to fundamental in-plane vibrational modes of the macrocycle. The SVLF spectra obtained at different excitation energy excess showed no evidence for trans1→trans2 phototautomerization in the S1 state. Also, no signs of tunneling splitting have been detected.

The localization of the internal hydrogens in HPC, a behavior opposite to that observed in porphycene, may be caused by several factors. First, lower symmetry of HPc dictates the lack of a symmetric double minimum potential, which is a condition for perfect coherent tunneling observed in Pc. Moreover, all four nitrogen atoms in HPc are chemically nonequivalent, which makes a synchronous concerted double hydrogen transfer (the mechanism proposed for Pc) very unlikely. In turn, single hydrogen transfers in HPc involve cis tautomers, the energies of which are significantly higher than those of the trans forms.

While symmetry is important, it does not warrant efficient tautomerization, as exemplified by porphyrin, in which the two minima corresponding to isoenergetic trans tautomers are separated by a high energy barrier. The barrier is determined by the properties of intramolecular hydrogen bonds: weak in porphyrin, very strong in porphycene. HPc represents a case intermediate between these two, as indicated by the values of NH stretching frequencies. 41 We plan to continue investigations on the role of symmetry in tautomerization, tunneling in particular, by studying corrphycene, another porphyrin isomer. This molecule is formally of the same symmetry as HPc (Cs), but it possesses two pairs of chemically equivalent inner cavity nitrogen atoms.

Another subject worthy of further studies relates to complexes HPc with water, or, possibly, alcohols. The shift of their 0-0 transition in comparison to that of the bare macrocycle, opposite to that observed in the case of porphycene, suggest an interesting possibility of switching from one tautomeric form to the other upon complexation. If corroborated, such an effect could be very attractive for investigating the details of intra-and intermolecular hydrogen bond interactions in the same molecule.

  Our DFT calculations show that in HPc these distances are: dN1-N2 ~ 260.4 and 259.5 pm, and dN3-N4 ~ 293.7 and 293.0 pm, for trans1 and trans2 forms, respectively. They are, on average, inbetween those of Pr (dN-N ~ 293.2 pm) and Pc (dN1-N2 = dN3-N4 ~ 265.5 pm), although a lower symmetry of HPc makes the situation more complicated. In turn, the average distance for transfer of two protons (calculated as [

Figures 6- 7

 7 Figures 6-7

  Measurements of LIF spectra and fluorescence depletion spectroscopy have shown that only one form of HPc is present in supersonic jet in the ground and lowest electronically excited states. It has been assigned to the lowest energy tautomer, trans1, based on the analysis of the experimental LIF and SVLF spectra and their comparison with theoretically predicted spectra

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .

 123456 Figure 1. Molecular structures of two trans tautomers of Pc (left) and HPc (right), and their double minimum potential energy surfaces (PES) for a phototautomerization reaction between tautomers. The PES for a hypothetical trans1 → trans2 reaction in HPc is asymmetric. Calculated barrier heights and energy level differences are given in kcal/mol. For details, see text.

Figure 7 .

 7 Figure 7. SVLF spectra of HPC complex with water upon excitation at (A) 0-0 band, (B) (0-0) + 143 cm -1 , (C) (0-0) + 292 cm -1 . Band positions are given in wavenumbers relative to the 0-0 or pseudo 0-0 band.

Table 1 .

 1 Ground and excited state (S1) vibrational frequencies of HPc obtained from LIF and SVLF spectra. Theoretically based frequencies correspond to those calculated for the trans1 tautomer. For complete list of unscaled vibrations in both trans tautomers, see Supporting Information.

	ṽ(S1) exp. ṽ(S1) calc. a	Assignment S1	ṽ(S0) exp. ṽ(S0) calc. a Assignment S0
	128	125	2 x 2A''	-
	138	135	1A'	140	138	1A'
	142	143	2A'		146	2A'
	144	140 b	2A''+3A''	-
	255	256	3A'	256	260	3A'
	257	248 b	3A'' + 6A''	-
	276	270	2 × 1A'	-
	280	278	1A' + 2A'	-
	284	286	2 × 2A'	-
	293	289	4A'	291	293	4A'
	333	329	5A'	336	333	5A'
	355	351	6A'	-
	375	367	7A'	-
	393	391	1A' + 3A'	-
	395	383	3A'' + 6A'' + 1A'	-
	397	399	2A' + 3A'	-
	399	391	3A'' + 6A'' + 2A'	-
	431	424	1A' + 4A'
	435	432	2A' + 4A'	-
	471	464	1A' + 5A'	477	471	1A' + 5A'

a scaling factor 0.969, b tentative assignment

Acknowledgments

This work was supported by the Polish National Science Centre grant 2014/15/B/ST4/05020, the PL-Grid infrastructure and computing grant G17-14 from the Interdisciplinary Centre for Mathematical and Computational Modeling of the Warsaw University. A.Z., S.P. and M.G. also acknowledge support from the CNRS and the Polish Academy of Sciences for the "Programmes Internationaux de la Cooperation Scientifique" (PICS 6363). The experiments performed in Orsay were conducted at the Centre Laser de l'Université Paris Sud (CLUPS) and we thank Dr. Michel Broquier for assistance and helpful discussions.

Supplementary Material

See Supplementary Material for the complete list of the calculated vibrational frequencies for both trans tautomers of HPc, and comparison of the experimental LIF spectrum with the simulated ones including Herzberg-Teller approximation.


Doriano Brogioli

Fabrizio Croccolo

Alberto Vailati

Asymmetric time cross-correlation of non-equilibrium concentration fluctuations in a ternary liquid mixture

Keywords: numbers: 47.10.+g: Fluid dynamics, 05.40.-a: Fluctuations phenomena, 05.70.Ln: Nonequilibrium thermodynamics, 66.10.Cb: Diffusion

Equilibrium phenomena are characterized by time symmetry. Thermodynamic fluctuations are also time-symmetric at equilibrium. Conversely, diffusion of a solute in a liquid in the presence of a gradient is a non-equilibrium phenomenon, which gives rise to long-range fluctuations with amplitude much larger than the equilibrium one for small enough wave number. In the case of diffusion in binary mixtures such fluctuations are time-symmetric, notwithstanding the fact that they are generated by a non-equilibrium condition. In this paper, we investigate diffusion of two solutes in a ternary liquid mixture by means of fluctuating hydrodynamics theory. We show that the time crosscorrelation function of the concentrations is not time-symmetric, hence showing that time-symmetry is violated for such non-equilibrium fluctuations. We discuss the feasibility of experiments aimed at the detection of the asymmetry of the cross correlation function of non-equilibrium concentration fluctuations in ternary mixtures, as envisaged in the Giant Fluctuations (NEUF-DIX) microgravity project of the European Space Agency.

I. INTRODUCTION

One of the features of systems at equilibrium is time symmetry: given a sequence of observations, it is not possible to say if they are presented in the forward or backward time direction [START_REF] Batrouni | Equilibrium and Non-Equilibrium Statistical Thermodynamics[END_REF]. Considering, in particular, a binary liquid mixture, thermodynamic fluctuations of the concentration are always present. Basically, they represent the Poisson noise arising when molecules are counted inside a given volume. The process describing molecules entering and exiting a volume is time-symmetric at the macroscopic scale. From the mesoscopic point of view the equilibrium fluctuations of the concentration of a solute can be described by using fluctuating hydrodynamics [START_REF] Landau | Fluid Mechanics[END_REF]: a fluctuation arises because of a random source, and is dissipated by diffusion. Although the two processes are not time-symmetric, they are however connected by the fluctuation-dissipation and linear response theorems, which safeguard the time-symmetry of the outcoming phenomena.

Under such circumstances, time-symmetry is a consequence of the fact that the process is isoentropic, and this feature prevents the identification of a preferential direction for time. In the presence of a non-equilibrium condition dissipative processes determine a progressive increase of entropy. If one takes into account the degrees of freedom of all the molecules, however, the system still exhibits microscopic time-symmetry. This is due to the fact that the equations of motion of the molecules are of second order in time, without any term containing time derivatives of odd order. Conversely, the evolution of the macroscopic degrees of freedom of the system is not timesymmetric, because the averaged microscopic degrees of freedom determine the presence of derivatives of odd or-der in the equations for the evolution of the macroscopic system [START_REF] Batrouni | Equilibrium and Non-Equilibrium Statistical Thermodynamics[END_REF]. A typical example of such system is a Brownian particle under the action of a constant external force like gravity or a thermophoretic force [START_REF] Andrieux | Entropy production and time asymmetry in nonequilibrium fluctuations[END_REF]. A reconciliation of time symmetry at the microscopic scale and time asymmetry at the macroscopic scale requires a thorough comparison of the probability distribution of finding a certain time evolution for the position of the particle in the presence of a forward non-equilibrium driving force and the probability distribution in the presence of a backward driving force. The presence of a non-equilibrium condition breaks the time-symmetry of these probability distributions, giving rise to a production of entropy proportional to the unbalance between the probability distributions [START_REF] Andrieux | Thermodynamic time asymmetry in non-equilibrium fluctuations[END_REF].

A signature of the presence of time-symmetry is provided by the behavior of the time correlation functions of fluctuations upon inversion of time. With this respect, the understanding of time-cross-correlations in a stochastic system is of wide general relevance, because an asymmetric behavior can underlie the presence of a causal relationship between the correlated variables. A meaningful example of a stochastic system which exhibits a timeasymmetry is represented by the fluctuations in returns of stocks in financial markets. In this case, the time crosscorrelation function between fluctuations is asymmetric whenever a causal relationship exists between the returns of different stocks [START_REF] Borysov | Cross-correlation asymmetries and causal relationships between stock and market risk[END_REF][START_REF] Roman | Modeling cross-correlations within a many-assets market[END_REF].

A model system suitable for the understanding of timesymmetry in non-equilibrium systems is represented by a binary mixture where a macroscopic concentration gradient determines a non-equilibrium diffusion process. In the last 20 years it has been shown that a non-equilibrium diffusion process is accompanied by long-range large-amplitude fluctuations of concentration. These fluctuations have been predicted theoretically [7][START_REF] Law | Fluctuations in fluids out of thermal equilibrium[END_REF][9][START_REF] Segrè | Nonequilibrium fluctuations in liquid mixtures under the influence of gravity[END_REF][START_REF] Vailati | Nonequilibrium fluctuations in time-dependent diffusion processes[END_REF][START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF][START_REF] Croccolo | Non-local fluctuation phenomena in liquids[END_REF], reported experimentally [START_REF] Vailati | q divergence of nonequilibrium fluctuations and its gravity-induced frustration in a temperature stressed liquid mixture[END_REF][START_REF] Vailati | Giant fluctuations in a free diffusion process[END_REF][START_REF] Brogioli | Universal behavior of nonequilibrium fluctuations in free diffusion processes[END_REF][START_REF] Brogioli | Giant fluctuations in diffusion processes[END_REF][START_REF] Croccolo | A light scattering study of non equilibrium fluctuations in liquid mixtures to measure the soret and mass diffusion coefficient[END_REF] and in numerical simulations [19][20][21][22]. Since the origin of such fluctuations is a nonequilibrium phenomenon determined by the presence of a macroscopic gradient [START_REF] Lansing | Diffusion: mass transfer in fluid systems[END_REF], the fluctuations are not compelled to follow the fluctuation-dissipation and linear response theorems. In the presence of a non-equilibrium steady state the fluctuation-dissipation theorem can be generalized by taking into account the total entropy production [START_REF] Seifert | Fluctuation-dissipation theorem in nonequilibrium steady states[END_REF][START_REF] Mehl | Experimental accessibility of generalized fluctuation-dissipation relation for nonequilibrium steady states[END_REF][START_REF] Andrieux | Experiments in stochastic thermodynamics: Short history and perspectives[END_REF], but so far we are not aware of attempts of application of the theorem to the case of non-equilibrium fluctuations in liquid mixtures.

In this paper, we investigate the time cross-correlations of the concentration fluctuations of two solutes in a ternary mixture. We show that a violation of timesymmetry is observable at the mesoscopic scale for fluctuations in this system, as evidenced by the asymmetry of the time cross-correlation function. In Sect. II we outline how non-equilibrium fluctuations in binary mixtures are time-symmetric. In Sect. III A, we show that the time cross-correlation between the concentrations of the two solutes is indeed not time-symmetric, its center of mass being displaced from ∆t = 0. In Sect. III B we give a qualitative account of the observed behavior. In Sect. IV, we discuss the feasibility of experiments showing the violation of the time-symmetry. Finally, in Sect. V we perform a critical comparison of our results with those of previous work in the field [27].

II. NON-EQUILIBRIUM FLUCTUATIONS IN BINARY LIQUID MIXTURES

Theoretical, experimental and numerical work performed in the last 25 years [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF][START_REF] Croccolo | Non-local fluctuation phenomena in liquids[END_REF] showed that Fluctuating Hydrodynamics [START_REF] Landau | Fluid Mechanics[END_REF] represents a reliable quantitative model to describe non-equilibrium fluctuations generated by small gradients. Fluctuating Hydrodynamics relies on a linearization of the Navier-Stokes equations to obtain a set of equations that, once supplemented with stochastic white noise terms, act as Langevin equations for the fluctuations of the relevant thermophysical variables. In this work we are interested mainly in modeling the non-equilibrium concentration fluctuations generated by a concentration gradient and we will assume that the system is isothermal, so that the non-equilibrium temperature fluctuations can be neglected. We will also neglect the contribution of temperature and concentration equilibrium fluctuations and the gravitational force.

A. Time-symmetry in binary mixtures

In binary mixtures of liquids, the composition is defined univocally by the concentration of one component. Therefore the only time-correlation function that can be calculated is the self correlation < δc * (t, q)δc(t + ∆t, q) >. Here the brackets indicate a time average and δc is the displacement of the local concentration c from the macroscopic average. By definition, this function is time-symmetric under stationary conditions: δc * (t, q)δc(t -∆t, q) = δc * (t ′ + ∆t, q)δc(t ′ , q) = δc * (t + ∆t, q)δc(t, q) (1)

where the first equality is obtained by substituting t = t ′ + ∆t and the second equality holds due to the time invariance of the process.

It is important to notice that this time-symmetry is simply a mathematical consequence of the definition of time correlation, and would hold also in the case of an apparently time-asymmetric signal. In principle, higherorder correlations could show a time-asymmetry. However, concentration fluctuations have a Gaussian statistics, as a consequence of the central limit theorem. Therefore, by means of Wick's theorem, we can relate any n-point correlation to the 2-point correlation, which is, as previously shown, time-symmetric. It follows that every n-point correlation of the concentration fluctuations shows a symmetric behaviour, and, more generally, we can conclude that all of them are actually timesymmetric.

B. Fluctuating hydrodynamics in binary mixtures

The Fluctuating Hydrodynamics equations for the concentration fluctuations in binary mixtures can be written as:

∂δc ∂t = D∇ 2 δc -δv z ∇c - 1 ρ ∇ • δJ (2)
where δc is the fluctuation of the concentration, δv z is the fluctuation of the vertical component of the velocity, D is the mass diffusion coefficient, ∇c is the macroscopic concentration gradient, ρ is the density, and δJ is a stochastic source term for concentration fluctuations. This equation describes the variation of concentration due to diffusion, advection and the source of fluctuations. We will neglect the source term δJ, which gives rise to the equilibrium fluctuations only; it is negligible under all the practical conditions where the amplitude of non-equilibrium fluctuations is much larger than that of equilibrium ones, i.e. small wave numbers.

Under non-equilibrium conditions concentration fluctuations are generated by the coupling of velocity fluctuations to the macroscopic concentration gradient, which gives rise to non-equilibrium concentration fluctuations [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF]. The relevant component of the velocity is δv z , the component perpendicular to the macroscopic concentration gradient. The correlation of the velocity fluctuations is [27,28]:

δv * z (ω, q) δv z (ω ′ , q ′ ) = 2 k B T ν ρ q 2 ω 2 + ν 2 q 4 (2π) 4 δ (ω -ω ′ ) δ (q -q ′ ) (3)
where q 2 = q 2 -(q • ∇c) 2 is the square of the component of the wave vector q perpendicular to the macroscopic concentration gradient.

C. Concentration-velocity cross-correlation in binary mixtures

The investigation of non-equilibrium fluctuations relies on linearized hydrodynamics, where second order fluctuations are neglected. However, it can be shown that second order fluctuations in concentration δc and velocity δv z give rise to mesoscopic mass currents δj = δv z δc. Quite interestingly, the cumulative contribution of the microscopic mass currents determined by non-equilibrium fluctuations accounts for the whole Fick's flux [START_REF] Brogioli | Diffusive mass transfer by nonequilibrium fluctuations: Fick's law revisited[END_REF][START_REF] Donev | Diffusive transport by thermal velocity fluctuations[END_REF][START_REF] Donev | A reversible mesoscopic model of diffusion in liquids: ù from giant fluctuations to fick's law[END_REF]. This term violates time-symmetry. However, it is not present unless renormalization is performed. Therefore, the investigation of cross-correlations between velocity fluctuations and concentration fluctuations is particularly meaningful, because it is strictly connected to the microscopic flux δj.

In the case of a binary liquid mixture the equation governing the evolution on non-equilibrium fluctuations is Eq. 2. Rewriting it in Fourier space we get:

iωδc = -q 2 Dδc -δv z ∇c (4)
By solving with respect to the concentration fluctuations, we get:

δc = - δv z ∇c iω + q 2 D (5)
The cross-correlation function of velocity (along ∇c 0 ) and concentration is expressed by:

δc (q, ω) δv z (q ′ , ω ′ ) • ∇c = - δv z (q, ω) δv z (q ′ , ω ′ ) -iω + q 2 D , (6)
By substituting the expression of the velocity correlation, Eq. 3, into Eq. 6:

δc (q, ω) δv z (q ′ , ω ′ ) = -δ (q -q ′ ) δ (ω -ω ′ ) ∇c K B T ν 8π 4 ρ q 2 (ω 2 + ν 2 q 4 ) (-iω + q 2 D) . (7)
We carry on the calculation by assuming that the diffusion time is much longer than the viscous time. this approximation holds true for Sc = ν/D ≫ 1, the socalled large Schmidt number approximation, a condition fulfilled by most binary liquid mixtures (more details about the validity of such approximation are discussed in Ref. [START_REF] Vailati | Nonequilibrium fluctuations in time-dependent diffusion processes[END_REF]). We obtain the approximated expression:

δc (q, ω) δv z (q ′ , ω ′ ) = -δ (q -q ′ ) δ (ω -ω ′ ) ∇c K B T ν 8π 4 ρ q 2 ν 2 q 4 (-iω + q 2 D) . (8)
The time correlation is obtained by Fourier transforming. The integrand has one pole at the positive imaginary part of ω, hence we get a correlation only for ∆t > 0:

δc (q, t) δv z (q ′ , t + ∆t) = -δ (q -q ′ ) ∇c K B T ν 8π 4 ρ q 2 ν 2 q 4 exp -∆tDq 2 H (∆t) . ( 9
)
This result has a clear physical interpretation: the concentration fluctuations in non-equilibrium conditions are generated by the coupling of velocity fluctuations with the macroscopic concentration gradient, hence the concentration fluctuation takes place only after the velocity fluctuation. Therefore, under such circumstances the causal relation between velocity fluctuations and concentration fluctuations determines the presence of a correlation between the two quantities.

In the next section we show that, at variance with the binary mixtures, mixtures of three or more components show an asymmetry of the time-correlations.

III. NON-EQUILIBRIUM FLUCTUATIONS IN TERNARY MIXTURES

We now consider the case of non-equilibrium fluctuations in a ternary liquid mixture. The determination of the time auto-correlations of non-equilibrium fluctuations in ternary mixtures has been dealt with in detail in a previous work [27]. A similar approach can be used to determine the time cross-correlation functions. In the Appendix we will present the results of this approach. In the following we will present a more direct determination of the time cross-correlation function of concentration fluctuations in a ternary mixture, which relies directly on the diagonalization of the hydrodynamic equations describing the non-equilibrium concentration fluctuations.

A. Correlation and cross-correlation of the concentration fluctuations

In analogy to Eq. 2, the Fluctuating Hydrodynamics equations for the concentration fluctuations in ternary mixtures can be written as:

∂δc 1 ∂t = D 11 ∇ 2 δc 1 + D 12 ∇ 2 δc 2 -δv z ∇c 1 - 1 ρ ∇ • δJ 1 ∂δc 2 ∂t = D 21 ∇ 2 δc 1 + D 22 ∇ 2 δc 2 -δv z ∇c 2 - 1 ρ ∇ • δJ 2 (10)
where δc i is the concentration fluctuations of the component i ∈ [1, 2], δv z is the fluctuation of the vertical component of the velocity, D i,j is the mass diffusion matrix, ∇c i is the macroscopic concentration gradient of component i, ρ is the density, and δJ i is a stochastic source term for concentration fluctuations. This equation describes the variation of concentration due to diffusion, advection and to the fluctuation source.

As we have done above for the binary mixture, we will neglect the source terms δJ i , because it gives rise to the equilibrium fluctuations only.

We diagonalize the matrix D, i.e. we find a transformation matrix T such that:

T -1 • D 1,1 D 1,2 D 2,1 D 2,2 • T = D1 0 0 D2 (11)
The diagonal elements D1 and D2 are the eigenvalues of the matrix D. Since the cross-diffusion terms are usually smaller than the self-diffusion terms, T is usually close to an identity and the eigenvalues D1 and D2 are close to the self-diffusion coefficients D 1 and D 2 . By using this transformation, we define the diagonalized concentration fluctuations δc i

T -1 • δc 1 δc 2 = δc 1 δc 2 (12)
and diagonalized macroscopic concentration gradients ∇c i

T -1 • ∇c 1 ∇c 2 = ∇c 1 ∇c 2 (13)
We thus rewrite Eqs. 10 as:

∂ δc 1 ∂t = D1 ∇ 2 δc 1 -δv z ∇c 1 ∂ δc 2 ∂t = D2 ∇ 2 δc 2 -δv z ∇c 2 (14)
The two equations are now decoupled and can be solved separately, similarly to what we have done in Sect. II for the fluctuations in a binary mixture. First we rewrite the equations in Fourier space:

iω δc i = -D1 q 2 δc i -δv z ∇c i (15)
By solving with respect to the concentration fluctuations, we get:

δc i = - δv z ∇c i iω + q 2 Di ( 16
)
From this expression, we calculate the self and cross correlations:

δc * i (q, ω) δc j (q ′ , ω ′ ) = ∇c i ∇c j δv z (q, ω) δv z (q ′ , ω ′ ) Di q 2 + iω Dj q 2 -iω (17)
By using Eq. 3:

δc * i (q, ω) δc j (q ′ , ω ′ ) = 2 k B T ν ρ ∇c i ∇c j q 2 (ω 2 + ν 2 q 4 ) Di q 2 + iω Dj q 2 -iω (2π) 4 δ (ω -ω ′ ) δ (q -q ′ ) (18)
and by using the definition of correlation:

δc * i (q, ω) δc j (q ′ , ω ′ ) T = Ĉi,j (q, ω) (2π) 4 δ (ω -ω ′ ) δ (q -q ′ ) (19)
we determine the correlation function C from Eq. 18:

Ĉi,j (q, ω) = 2 k B T ν ρ ∇c i ∇c j q 2 (ω 2 + ν 2 q 4 ) Di q 2 + iω Dj q 2 -iω (20)
Again, we make use of the large Schmidt number approximation [START_REF] Vailati | Nonequilibrium fluctuations in time-dependent diffusion processes[END_REF] in its straightforward extension to ternary mixtures:

Sc i = ν/D i ≫ 1, so that the result is: Ĉi,j (q, ω) = 2 k B T ρνq 4 ∇c i ∇c j q 2 Di q 2 + iω Dj q 2 -iω (21)
The time-correlation is then obtained by Fourier transforming in ω:

Ĉi,j (q, ∆t) = 2 k B T ρνq 4 ∇c i ∇c j q 2
Di q 2 + iω Dj q 2 -iω e iω∆t dω (22) In the case of the autocorrelation, the two decay times (forward and backward) equal the diffusion decay time of the component under consideration.

The integration can be easily performed by means of the method of the residues (see Fig. 1). The integrand has two poles on the imaginary axis of ω, one at positive and one at negative imaginary part, with residues R + and R -:

R + = i e -D1 q 2 ∆t D1 + D2 q 2 R -= i e D2 q 2 ∆t D1 + D2 q 2 . ( 23
)
Re( ) ω

) ω Im( R + R - ∆t>0 ∆t<0 FIG. 1.
Residues of the integrand in Eq. 22 and integration paths.

The integration is performed along different paths in the case of ∆t > 0 and ∆t < 0, as shown in Fig. 1. In the two different cases, a different pole is enclosed by the path, and the corresponding residue (with either D1 or D2 ) appears in the result. By using the Heaviside step function H the correlation function can be expressed as:

Ĉi,j (q, ∆t) = 2 k B T ρνq 4 ∇c i ∇c j H (∆t) e -∆t D1q 2 + H (-∆t) e ∆t D2q 2 (24)
Therefore, the correlation function is composed by two exponential decays, with different characteristic decay times forward and backward in time, as shown in Fig. 2.

B. Origin of the time-asymmetry

In this section, we discuss the qualitative reason why the cross-correlation is time-asymmetric, and in particular, why it shows an exponential decay with different time constants forward and backward in time.

In our system, the observable quantities (δc 1 and δc 2 ) are fully determined by the liquid velocity v z . In particular, the concentration fluctuation at time t 0 is determined by the values of v z at times t < t 0 ; this can be seen as a consequence of the cause-effect relation between liquid velocity and concentration fluctuations discussed in Sect. II C. We can thus write δc v (-∞,t] z , t, q , evaluated at time t and wave vector q, as deterministically given as a function of v z in the interval (-∞, t].

The correlation function becomes: The ratio between the two diffusion coefficients is 3.

δc 1 (t, q) * δc 2 (t + ∆t, q) = δc 1 v (-∞,t] z , t, q * δc 2 v (-∞,t+∆t] z , t + ∆t, q P v (-∞,t+∆t] z D v (-∞,t+∆t] z ( 25
)
where P represents the probability of a given v z and •D [v z ] is the functional integral over the function v z . In the case ∆t > 0, we can separate the interval (-∞, t+∆t] into the two separate intervals (-∞, t] and (t, t + ∆t]:

δc 1 (t, q) * δc 2 (t + ∆t, q) = δc 1 v (-∞,t] z , t, q * δc 2 v (-∞,t] z , v (t,t+∆t] z , t + ∆t, q P v (-∞,t] z P v (t,t+∆t] z |v (-∞,t] z D v (-∞,t] z D v (t,t+∆t] z (26)
where the symbol P [A|B] represents the conditional probability. By multiplying and dividing by δc 2 (t, q):

δc 1 (t, q) * δc 2 (t + ∆t, q) = δc 1 v (-∞,t] z , t, q * δc 2 v (-∞,t] z , t, q δc 2 v (-∞,t] z , v (t,t+∆t] z , t + ∆t, q δc 2 v (-∞,t] z , t, q P v (-∞,t] z P v (t,t+∆t] z |v (-∞,t] z D v (-∞,t] z D v (t,t+∆t] z (27)
This can be rewritten as:

δc 1 (t, q) * δc 2 (t + ∆t, q) = g v (-∞,t] z , t, q f 2 v (-∞,t] z , t, ∆t, q P v (-∞,t] z D v (-∞,t] z ( 28
)
where

g v (-∞,t] z , t, q = δc 1 v (-∞,t] z , t, q * δc 2 v (-∞,t] z , t, q f 2 v (-∞,t] z , t, ∆t, q = δc 2 v (-∞,t] z , v (t,t+∆t] z , t + ∆t, q δc 2 v (-∞,t] z , t, q P v (t,t+∆t] z |v (-∞,t] z D v (t,t+∆t] z ( 29
)
We thus see that, for ∆t > 0, the correlation function reflects the behavior of δc 2 with time. An analogous calculation with ∆t < 0 gives an analogous expression, with δc 1 playing the same role. Therefore, the time-asymmetry is a consequence of the cause-effect relation: both δc 1 and δc 2 are determined by v z , and the effect of v z at a given time can be seen only at subsequent times.

The center of mass of the cross-correlation function, Eq. 24, is not at ∆t = 0, as visible in Fig. 2. This is a violation of the time-symmetry. As mentioned above, the concentration fluctuations of the two solutes are generated simultaneously by a velocity fluctuation, but then they relax at different rate, depending on the diffusion coefficient. Obviously, the solute with larger diffusion coefficient relaxes faster than the other. The center of mass of the cross-correlation function is displaced accordingly.

IV. APPLICATIONS TO EXPERIMENTS

Non-equilibrium concentration fluctuations arise during diffusion processes in multi-component mixtures. In the presence of a macroscopic concentration gradient the coupling of spontaneous thermal velocity fluctuations gives rise to non-equilibrium concentration fluctuations. On Earth large-wavelength fluctuations are strongly influenced by the gravity force, which can give rise to their stabilization or destabilization, depending on whether the macroscopic density profile associated to the concentration gradient is gravitationally stable [START_REF] Segrè | Nonequilibrium fluctuations in liquid mixtures under the influence of gravity[END_REF][START_REF] Vailati | q divergence of nonequilibrium fluctuations and its gravity-induced frustration in a temperature stressed liquid mixture[END_REF] or not [START_REF] Giavazzi | Scaling of the spatial power spectrum of excitations at the onset of solutal convection in a nanofluid far from equilibrium[END_REF][START_REF] Croccolo | Effect of a marginal inclination on pattern formation in a binary liquid mixture under thermal stress[END_REF].

In the case of binary liquid mixtures a stable configuration can be obtained by bringing two miscible fluids into contact, so that the denser fluid is below the less dense one [START_REF] Vailati | Giant fluctuations in a free diffusion process[END_REF][START_REF] Brogioli | Universal behavior of nonequilibrium fluctuations in free diffusion processes[END_REF][START_REF] Fabrizio Croccolo | Nondiffusive decay of gradient-driven fluctuations in a free-diffusion process[END_REF]. Alternatively, one can use an external temperature gradient to induce the concentration gradient through the Soret effect [START_REF] Vailati | q divergence of nonequilibrium fluctuations and its gravity-induced frustration in a temperature stressed liquid mixture[END_REF][START_REF] Croccolo | A light scattering study of non equilibrium fluctuations in liquid mixtures to measure the soret and mass diffusion coefficient[END_REF]35,[START_REF] Vailati | Study of the q divergence of nonequilibrium fluctuations in a stressed fluid[END_REF]. In the case of a sample with a positive Soret coefficient S T , the imposition of a temperature gradient ∇T induces a steady-state concentration gradient ∇c = -S T c(1 -c)∇T , where c is the concentration of the sample. Non-equilibrium fluctuations induce fluctuations in the index of refraction, which can be picked up by using either small angle light scattering techniques [START_REF] Vailati | Giant fluctuations in a free diffusion process[END_REF] or near field techniques such as quantitative shadowgraphy [START_REF] Brogioli | Universal behavior of nonequilibrium fluctuations in free diffusion processes[END_REF][START_REF] Steven | Physical optics treatment of the shadowgraph[END_REF][39]. Both methods allow to determine the time autocorrelation function of the correlation fluctuations [START_REF] Fabrizio Croccolo | Nondiffusive decay of gradient-driven fluctuations in a free-diffusion process[END_REF].

Per il quantitative shadowgraph ho aggiunto un riferimento al nostro articolo del 2000. Avete qualcosa in contrario?

The experimental investigation of non-equilibrium fluctuations in ternary mixtures is problematic, because on Earth the presence of the gravity force gives rise to convective instabilities under rather generic conditions. This is due to the fact that, even in the configuration where the density profile is gravitationally stable, solutes diffuse at different rates and this can give rise to a local destabilization of the density profile. For this reason a systematic investigation of mass transfer in ternary liquid mixtures requires microgravity conditions. The mass transfer in ternary liquid mixtures has been investigated extensively in microgravity in the framework of the DCMIX project of the European Space Agency [40][41][42]. The DCMIX experiments were hosted inside the SODI facility of ESA, where a two color Mach-Zehnder interferometer allowed to recover the Soret coefficients and the eigenvalues of the diffusion matrix. Thermal gradient cells for the investigation on non-equilibrium fluctuations in ternary systems require a different geometry with respect to that adopted in DCMIX. Indeed, one wants to access the sample optically by looking in the direction parallel to the concentration gradient. This kind of configuration was used during the GRADFLEX experiment, which flew for two weeks aboard the FOTON M3 spacecraft [START_REF] Vailati | Fractal fronts of diffusion in microgravity[END_REF][START_REF] Takacs | Thermal fluctuations in a layer of liquid cs2 subjected to temperature gradients with and without the influence of gravity[END_REF][START_REF] Cerbino | Dynamic scaling for the growth of non-equilibrium fluctuations during thermophoretic diffusion in microgravity[END_REF][46]. GRADFLEX contained a thermal gradient cell and a shadowgraph diagnostics that allowed to grab images of the phase perturbations determined by non-equilibrium fluctuations. The experiments allowed to attain a first important confirmation of the fact that in the absence of gravity non-equilibrium fluctuations grow up to the size of the container hosting the sample. The Giant-Fluctuations (NEUF-DIX) project of ESA will adopt a configuration similar to GRADFLEX to investigate non-equilibrium fluctuations in multi-component complex liquids [47]. The facility will comprise 6 thermal gradient cells and shadowgraph diagnostics. The use of two light sources, one in the red and one in the blue region of the visible spectrum will allow to decouple the contributions to the fluctuations determined by the two independent components of the investigated mixtures. The cross-correlation of the shadowgraph images obtained with the two light sources will allow to perform an experimental check of the asymmetry of the time cross-correlation function of non-equilibrium fluctuations in ternary mixtures. Suitable samples will include ternary mixtures including either a polymer or a colloid as a third component and ternary mixtures of biological relevance, such as protein solutions.

Qui e' necessaria una connessione con gli esperimenti The various components of the mixtures have different refraction index and different disperions, thus the transfer function of the shadowgraph technique ... with a suitable procedure it is possible to calculate ... Questa e' la frase di Jose' The use of two-color shadowgraphy and light scattering for separating the contributions of the various compoments of the mixture requires to re-work the underlying physical optics theory, taking into account the presence of two wavelength. This study is beyond the scope of the present paper and will be performed as a part of the above-mentioned projects.

Qui inizia la parte banale di connessione tra indici di rifrazione e concentrazioni diagonali

In order to show the feasibility of the detection of the time-asimmetry of the fluctuations, here we use a simplified approach, i.e. we assume that the optical technique give access to the fluctuation of the refraction indices at the two wavelengths, [δn 1 , δn 2 ]. We approximate with a linear relationship, as usually observed:

δc 1 δc 2 = R • δn 1 δn 2 (30)
with a given matrix R. Both R and T are experimentally measurable, thus, from the optically measured [δn 1 , δn 2 ] it is possible to calculate the diagonalized concentrations:

δc 1 δc 2 = T • R • δn 1 δn 2 (31)
It is possible to first calculate the diagonalized concentrations δc 1 , δc 1 and then calculate their correlation function Ĉi,j , which shows the time-asymmetry. An alternative approach could be to calculate the correlation function of the refraction indices at the two wavelengths, C n i,j ; it is connected to the correlation function of the diagonalized concentration by the following equation:

C n i,j = (T • R) -1 Ĉi,j (T • R) -1 T (32)
From this equation, we see that the cross-correlation of the refraction index is a linear combination of the autoand the cross-correlations of the diagonalized concentrations, hence it will appear more time-symmetric than the cross-correlation of the diagonalized concentration.

V. CONCLUSION

In this work we have analyzed the cross correlations of non-equilibrium concentration fluctuations of independent components in a multi-component mixture. In the trivial case of binary mixtures, the number of independent components is only one, so that one can only investigate auto-correlation, which is symmetric, notwithstanding the entropy production during the nonequilibrium process. In the case of a ternary mixture, the cross-correlations are investigated and they appear to be asymmetric in time. An experimental verification would require a measurement technique able to separate the contribution of different components. This is in principle possible with the two-wavelength shadowgraph apparatus that is being developed within the NEUF-DIX project. An experimental verification of the timeasymmetry will then be performed in a near future.

Appendix

In this section we will derive the time cross-correlation function of non-equilibrium concentration fluctuations in a ternary mixture by using a procedure similar to that adopted in Ref. [27] to determine the auto-correlation function. At variance with Ref. [27] we will take into account the contribution of the imaginary part of the time correlation function. These contributions can be neglected when determining the auto-correlation function, but become significant in the case of the cross-correlation function. Following Ref. [27] the Fluctuations Hydrodynamics equations describing non-equilibrium concentration fluctuations in a ternary mixture are:

∂δc 1 ∂t = D 11 ∇ 2 δc 1 + D 12 ∇ 2 δc 2 -δv z ∇c 1 - 1 ρ ∇ • δJ 1 ∂δc 2 ∂t = D 21 ∇ 2 δc 1 + D 22 ∇ 2 δc 2 -δv z ∇c 2 - 1 ρ ∇ • δJ 2 (A.1)
In Fourier space:

iωδc 1 = -D 11 q 2 δc 1 -D 12 q 2 δc 2 -δv z ∇c 1 + F 1 iωδc 2 = -D 21 q 2 δc 1 + D 22 q 2 δc 2 -δv z ∇c 2 + F 2 (A.2)
This expression can be rewritten as:

iωδc + q 2 D • δc = -δv z ∇c + F (A.3)
where δc = (δc 1 , δc 2 ), ∇c = (∇c 1 , ∇c 2 ), F = (F 1 , F 2 ), and D is the matrix of diffusion coefficients. We decompose the concentration fluctuation as the sum of an equilibrium and a non-equilibrium contribution:

δc = δc E + δc N E (A.4)
where

iωδc E + q 2 D • δc E = F (A.5) and iωδc N E + q 2 D • δc N E = -δv z ∇c (A.6)
Since F and δv z are not correlated, we can evaluate separately the contributions of the two terms, and we now focus on the non-equilibrium part δc N E . By calculating δc N E :

δc N E = -iω + Dq 2 -1 • ∇cδv z (A.7)
The correlation is:

δc N E (ω, q) * δc N E (ω ′ , q ′ ) T = -iωI + Dq 2 -1 • ∇c • ∇c T • iωI + Dq 2 -1T δv * z (ω, q) δv z (ω ′ , q ′ ) (A.8) The correlation of δv z is: δv * z (ω, q) δv z (ω ′ , q ′ ) = 2k B T q 2 νρq 4 (2π) 4 δ (ω -ω ′ ) δ (q -q ′ )
(A.9) Using this correlation, we get:

δc N E (ω, q) * δc N E (ω ′ , q ′ ) T = -iωI + Dq 2 -1 • ∇c • ∇c T • iωI + Dq 2 -1T 2k B T q 2 νρq 4 (2π) 4 δ (ω -ω ′ ) δ (q -q ′ ) (A.10)
By using the definition of correlation:

δc N E (ω, q) * δc N E (ω ′ , q ′ ) T = C N E (ω, q) (2π) 4 δ (ω -ω ′ ) δ (q -q ′ ) (A.11)
we get:

C N E (ω, q) = 2k B T q 2 νρq 4 -iωI + Dq 2 -1 • ∇c • ∇c T • iωI + Dq 2 -1T (A.12)
Now we diagonalize the matrix D. We consider the case in which D has two real eigenvalues, although an imaginary part could be present. The eigenvalues can be calculated analytically (Eq. 18 of Ref. where D is a diagonal matrix with elements D1 and D2 . By using this transformation:

C N E (ω, q) = 2k B T q 2 νρq 4 -iωI + T • D • T -1 q 2 -1 • ∇c• ∇c T • iωI + T • D • T -1 q 2 -1T (A.15)
We rewrite this last equation as:

C N E (ω, q) = 2k B T q 2 νρq 4 T • -iωI + Dq 2 -1 T -1 • ∇c• ∇c T • T -1T • iωI + Dq 2 -1T • T T (A.16)
By introducing the projectors P i : P 1 = 1 0 0 0 (A.17)

P 2 = 0 0 0 1 (A.18)
The correlation becomes:

C N E (ω, q) = 2k B T q 2 νρq 4 T • P 1 -iω + D1 q 2 + P 2 -iω + D2 q 2 T -1 • ∇c• ∇c T • T -1T • P 1 iω + D1 q 2 + P 2 iω + D2 q 2 • T T (A.19)
By distributing the products on the sum:

C N E (ω, q) = 2k B T q 2 νρq 6 i,j

A i,j

Di + Dj q 2

-iω + Di q 2 iω + Dj q 2 (A. 20) where the matrices A i,j are defined as:

A i,j = T • P i T -1 • ∇c • ∇c T • T -1T • P j • T T Di + Dj (A.21)
We now compare our results with those of Ref. [27].

The real part of C N E can be written as:

C N E (ω, q) = 2k B T q 2 νρq 6 i,j

A i,j Di q 2 ω 2 + D2 i q 4 + Dj q 2 ω 2 + D2 j q 4

(A.22)

The sum can be rearranged as:

C N E (ω, q) = 2k B T q 2 νρq 6

A 1 Di q 2 ω 2 + D2 i q 4

+ A 2 Dj q 2 ω 2 + D2 j q 4 (A.23)

where This expression corresponds to Eqs. 20-23 of Ref. [27]. The result given in Ref. [27] is thus correct for the autocorrelations, which are real, while it neglects a significant imaginary part in the calculation of the cross-correlation.

A 1 = 2A
In practically relevant cases, D 1,2 = D 2,1 = 0; this means that the eigenvalues D correspond to the diffusion coefficients D 1 and D 2 , and the transformation matrix T is the identity.

We calculate the correlations for this case:

C N E 1,1 (ω, q) = 2k B T q 2 νρq 6 ∇c 2 1 2D 1 2D 1 q 2 ω 2 + D 1 q 2
(A.26)

C N E 1,2 (ω, q) = 2k B T q 2 νρq 6 ∇c 1 ∇c 2 D 1 + D 2
(D 1 + D 2 ) q 2 (-iω + D 1 q 2 ) (iω + D 2 q 2 ) (A.27)

C N E 2,2 (ω, q) = 2k B T q 2 νρq 6 ∇c 2 1 2D 2
2D 2 q 2 ω 2 + D 2 q 2 (A.28)

C N E 2,1 (ω, q) = 2k B T q 2 νρq 6 ∇c 1 ∇c 2 D 1 + D 2 (D 1 + D 2 ) q 2 (iω + D 1 q 2 ) (-iω + D 2 q 2 ) (A.29)
We see that the self power-spectra of the concentration fluctuations are real with a Lorentzian shape. In this case the determination of the auto-correlation function is straightforward and yields a single exponential decay. Conversely, the cross-spectra exhibit two poles and the determination of the cross-correlation function requires the procedure outlined in Sect. III A.

FIG. 2 .

 2 FIG. 2. Example of correlation function. The axis are arbitrary.The ratio between the two diffusion coefficients is 3.

11 + 2 + 4D 12 D 21 (A. 13 )

 1122113 D 22 ± (D 11 -D 22 ) The transformation T diagonalizes D: T -1 • D • T = D (A.14)

  1,1 + A 1,2 + A 2,1 (A.24) A 2 = 2A 2,2 + A 1,2 + A 2,1 (A.25)

ACKNOWLEDGMENTS

We thank J. M. Ortiz de Zárate for useful discussions. F. C and A. V. acknowledge partial financial support from the European Space Agency.