Matteo Sabato

Fabrizio Croccolo

Marina Carpineti

Alberto Vailati

Kinetics of growth of non-equilibrium fluctuations during thermodiffusion in a polymer solution

Keywords: XX, XX, XX No code given

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Kinetics of growth of non-equilibrium fluctuations during thermodiffusion in a polymer solution

Marina Carpineti,

Introduction

Diffusion processes in liquid mixtures under the action of a temperature or concentration gradient are accompanied by giant non-equilibrium concentration fluctuations [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF][START_REF] Croccolo | [END_REF]. These fluctuations have been extensively described and investigated in theoretical and experimental studies performed under ideal conditions, typically during a thermal diffusion process at steady-state [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF]3,4] or during quasistationary processes, such as free diffusion [5][6][7][8]. A key open question is how do the fluctuations develop in a binary liquid mixture starting from a condition of uniform concentration and leading to a steady macroscopic concentration profile. The kinetics of growth of the concentration gradient under these conditions is of great relevance for the determination of the transport coefficients of mixtures. A huge effort has been devoted to the development of methods suitable to determine the time evolution of the macroscopic concentration gradient in multi-component mixtures. This includes the recent development of molecular dynamics simulations, which have been used to determine the transport coefficients during a thermodiffusion process in a binary mixture [9]. Of particular rele-Send offprint requests to: marina.carpineti@unimi.it vance is the investigation of the transport coefficients of ternary mixtures in the absence of gravity on the International Space Station (ISS), performed in the framework of the SODI-DCMIX project of the European Space Agency [10][11][12][13].

Conversely, so far the investigation of the growth of fluctuations during the development of the macroscopic state has been very limited. A theoretical model based on Fluctuating Hydrodynamics provides an effective description of the statistical properties of the non-equilibrium fluctuations (NEFs) for systems where the macroscopic state changes much slower than the fluctuations [14]. Moreover, recent simulations [15] describe the onset of nonequilibrium fluctuations induced by thermophoretic diffusion in microgravity, predicting a spinodal-like growth [16][17][18][19] of the fluctuations, leading to a peaked structure factor with scale invariance at large wave vectors. A difficulty in investigating transient diffusion processes on Earth is that gravity strongly affects the behaviour of long wavelength fluctuations, limiting their amplitude below a typical cut-off size, and making it difficult to observe a peaked structure factor. In principle the effect of gravity can be reduced by using a thin sample [20], but this choice limits the amplitude of the light scattered by non-equilibrium fluctuations, thus making difficult to characterize them during a transient diffusion process. Of course another option is to perform experiments in microgravity conditions and this is one of the purposes of the Giant Fluctuations (formerly NEUF-DIX) project of ESA [21]. The present work performed on earth is intended as preliminary to the future tests in microgravity conditions. We present the results of a study performed on Earth at the onset of non-equilibrium concentration fluctuations induced by the Soret effect in a solution of a polystyrene polymer in toluene. Transient processes are usually very fast and it is difficult to collect a sufficient amount of data to obtain a good statistical characterization of them. As suggested in [15], a more accurate statistics can be achieved by increasing the time necessary for the development of a concentration gradient induced by thermophoresis. To slow down the kinetics we chose to use a solution of a high molecular weight polymer. The concentration of polymer is chosen above the overlap concentration to guarantee that the amplitude of the scattered light is large enough to be detected.

We use a quantitative shadowgraph technique [22,6,7], particularly effective for studying this kind of phenomena due to its extremely high sensitivity at small wave vectors. Shadowgraph relies on the detection in the near field of the interference between the transmitted beam and the scattered light. Depending on their wave vector, concentration fluctuations produce changes of the index of refraction, which appear as modulations of light intensity on the images. With respect to light scattering, Shadowgraph makes it possible to directly visualize the sample, but also to recover the structure factor from the images, collecting light at small wave vectors q without problems of stray light rejection and alignment that affect small angle light scattering. The only disadvantage of the technique is the presence of a strongly q-dependent transfer function, which usually requires a precise calibration to determine the static structure factor of the fluctuations (See for example ref [23]).

In this work we have focused our attention on the study of the kinetics of growth of non-equilibrium concentration fluctuations. As we will show, the method that we have used allows to get rid of the shadowgraph transfer function without performing calibration measurements.

We will show that the growth rate of concentration fluctuations exhibits a nearly q-independent power law behaviour R f (t) ∝ t -α , at variance with the diffusive growth of the amplitude of the concentration modes c(q, t) ∝ 1 -exp(-Dq 2 t), whose characteristic time strongly depends on the wave vector.

Non-equilibrium fluctuations in diffusion processes

NEFs play a fundamental role in all diffusive processes [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF] due to the fact that they perturb significantly the macroscopic state. Indeed, it can be shown that the microscopic mass flows generated by non-equilibrium fluctuations are responsible for the Fickean diffusive mass transfer [24]. Moreover, it has been predicted recently that non-equilibrium fluctuations can significantly modify the interactions between macroscopic bodies embedded in a fluid [25][26][27][28][29], a result that has not yet been confirmed experimentally. Linearized fluctuating hydrodynamics gives a quite complete theoretical description of NEFs in binary liquid mixtures when ideal conditions are realized, as for example stationary states, small gradients, and diluted systems [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF][30][31][32][33]. Theoretical work provided evidence that the non-equilibrium fluctuations are long range correlated, their static structure factor diverging as q -4 as the wave vector q goes to zero. At large wave vectors the fluctuations relax diffusively and they are not affected by gravity. However, at small wave vectors gravity frustrates fluctuations, because at these length scales buoyancy determines a faster relaxation than diffusion [34]. The separation between these two regimes of relaxation is reflected by the presence of a roll-off in the static structure factor of the non-equilibrium fluctuations at the roll-off wave vector [14]:

q ro = βg∇c νD 1 4 (1)
where g is the acceleration of gravity, c is the weight fraction of the denser component ν the kinematic viscosity, D the diffusion coefficient and β the solutal expansion coefficient. For q < q RO the structure factor is almost constant. These predictions found confirmation in many experiments performed on Earth [3,4,6,7,[35][36][37][38][39] Further experiments allowed to establish that under microgravity conditions the only limit to the growth of non-equilibrium fluctuations is determined by the size of the cell containing the sample, which imposes a finite size limit [23,15,40]. Notwithstanding the good understanding of non-equilibrium fluctuations at steady state in the linear regime, recent experiments have shown that a deeper understanding of non-equilibrium fluctuations under more general conditions would be highly desirable [8,15,41,42].

Our attention is focused on the determination of the kinetics of growth of the concentration fluctuations' modes during the transient leading to a stationary state and on the comparison of the results with theoretical predictions and preliminary experimental data [14,15]. Starting from a condition where the concentration is uniform, non-equilibrium fluctuations can be induced by taking advantage of a thermodiffusion process. When a temperature gradient ∇T is suddenly applied to a liquid mixture, previously kept at fixed temperature, a mass flow j = -ρD[∇c + S T c(1 -c)∇T ] is induced due to Soret effect, until a steady state is reached when a macroscopic constant concentration gradient ∇c s = -S T c(1 -c)∇T is formed inside the sample. Assuming that the system is in a gravitationally stable configuration, it is possible to describe how the concentration profile evolves in time, writing it as a function of the dimensionless height ζ = z/a and time t [14,[START_REF] Crank | The Mathematics of Diffusion[END_REF] (Fig. 1) :

c(ζ, t) = c 0 + a∇c s 1 2 -ζ - 2 π 2 ∞ j=1 1 j 2 [1 -(-1) j ] × cos(jπζ)exp - j 2 π 2 D a 2 t ( 2
)
where z is the vertical coordinate, a is the vertical thickness of the sample and c 0 is the initial uniform concentration. Theory and preliminary experiments performed on Earth show that during a transient thermal diffusion process the static structure factor of concentration fluctuations has features similar to those observed at steady state, namely a q -4 decay at large q vectors and a rolloff at smaller wave vectors. It can be shown [14] that the structure factor can be written as

S(q, t) = S eq   1 + ∇c(t) ∇c g -1 1 1 + q qro(t) 4    (3)
where S eq is the equilibrium static structure factor, ∇c g is the equilibrium concentration gradient induced by sedimentation, and q ro (t) is the roll-off wave vector defined in Eqn. 1, where now the concentration gradient ∇c is time dependent due to the transient condition.

Notice that at the impermeable horizontal boundaries the mass flow must be zero. Therefore, following Archibald [START_REF] Tanford | Physical Chemistry of Macromolecules[END_REF]14], the concentration gradient at the boundaries must reach instantaneously its steady state value ∇c s . As a consequence, the concentration profile shown in Fig. 1 can be roughly divided into three regions: two boundary layers where the gradient has achieved approximately the steady state value ∇c s and one central portion where the gradient is zero. Since the structure factor is proportional to ∇c below q ro and to ∇c 2 at larger wave vectors (see Eqn. 1) the only regions where its amplitude is significantly large are the boundary layers.

Under typical experimental conditions S(q) S eq and ∇c ∇c g and the structure factor of fluctuations occurring in the boundary layers can be approximated as

S(q) = S eq ∇c s ∇c g 1 1 + q qro 4 (4)
Equation 4 does not depend on time. However, in any real light scattering and shadowgraph experiment the structure factor is integrated across the sample thickness. As outlined above, in a transient thermal diffusion experiment only the boundary layer regions contribute significantly to it, and their growth gives rise to a time dependence of the integrated structure factor S c (q, t): S c (q, t) = = S eq ∇c s ∇c g

1 1 + q qro 4 δ(t) 0 dz + a a-δ(t) dz = = S eq ∆c(t) ∇c g 1 1 + q qro 4 (5)
Here δ(t) is the boundary layer thickness and ∆c(t) = 2∇c s δ(t).

From Eqn. 5 it can be appreciated that the structure factor is proportional to the difference between the concentrations at the two cell boundaries ∆c (see Fig. 1) increases in time. On the contrary, the roll-off position is almost stationary, because it depends on the magnitude of the concentration gradient near the boundaries, which reaches the steady-state almost immediately after the beginning of the process [14].

A different behavior is predicted for measurements performed under microgravity conditions starting from a configuration where the concentration is uniform [15]. In the absence of buoyancy driven effects, the structure factor is expected to exhibit a peak at finite wave vector that moves in time to smaller q vectors, while its amplitude increases. This behaviour is strongly reminiscent of that observed during spinodal decomposition processes, where a dominant mode in the concentration flutuations depends on the fact that the process is driven by a negative diffusion coefficient [16][17][18]15]. For the growth of NEFs, however, the presence of a dominant mode in the fluctuations is originated by the competition between the diffusive and Soret fluxes [15]. During the transient, the mass flow is dominated by the Soret flux, which gives rise to a partial demixing of the sample, a situation that mirrors, at least qualitatively, the demixing process occurring during a spinodal decomposition. Some preliminary results collected in microgravity show fair agreement with the theoretical predictions [15]. However, in order to confirm the presence of a peak at small wave vectors, further efforts are required.

Material and methods

The cell is formed by two sapphire windows pressed against an o-ring gasket, 26 mm inner diameter, delimiting the sample volume, while two spacers of thickness 1.3 mm placed between the windows guarantee a well defined sample thickness. The sample is thermalized using two ring shaped Peltier elements thermally coupled to the windows, and controlled through two Proportional-Integral-Derivative servo-controls with an accuracy of 0.01 • C.

Two needles inserted in the o-ring allow to fill the cell. In filling the cell it is important to carefully remove any residual air, as the presence of bubbles strongly affects the thermal transport through the sample. The filled cell is carefully leveled horizontally, after checking that all the optical components are well aligned along the vertical optical axis of the setup.

The onset of the fluctuations is a quite elusive process and in order to study it, it is very important to slow down the kinetics of the process. For this purpose we chose to use a polymer with large molecular weight. M w = 7.06 × 10 5 g mol -1 . The polymer is diluted in toluene with a mass fraction w = 0.02, corresponding to a concentration c 0 = 0.0173 g cm -3 (approximately three times the overlap concentration c o = 0.0057 g cm -3 ) and to a collective diffusion coefficient D = 4.3 × 10 -7 cm 2 s -1 [START_REF] Zhang | [END_REF]46]. The choice to work above the overlap concentration is dictated by the need of having a trade-off between a slow dynamics and a significant amount of light scattered by the sample. We stress that at this concentration the collective diffusion coefficient, describing the rate at which concentration fluctuations decay, is independent of the molecular mass of the polymer, but is still extremely low. Before starting a measurement cycle, the sample is initially kept at a uniform temperature of 25 • C for some hours to reach an equilibrium state. To avoid convective motions caused by small residual gradients, we impose a small temperature difference of 0.1 • C between the two Peltier elements, with the above temperature higher than the below one. We collect images of the sample in the equilibrium state (200 frames at a rate of 1 s -1 ) to characterize the camera background.

The measurement cycle is started by applying a temperature difference ∆T = 20 • C, heating the upper face at 35 • C, and cooling the lower one at 15 • C. After the imposition of a temperature difference between the sapphire windows, a constant temperature gradient develops inside the cell with a characteristic time much faster than the one needed to attain a steady concentration profile. This is due to the fact that the thermal diffusivity of the sample is two orders of magnitude larger than its diffusion coefficient. We can estimate a time τ ≈ 100 s for the development of the thermal gradient. The sample has a positive Soret coefficient S T = 0.238 K -1 [START_REF] Zhang | [END_REF], and therefore the mass flux is in the direction of the colder temperature, with stabilizing effect with respect to convective motions. When the steady state is approached, the mass flux vanishes leading to a constant concentration gradient ∇c s across the whole cell . We can estimate the time necessary for the system to reach the stationary state to be about τ = a 2 /π 2 D ≈ 11.5 hours.

We study the fluctuations with a highly sensitive shadowgraph technique [7,23] whose setup is very simple, easy to align and extremely versatile. The light source is a super-luminescent diode (SLD) Superlum SLD 261-MP with wavelength λ = 670 µm and bandwidth (FWHM) of 7.5 nm. This source is definitely more suitable for a shadowgraph experiment than a laser, as the larger bandwidth prevents fringe formation coming from the interference of beams reflected by optical surfaces. The SLD is coupled to a monomode fiber. An achromatic doublet placed before the sample cell collimates the diverging beam coming out of the fiber.

A scientific CMOS detector (IDT NX4-S1) 13.9 mm × 13.9 mm with a resolution of 1024×1024 pixels is placed at a distance of 41 cm from the cell. The collimation of the beam is carefully calibrated to give the exact correspondence between pixels and lengths in real space 1 pixel = 13.68 µm. One advantage of the shadowgraph diagnostics is that the scattered light is superimposed with the transmitted beam onto the sensor, which receives all the light coming from the cell falling inside the solid angle subtended by the sensing area. The transmitted beam acts as a local oscillator and interferes both with the scattered and the stray light coming from spurious reflections, imperfections in the optics, or dust particles along the optical path. As a results, the intensity of the light collected by the detector is linearly related to the amplitude of the scattered field I(x, t)

∝ |E 0 | 2 + 2Re {E 0 E * s (x, t)}
, where E 0 is the amplitude of the transmitted field and E s (x, t) the scattered field. After the imposition of the temperature gradient to the sample, we start acquiring a series of 900 shadowgraph images with a sampling period of 4s. The number of images was dictated by the storage capacity of the CMOS camera, while the sampling time was chosen to guarantee a good statistical accuracy in the initial stages of the process. It must be pointed out that the data are intrinsically noisy during the transient and can be reduced only by performing time averages across a time much larger than that needed for the relaxation of the fluctuations. For non-equilibrium fluctuations the slowest mode corresponds to the roll-off wave vector, and for our experimental sample the slowest fluctuations relax with a time constant τ ro = (Dq ro ) -1 ≈ 20 s. The reproducibility of the process was tested by performing three independent runs under the same experimental conditions, which yielded comparable results. Each shadowgraph image contains several contributions to the scattered field E s (x, t): a static background E B (x), equilib- rium temperature fluctuations E T e (x, t), equilibrium concentration fluctuations E ce (x, t), non-equilibrium temperature fluctuations E T ne (x, t), non-equilibrium concentration fluctuations E cne (x, t) and a dynamic background due to the camera noise E D (x, t). All the dynamic terms have zero mean.

The average of a set of images I 0 (x) = I(x, t) t allows to recover the background term E B (x), which can be eliminated from each image by subtraction so to obtain the normalized intensity distribution [15]:

i(x, t) = I(x, t) -I 0 (x) I 0 (x) (6)

Results and discussion

A first information on the growth of non-equilibrium fluctuations can be obtained from the time evolution of the variance of the shadowgraph images of the fluctuations (Fig. 2) var[i(x, t)], which is proportional to the integral of the scattered light over the solid angle subtended by the detector

var [i(x, t)] = x i(x, t) 2 N ∝ var [E T ne (x, t)] + var [E cne (x, t)] + var [E D (x, t)] . (7)
Here we have omitted the contributions due to equilibrium fluctuations because under the conditions of interest for our experiment their amplitude is much smaller than that of non-equilibrium fluctuations and can thus be neglected. Following the initial phase where the temperature profile reaches a steady state, the signal starts growing until it finally reaches a plateau. The growth of the variance of the scattered light provides a first evidence of the emergence of the non-equilibrium fluctuations in an originally unperturbed sample. A first question that arises is how the growth of the fluctuations is related to the development of a macroscopic concentration profile, starting from an originally uniform concentration. As we will see shortly, a meaningful parameter reflecting the growth of the macroscopic state is the concentration difference ∆c(t) generated by the Soret flow across the sample. This quantity can be calculated theoretically from Eqn. 2:

∆c(t) = c(0, t) -c(1, t) = = a∇c s 1 - 4 π 2 ∞ j=1 1 j 2 [1 -(-1) j ] × exp - j 2 π 2 D a 2 t (8)
In Fig. 2 we have superimposed the time evolution of the concentration difference ∆c(t) to the experimental results for the variance of the fluctuations. The dashed line is obtained from Eqn. 8 using the reference value for D [START_REF] Zhang | [END_REF], the only free parameter being a multiplicative constant, while the solid line is a fit obtained by also using the diffusion coefficient as a free parameter. One can appreciate that the growth of ∆c(t) is in good agreement with the growth of the variance of the shadowgraph images. Therefore, the variance of the non-equilibrium fluctuations at the large length scales where the fluctuations are strongly affected by gravity is intimately related to the dynamics of growth of the concentration difference, which in turn allows the determination of the diffusion and Soret coefficient. Although this method for the determination of the diffusion coefficient is very preliminar, this approach, when properly developed, could represent a suitable alternative to Mach-Zehnder Interferometry, Electronic Speckle Pattern Interferometry or Beam Deflection to recover the transport coefficient of a binary liquid mixture during a thermodiffusion experiment.

A more refined assessment of the kinetics of growth of non-equilibrium fluctuations involves the determination of their structure factor, which is directly related to the structure factor of the shadowgraph images S S (q, t), where q is the transferred wave vector: S S (q, t) = T (q)[S T (q, t) + S c (q, t)] + S D (q, t)

where S T (q, t) and S c (q, t) are the power spectra of temperature and concentration fluctuations respectively, S D (q, t) is the power spectrum of the dynamic camera noise and T (q) is the transfer function of the shadowgraph method [7,23,15,47] Fig. 3. Time evolution of the static structure factor T (q)Sc(q, t) of non-equilibrium concentration fluctuations taken during the transient phase. The first curve (lower one) is taken at t = 180 s after the imposition of the temperature gradient, while the last one at t = 3600 s. The solid vertical line shows the theoretical roll-off wave vector qRO at steady state.

In order to compare our results with the data presented in Ref. [15] we study the structure factor of the images at various time. To get rid of the contribution of the temperature fluctuations and of the dynamic camera noise we evaluate the power spectrum S 0 (q) of the shadowgraph images grabbed about 100s after the imposition of the gradient. Due to the large separation of the timescales for the growth of fluctuations, after this time the temperature fluctuations are fully developed, while the concentration fluctuations have barely started to grow and S 0 (q) = T (q)S T (q, t) + S D (q, t). The subtraction of S 0 (q) from S S (q, t) allows to isolate the contribution of nonequilibrium fluctuations, (Fig. 3): S S (q, t) -S 0 (q) = T (q)S c (q, t)

From Fig. 3 we observe that the structure factor grows rapidly in the first hour, and slows down during the following part of the process. To characterize the transient regime we analyzed, for various runs, the first 900 images collected every 4 seconds after applying the temperature difference across the cell.

The typical oscillations due to the shadowgraph transfer function are visible, superimposed to the static structure factors [40]. As the signal due to non-equilibrium concentration fluctuations is very small during the early stages of growth, data are particularly noisy at small wave vectors, and it is difficult to make any statement on the presence of a peak predicted in Ref. [15].

From Eqn. 1, we can estimate the steady state roll off for our sample to be q RO = 334 cm -1 (being β = 0.373, ν = 4.91 × 10 -2 cm 2 /s , ∇c s = 0.71 cm -1 , and D = 4.3 × 10 -7 cm 2 s -1 ). Its position is made evident in Fig. 3 with the straight vertical line. q RO seems slightly underestimated with respect to the actual roll off position for all the curves. Actually Eqn. 1 is obtained in the hypothesis of dilute samples and at the stationary state, when a constant gradient is present in the cell, and a vertical displacement of a volume of sample creates a density mismatch.

The time evolution of the structure factors shown in Fig. 3 is directly related with the evolution of the variance of shadowgraph images shown in Fig. 2. More in detail, the integral of the structure factor at a given time over the accessible q-range is directly proportional to the variance of the shadowgraph images. A more refined analysis involves the experimental determination of the growth rate as a function of wave vector of the structure factor of the fluctuations S c (q, t): R f (q, t) = S S (q, t + ∆t) -S S (q, t) [S S (q, t) -S 0 (q)]∆t = = S c (q, t + ∆t) -S c (q, t) S c (q, t)∆t

As it can be appreciated from Eqn. 11, this method for analysing the kinetics of growth of the fluctuations allows to get rid completely of the shadowgraph transfer function without the need of performing calibration measurements. By combining equations 5 and 11 one finds the remarkable result that the growth rate R f of the structure factor of the fluctuations as a function of wave vector is exactly equal to the growth rate R ∆c of the concentration difference ∆c. Using Eq. 2 and 11 we can evaluate theoretically the growth rate of the concentration difference ∆c as

R ∆c (t) = ∆c(t + δt) -∆c(t) ∆c(t)δt = = 4D ∞ j=1 [1 -(-1) j ]exp -j 2 π 2 D a 2 t a 2 1 -4 π 2 ∞ j=1 1 j 2 [1 -(-1) j ]exp -j 2 π 2 D a 2 t (12)
In Fig. 4 we show the comparison between the time evolution of the growth rate of fluctuations defined in Eqn. 11 for different wave vector ranges.

At variance to what expected in a diffusive process, where the growth of the amplitude of the concentration modes is strongly dependent on the wave vector q, being c(q, t) ∝ [1 -exp(-Dq 2 t)], we notice that the data for the growth rate of non-equilibrium fluctuations approximately collapse onto the same curve, without evidence of any dependence on the particular q value. Figure 4 shows the superposition of the growth rate of fluctuations and the rate theoretically calculated from the concentration profile as indicated in Eqn. 12 and the experimental rate of growth of non-equilibrium concentration fluctuations. The dashed line is obtained from Eq.12 The two lines are calculated using Eqn. 12 without adjustable parameters: the dashed line with the nominal diffusion coefficient Dn while the solid line with D = 2.5Dn. One can appreciate that the growth rate during the onset of the fluctuations is independent on the diffusion coefficient, that becomes significant only at large times, as shown by the theoretical curves.

without any adjustable parameters and using the nominal diffusion coefficient (D = 4.3 × 10 -7 cm 2 s -1 ), while the solid line is obtained using the value of the diffusion coefficient recovered from the fitting of the variance shown in Fig. 2 (D = 10.8 × 10 -7 cm 2 s -1 ). One can appreciate that at small t the rate of growth does not show the presence of any typical time scale and follows a powerlaw behavior R c (q, t) ∝ 1/t. This behavior is independent from the value of the diffusion coefficient. At larger t, the finite thickness of the sample determines a cut-off, which strongly depends from the value of the diffusion coefficient.

The overall good agreement between the growth rates shown in Fig. 4 confirms the theoretical result outlined above that the growth of fluctuations at the mesoscopic scale is directly related to the development of the concentration profile at the macroscopic scale. A more complete and refined understanding of the kinetics of growth of non-equilibrium concentration fluctuations would require performing experiments under microgravity conditions, where the amplitude of non-equilibrium fluctuations at small wave vectors is not limited by the presence of gravity. This represents one of the main goals of the Giant-Fluctuations (NEUF-DIX) project of the European Space Agency, which will be hosted on the International Space Station during the next years [21].

Fig. 1 .

 1 Fig. 1. Time evolution of the concentration profile plotted vs the normalized height ζ in a thermal diffusion process. Concentration profiles have been obtained numerically from Eqn. 2 .

Fig. 2 .

 2 Fig. 2. Time evolution of the variance of shadowgraph images on non-equilibrium concentration fluctuations. The dashed line represents the theoretical time evolution of the concentration difference across the sample obtained from Eqn. 8 using the reference value of the diffusion coefficient D = 4.3×10 -7 cm 2 s -1 . The solid line is the best fit of the experimental data with Eqn. 8, yielding a value of the diffusion coefficient 2.5 times larger than the reference one.

Fig. 4 .

 4 Fig. 4. Time evolution of the growth rate of non-equilibrium concentration fluctuations for different ranges of wave vectors.The two lines are calculated using Eqn. 12 without adjustable parameters: the dashed line with the nominal diffusion coefficient Dn while the solid line with D = 2.5Dn. One can appreciate that the growth rate during the onset of the fluctuations is independent on the diffusion coefficient, that becomes significant only at large times, as shown by the theoretical curves.

Acknowledgements

We thank W. Köhler and J. M. Ortiz de Zárate for useful discussion. Work partially supported by the European Space Agency. 6 Author contributions M. C., F. C., and A. V. conceived the study. M. S. performed measurements. M. C., M. S. and A. V. analyzed results. M. C. and A. V. wrote the paper. All the authors contributed to the discussion of the results.


Ariel Pérez-Mellor

Ariel F Pérez -Mellor

Anne Zehnacker
email: anne.zehnacker-rentien@u-psud.fr

Dr Mr A F Pérez-Mellor

Vibrational Circular Dichroism of a 2,5-Diketopiperazine (DKP) Peptide: Evidence for Dimer Formation in cyclo LL or LD diphenylalanine in the solid state

Keywords:

HAL is

Introduction

Homochirality is pervasive in life chemistry. It plays a major role in the structure of biomolecules, which are, for most of them, composed of elementary blocks of the same handedness. Cornerstone examples are peptides and proteins, which are built from naturally-occurring L amino-acids. Exceptions to homochirality, resulting for example from post translational modifications, [START_REF] Buczek | Post-translational amino acid isomerization -A functionally important D-amino acid in an excitatory peptide[END_REF][START_REF] Cotter | Posttranslational conversion of L-serines to D-alanines is vital for optimal production and activity of the lantibiotic lacticin 3147[END_REF] have important consequences in the function of biomolecules, due to a change in both intra and intermolecular interactions, for example in protein folding [START_REF] Kwiecinska | Chirality and protein folding[END_REF] or activity. [START_REF] Nielsen | Effects of chirality at Tyr13 on the structure-activity relationships of omegaconotoxins from Conus magus[END_REF] These changes are often related to a change in the biopolymer structure, as observed in the Trp-cage mini-protein, the structure of which is strongly modified by the change of chirality of a single aminoacid. [START_REF] Adams | Electron capture dissociation distinguishes a single D-amino acid in a protein and probes the tertiary structure[END_REF] The effects of modifying the handedness of one monomer in a biopolymer have been studied at the molecular level in model systems, both neutral molecules as well as protonated or sodiated species. [START_REF] Abo-Riziq | Discrimination between diastereoisomeric dipeptides by IR-UV double resonance spectroscopy and ab initio calculations[END_REF][START_REF] Dean | Conformation-Specific Spectroscopy and Populations of Diastereomers of a Model Monolignol Derivative: Chiral Effects in a Triol Chain[END_REF][START_REF] Gloaguen | Intramolecular recognition in a jetcooled short peptide chain: gamma-turn helicity probed by a neighbouring residue[END_REF][START_REF] Alauddin | Intrinsic Folding Proclivities in Cyclic b-Peptide Building Blocks:Configuration and Heteroatom Effects Analyzed by Conformer-Selective Spectroscopy and Quantum Chemistry[END_REF][START_REF] Lepere | Chirality-dependent structuration of protonated or sodiated polyphenylalanines: IRMPD and ion mobility studies[END_REF] Although often weak in short neutral or protonated peptides, the effects are more pronounced in long protonated poly-alanine strands whose helical shape is disrupted when reverting the chirality of one residue, [START_REF] Sudha | Left-handed and ambidextrous helices in the gas phase[END_REF] in sodiated peptides, [START_REF] Lepere | Chirality-dependent structuration of protonated or sodiated polyphenylalanines: IRMPD and ion mobility studies[END_REF][START_REF] Dunbar | Conformations and vibrational spectroscopy of metal-ion/polylalanine complexes[END_REF][START_REF] Dunbar | Chirality-Induced Conformational Preferences in Peptide-Metal Ion Binding Revealed by IR Spectroscopy[END_REF] or in cyclically constrained β-peptides with chirality-dependent intramolecular hydrogen bonding pattern. [START_REF] Alauddin | Intrinsic Folding Proclivities in Cyclic b-Peptide Building Blocks:Configuration and Heteroatom Effects Analyzed by Conformer-Selective Spectroscopy and Quantum Chemistry[END_REF] Solution-phase studies of the way diastereomerism impinges the structure of peptides rely on various structure-sensitive techniques such as NMR, [START_REF] Saavedra | Conformation and Chiral Effects in alpha,beta,alpha-Tripeptides[END_REF] and vibrational spectroscopy. [START_REF] Vass | Vibrational spectroscopic detection of beta-and gamma-turns in synthetic and natural peptides and proteins[END_REF][START_REF] Martinek | Effects of the alternating backbone configuration on the secondary structure and self-assembly of beta-peptides[END_REF] Of special interest are the chiroptical methods like Vibrational Circular Dichroism (VCD) [START_REF] Kubelka | Differentiation of beta-sheetforming structures: Ab initio-based simulations of IR absorption and vibrational CD for model peptide and protein beta-sheets[END_REF][START_REF] Bochicchio | Polyproline II structure in proteins: Identification by chiroptical spectroscopies, stability, and functions[END_REF][START_REF] Eker | Tripeptides adopt stable structures in water. A combined polarized visible Raman, FTIR, and VCD spectroscopy study[END_REF][START_REF] Ducasse | Density functional theory calculations and vibrational circular dichroism of aromatic foldamers[END_REF][START_REF] Nafie | Vibrational circular-dichroism[END_REF][START_REF] Stephens | Theory of vibrational circular-dichroism[END_REF] or Raman Optical Activity (ROA). [START_REF] Freedman | VIBRATIONAL OPTICAL-ACTIVITY OF OLIGOPEPTIDES[END_REF][START_REF] Blanch | Is polyproline II helix the killer conformation? A Raman optical activity study of the amyloidogenic prefibrillar intermediate of human lysozyme[END_REF][START_REF] Mccoll | A study of alpha-helix hydration in polypeptides, proteins, and viruses using vibrational Raman optical activity[END_REF] VCD is indeed very sensitive to minute conformational changes, [START_REF] Bouchet | Enantioselective Complexation of Chiral Propylene Oxide by an Enantiopure Water-Soluble Cryptophane[END_REF][START_REF] Heshmat | The Importance of Large-Amplitude Motions for the Interpretation of Mid-Infrared Vibrational Absorption and Circular Dichroism Spectra: 6,6 '-Dibromo-1,1 'binaphthalene -2,2 '-diol in Dimethyl Sulfoxide[END_REF] hence a very good probe of molecular conformation. [START_REF] Merten | Strong Solvent-Dependent Preference of Delta and Lambda Stereoisomers of a Tris(diamine)nickel(II) Complex Revealed by Vibrational Circular Dichroism Spectroscopy[END_REF][START_REF] Merten | Solvent-induced conformational changes in cyclic peptides: a vibrational circular dichroism study[END_REF] It has been widely applied to peptides, as a powerful tool for determining their secondary structure, in particular in the Amide I (C=O stretch) region. 30-36 37-39 β-sheets and coils have for example different amide I spectra from that of the α-helix. [START_REF] Baumruck | Conformational study of sequential lys-based and leu-based polymers and oligomers using vibrational and electronic CD-spectra[END_REF] VCD is also very sensitive to the aggregation state and allows the characterisation of various size entities ranging from dimers [START_REF] Buffeteau | Vibrational absorption and circular dichroism studies of (-)-camphanic acid[END_REF] to β-sheets or fibrils formation, [START_REF] Marty | Aggregates from Perylene Bisimide Oligopeptides as a Test Case for Giant Vibrational Circular Dichroism[END_REF] as well as selfassemblies in solution or in the solid state. [START_REF] Moreno | Self-Assembly Structures of 1H-Indazoles in the Solution and Solid Phases: A Vibrational (IR, FIR, Raman, and VCD) Spectroscopy and Computational Study[END_REF] Diketopyperazine (DKP) peptides are an important class of dipeptides with applications in pharmacology as HIV antiviral drugs, or anticancer therapy. [START_REF] Sinha | Synthesis and antiviral properties of arabino and ribonucleosides of 1,3-dideazaadenine, 4-nitro-1,3dideazapurine and diketopiperazine[END_REF][START_REF] Grande | Small molecules anti-HIV therapeutics targeting CXCR4[END_REF][START_REF] Martins | Diketopiperazines: biological activity and synthesis[END_REF][START_REF] Nicholson | NPI-2358 is a tubulindepolymerizing agent: in-vitro evidence for activity as a tumor vascular-disrupting agent[END_REF] The DKP ring also is a decomposition product of the well-known sweetener aspartame or dipeptide drugs. [START_REF] Berset | Stability considerations of aspartame in the direct analysis of artificial sweeteners in water samples using high-performance liquid chromatography-tandem mass spectrometry (HPLC-MS/MS)[END_REF][START_REF] Lin | Simultaneous formation and detection of the reaction product of solid-state aspartame sweetener by FT-IR/DSC microscopic system[END_REF][START_REF] Lin | Advances in simultaneous DSC-FTIR microspectroscopy for rapid solid-state chemical stability studies: Some dipeptide drugs as examples[END_REF] These cyclic dipeptides result from the intramolecular peptide coupling of a linear α dipeptide, with concomitant dehydration. [START_REF] Borthwick | 2,5-Diketopiperazines: Synthesis, Reactions, Medicinal Chemistry, and Bioactive Natural Products[END_REF][START_REF] Capasso | Mechanism of 2,5-dioxopiperazine formation[END_REF][START_REF] Basiuk | THE GAS SOLID-PHASE 2,5-DIOXOPIPERAZINE SYNTHESIS -CYCLIZATION OF VAPOROUS DIPEPTIDES ON SILICA SURFACE[END_REF] They have been the subject of many conformational studies, since the first elucidation of their crystal structure. [START_REF] Corey | The Crystal Structure of Diketopiperazine[END_REF] The DKP ring may indeed adopt several conformations depending on the nature of the residues, with either planar, boat, or chair conformation of the ring. While it remains planar for very small residues like glycine, it undergoes an out of plane distortion for those that are bulkier, with pseudo axial or pseudo equatorial position of the substituents. [START_REF] Karle | X-Ray Analysis: Conformation of Peptides in the Crystalline State[END_REF] Bulky residues like phenylalanine or tyrosine often adopt a so-called flagpole position, [START_REF] Kopple | Conformation of cyclic peptides. The folding of cyclic dipeptides containing an aromatic side chain[END_REF][START_REF] Lin | Crystal structures and conformations of the cyclic dipeptides cyclo-(Glycyl-Ltyrosyl) and cyclo-(L-seryl-L-tyrosyl) Monohydrate[END_REF] unless steric hindrance decides otherwise. Indeed, when the DKP peptide contains two bulky substituents, it also appears that one of them is folded over the ring while the second one is extended. [START_REF] Benedetti | Conformational studies on peptides. X-ray structure determinations of six N-methylated cyclic dipeptides derived from alanine, valine, and phenylalanine[END_REF] DKP dipeptides have restricted conformational freedom, a fact that provides useful model systems for the study of electronic or vibrational coupling between the two residues. Indeed, the question has been raised as to whether the electronic transitions located on the two residues are coupled, if this coupling depends on the ring geometry, and if it can be evidenced in electronic circular dichroism using the so-called exciton coupling model. 59 60,61 From a vibrational point of view as well, DKP dipeptides allow probing the coupling between the Amide I and II modes in a system of well-defined geometries. This will allow shedding light on the factors that defines the band shapes in IR absorption and VCD in systems that show several amide modes potentially coupled. 38 62-65 Like in electronic circular dichroism, coupling between two vibration modes has been proposed to interpret the bisignate signature observed, which consists of two bands of opposite sign. [START_REF] Taniguchi | Exciton Chirality Method in Vibrational Circular Dichroism[END_REF] However, bisignate band shapes can also be accounted for by the presence of several conformers, with close-lying absorption bands with opposite VCD. [START_REF] Covington | Determination of the Absolute Configurations Using Exciton Chirality Method for Vibrational Circular Dichroism: Right Answers for the Wrong Reasons?[END_REF] A generalised coupled oscillator (GCO) model has been recently proposed to interpret the shape of VCD spectra in both symmetrical and asymmetric molecules. [START_REF] Nicu | Revisiting an old concept: the coupled oscillator model for VCD. Part 1: the generalised coupled oscillator mechanism and its intrinsic connection to the strength of VCD signals[END_REF] We have undertaken the study of dipeptides with identical or opposite chirality, both in the gas phase and in the condensed phase, with the aim of understanding how homo or heterochirality of the residues impinges their structure and can be identified by its spectroscopic signature. [START_REF] Lepere | Chirality-dependent structuration of protonated or sodiated polyphenylalanines: IRMPD and ion mobility studies[END_REF]67 We shall focus here on cyclo diphenylalanine, denoted cyclo Phe-Phe hereafter, which has attracted special attention due to its spontaneous formation by heating the parent Phe-Phe linear dipeptide. [START_REF] Nitecki | Simple route to sterically pure diketopiperazines[END_REF][START_REF] Jeziorna | Cyclic Dipeptides as Building Units of Nano-and Microdevices: Synthesis, Properties, and Structural Studies[END_REF] Both cyclo LPhe-LPhe and cyclo LPhe-DPhe will be studied in the solid phase. The presence of bulky aromatic rings is expected to introduce steric constraints and to increase stabilizing dispersive interactions within the peptide. [START_REF] Barbu-Debus | Chiral recognition in jet-cooled complexes of (1R,2S)-(+)-cis-1-amino-2-indanol and methyl lactate: on the importance of the CH center dot center dot center dot pi interaction[END_REF][START_REF] Scuderi | The role of weak hydrogen bonds in chiral recognition[END_REF] Last, the Phe aromatic rings can act as weak hydrogen bond acceptors. The molecules under study are shown in Figure 1.

Materials and Methods

EXPERIMENTAL SET-UP

The vibrational IR absorption and VCD spectra were measured using a FTIR spectrometer (Vertex 70 Bruker) equipped with a VCD module (PMA 50 Bruker). The signal was measured by a MCT IR detector with a BaF 2 window, cooled with liquid nitrogen. A spectral resolution of 4 cm -1 was used for both absorption and VCD spectra in the fingerprint region. The absorption spectra were repeated over the whole 800-3700 cm -1 range at a 1 cm -1 resolution. The IR radiation was first polarized with a linear polarizer then modulated by a 50 kHz ZnSe photoelastic modulator (Hinds). A low-pass filter cutting at 2000 cm -1 was added before the linear polarizer to increase the dynamical response of the detector when measuring the fingerprint region. The signal of the MCT detector was demodulated using a lock-in amplifier (Stanford Research Systems SR 830). The alignment of the spectrometer was carefully checked by checking the mirror-image relation between the VCD spectra of the two enantiomers of camphor (0.3M in CCl 4 ). The samples were prepared by grinding 10mg of molecule and 3.5g of KBr in a mixer mill (MM 400 Retsch) at 20Hz during 1hr. The samples were then gently heated at 80°C to eliminate most of the water. The artefacts due to the birefringence of the KBr pellets were eliminated by following the procedure proposed by Merten, [START_REF] Merten | Vibrational circular dichroism spectroscopy of solid polymer films: Effects of sample orientation[END_REF] derived from that introduced by Buffeteau. [START_REF] Buffeteau | Vibrational circular dichroism in general anisotropic thin solid films: Measurement and theoretical approach[END_REF] It consists in rotating the sample in the plane perpendicular to the light propagation axis for each side (front F or back B) of the pellet and averaging the spectra obtained for each position of the pellet at 0° and 90°. For the sake of better averaging and artefact control, the procedure has been extended to 180° and 270°. The measure was obtained by adding 4200 scans for each sample position.

I VCD (ν ̃) = 1 8 ∑ I VCD (ν ̃; n ⋅ 90°; F) + I VCD (ν ̃; n ⋅ 90°; B) 3 n=0
The dipeptides were purchased from GeneCust-Luxembourg (98% purity) and used without further purification.

THEORETICAL METHODS

A first exploration of the potential energy surface was performed for both diastereomers using the Macromodel program included in the Maestro suite. [START_REF]MacroModel version 9.8[END_REF] The conformational mobility is limited by the cyclic nature of the system, and only ~10 different monomers were calculated with energy within 21kJ/mol with the OPLS_2005 force field used. They were further optimized at the B3LYP/6-311++g(d,p) level of theory, including the D3 empirical dispersion factor, using Gaussian 09. [START_REF] Grimme | A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu[END_REF][START_REF] Frisch | Gaussian 09, Revision D.01[END_REF] The vibrational IR absorption and circular dichroism spectra were simulated by calculating the harmonic frequencies at the same level of theory and convoluting the scaled frequencies by a Lorentzian line shape of 8 cm -1 FWMH, chosen to reproduce the experimental bandwidth. The calculated frequencies were scaled by 0.952. This scaling factor gives indeed a good agreement between the experimental and calculated amide stretch A of cyclo Phe-Phe in the gas phase. 67 The stability of the monomer is given in terms of Gibbs energy ΔG at 298.15 K (in kcal/mol) relative to the most stable conformation.

No significant change was observed in the relative stability nor in the calculated IR and VCD spectra of the monomers when using the smaller basis set 6-31g(d,p), as shown in Table S1 and Figure S1 of the supporting information. The 6-31g(d,p) basis set was therefore used for the dimers; the same procedure was followed, using the same scaling factor as for the monomers.

Results and Discussion

EXPERIMENTAL RESULTS

The IR absorption spectra of the two diastereomers are shown in Figure 2. They display a similar pattern in the Amide I and II region, with an intense feature centred around 1670 cm -1 , readily assignable to the C=O stretch. This feature is composed of a doublet in cyclo LPhe-LPhe at 1663-1677 cm -1 while it is a single band located at 1677 cm -1 in cyclo LPhe-DPhe, with however a shoulder in the low-energy side that shows that it corresponds to the superposition of two bands. The Amide II (β(NH) bend) region of cyclo LPhe-LPhe displays a broad feature with two maxima at 1462 cm -1 and 1454 cm -1 , and a side band at 1498 cm -1 . In cyclo LPhe-DPhe, similar features are observed. However, the cyclo LPhe-DPhe spectrum extends slightly more, with additional side bands at 1440 and 1486 cm -1 . The hydride stretch region shows a complicated pattern resulting from the superposition of the 14 ν(CH) stretches, and two bands assigned to the ν(NH) stretches at 3205 / 3316 cm -1 and 3189 /3326 cm -1 for cyclo LPhe-LPhe and cyclo LPhe-DPhe, respectively. One can deduce from the similitude between the absorption spectra that similar structures exist for cyclo LPhe-LPhe and cyclo LPhe-DPhe. Their shape, in particular the presence of two ν(NH) bands with different frequencies, indicates that the sample contains NH groups in two sorts of local environment. Similarly, the doublet observed in the ν(CO) stretch region of cyclo LPhe-LPhe indicates that the cyclo LPhe-LPhe sample contains CO groups in at least two distinct local environments.

The experimental VCD spectra recorded in the fingerprint region are shown in Figure 3. They totally differ from each other. While cyclo LPhe-LPhe shows intense VCD signal in the ν(CO) stretch region, of the order of 3.5•10 -4 , and weaker bands in the β(NH) bend region, no signal is detected for cyclo LPhe-DPhe in the same experimental conditions, which easily allow detecting signals down to 10 -5 . The absence of signal for cyclo LPhe-DPhe might arise from different possibilities, including centrosymmetric structures, which will be discussed later. The cyclo LPhe-LPhe spectrum is characterised by a bisignate signature in the ν(CO) region, with an intense positive peak at low energy followed by a negative peak of similar intensity, a point to which we shall return in the discussion. The broad β(NH) band at 1454/1462 cm -1 also shows a bisignate signature, although of opposite sign and much weaker intensity than that due to the ν(CO) stretches.

THEORETICAL RESULTS

Calculated structures and assignment for cyclo LPhe-LPhe

The experimental IR spectrum of cyclo LPhe-LPhe has been compared first to that simulated for the most stable calculated structure cLL-A, which is the conformer observed in the gas phase in jet-cooled conditions. 67 This structure is very close to that observed in the cyclo LPhe-LPhe crystal, with one Phe folded over the DKP ring and the other one extended. [START_REF] Gdaniec | STRUCTURE OF CYCLO(-L-PHENYLALANYL-L-PHENYLALANYL-)[END_REF] Only a mild agreement is obtained for the IR spectrum, as shown in Figure 4. In particular, the energy gap between the amide I and II bands is by far too large in the calculated spectrum; this is the case for all the calculated monomers. Moreover, the calculated VCD spectrum of the most stable structure fails to reproduce the bisignate signature. A bisignate signature is only obtained in symmetrical structures such as cLL-D and cLL-E, shown in Figure 4, with however the wrong sign.

The existence of dimers could explain the observation of two different ν(CO) and two different ν(NH) bands. The most stable calculated (cLL) 2 -I dimer satisfactorily reproduces the experimental spectroscopic results (see Figure 5). The most stable (cLL) 2 -I dimer displays two strong hydrogen bonds bridging the CO and NH groups of the two subunits, with CO•••HN distances as short as 1.81 Å. Due to the steric hindrance induced by the aromatic rings, the double hydrogen bridge shows a small deviation from planarity (ONON dihedral of 8°). The DKP geometry in both monomers consists in a flattened boat with the C β atoms in pseudoaxial positions. The deviation from planarity of the DKP ring is weak. The calculated CCNC dihedral amounts indeed to -14° in the dimer vs. -12° in the gas phase monomer. This value is very close to that measured in the crystal (-11°). 67,[START_REF] Gdaniec | STRUCTURE OF CYCLO(-L-PHENYLALANYL-L-PHENYLALANYL-)[END_REF] The small difference between the monomer and the dimer is not significant and might result from differences in the methods used. Interestingly, the two subunits are not totally identical to each other, which results to a slightly asymmetric hydrogen bond bridge with one hydrogen bond farther to linearity (165°) than the other (175°). One of the two moieties is similar to that observed in the gas phase and in the crystal, with one aromatic ring bent over the DKP ring in a "flag pole" position while the other one is in an extended position. [START_REF] Gdaniec | STRUCTURE OF CYCLO(-L-PHENYLALANYL-L-PHENYLALANYL-)[END_REF] This structure corresponds to NCCC dihedral angles of -58° for the extended Phe and 65° for the folded one. The second subunit is much more symmetrical with Phe substituents located on the same side of the DKP ring, as often observed when the two residues interact with each other. [START_REF] Li | Absolute Configuration of a Cyclic Dipeptide Reflected in Vibrational Optical Activity: Ab Initio and Experimental Investigation[END_REF] The corresponding dihedral angles are 73° and 59°, respectively. Such a structure allows Phe-Phe stacking interaction to take place, similar to what is observed in cyclo LArg-LTrp. [START_REF] Li | Determination of Absolute Configuration and Conformation of a Cyclic Dipeptide by NMR and Chiral Spectroscopic Methods[END_REF] This contrasts with what is observed in the neat crystal, in which all the monomers have an identical structure. [START_REF] Gdaniec | STRUCTURE OF CYCLO(-L-PHENYLALANYL-L-PHENYLALANYL-)[END_REF] It should be noted that attempts at reproducing the VCD spectrum with a symmetrical structure identical to the crystal sub-unit and optimised in the gas phase failed, as shown in Figure S2 of the supporting information. As both free and bound ν(CO) and ν(NH) stretching modes appear in the experimental spectrum, its interpretation in terms of trimers, tetramers, or larger oligomers of cyclo LPhe-LPhe cannot be ruled out. Information can be gained however from the comparison between the relative intensity of the doublet experimentally observed in the ν(CO) stretch region and the shape of the spectrum calculated for clusters of different sizes. For the sake of computational efficiency, we have optimised the structures of monomers, dimers, trimers and tetramers of the model system (LL) cyclo Ala-Ala, at the B3LYP/6-31g(d,p) level of theory. The simulated spectra, displayed in Figure S3 of the supporting information, show that the calculated intensity ratio between the high-energy and low-energy components of the doublet decreases when the cluster size increases. This ratio reflects indeed the ratio between free (high energy band) and bound (low energy band) CO groups, which scale like (n-1), n being the number of units in the oligomer. The best agreement with the experiment is clearly obtained with the dimer structure, which shows similar intensities for the two components of the doublet as experimentally observed. This observation allows ruling out the existence of a long oligomeric chain of cyclo LPhe-LPhe.

Frequency analysis for the cyclo LPhe-LPhe dimer

The main observed vibrational bands are collected in Table 1, together with the calculated frequencies, IR absorption intensities, oscillator strengths, and proposed assignment. In the calculated (cLL) 2 -I dimer, the free ν(NH) stretches are not coupled to each other, nor are the free ν(CO) stretches, which appear as the high-energy component of the doublet experimentally observed at 1662-1678 cm -1 . The two free ν(CO)s have a negative VCD signal that both contribute to the observed negative component of the bisignate doublet. The bound CO stretches are strongly coupled together, which results in a strongly allowed asymmetric combination and an almost completely forbidden, symmetric, one. The allowed ν(CO) has a strong positive VCD signal while that of the forbidden ν(CO) is zero. The bisignate signature observed in the ν(CO) region is therefore not due to exciton coupling but to superimposed contributions of different ν(CO)s. Like the bound ν(CO) stretches, the bound β(NH) bends are strongly coupled to each other. They appear however weakly in the spectrum due to limited oscillator strength. In contrast to the free stretches, the free β(NH) bends are strongly coupled to each other and to the β(CH) or β(CH 2 ) bends. They appear as a bisignate doublet in the VCD spectrum at 1452 and 1462 cm -1 . In conclusion, the two bisignate doublets which appear in the VCD spectrum of cyclo LPhe-LPhe have completely different origins: while the weak-intensity bisignate signature around 1450 cm -1 is due to exciton coupling, the strong one around 1670 cm -1 is due to ν(CO)s in different environments, which show contributions of opposite sign. TABLE 1 Frequencies assignment for cyclo LPhe-LPhe. ( * ) The theoretical frequencies are not scaled. The position of experimental frequencies was determined using the second derivative criterion.

Calculated structures and assignment for cyclo LPhe-DPhe

The case of cyclo LPhe-DPhe is more delicate to interpret as no VCD signal is observed. The presence of a single feature in the ν(CO) stretch region might suggest the presence of monomers. However, the calculated monomer structures do not reproduce the experimental IR spectrum, in particular in the region of the ν(NH) stretch, where both free and bound NH appear. Moreover, all of them display non zero VCD spectrum (see Figure S4 of the supporting information). We therefore calculated the cyclo LPhe-DPhe dimers, as we did for cyclo LPhe-LPhe. There are two possibilities for obtaining a system with no optical activity. The first one is the co-existence of two enantiomers with opposite VCD spectra that cancel out, a point to which we shall return later.

The second possibility is the formation of a centrosymmetric dimer, for example the calculated (cLD) 2 -sym dimer shown in Figure 6a). It displays the same double hydrogen bridge pattern as the cyclo LPhe-LPhe dimer. However, the stereochemistry of cyclo LPhe-DPhe allows for the formation of a centrosymmetric dimer with no optical activity. This is an example of what is called "la coupe du roi", a non-chiral system built from two chiral moieties of opposite chirality, 79 a phenomenon also observed in a jetcooled isolated dimer. This centrosymmetric dimer is however not the most stable one; its relative energy amounts to 1.77 kcal/mol relative to the most stable form. The most stable dimers obtained after the exploration and optimization steps belong to the C 1 symmetry point group, due to the position of the Phenylalanine residues. As seen for the most stable of them ((cLD) 2 -I shown in Figure 6b), a weak CH-π interaction takes place between the C α H of one of the residues and the benzene ring of the other, which breaks the symmetry of the system. As cyclo LPhe-DPhe contains Phe residues, which differ by nothing but their chirality, it is not possible to distinguish which of LPhe or DPhe aromatic ring acts as a hydrogen bond acceptor and which is "free". For this reason two transient enantiomers are expected for cyclo LPhe-DPhe, namely cyclo LPhe(donor)-DPhe(acceptor) and cyclo DPhe(donor)-LPhe(acceptor). They show equal populations hence contributions to the IR absorption spectrum, shown in Figure S5, and opposite contributions to the VCD spectrum. Switching from one of the transient enantiomers to the other corresponds to a switch between "free" and "H-bonded" aromatic rings. The enantiomers can formally be seen as cyclo LPhe(donor)-DPhe(acceptor) and cyclo DPhe(donor)-LPhe(acceptor), but the asymmetric carbons themselves of course do not change chirality. What physically happens is that LPhe(donor) changes to LPhe(acceptor). This process does not involve cleavage and formation of chemical bonds but inversion of the DKP ring and rotation of the phenethyl substituents. A possible reaction path has been calculated between the two transient enantiomers of the lower-energy cyclo LPhe-DPhe monomer and is shown in Figure S6. The calculated transition state is centrosymmetric and corresponds to a barrier of 2.1 kcal/mol only. Although these calculations are carried out in the gas phase, we expect that interconversion between the two transient enantiomers also happens during the sample preparation.

On the basis of the experimental finding, it is not possible to decide between the two possibilities, either centrosymmetric dimer, or two transient enantiomers. We tend to favour the second one because the calculated structure is lower in energy in the gas phase. However, the energy ordering could be modified in the condensed phase.

Conclusion

The two diastereomers of cyclo diphenylalanine has been studied in the solid state, in order to shed light on its aggregation state when embedded in KBr pellets. To this end, samples of finely grinded cyclo LPhe-LPhe and cyclo LPhe-DPhe in KBr have been studied by IRTF as well as VCD spectroscopy, combined with quantum chemical calculations. For simple aromatic molecules like 1H-indazoles, it has been proposed that calculations resting on a description of the system in terms of a small cluster that reflects the elementary cell give satisfactory agreement with the experimental VCD spectrum, for both band position and sign. [START_REF] Moreno | Self-Assembly Structures of 1H-Indazoles in the Solution and Solid Phases: A Vibrational (IR, FIR, Raman, and VCD) Spectroscopy and Computational Study[END_REF] In the cyclo LPhe-LPhe studied here however, the sample does not exist in the form it possesses in crystals, namely, infinite chains involving molecular units bridged by two NH…O hydrogen bonds. Here, the unit cell is the dimer, as can be deduced from the observation of both bound and free ν(CO) and ν(NH) stretches. The DKP geometry in both monomers consists in a flattened boat with the C β atoms in pseudoaxial positions, with a deviation from planarity of the DKP ring of the same order as in the gas phase or the crystal. 67,[START_REF] Gdaniec | STRUCTURE OF CYCLO(-L-PHENYLALANYL-L-PHENYLALANYL-)[END_REF] The two moieties of the dimer are not equivalent in terms of phenyl ring arrangement, which contrasts to the neat solid. Such a dissymmetry has been observed already for supramolecular structures in solution; for example the tetramer of a chiral dicarboxylic acid is composed of three identical monomers plus one. [START_REF] Urbanova | Determination of molecular structure in solution using vibrational circular dichroism spectroscopy: the supramolecular tetramer of S-2,2 '-dimethyl-biphenyl-6,6 '-dicarboxylic acid[END_REF] This structure accounts well for the bisignate signatures observed in the ν(CO) and β(NH) regions of the VCD spectrum. The higher-energy positive component of the intense doublet observed in the ν(CO) region has been assigned to the superimposed contribution of the two free ν(CO) stretches while the lower-energy negative component has been assigned to the antisymmetric combination of the bound ν(CO) stretches. In contrast, the two components of the much weaker bisignate signature observed in the β(NH) range are due to the symmetric and asymmetric combinations of the bound β(NH) bends, respectively. Dimers of rigid DKP peptides provide therefore interesting systems for studying the effects of mode coupling due to transition dipole coupling (TDC) as well as through-bond effects, including hydrogen bonds. Other DKP dipeptides are currently under study both in the gas and in the solid phases.

Graphical Abstract

FIGURE 1

 1 FIGURE 1 Molecules under study. Left: cyclo LPhe-LPhe. Right: cyclo LPhe-DPhe.

FIGURE 2

 2 FIGURE 2 Experimental infrared absorption spectra, recorded at a resolution of 1 cm -1 . Top: cyclo LPhe-LPhe. Bottom: cyclo LPhe-DPhe. The intensity in the region of the hydride stretch has been multiplied by 4.

FIGURE 3

 3 FIGURE 3 Experimental VCD spectra in the fingerprint region, recorded at a resolution of 4 cm -1 . Top: cyclo LPhe-LPhe. Bottom: cyclo Lphe-DPhe.

FIGURE 4

 4 FIGURE 4 Comparison between experimental and simulated IR and VCD spectra in the fingerprint region for the most stable monomers of cyclo LPhe-LPhe at the DFT-D3/B3LYP/6-311++G(d,p) level of theory. The simulated spectra were obtained by convoluting the harmonic frequencies by a Lorentzian shape with a FWHM=8 cm - 1 .

FIGURE 5

 5 FIGURE 5 Comparison between experimental and simulated IR

80

 80

FIGURE 6 a

 6 FIGURE 6 a) Centrosymmetric structure c(LD) 2 -sym of the cyclo LPhe-DPhe dimer. b) Most stable structure c(LD) 2 -I of the cyclo LPhe-DPhe dimer.

Acknowledgements

We acknowledge the use of the computing facility cluster GMPCS of the LUMAT federation (FR LUMAT 2764).

Supporting information

Additional supporting information may be found in the online version of this article at the publisher's website.