Aurélien Perera

Martina Poºar

Bernarda Lovrin£evi¢

Camel back shaped Kirkwood-Bu Integrals

Some binary mixtures, such as specic alcohol-alkane mixtures, or even water-tbutanol, exhibit two humps camel back shaped KBI. This is in sharp contrast with usual KBI of binary mixtures having a single extremum. This extremum is interpreted as the region of maximum concentration uctuations, and usually occurs in binary mixtures presenting appreciable micro-segregation, and corresponds to where the mixture exhibit a percolation of the two species domains. In this paper, it is shown that two extrema occur in binary mixtures when one species forms "metaparticle" aggregates, the latter which act as a meta-species, and have their own concentration uctuations, hence their own KBI extremum. This "meta-extremum" occurs at low concentration of the aggregate-forming species (such as alcohol in alkane), and is independant of the other usual extremum observed at mid volume fraction occupancy. These systems are a good illustration of the concept of the duality between concentration uctuations and micro-segregation.

Introduction

The so-called Kirkwood-Bu integrals (KBI) [1,2] are dened as the integrals of the species-species pair correlation functionsG ab = 1 Ω ´dX 1 dX 2 [g ab (X 1 , X 2 ) -1] where X i is the set of position, and if required, the orientations, of molecule i, where a and b designates the species indexes (Ω = V ω 2 , where ω is the angular integral, equal to 4π or 8π 2 , depending on the symmetry of the molecules).

It can be shown [3,4] that this integral is identical to that involving the pair correlation between any two atoms belonging to each molecules. Following this, the KBI are more simply dened as

G ab = 4π ˆ∞ 0 drr 2 [g iaj b (r) -1] (1)
1 where i a and j b designate any two atoms i and j of , respectively, species a and b, and g iaj b (r) the atom-atom pair correlation function.The Kirkwood-Bu theory relates specic thermodynamic properties, such as the compressibility, or the partial molar volumes, for example, to be related to these integrals, thus providing an appealing link between macroscopic measurable properties to the microscopic structure, albeit in an integrated form, where all microscopic details are lost. Ben-Naim [5], Matteoli and Lepori [6] and other authors [7,8,9,[START_REF] Smith | On the kirkwood-bu inversion procedure[END_REF][START_REF] Shulgin | Kirkwood-bu integrals in aqueous alcohol systems: comparison between thermodynamic calculations and x-ray scattering experiments[END_REF] have managed to invert these relations, providing a way to calculate these integrals from the experimentally available data on compressibility, partial molar volume and vapour pressure or Gibbs free-energy. The corresponding expressions for a binary mixture (a, b) are well known [START_REF] Perera | Kirkwood-bu integrals of alcohol binary mixtures[END_REF]:

G aa = G ab + 1 x b Vb D -V (2)
G ab = χ T k B T - Va Vb V D (3)
were x b is the mole fraction of component b, Vc is the partial molar volume of component c (c = a or b), V is the molar volume, χ T is the isothermal compressibility (with T the temperature and k B the Boltzmann constant) and D is related to the concentration uctuations and given by D = x i ∂βµ i ∂x i T P

where µ i is the chemical component of species i and β = 1/k B T is the Boltzmann factor.

Fig. 1 of the seminal Matteoli and Lepori paper [6] displays KBI for various types of aqueous mixtures, and it can be seen that most of them have a single extremum, which is positive for like KBI G aa and negative for then unlike G ab with (a = b). This extremum has its origin in the shape of the coecient D above, which is often a U-like shaped curve with a single minimum. An example of such typical KBI and D are given in Fig. 1, and other similar examples can be found in various place in the literature such as [START_REF] Donkersloot | The structure of binary liquids. the kirkwoodbu theory of liquid mixtures, illustrated on the basis of the systems water/methanol, water/ethanol, and cyclohexane/2,3-dimethylbutane, as a link between thermodynamic data and x-ray and neutron scattering results[END_REF][START_REF] Nishikawa | Small-angle x-ray scattering study of uctuations in ethanol and water mixtures[END_REF][START_REF] Marcus | Preferential solvation in mixed solvents x. completely miscible aqueous co-solvent binary mixtures at 298.15 k[END_REF][START_REF] Marcus | Preferential solvation in mixed solvents. part 11. eight additional completely miscible aqueous co-solvent binary mixtures and the relationship between the volume-corrected preferential solvation parameters and the structures of the co-solvents[END_REF][START_REF] Perera | Kirkwood-bu integrals of alcohol binary mixtures[END_REF]. Note that, while for a given mixture, the D coecient has a single extremum, the corresponding extrema in the KBI are not necessarily at the same x-position, as illustrated in Fig. 1, as Eqs. (2,3) tend to alter this position from that of D.

A single extremum for D is expected for a curve which reaches the same value at both end points. In fact, Eq.( 4) can be shown (see Appendix) to be related to the stability limit criteria for a binary mixture, that is precursor for demixing, namely

D ∝ C 11 C 22 -C 2 12 ( 5
)
where C ij = δ ij -√ ρ i ρ j cij (k = 0) , with the cij (k = 0) being the integrals of the direct correlation functions and i,j refer to species index.Indeed, when D → 0 the mixture becomes mechanically unstable [3], such when approaching curves for aqueous-acetone from Ref. [START_REF] Perera | On the evaluation of the kirkwood-bu integrals of aqueous acetone mixtures[END_REF]; blue curves for aqueous-2propanol from Ref. [START_REF] Perera | Kirkwood-bu integrals of alcohol binary mixtures[END_REF]. Circles for Water-water KBI, squares for cross water-solute KBI, and triangles for solute-solute KBI. The inset shows the corresponding D (Eq.( 4)) with same color codes, both showing a single minimum.

a spinodal or a critical point. Therefore, a single extremum in D can be physically interpreted as the point where concentration uctuations are maximal, and with the eventuality that when these grow beyond the mechanical stability, the mixture undergoes demixing. Subsequently, the extremum in the KBI is a signature of maximal concentration uctuations [START_REF] Nishikawa | The solution chemistry of mixing states probed via uctuations: a direct description of inhomogeneity in mixing[END_REF] . Since many binary mixtures, such as aqueous mixtures, for example, show stable local nano-segregation of constituents [START_REF] Dixit | Molecular segregation observed in a concentrated alcohol-water solution[END_REF][START_REF] Guo | Molecular structure of alcohol-water mixtures[END_REF][START_REF] Allison | Clustering and microimmiscibility in alcohol-water mixtures: Evidence from moleculardynamics simulations[END_REF][START_REF] Mijakovi¢ | Ethanol-water mixtures: ultrasonics, brillouin scattering and molecular dynamics[END_REF][START_REF] Poºar | Micro-heterogeneity versus clustering in binary mixtures of ethanol with water or alkanes[END_REF], the extremum can be interpreted as maximal amplitude in local segregation [START_REF] Nishikawa | The solution chemistry of mixing states probed via uctuations: a direct description of inhomogeneity in mixing[END_REF].

In this context, it came as a surprise that Jan Zielkiewicz published experimental KBI for various alcohols-heptane mixtures [START_REF] Zielkiewicz | Kirkwood-bu integrals in the binary and ternary mixtures containing heptane and aliphatic alcohol[END_REF], which clearly showed the existence of two extremum. In Fig. 2 of his paper, he shows KBI written as ρ a G ab where ρ a = N a /V is the partial density of species a (N a is the number of particle in volume V ), which amplies the second extremum, the resulting KBI looking like camel back shaped, as opposed to the usual single extremum KBI which are dromadary camel back shaped.

The purpose of this paper is to understand the origin of this dual extremum in terms of uctuations. The driving idea is the following. It is well known that alcohols generally tend to cluster their hydroxyl groups into linear patterns, forming chains, loops, etc... [START_REF] Narten | Hydrogen bonding in liquid methanol and ethanol determined by x-ray diraction[END_REF][START_REF] Vahvaselkä | Determination of liquid structures of the primary alcohols methanol, ethanol, 1-propanol, 1-butanol and 1-octanol by x-ray scattering[END_REF][START_REF] Karmakar | On the structure function of liquid alcohols at small wave numbers and signature of hydrogenbonded clusters in the liquid state[END_REF][START_REF] Tom²i£ | Structural properties of pure simple alcohols from ethanol, propanol, butanol, pentanol, to hexanol: Comparing monte carlo simulations with experimental saxs data[END_REF][START_REF] Poºar | On the x-ray scattering pre-peak of linear mono-ols and the related microstructure from computer simulations[END_REF][START_REF] Bolle | Isomeric eects in structure formation and dielectric dynamics of dierent octanols[END_REF] When put in small concentration in an inert alkane solvent, such as heptane, for example, small chain-micelles of alcohols form and act as independant meta-particles. Herein, we will adopt the convention of using the wording meta to designate local aggregated assemblies which act as micelle-like objects, and cannot be considered as mere local concentration uctuations. Consequently, such exhibit their own concentration uctuations, which corresponds to one of the observed extrema in the corresponding KBI. With increasing concentration, the frontier between well separated micelles and alcohol micro-segregated domains become loose, and the mixture becomes a simple micro-segregated mixture, which has its own independent maximum in concentration uctuations. It is not obvious that such scenario should happen, and the purpose of this paper is to use computer simulation to provide evidence that this scenario is correct, principally through the analysis of pair correlation functions and the KBI.

Theoretical and simulation details

The study of equilibrium locally micro-segregated mixtures has proven to be very dicult in the past decades, particularly in the case of aqueous mixtures.

Perhaps a canonical example is that of aqueous tert-butanol (TBA) mixtures, for which the force eld induced slow demixing at very small TBA concentrations around x TBA ≈ 0.01 was shown [START_REF] Lee | A new force eld for atomistic simulations of aqueous tertiary butanol solutions[END_REF][START_REF] Banerjee | Fluctuating micro-heterogeneity in water-tert-butyl alcohol mixtures and lambda-type divergence of the mean cluster size with phase transition-like multiple anomalies[END_REF][START_REF] Gupta | Aggregation in dilute aqueous tert-butyl alcohol solutions: Insights from large-scale simulations[END_REF][START_REF] Overduin | Comparison of simulation and experimental results for a model aqueous tert-butanol solution[END_REF][START_REF] Overduin | Structural behavior of aqueous t-butanol solutions from large-scale molecular dynamics simulations[END_REF] to rule out many classically robust force eld models such as OPLS [START_REF] Jorgensen | Optimized intermolecular potential functions for liquid hydrocarbons[END_REF][START_REF] Jorgensen | Optimized intermolecular potential functions for liquid alcohols[END_REF] or TraPPE [START_REF] Martin | Transferable potentials for phase equilibria. 1. united-atom description of n-alkanes[END_REF][START_REF] Chen | Monte carlo calculations for alcohols and their mixtures with alkanes. transferable potentials for phase equilibria. 5. united-atom description of primary, secondary, and tertiary alcohols[END_REF].

One may even consider this problem to remain unsolved. Similarly, the watertetrahydrofuran mixture remains dicult to simulate, in particular for putting into evidence the loop coexistence phase diagram [START_REF] Matou² | Liquid-liquid equilibrium in the system water-tetrahydrofuran[END_REF][START_REF] Narayanan | Reentrant phase transitions in multicomponent liquid mixtures[END_REF][START_REF] Brovchenko | Simulation of the liquid-liquid coexistence of the tetrahydrofuran+water mixture in the gibbs ensemble[END_REF]. One of the principal obstacle for successfully simulating such mixtures is the fact that both water and the alcohol tend to form hydrogen bonded clusters, and it would seem that it is precisely the competition between these cluster formation which tend to drive the simulated mixture into demixing, often after unusually very long times.

These problems might be avoided if water is replaced by a more inert solvent such as an alkane. In a previous work [START_REF] Poºar | Simple and complex disorder in binary mixtures with benzene as a common solvent[END_REF], we have studied ethanol-benzene mixtures and found that the description of the strong micro-heterogeneity (MH) in the low ethanol content region posed statistical issues which necessitated large size simulations. We expect to nd similar issues in the present study.

Herein, we focus on the ethanol-heptane mixtures, simulated in the following range of ethanol mole fractions: 0.02, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9. All of the mixtures contain N=16000 particles, in order to properly describe extented MH. The program package Gromacs was used to perform molecular dynamics simulations [START_REF] Pronk | Gromacs 4.5: a high-throughput and highly parallel open source molecular simulation toolkit[END_REF][START_REF] Abraham | Gromacs: High performance molecular simulations through multi-level parallelism from laptops to supercomputers[END_REF]. The simulation protocol has been the same for all mole fractions of ethanol. The initial random congurations of 16000 molecules were created with the program Packmol [START_REF] Martínez | Packing optimization for automated generation of complex system's initial congurations for molecular dynamics and docking[END_REF]. These initial congurations were rst energy minimized and then equilibrated in the NpT ensemble 5 ns. The length of the production runs varied for each mole fraction of ethanol and can be found in Table 1. The shortest production runs of 10 ns sampled on average 1000 congurations, whereas the longest runs of 35 ns sampled more than 3500 congurations.

The TraPPE forceeld for heptane [START_REF] Martin | Transferable potentials for phase equilibria. 1. united-atom description of n-alkanes[END_REF] and ethanol [START_REF] Chen | Monte carlo calculations for alcohols and their mixtures with alkanes. transferable potentials for phase equilibria. 5. united-atom description of primary, secondary, and tertiary alcohols[END_REF] was used throughout the range of ethanol mole fractions. The mixtures were simulated at T = 300 K and p = 1 bar. Temperature was maintained constant using mostly the v-rescale [START_REF] Bussi | Canonical sampling through velocity rescaling[END_REF] or NoseHoover [START_REF] Nose | A molecular dynamics method for simulations in the canonical ensemble[END_REF][START_REF] Hoover | Canonical dynamics: Equilibrium phase-space distributions[END_REF] ParrinelloRahman barostat [START_REF] Parrinello | Crystal structure and pair potentials: A molecular-dynamics study[END_REF][START_REF] Parrinello | Polymorphic transitions in single crystals: A new molecular dynamics method[END_REF]. The temperature algorithms had a time constant of 0.2 ps, while the pressure algorithm was set at 2 ps. The integration algorithm leap-frog [START_REF] Hockney | The potential calculation and some applications[END_REF] was used at every time-step of 2 fs. The short-range interactions were calculated within the 1.5 nm cut-o radius. The electrostatics were handled with the PME method [START_REF] Darden | Particle mesh ewald: An n*log(n) method for ewald sums in large systems[END_REF], and the constraints with the LINCS algorithm [START_REF] Hess | Lincs: A linear constraint solver for molecular simulations[END_REF].

The snapshots were made with VMD [START_REF] Humphrey | VMD Visual Molecular Dynamics[END_REF].

3 Results

Experiments

The experimental KBI data for the ethanol-heptane mixtures scanned from Zielkiewicz paper are reported as lines in the main panel of Fig. 2. In order to make sure that these results are fully consistent with one another, we have inverted Eqs (2,3) in order to extract D, using the following assump-tions. The partial molar volumes have been replaced by the molar volume of each species, thus making these quantities independent of the concentrations.

The isothermal compressibility term has been neglected (assuming the incompressibility of the liquid mixtures). Finally, the excess volume of the mixture has been set to be a linear function of the pure liquid volumes, hence neglecting the excess volumes. These are usually found to be less that 0.3cm^3/mol. All 3 assumptions can be justied only by the resulting KBI (as was proven by us for aqueous alcohol mixtures in Ref. [START_REF] Perera | Kirkwood-bu integrals of alcohol binary mixtures[END_REF][START_REF] Mijakovi¢ | Ethanol-water mixtures: ultrasonics, brillouin scattering and molecular dynamics[END_REF]). From Eqs(2,3) we have extracted the following equivalent expressions for D

D ab = - V a B b V G ab (6)
D aa = x b V b -x a V a f 1 -f 2 (7)
with

f 1 = x a x b (G aa -G bb ) and f 2 = V (x a -x b ).
The resulting values are reported in Fig. 3.

Figure 3: Dual minima D for the ethanol-heptane mixtures, as calculated from the experimental results of Ref. [START_REF] Zielkiewicz | Kirkwood-bu integrals in the binary and ternary mixtures containing heptane and aliphatic alcohol[END_REF] (see Eqs. (6,7) in the text).

It is seen that they are quite similar, with the denominator in Eq.( 7) producing a singularity at x Eth ≈ 0.7. The dots represent the values which have been selected to best represent a good compromise between the two sets. The most prominent feature is the very apparent existence of the 2 extrema, one at x Eth ≈ 0.2 and the other at x Eth ≈ 0.5. Interestingly, both evaluations of D coincide almost perfectly in the entire range of the minimum part of D.

Using these values of the extracted D function, we return to evaluate back the 3 KBI integrals, by still preserving the 3 approximations mentioned above. The resulting values are reported in Fig. 2 as dots, and are seen to nicely superpose to the original data. This simple exercise proves that a single form of D, which must be quite close to the one reported in the inset, must have served in Ref. [START_REF] Zielkiewicz | Kirkwood-bu integrals in the binary and ternary mixtures containing heptane and aliphatic alcohol[END_REF] to calculate the KBI, and moreover, that the use of true partial molar volumes, compressibilities and excess volume did not aect much the nal shapes. We also prove that the 2 extrema of the KBI originate from the 2 extrema observed in D. Detailed studies of snapshots for the entire concentration range shows that ethanol molecules start to spontaneously form these pentameric micelles at the smallest ethanol concentrations studied herein, which is x = 0.02. These pentameric micelle structures persist until x = 0.3, after which larger aggregated structures take over. The visual inspection of formed structure tends to conrm that there are 2 regime of clustering, pentameric ethanol micelles for x < 0.3 and larger clusters for higher concentrations. We now conrm this through cluster and pair correlation function studies.

Cluster structure

The cluster size probability distributions were calculated as:

P n = k s (n, k) k n s (n, k) (8)
where P n is the probability for the cluster formed of n sites, s (n, k) represents the number of clusters of the size n in the conguration k. P n is obtained by averaging the number s (n, k) of clusters of size n over several such congurations. The cut-o distance (r c ) for a calculation depends on the rst minimum in the pair correlation function of a particular site. For the oxygens in ethanol, the r c = 3.7 Å. Fig. 6 shows the size s dependence of the cluster distributions P (s) in 2 dierent ways. Left panel (a) shows the distribution as function of the cluster size, and for dierent ethanol concentrations. What is apparent is the pentamer peak, which exists for all concentrations. In addition, we observe the appearance of a shoulder-like feature for large clusters, showing that the probability of larger clusters is not negligible for ethanol concentrations above x > 0.3. Moreover, while the probability of monomers is higher than that of pentamers for x < 0.3, and becomes comparable or lower for x > 0.3. The right panel (b) shows P (s) as function of ethanol concentration, and for dierent types of n-mers. What is strikingly apparent here is that, for the entire concentration range, there is a sharp fall of P(s) from monomer to dimers and 3-mers, followed by a second dramatic increase for 4-mers and 5-mers, followed by 6-mers, while higher n-mers are the same level as monomers and dimers. An important feature is that the maximum of the curves seems to saturate around x = 0.3, indicating a turnover of cluster structures above this concentration.

The cluster study complements the visual inspection of Fig. 5 by showing more details, such as for example the existence of 4-mers at small concentrations, and not only pentamers, as the visual inspection may suggest.

Atom-atom pair correlation functions

In order to further conrm the sharp separation of cluster structure highlighted in the previous sections, we show in Fig. 7 the evolution of 3 typical pair correlation functions as a function of ethanol concentration. The right panel shows the ethanol oxygen-oxygen correlation function g OO (r) for all the ethanol con-centrations calculated. Focusing on the second peak, which represent second neighbour correlations, hence the inuence of clustering beyond the rst neighbour peak, we clearly a dierence between two successive concentrations: the gap is wider for x < 0.3. In order to facilitate this observation, the curve for x=0.3 is marked in thicker orange color. In order to further conrm this trend, we examine in the middle panel of Fig. 7 the cross species correlations g OM1 (r) between the ethanol oxygen and the heptane rst(last) carbon group termed M1. These correlations are seen to be less concentration dependent than the previously examined g OO (r) correlations. This is expected, since heptane site are not charged, hence only Lennard-Jones like correlations exist, which are less prominent then for those between charged groups [START_REF] Poºar | Simple and complex disorder in binary mixtures with benzene as a common solvent[END_REF]. Nevertheless, we observe that for x < 0.3 all correlation functions are nearly superposed (below the thick orange curve), and start to show appreciable dierences only when x > 0.3. Again, this is fully consistent with the existence of same type of clusters for x<0.3. Indeed, is clustered objects are the same, the correlations between the carbon sites of heptane and the meta objects would be nearly similar, and very weakly dependent on x. In contrast, if the clustered objects are very dierent in shape, and if this depends strongly on ethanol concentration, we would indeed expect a larger concentration dependence of g OM1 (r).

The last right panel Fig. 7 shows a weak concentration correlations between the heptane last/rst carbon site g M1M1 (r), very similar to that observed for g OM1 (r). First of al these are Lennard-Jones like correlations, and correlations below x<0.3 are also nearly superimposed, witnessing the same insensitivity to concentration in presence of same type of aggregates of the ethanol.

If we reconsider now the experimental KBI in the light of this sharp separation of the behaviour of the correlations for x < 0.3 and for x > 0.3, we obtain the result shown in Fig. 8, where the thick orange vertical line at x=0.3 perfectly separates the concentrations under the rst KBI extremum for x< 0.3, where mostly pentameric ethanol micelles are observed, from the concentrations under the second weaker peak at right for x < 0.3, for which usual micro-segregation of the 2 species is observed. In order to fully conrm this picture of sharp separation between a meta-object mixture and a usual mixture, we need to calculate the KBI from the simulations and reproduce the same KBI shapes as observed in experiments.

Kirkwood-Bu integrals from simulations

The evaluation of the KBI through computer simulation requires that the asymptotes of the various site-site correlations are well dened and converging to 1 as expected. Recent investigations have shown that there are 2 major obstacles. The rst obstacle is that computer simulations conducted in the Canonical or Isobaric ensemble cannot lead to the proper asymptote 1, this value being reached only in the Grand Canonical ensemble [START_REF] Lebowitz | Long-range correlations in a closed system with applications to nonuniform uids[END_REF][START_REF] Salacuse | Finite-size eects in molecular dynamics simulations: Static structure factor and compressibility. i. theoretical method[END_REF][START_REF] Salacuse | Finitesize eects in molecular dynamics simulations: Static structure factor and compressibility. ii. application to a model krypton uid[END_REF]. The second obstacle concerns the existence of micro-segregation, whose spatial extent and slow kinet-ics alters the statistics of the correlations at large atom-atom separations. Both cases are illustrated below, where we examine both the tail of the correlation functions g ab (r),but also the so-called running KBI dened as [1]:

G ab (r) = 4π ˆr 0 ds s 2 [g ab (s) -1] (9)
which, when r is large enough, is expected to converge to the KBI dened in Eq.( 1).

In a rst example, we examine the case of ethanol concentration x = 0.7, for which we have shown above that micro-segregation is dominant. The typical order parameter for micro-segregation is g OM1 (r), the cross species pair correlation between the ethanol oxygen atom and the rst/last carbon group atom of heptanol. Fig. 9 shown this function in the main panel for 3 dierent runs of 5ns each. The strong depletion between adverse ethanol and heptane nano-domains is clearly visible, as all 3 curves stay below 1 until 20 Å or so. On this scale, all 3 curves seem nearly similar, which is what is expected from statistics. However, the zoom on tail region in the upper inset shows that there are visible dierences between the 3 runs, as the segregation of domain is not the same in each of the 3 runs. What is observed is the typical domains oscillatory correlations, with half-period about 20 Å, which is the depletion extent. All 3 curves oscillated about the asymptote 1, hence the rst obstacle mentioned is not really apparent. The lower inset shows the corresponding running KBI G OM1 (r), as well as the expected experimental value. It is seen that the various runs have an oscillatory asymptotic feature, witnessing the domains alternation, but oscillate around the experimental value. This result shows the rather excellent agreement between the experimental KBI and the calculated one.

In the second example, we examine the case of ethanol concentration x = 0.2, for which we expect an homogeneous distribution of ethanol micelles in the midst of heptane solvent. In this case, micro-segregation is very dierent in that it generates these micelles. Fig. 10 shows the same order parameter g OM1 (r) in the main panel, and for 5 dierent runs of 5ns each.

This function again shows appreciable depletion, but of the size of the ethanol micelles, which is more about 12 Å. But now, the zoom of the tail in the upper inset shows that there are no domain oscillations, which is indeed expected if the ethanol droplets are homogeneous distributed. However, we observe that all the asymptote are clearly shifted upwards from 1. This is a direct manifestation of the rst obstacle mentioned above, which occurs here since our calculations are in the isobaric ensemble instead of the Grand canonical ensemble. It can be shown that the asymptote of the correlation functions has the following form

lim r→∞ g ab (r) = 1 - ab N ( 10
)
where N is the number of particles in the simulation box, and the value of ab depend both on species a,b and the statistical ensemble [START_REF] Lebowitz | Long-range correlations in a closed system with applications to nonuniform uids[END_REF][START_REF] Salacuse | Finite-size eects in molecular dynamics simulations: Static structure factor and compressibility. i. theoretical method[END_REF][START_REF] Salacuse | Finitesize eects in molecular dynamics simulations: Static structure factor and compressibility. ii. application to a model krypton uid[END_REF]. For like correlations the shift is downwards from 1, and for unlike it is upwards [START_REF] Poºar | Simple and complex disorder in binary mixtures with benzene as a common solvent[END_REF]. The RKBI are shown in the lower inset, and are seen to lie above the experimental value, and not to have the expected at asymptote for a proper dentition of the KBI value for G OM1 . In order to correct for this spurious behaviour, we simply multiply the pair correlation function with the appropriate coecient γ ab = 1/ 1ab N before applying Eq.( 10) to ḡOM1 (r) = γ OM1 g OM1 (r), which now has the correct asymptote 1. The asymptote-corrected RKBI are computed using the expression

Ḡab (r) = 4π ˆr 0 ds s 2 [γ ab g ab (s) -1] (11)
and the corresponding KBI are given by

Ḡab = lim r→∞ Ḡab (r) ≈ Ḡab (L/2) ( 12
)
where L is the box size. The above approximation is appropriate if a at asymptote is reached for Ḡab (r) before r < L/2. For each correlation function an initial value of ab /N is guessed, Eq.( 11) applied and we visually test if the Ḡab (r) function is becoming more horizontal. If not, a new guess of ab /N is used, and the procedure repeated. At some point, the change of horizontality shifts in the opposite direction, and this determines the optimum ab /N value. Clearly, this empirical procedure is not obvious when the functions Ḡab (r) have large domain oscillations, such as in Fig. 11. We have successfully used this empirical shift methodology in several of previous our works in obtaining very good agreement with the experimental data [4,[START_REF] Perera | On the evaluation of the kirkwood-bu integrals of aqueous acetone mixtures[END_REF][START_REF] Mijakovi¢ | Ethanol-water mixtures: ultrasonics, brillouin scattering and molecular dynamics[END_REF][START_REF] Poºar | Simple and complex disorder in binary mixtures with benzene as a common solvent[END_REF] .

The corresponding results are shown in Fig. 11. conrms that the correct asymptote 1 has been set by the operation above, the lower inset shows that all the RKBI have now at asymptotes, even though it is not quite the value set by the experiments. This discrepancy is expected, since the ethanol aggregation in the real mixtures may not be well represented by the model simulations.

For the calculations of the KBI, from the cross KBI G OM1 we extract D from Eq.( 6) and calculate the like KBI G OO and G M1M1 using Eq. (7). Indeed, the simulations results for these KBI do not often converge to the consistency for D in Eqs. (7,6)), and the above trick is one way to get around this problem.

The origin of the inconsistency is due to system size problems, since the like domain statistics require larger simulations, with N=32000 particles or more [START_REF] Gupta | Aggregation in dilute aqueous tert-butyl alcohol solutions: Insights from large-scale simulations[END_REF][START_REF] Overduin | Comparison of simulation and experimental results for a model aqueous tert-butanol solution[END_REF][START_REF] Overduin | Structural behavior of aqueous t-butanol solutions from large-scale molecular dynamics simulations[END_REF]. When these operations are conducted for all ethanol concentrations, we obtain the result shown in Fig. 12, which shows a remarkable agreement with the experimental results.

The very low ethanol concentrations for x = 0.02 and x = 0.05 pose particular problems for the proper evaluation of the asymptotes. Indeed, the corresponding RKBI have very distorted shapes, which cannot be arranged to look at by simple shifting of the asymptotes of the pair correlation functions. These distortions, which vary considerably from one run the other, witness the strong kinetics of the ethanol micelle formation. Since these micelles are about 15 Å in diameter, while the half box length extends to 70 Å, if the micelles were replaced by soft spheres of same diameter, the statistics on the tail of the soft sphere correlations would be much better. Therefore, it is really the micelle-monomer exchange which aect those statistics at very small ethanol concentrations.

Even though the calculated KBI shown in Fig. 12 are in good agreement with the experimental ones, and more importantly, allowing to explain the experimental evidence of the 2 extrema, one could ask if the techniques used in Eqs. [START_REF] Shulgin | Kirkwood-bu integrals in aqueous alcohol systems: comparison between thermodynamic calculations and x-ray scattering experiments[END_REF][START_REF] Perera | Kirkwood-bu integrals of alcohol binary mixtures[END_REF] are the appropriate ones. Conversely, one may question if other methodologies developped for the calculations of the KBI are suitable for the demanding cases of very large heterogeneity, such as that studied in the present work. We examine this issue in the Supplementary Information (SI). The fact that other methodologies often give results similar to that proposed here ,demonstrates that the current method is equally appropriate, with the added advantage of being intuitive and very simple to implement.

Discussion and Conclusion

It is not a priori obvious that the segregation of ethanol in small concentration in oil-like solvent should evolve in a discontinuous manner into the microsegregation as this concentration is increased, leading to the separation illustrated in Fig. 8. For instance, in a previous study of ethanol-benzene mixtures [START_REF] Poºar | Simple and complex disorder in binary mixtures with benzene as a common solvent[END_REF], we have found a single extremum shaped KBIs. Yet, this study has shown that small micelle-like aggregates at low ethanol mole fraction mixtures. This dierence suggests that closed carbon group molecules, such as benzene, aect dierently the cluster structural changes from micelle to domain, than chainlike shaped alkanes. Indeed, these latter molecules can more easily merge with the micellar alkane corona than the disc-shaped benzene molecules. In order to explain this apparent discrepancy, we hypothetize here that it is solvent shape induced depletion entropic eects [START_REF] Asakura | On interaction between two bodies immersed in a solution of macromolecules[END_REF][START_REF] Sapir | Origin of enthalpic depletion forces[END_REF][START_REF] Sapir | Is the depletion force entropic? molecular crowding beyond steric interactions[END_REF] which aect the structural transition between the ethanol micelles regime and the micro-segregated domain regime. Following this hypothesis, chain-like solvent would allow a smoother transition between micelles and domains, hence clearly separating the two manifestations in terms of uctuations, thus leading to separate uctuation regimes.

In the abstract, we mention that water tert-butanol mixtures equally show two extrema in KBI. This was indeed reported our previous work [START_REF] Perera | Kirkwood-bu integrals of alcohol binary mixtures[END_REF] , where the KBI calculated by us (see Fig. 6b in Ref. [START_REF] Perera | Kirkwood-bu integrals of alcohol binary mixtures[END_REF]) were obtained by Y. Koga from the vapour pressure measurements of the chemical potentials (see the plot of D in Fig. 6a of Ref. [START_REF] Perera | Kirkwood-bu integrals of alcohol binary mixtures[END_REF]) . The double extrema is absent from the KBI data from other authors (see Fig. 6b in Ref. [START_REF] Perera | Kirkwood-bu integrals of alcohol binary mixtures[END_REF]), but present in the SANS data obtained in the same work. Interestingly, in boths cases, the double extrema occurs clearly only for the water-water KBI. In addition, the equivalent of the micelle extrema -in the present case it would be tbutanol micellar aggregates, occurs for very small tbutanol concentrations in the interval 0.1 < x < 0.2, which is similar to that observed in the present work for an entirely dierent system. Interestingly, the aggregation in aqueous tbutanol is not clearly understood to date, despite several experimental [START_REF] Winkler | Impact of solvent structuring in water/tert-butanol mixtures on the assembly of silica nanoparticles to aerogels[END_REF][START_REF] Sedlák | On the origin of mesoscale structures in aqueous solutions of tertiary butyl alcohol: The mystery resolved[END_REF] and simulation [START_REF] Banerjee | Fluctuating micro-heterogeneity in water-tert-butyl alcohol mixtures and lambda-type divergence of the mean cluster size with phase transition-like multiple anomalies[END_REF][START_REF] Gupta | Aggregation in dilute aqueous tert-butyl alcohol solutions: Insights from large-scale simulations[END_REF][START_REF] Overduin | Structural behavior of aqueous t-butanol solutions from large-scale molecular dynamics simulations[END_REF][START_REF] Cerar | Structural, rheological and dynamic aspects of hydrogen-bonding molecular liquids: Aqueous solutions of hydrotropic tert-butyl alcohol[END_REF] investigations.

From these two points, it appears that the topic of the existence of camel back shaped KBI deserves further investigations as to the conditions where it might occur.

To conclude, the present study illustrates the dierence between concentration uctuations and micro-segregation both in the microscopic and macroscopic thermodynamic level. It is generally believed that micro-segregation would cor-respond to a more microscopic k = 0 part of the concentration uctuations, related to the nite extent of segregated domains, while concentration uctuations would be their thermodynamic limit, which correspond to k = 0. The present study shown that the KBI, which are related to the macroscopic k = 0 limit of the structure factors, themselves contain the dierence in both manifestations. This was illustrated in Fig. 8, through the sharp separation between concentration uctuations of meta-objects (the ethanol micelles) for x < 0.3 and the micro-segregation regime for x > 0.3. To be more specic, while micro-segregation concerns the initial microscopic objects, namely ethanol and heptane molecules, the concentration uctuations described here concern the meta-objects formed by the ethanol micelles, themselves oating in molecular heptane solvent. This study has been possible, precisely because heptane is an inert solvent, due to its uncharged carbon group sites, which cannot form associated entities. In this context, heptane concentration uctuations are very small, and the neutrality of this oil-solvent enhances the charge association of ethanol molecules. This is not possible to observe with water-solvent and small alcohol molecules, because both species tend to associate. In order to observe a similar phenomena in water, one requires much larger solutes molecules, such as surfactants, which can form micelles and other self-assembled objects.

The present study also illustrates the importance of the pair correlation functions and their asymptotes, and through these quantities, the issues related to domain segregation and concentration uctuations in nite size simulations, in particular through the Lebowitz-Percus shift of the asymptotes, illustrate here for the uctuations between meta-objects, as opposed to the usual illustration for molecular objects.

Appendix

Eq.( 2) in the Kirkwood-Bu paper [1] expresses the derivative of the chemical potential µ i of species i with respect to its mole fraction x i in terms of the KBI, which we rewrite below using a trivial rearrangement of the original equation as:

∂βµ i ∂x i T,P = 1 x i [1 + ρ j x i (G 11 + G 22 -2G 12 )]
The Ornstein-Zernike equation for a binary mixture can be written in a matrix equation as

S 11 S 12 S 12 S 22 C 11 C 12 C 12 C 22 = I
where I is the identity matrix, and where the structure matrix elements are S ij (k) = δ ij + √ ρ i ρ j hij (k) and the C-matrix elements are dened in terms of the direct correlation functions as

C ij (k) = δ ij - √ ρ i ρ j cij (k),
S ij = δ ij + √ ρ i ρ j G ij .
Therefore, one can rewrite the equation above as (using

ρ i = x i ρ) ∂βµ i ∂x i T,P = 1 x i √ x 1 x 2 [S 11 + S 22 -2S 12 ]
which can be rewritten by using the OZ equation as

∂βµ i ∂x i T,P = det C x i √ x 1 x 2 [C 11 + C 22 + 2C 12 ]
Using Eq.( 4) which denes the KBI coecient D, one has the relation relating D to the determinant of C, which controls the stability limit of a mixture [3],

as written in Eq.( 5)

D = α C 11 C 22 -C 2 12 with α = 1/( √ x 1 x 2 [C 11 + C 22 + 2C 12 ]).
First, the statistics on the aggregated entities show appreciable domain oscillations, which are genuine features, and may require excessive long simulation to stabilize. Second, small uncertainties in the large distance behaviour of the g(r ) are amplified by the integration procedure. These results show that very large systems are not necessarily a good strategy when dealing with strongly heterogeneous systems with inner large association-dissociation kinetics.

B -Finite size scaling analysis

Finite size analysis has been originally [1] used in the context of second order phase transitions for the following reason. In this type of phase transition, the correlation length ξ diverges, hence may become larger than the simulation box size L . It necessary to find the way ξ scales with L in order to predict the distribution of the order parameter with L . In the present context which is far from any demixing phase transition, the correlation is about few atom size ( ξ≈5 Å), much smaller the box size L≈135 Å), and such considerations may appear as remote. Yet, the KBI are a measure of the concentration fluctuations, since , alternatively to the definition in Eq.( 1), they are also defined as [2]

G ab =V ⟨ N a N b ⟩-⟨ N a ⟩ ⟨ N b ⟩ ⟨ N a ⟩ ⟨ N b ⟩ -V δ ab ⟨ N b ⟩ (1B)
where N x is the number of particles of species x . However, in the context of simulations when the total number of particles is fixed, the total fluctuation is zero (-ie-⟨N a N b ⟩-⟨N a ⟩ ⟨N b ⟩=0 . This is the case in canonical and isobaric ensembles, the latter which we have used in this work.

Hence, the formula above can only give the ideal gas result for the KBI. However, one can compute local fluctuations within smaller boxes of size L i < L embedded within the simulation box, and accumulate statistics while varying the sizes L i , hence estimate the size dependence of local G ab by using Eq.(1a). While these values will be strongly distorted for L/2< L i < L , one can expect that there will be a scaling with smaller sizes L i < L/2 , which would allow to extract the correct KBI by extrapolating to larger L i . This is exactly what was found in Ref. [3], with the following scaling relation for intermediate values of the L i G ab

(i) =G ab + α ab L i (2B)
Below, we have applied this method to the 2 cases studied in detail in Section 3.3, namely for ethanol concentrations x=0.2 and x=0.7 . We have followed the procedure described in Ref. [4]. We started with small boxes of size L 0 =15 Å, and computed cross fluctuations between the oxygen sites of ethanol (O site) and first carbon group of heptane (M 1 site), using Eq.(1B). This was done for 100 random choices of the box center within the main box. This operation was repeated for several choices for L i =L 0 +δ L , with δ L=5 Å, which represents about 15-17 points before reaching L/ 2 where distortions appear (as one moves towards the zero global fluctuation regimes). The relation (2B) is indeed well verified, as can be seen in Fig- 1B and Fig- 2B below.

The finite size scaling applied to the case of x=0.7 is shown in Fig. 1B. The main panel shows, in addition to the scaling analysis, the RKBI of Fig. 9. The inset shows the 1/r plot, which allows to verify the extent of the applicability of Eq.(2B). It is seen that the scaling analysis tends to predict KBI in close agreement to that guessed from the KBI. Fig. 1B shows a similar analysis for x= 0.2 , but for the asymptote shifted KBI. The close agreement demonstrates that the asymptote shift method produces correct KBI values with a simpler empirical approach. One of the data for the N=32000 particles is also shown as brown curve and symbols. Again, a rather good agreement with previous smaller size data is observed

The extrapolation 1/ L i →0 shown in the insets predicts values for the KBI which are very similar to those shown in Fig. 9 and Fig. 11, showing the same dispersion. This is not a surprise, since each calculations have been performed using the same trajectories as for the calculations of the g OM 1 (r ) reported in section 3.3. It is particularly interesting for the case of x=0.2 , since the newly extrapolated KBI are consistent with the KBI obtained the corrected asymptotes as in Fig. 11, and not those of Fig. 10. Again, this is not a surprise, since the work of Lebowitz and Percus in the 70's have demonstrated [5] that correlation functions computed in other ensembles than the Grand Canonical ensemble (GCE) (which allows for total particle number fluctuations, due to the coupling to a particles reservoir), should not tend asymptotically to 1, but rather obey Eq. [START_REF] Smith | On the kirkwood-bu inversion procedure[END_REF]. In particular, it should not obey Eq.( 1), and this is exactly what the current calculations have confirmed. However, the results shown in the SI equally demonstrate that calculations based on shifting the asymptote, as in Eqs. [START_REF] Shulgin | Kirkwood-bu integrals in aqueous alcohol systems: comparison between thermodynamic calculations and x-ray scattering experiments[END_REF][START_REF] Perera | Kirkwood-bu integrals of alcohol binary mixtures[END_REF] are a proper way to use Eq.(1B) in the isobaric ensemble, since the predict KBI very similar to those of the finite scale analysis.

C -Alternate computational methodology for evaluating KBI from simulations

The comparison conducted in Section B of this SI document confirms that the incorrect behaviour of the asymptote of the g ab (r ) , is a fundamental problem for the computation the KBI from simulations conducted in ensembles other than the GCE, since Eq.( 1) cannot be used. A similar conclusion was equally reached in Ref. [4] where the authors proposed a formula to compute the asymptote shift, instead of the empirical numerical approach used here. A different method was proposed in Ref [6] which consists in weighting the integral in the KBI by a function which takes into account cross correlations between the domains surrounding the two particles 1 and 2 for which the separation is r=|⃗ r 1 -⃗ r 2 | . The important point we would like demonstrate here, is that this method is not suited to predict the KBI, unless the asymptote is corrected prior to using it. According to this method the KBI are given by the expression

G ab =lim R →∞ G ab (w) (R)=lim R →∞ 4 π ∫ 0 R dr r 2 [g ab ( r)-1]w R (r ) (1C)
The distance dependent weight w R (r) accounts for the overlap volume between the 2 spheres centered around particles 1 and 2, which leads to the expression

w R (r)=1- 3 2 r R + 1 2 ( r R ) 3 (2C)
An alternative expression, valid for large R , is

w R (r)=1-( r R ) 3 (3C)
A first remark about Eq.(1C) is that it has the same problem as Eq.( 1) with respect to the asymptote, since it uses g ab (r )-1 , which is not appropriate in constant N ensembles. In other words, it does not cure the fundamental problem of the computation of the KBI noted above. A second remark concerns how the weight function w R (r) affect the integral in Eq.(1C) as compared to that in Eq. (1). In both expressions, w R (r) varies smoothly from 1 at r=0 , to 0 for r=R . Hence, it will affect the value of the running integral G ab (r ) close to r=R ,to differ from the unweighted one, except when R is large enough such that g ab (r )-1≈0 . This damping effect makes the running integral to converge faster to KBI value, and in particular at intermediate R to differ from those obtained using the unweighted expression. From the expressions above, we see that the damping will be faster for Eq.(3C) than for Eq.(2C), which is indeed verified below.

Before using Eqs.(1C-3C) in the present context, and to illustrate the two remarks above, we have calculated the KBI for a simple Lennard-Jones liquid, for the reduced density ρ * =(N /V )σ 3 =0.8 and reduced temperature T * =T /(ϵ/k B )=1.5 , in a simulation with N=16000 particles. The RKBI function G(r ) , as calculated from the 3 methods, namely the asymptote shifting method of the present work and the 2 weighting methods Eq.(2C) and Eq.(3C), are shown in Fig. 1C, and for both the cases when the asymptote of g(r ) is un-shifted or shifted. This figure is very similar to Fig.(1a) in Ref. [6]. The G(r ) for the original data, calculated using Eq.( 9) is shown in dashed orange lines and that using Eq.( 11) is shown in red lines. The asymptote shift is empirically found to be extremely small (since ϵ N =0.0352 ). Although the difference between the 2 curves in invisible in the main panel, it is very apparent in the zoom shown in the inset, the orange curve for G(r ) of the un-shifted g(r ) showing this typical tendency of curving downward.

When using Eq.(2C), the G(r ) for the un-shifted (thick dashed pink curve) and the shifted one (thick green curve) show a very good damping of the oscillations at all distances, but very slow convergence, as illustrated in the inset. By contrast, G(r ) obtained by Eq.(3C) for the shifted g(r ) (thick black curve) converge faster, and the agreement with the method of Eq.( 11) is very good, with the advantage of being less noisy, precisely because of the damping. However, if Eq. (3C) is applied to the un-shifted g(r ) (thick dashed blue curve), it curves downward, just as the orange curve. This plot demonstrates clearly that it is the asymptote shift which ensures the obtention of the correct KBI. The weighting methodology could be useful to spot the incorrect asymptote, since it damps the noise in the G(r ) . Next, in Fig. 2C, we compare the RKBI for ethanol concentration x=0.7 in a plot similar to the lower inset of Fig. 9, the 3 different runs depicted with different colors.

The RKBI obtained with Eq.(2C) (dashed lines) are clearly the worst of the 3, and this is consistent with the results for the LJ fluid shown above. Those computed with Eq.(3C) (thick lines) are better than those reported in Fig. 9, in the sense that they flatten out better, thus allowing for a better estimate. However, these estimates are not so different than those predicted in this work and the total estimate is very close to the previous one, both predictions being indistinguishable on the scale of the final plot Fig. 12.

The same analysis for the case of x=0.2 is done below in Fig. 4C, first for the case of the unshifted asymptotes (as in Fig. 10), and second for the case of the shifted ones (Fig. 11). To make the plots less cluttered, only 3 of the 5 runs have been analyzed. The conclusion of these 3 calculations can be expressed as follows. Eq.(3C) produces better KBI results than the original expression Eq.(2C), the latter which was based on the physical picture of allowing for cross correlations between the 2 particles of interest, itself based on the claim of better mathematical treatment of the initial expression for the KBI integral [6]. In view the results shown here, but also in the original papers [3,6,7], one may raise some doubts on the entire procedure. Indeed, both in Ref. [3] and [7], the validity of Eqs.(1C-3C) have been successfully tested for a model g(r ) which has the correct asymptote of 1, valid for the GCE but for the N-constant ensembles for which the entire procedure is supposed to apply. The dashed blue curve in Fig. 1C clearly shows that a g(r) taken from N-constant ensemble will not produce the proper KBI. When using properly shifted asymptotes, Eq.(3C), which is an approximation of Eq.(2C) valid for large separations, produces better results by damping the oscillating RKBI asymptotes. However, this forced damping may produce artifacts for mixtures where the oscillations are realistic physical features witnessing the alternance of segregated domains, such in the case of aqueous-1propanol mixtures [8], and possibly in more complex soft matter liquids. These results may not be so surprising, since this methodology was generally tested in less demanding cases, where the asymptote of the correlations are well defined, because the corresponding liquids are very homogeneous at microscopic scales. The cases discussed here present heavy domain segregation, hence asymptote distortions are important physical manifestations which cannot be dismissed [9]. The method proposed in the present work (and also in our previous ones), although empirical, has the advantage of not altering the influence of domain formation statistics on the pair correlation functions.

Figure 1 :

 1 Figure 1: Examples of single extremum KBI. Red curves experimental KBI

Figure 2 :

 2 Figure 2: Experimental camel back shaped KBI from Ref.[24] for the ethanolheptane mixtures, reproduced as lines, with symbols for recalculated values (see text) for consistency check.

3. 2

 2 Computer simulations 3.2.1 Snapshots When mixtures present strong local heterogeneity, it is often instructive to look at snapshots in order to visually appreciate the nature and the extent of the concentration dependence of the spatial segregation. Fig.4 shows snapshots of 4 typical concentrations of ethanol, with dierent styles highlighting the morphological changes in the ethanol clustering in heptane solvent.

Figure 4 :Figure 5 :

 45 Figure 4: Snapshots of the ethanol-heptane mixtures for 4 typical ethanol concentrations. Ethanol oxygen, hydrogen and carbon groups, in red, white and cyan, respectively, and heptane carbon groups in blue. Heptane is shown as ghost pale blue for x = 0.1 and 0.2. Ethanol carbon groups are also shown as ghost pale blue for x = 0.2. (see text for details).

Fig. 5

 5 Fig.5 shows a zoom on the hydroxyl group clustering for the same 4 ethanol concentrations highligted in Fig.4. For x = 0.1 one sees that the small ethanol droplets of Fig.4 consist in fact in pentameric rings of hydroxyl groups, forming ethanol pentameric micelles. This structure vanishes for the other higher ethanol concentrations shown in Fig.5, replaced by more complex cluster structures, such as globules and various type of chains conformations..

Figure 6 :

 6 Figure 6: Cluster distribution probability function P (s) as function of (a) cluster size s and (b) ethanol concentration. (See text for details).

Figure 7 :

 7 Figure 7: Ethanol concentration dependance of selected atom-atom pair correlation functions. (a) Ethanol oxygen-oxygen pair correlation functions g OO (r); (b) ethanol oxygen heptane 1st carbon group cross correlations g OM1 (r): (c) heptane 1st carbon group correlations g M1M1 (r). The ethanol concentration x = 0.3 is marked at thicker orange line.

Figure 8 :

 8 Figure 8: Illustration of the two ethanol concentration regions for each extremum of the KBI, separated by x = 0.3 (vertical orange line). Left part for x < 0.3 concerns ethanol micellar entities, while right part x > 0.3 concerns standard micro-segregated ethanol-heptane domains.

Figure 9 :

 9 Figure 9: Long range tail contribution for ethanol-heptane cross correlation function g OM1 (r) for ethanol concentration x = 0.7 shown in the main panel for 3 dierent runs of 5ns each. The upper inset is a close-up of the tail part. The lower inset shows the respective RKBI G OM1 (r) functions, along with the experimental KBI (in blue).

Figure 10 :

 10 Figure 10: Long range tail contribution for ethanol-heptane cross correlation function g OM1 (r) for ethanol concentration x = 0.2 shown in the main panel for 5 dierent runs of 5ns each. The upper inset is a close-up of the tail part. The lower inset shows the respective RKBI G OM1 (r) functions, along with the experimental KBI (in purple).

Figure 11 :

 11 Figure 11: Tail corrected version of pair correlation functions shown Fig.10 (see text)

Figure 12 :

 12 Figure 12: Comparison between the calculated KBI from simulations versus the experimental ones [24].

  with the Fourier transform notation and denition f (k) = ´d rf (r) exp(i r. k). Since the KBI are dened as in Eq.(1), one has G ij = hij (k = 0), and

Fig. 1B .

 1B Fig.1B. Finite size scaling analysis of KBI for x=0.7, for the 3 independent runs shown in Fig.9 with the same 3 color conventions. The inset shows the 1/r dependence which obeys Eq.(2B). Corresponding RKBI are shown both in the main panel and inset. Close dots correspond to points r<L/2, and open dots for r>L/2. The horizontal blue line is the experimental value.

Fig. 2B .

 2B Fig.2B. Finite size scaling analysis of KBI for x=0.2, for the independent runs shown in Fig.11 with the same color conventions. In addition, one result for the N=32000 particle of Fig.1A is equally shown as brown lines and symbols. Line and symbol conventions are as in Fig.1B.

Fig. 1C .

 1C Fig.1C. Comparison of the calculation methods for the RKBI for a Lennard-Jones liquid (see text). The lower inset is a zoom on the tail behaviour of the various G(r) of the main panel. In both plots, the thin horizontal blue lines serves as a guide for the correct KBI value.

Fig. 2C .

 2C Fig.2C. Comparison of the calculation methods for the RKBI for x=0.7 for 3 different runs depicted with 3 colors (black, red, green). The data from Eq. (2C) are shown as thin dashed lines, that for Eq.(3C) in thick lines, and that from Fig.9 in thin lines. The horizontal blue line is the data the experimental KBI value.

Fig. 3C .

 3C Fig.3C. Comparison of the calculation methods for the RKBI for x=0.2 for 3 different runs, and computed from the un-shifted correlation functions. The horizontal blue line is the data the experimental KBI value. Line conventions as in Fig.3C.

Fig. 4C .

 4C Fig.4C. Comparison of the calculation methods for the RKBI for x=0.2 for 3 different runs, and computed from the shifted correlation functions. The horizontal blue line is the data the experimental KBI value.

Table 1 :

 1 Production run times for the simulated ethanol-heptane mixtures.

	thermostat, while pressure was maintained with the

Supplementary Information for : Camel back shaped Kirkwood-Buff integrals Aurélien Perera (1) , Martina Požar (2) and Bernarda Lovrinčević (2) (1) Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Sorbonne Université, 4 Place Jussieu, F75252, Paris cedex 05, France.

(2) Department of Physics, Faculty of Science, University of Split, Ruđera Boškovića 33, 21000, Split, Croatia.

In this SI document, we consider the size dependence of the KBI, as well two technical issues concerning the evaluation of macroscopic fluctuations within finite size simulation boxes, and which could potentially affect the KBI computed in the work. The analysis shows that these methods are equally affected by the strong local heterogeneity, and do not improve over the asymptote shifting method used herein.

A -System size dependence of the KBI Although rather large system size was used in the simulations (N=16000 molecules), leading to calculated KBI in good agreement with the experimental data, and allowing to support the idea of duality in aggregation and fluctuations, we have nevertheless considered useful to simulate larger systems of N=32000 molecules, doubling the previous size. We have focused on ethanol concentration x= 0.2 which corresponds to the middle of the ethanol "micelle" region, where competition between aggregation and fluctuation may be the most challenging to reproduce. Two independent runs of 5ns each were performed, after the initial 10ns equilibration. The two calculated RKBI G OM 1 (r ) , both for the asymptote shifted g OM 1 (r ) data (main panel) and nonshifted data (inset) are shown below in Fig. 1A. The data from Fig. 11 is equally shown as green curves, for comparison purpose.

It is seen that both new curves has quite an appreciable horizontal middle range, pointing toward the same mean value suggested by the smaller systems. When averaging the 2 curves, the resulting orange curve shows a rather excellent agreement with the mean value obtained by the smaller system calculations. The large deviations at the end of the half-box size are expected for 2 reasons.


Product Owner Suzanne Billot Digital

Christian Dior Couture

Gaëlle Pantin-Sohier
email: gaelle.pantin-sohier@univ-angers.fr

Romain Sohier
email: rsohier@em-normandie.fr

Alice Sohier
email: alice.sohier@univ-rouen.fr

DOES THE RAP WEAR DIOR? THE IMPACT OF RAP ENDORSERS ON LUXURY BRAND PERSONALITY

1.

Introduction Luxury brands spend millions of dollars for their image in advertising (US spend $ 4,93 billions) 1 . One solution which has been used to capture consumers, increase effectiveness of the campaign or gain brand attitude is the celebrity endorsement [START_REF] Schimmelpfennig | Fifty years of celebrity endorser research: Support for a comprehensive celebrity endorsement strategy framework[END_REF], as for example alliance with movie stars (Robert Pattinson for Dior2 ). As brands have personal and social meanings for consumers [START_REF] Mccracken | Who Is the Celebrity Endorser? Cultural Foundations of the Endorsement Process[END_REF] and can extend the self [START_REF] Belk | Possessions and the Extended Self[END_REF], companies choose celebritie's endorsers to share their values with consumers. Companies try to create the match-up hypothesis [START_REF] Kamins | An Investigation into the "Match-up" Hypothesis in Celebrity Advertising: When Beauty May Be Only Skin Deep[END_REF], that is characterized by a congruence between brand and the endorser, based on credibility, social meanings or attractivity [START_REF] Bergkvist | Cause-related marketing persuasion research: An integrated framework and directions for further research[END_REF] . Endorsers can support or modify brand personality [START_REF] Ambroise | L'endossement par les célébrités: Clarifications conceptuelles, synthèse critique et perspectives de recherche[END_REF]) that may positively affect brand attitude and intention behaviors. On the other hand, it is also possible to create incongruence to increase brand image, brand effectiveness or sales [START_REF] Lee | The Impact of Celebrity-Product Incongruence on the Effectiveness of Product Endorsement[END_REF]. For example, brands can select an influencer that has no interest on the brand to reach the influencer's community and revitalize the brand image [START_REF] Breves | The Perceived Fit between Instagram Influencers and the Endorsed Brand: How Influencer-Brand Fit Affects Source Credibility and Persuasive Effectiveness[END_REF]. [START_REF] Fleck | Thirty Years of Conflicting Studies on the Influence of Congruence as perceived by the Consumer: Overview, Limitations and Avenues for Research[END_REF] propose that a moderate mismatch between a sponsor and the sponsored brand should be more effective than a match and is preferable for strongest brand attitudes. Our article is following this idea to "work with incongruence" for luxury communication campaigns as it is suggested by [START_REF] Ambroise | L'endossement par les célébrités: Clarifications conceptuelles, synthèse critique et perspectives de recherche[END_REF]. Previous research has mainly focused on the causes of celebrity endorsement (attractiveness, credibility) and on the concept of congruence but few of them are focusing on the impact of incongruent pairs, which is supposed to led to negative impact on the brand perception and company's benefits. The aim of this research is therefore to extend the current knowledge by examining the effects of celebrity's incongruence on brand personality of luxury brand. In this article, we specifically study the impact for a luxury brand to use a rap artist endorser on brand personality. Luxury is mainly considered as elitist and rap is a musical style that is more popular and "from the street" [START_REF] Dion | Managing Status: How Luxury Brands Shape Class Subjectivities in the Service Encounter[END_REF]). At first sight, and following endorser model effects [START_REF] Wong | Toward a Process-Transfer Model of the Endorser Effect[END_REF], the association between luxury brand and a rap artist will lead to negative impacts on brand personality and brand attitude. However, as noted by Burgh-Woodman (2020), it is possible to "reinvente luxury in the streets" with an assemblage between luxe and rap [START_REF] Rokka | Heterotopian selfies: How social media destabilizes brand assemblages[END_REF]. Such research might provide answers to understand the role of "non logical" associations and to find news strategies for luxury brands to revitalize their image, renew the brands and to target other/new consumers without diluting the original brand personality. We first present the luxury brands meanings, then the conceptual link between celebrity endorsement and brand personality, and finally the rap culture. Experimental method is presented, and we discuss the results.

Research background & conceptual framework

1.

Luxury brands meanings Originally for an elite, the luxury market changed over the years. The year 2000 balanced the affinities towards the brand while the current generation attaches importance to the values and authenticity [START_REF] Roux | Le luxe au temps des marques[END_REF]. It was at the beginning a niche sector which is today an actively growing sector, targeting an expanding clientele [START_REF] Kapferer | Kapferer on Luxury: How Luxury Brands Can Grow Yet Remain Rare[END_REF]. The luxury sector has some specificities [START_REF] De Barnier | Do consumers perceive three levels of luxury? A comparison of accessible, intermediate and inaccessible luxury brands[END_REF] which vary in terms of personal and interpersonal values [START_REF] Vigneron | A Review and a Conceptual Framework of Prestige-Seeking Consumer Behavior[END_REF]. Price is considered as the first factor which explains the luxury product [START_REF] Ko | What is a luxury brand? A new definition and review of the literature[END_REF]. However, we have also to consider social and psychological meanings. One specific dimension of luxury is the rarity and uniqueness which can explain why people want to possess luxuries [START_REF] Vigneron | Measuring perceptions of brand luxury[END_REF]. From a personal perspective, the feeling to possess a unique product is hedonic and provides a strong emotional response. Furthermore luxury brands are fundamentally different from mass-markets because luxury is considered as an exclusive privilege for elite and a way to share a specific status [START_REF] Han | Signaling Status with Luxury Goods: The Role of Brand Prominence[END_REF]. This is effective in several categories such as cars, handbag or watches. We can also consider the renown of the luxury brand (high quality, excellence, history, reputation) [START_REF] Kapferer | Why are we seduced by luxury brands?[END_REF]. Depending on these dimensions, people will be attracted to consume luxury goods with their specific needs and their identity. Even if the luxury market still "play" with these dimensions, with specific services, privileges, uniqueness feelings, high customer relationship management (Holmqvist & al., 2020a;Wirtz & al., 2020), it is also faced to new challenges. With the democratization of luxury or the luxury bandwagon effect [START_REF] Kastanakis | Between the mass and the class: Antecedents of the "bandwagon" luxury consumption behavior[END_REF], brands use various strategies to revitalize their image, target new audience, diversify their offer. Luxury consumption is no longer restricted to the traditional elite and includes new consumers, younger, modern. Luxury brands can use a masstige strategy [START_REF] Truong | New luxury brand positioning and the emergence of Masstige brands[END_REF]. From a communication perspective, they extend their presence on online social media [START_REF] Song | Celebrity endorsements for luxury brands: Followers vs. nonfollowers on social media[END_REF] to diffuse quickly brand messages and capture attention. They also use new types of endorsers like rap artists3 . Even if it seems inappropriate and result to an incongruent endorsement, brands sometimes make this choice: Pharrel Williams with Chanel, Moha La Squale with Lacoste, Asap Rocky with Dior. In this article, we focus on this new strategy of celebrity endorsement in order to know if it is efficient for the luxury brands.

2.

Celebrity endorsment Several brands use celebrity endorsement [START_REF] Knoll | The effectiveness of celebrity endorsements: A meta-analysis[END_REF] ) which impacts the attitude towards the brand in various ways such as attractiveness and likability of celebrity, product expertise category and/or celebrity-brand fit [START_REF] Kahle | Physical Attractiveness of the Celebrity Endorser: A Social Adaptation Perspective[END_REF][START_REF] Kamins | Congruence between spokesperson and product type: A matchup hypothesis perspective[END_REF]. Celebrity endorsement can be understood as an alliance between a celebrity and a corporate brand which can influence brand equity, purchase attention, brand attitude [START_REF] Albert | Consumer, brand, celebrity: Which congruency produces effective celebrity endorsements[END_REF][START_REF] Choi | It Is a Match: The Impact of Congruence between Celebrity Image and Consumer Ideal Self on Endorsement Effectiveness[END_REF]. Previous research show the importance of the fit between brands and celebrities [START_REF] Knoll | The effectiveness of celebrity endorsements: A meta-analysis[END_REF]. Fit, or congruence, refers to the similarity between celebrity and brand and it leads to a positive attitude towards the brand [START_REF] Choi | It Is a Match: The Impact of Congruence between Celebrity Image and Consumer Ideal Self on Endorsement Effectiveness[END_REF]. Brand congruency can be seen with a tridimensional view [START_REF] Albert | Consumer, brand, celebrity: Which congruency produces effective celebrity endorsements[END_REF] : brand-consumer congruency, celebrity-consumer congruency and celebrity-brand congruency. Consumer will perceive brand image by several ways as for example the association between the ingredient brands and people/ads who endorse this brand [START_REF] Aaker | Consumer Evaluations of Brand Extensions[END_REF]. For example, Fiat choose Leonardo Di Caprio for its last ad campaign for « the electric Fiat 500 ». He is well-know with his engagements for environment, the association Di-Caprio/Fiat is said congruent for the electric. Several studies define this fit or congruence by the match-up hypothesis (Kamins,1990) that assumes the effectiveness of celebrity endorsements is partially dependent on the degree of perceived fit between an endorsed object and the respective celebrity [START_REF] Erdogan | Celebrity Endorsement: A Literature Review[END_REF] and that the fit is a moderator of the effects of celebrity endorsement on brand attitude. At the opposite, the incongruence can be defined by the inconsistency between pre-existing consumer expectations and new information [START_REF] Fleck | Celebrities in Advertising: Looking for Congruence or Likability[END_REF]. The incongruence impacts the attention, memory and attitude of the consumer [START_REF] Lee | The Impact of Celebrity-Product Incongruence on the Effectiveness of Product Endorsement[END_REF]. If there is no fit (or lack of fit), it can imply a negative evaluation of the brand [START_REF] Campbell | A risk of meaning transfer: Are negative associations more likely to transfer than positive associations?[END_REF].

Previous research have mainly shown that the fit is essential and very few research have demonstrated that incongruence, or a moderated mismatch could be preferable for brand attitude [START_REF] Jagre | Sponsorship and Congruity Theory: A Theoretical Framework For Explaining Consumer Attitude and Recall of Event Sponsorship[END_REF]. Regarding congruence (or incongruence) of celebrities, the concept of brand personality remains important [START_REF] Pradhan | When fan engagement with sports club brands matters in sponsorship: Influence of fan-brand personality congruence[END_REF]. Indeed, the celebrity is a human brand [START_REF] Hofmann | Talent or popularity: What drives market value and brand image for human brands[END_REF] which whom consumer can be attached. The brand also has a personality (Aaker, 1997, p. 347) which are divided in five dimensions : sincerity, excitement, competence, sophistication, and ruggedness. In this way, consumers choose brands that express who they are/ want to be or they are not/do not want to be [START_REF] Swaminathan | When Brand Personality Matters: The Moderating Role of Attachment Styles[END_REF]. The choice of brands, based on brand-personality process, is dependent of several attributes (cheerful, imaginative, up-to-date, reliable, …) and the endorser that will reflect the meanings of the brand. As consumer see various ads, campaigns, several strategies can be adopted by brands such for example to be incongruous with the endorser to amaze consumers and mark their memory or to be congruent with the endorser so that consumers view it in a positive way. In the specific case of luxury brands, we choose to focus on incongruency and new endorsement strategies implying rap singers. Next, we will explain why Rap culture can be considered as incongruent with luxury meanings.

3.

The rap culture Very few research have studied rap or hip-hop culture regarding celebrity endorsement [START_REF] Ferguson | Yo, DJ, That's My Brand: An Examination of Consumer Response to Brand Placements in Hip-Hop Music[END_REF], especially in luxury sector. Even if endorsement has been widely used in luxury sector, new endorsement strategies appear as digital influencers associations or rap artists. So, the question is to know if a luxury brand (considered as elitist) will gain from an association with rap artists who are historically associated with popular culture. We are wondering if congruence between rap celebrities and luxury brands could have a positive impact on brand personality. Historically, rap has emerged in the working-class neighborhoods of large cities in the northern United States [START_REF] Hebdige | That's the joint! The hip-hop studies reader[END_REF]. At the same time, a larger culture called hip-hop has developed around rap, largely influenced by music and its artists [START_REF] Motley | The global hip-hop Diaspora: Understanding the culture[END_REF]. This culture, which brings together several elements -fashion , graffiti art, music, language -and all the experiences shared by participants and consumers [START_REF] Hebdige | That's the joint! The hip-hop studies reader[END_REF], has spread around the world mainly to marginalized populations. In fact, hip-hop culture is linked not only by an awareness and proximity to hip-hop elements, but also by a feeling of "connective marginality" -a paradigm linking culture, class and historical oppression in young people around the world [START_REF] Osumare | Beat Streets in the Global Hood: Connective Marginalities of the Hip Hop Globe[END_REF]. Thus, hip-hop artists worldwide are inspired by African-American hip-hop ( resistance to oppression and struggles against discrimination and racism [START_REF] Motley | The global hip-hop Diaspora: Understanding the culture[END_REF]. Then they bring a productivity and a cultural originality specific to the receiving country, which echoes the pains, struggles and political problems they experience at the national level [START_REF] Osumare | Beat Streets in the Global Hood: Connective Marginalities of the Hip Hop Globe[END_REF]. Since its development in the 1980's, rap music has been the subject of much debate as to its content [START_REF] Dixon | Rap music and rap audiences: Controversial themes, psychological effects and political resistance[END_REF] ). The controversies have mainly focused on the messages sent to the public -violence, drugs, misogyny, materialism - (Conrad & al., 2009), their reception by the audience and their impact on the social behaviors of young listeners ; [START_REF] Miranda | Rap Music Genres and Deviant Behaviors in French-Canadian Adolescents[END_REF]. These controversies disrupt the artistic legitimacy of rap music [START_REF] Hammou | Mainstreaming French rap music. Commodification and artistic legitimation of othered cultural goods[END_REF]. Although many initiatives have succeeded in having rap recognized as one of the main forms of popular art of the late twentieth century [START_REF] Forman | That's the Joint!: The Hip-Hop Studies Reader (Routledge)[END_REF], disputes about its status still are significant [START_REF] Hammou | Mainstreaming French rap music. Commodification and artistic legitimation of othered cultural goods[END_REF]. According to Armstrong (2001), this negative perception of rap probably stems from too much media visibility of negative rap music (gangsta) compared to "positive and constructive" rap [START_REF] Tyson | Hip Hop Therapy: An Exploratory Study of a Rap Music Intervention with At-Risk and Delinquent Youth[END_REF][START_REF] Tyson | Rap Music in Social Work Practice with African-American and Latino Youth[END_REF]. In parallel, some research indicates that rap can have a positive impact on the population, if it is viewed as a tool of resistance, self-esteem, respect, (Dixon & al., 2009), empowerment and self-affirmation of individuals [START_REF] Tyson | Rap-music Attitude and Perception Scale: A Validation Study[END_REF]. In addition to the music market, rap is now recognized as a major player in marketing. Rap artists like brands and they use product placements in music videos [START_REF] Burkhalter | Advertising to the beat: An analysis of brand placements in hip-hop music videos[END_REF] and mention brand names in songs 4 [START_REF] Ferguson | Yo, DJ, That's My Brand: An Examination of Consumer Response to Brand Placements in Hip-Hop Music[END_REF]. [START_REF] Martin | Violence and consumption imagery in music videos[END_REF] noticed that nearly half of rap clips contained specific brand references. It also seems to be the most appropriate musical style for brand placement (de Gregorio & Sung, 2009). However, since rap artists are not unanimously popular, there is a lot of potential for positive or negative attitude. Thus, if properly framed, rap clip videos provide an ideal environment for integrating brands and products [START_REF] Schemer | Does "Passing the Courvoisier" always pay off? Positive and negative evaluative conditioning effects of brand placements in music videos[END_REF]. Beyond that, the congruence between the brand and the rap artist must also be taken into account because the identity of hip-hop or rap culture can have an impact on the affective responses to congruent brands [START_REF] Ferguson | Yo, DJ, That's My Brand: An Examination of Consumer Response to Brand Placements in Hip-Hop Music[END_REF]. Regarding luxury brands, a priori incongruent because opposed to hip hop culture (elitist vs popular) (Holmqvist & al., 2020b;;[START_REF] Seo | Constructing a typology of luxury brand consumption practices[END_REF], the placements of their brands are legion in hip-hop and rap videos and seem to be a good way for brands to reach a young or new audience [START_REF] Burkhalter | Advertising to the beat: An analysis of brand placements in hip-hop music videos[END_REF]. Beyond that, the endorsement of a rap / hip-hop celebrity by luxury houses offers new perspectives to analyze the match-up hypothesis. Consistent with our theoretical background, we should expect a public controversy, or a negative brand attitude caused by a perceived incongruence. Indeed, luxury is associated with a high status and a social class [START_REF] Han | Signaling Status with Luxury Goods: The Role of Brand Prominence[END_REF][START_REF] Roux | Values associated with luxury brand consumption and the role of gender[END_REF] while rap is more linked to the street and popular culture which seem far from this idea of elitism. We also assume the main hypothesis that: H1: Non-Matching/incongruent endorsers (vs neutral) elicit less favorable brand personality traits (a: excitement, b: sincerity, c: competence, d: sophistication and e: ruggedness).

Main experiment

In this study, we examine the influence of celebrity endorsement on brand personality and measure the degree to which the celebrity incongruency elicits less favorable personality traits. A within-subjects experiment (neutral condition versus incongruent condition) was designed to validate the predictions of this study. The experimental conditions were implemented through four corresponding print advertisements. Participants were exposed to two ads that included a branded product endorsed by a celebrity (for a description of the ad copy and images, see Appendix A1&2).

To test the research model and its associated hypotheses, the authors conducted a two-phase study with 129 consumers (64 for the first pair: neutral model with brand A and incongruent model with brand A and 65 for the second one: neutral model with brand B and incongruent model with brand B). The participants (129) were told that they would be evaluating test advertisements involving celebrity endorsements for luxury brands and were asked to complete an online questionnaire. The questionnaire began with a short study introduction, after which participants were asked to evaluate the brand personality twice (firstly in the neutral celebrity condition and secondly in the incongruent celebrity condition). In order to focus the attention of the respondents on the incongruent condition, we noticed that the second endorser was a rap singer without naming it. Reversely, we gave no indication concerning the celebrity in the first condition. We supposed that this information is sufficient to create specific meanings and convey new brand personality traits. This was followed by questions about whether the association between the brand and the rap singer was congruent in order to check if our choice of incongruent celebrity was verified. The research design features (1) two brands and four celebrities, which improves ecological validity and (2) a sample of consumers aged from 21 to 73. The celebrity selection involved two steps. First of all, this celebrity should be a rap singer because rap is a popular culture which is incongruent with luxury brands. In order to generalize our results, we decided to replicate the same study with two conditions and two different rap singers. So, for each ad, we had an ad (with an unknow endorser: neutral condition) and another ad (with a well-known rap singer: incongruent condition). We selected rap artists with two criteria. The first one is their popularity (their image and their music), so we selected the number of followers on Instagram and Youtube, also the number of views and likes in Youtube which are good indicators for a positive word-of-mouth and engagement [START_REF] De Vries | Examining the drivers and brand performance implications of customer engagement with brands in the social media environment[END_REF]. Indeed, the number of followers on social media is important for brand attitude [START_REF] Veirman | Marketing through Instagram influencers: The impact of number of followers and product divergence on brand attitude[END_REF] because the number of followers positively affected the influencer's likability. The second one is based on the style of music. Rap artist are well known to have a flow, to have a pace, a rhythm, and "punchlines". Based on their fashion style, their music, we then selected Kaaris and Eddy de Pretto. We selected two luxury brands than are valued in the top 10 luxury brands5 all over the world and which have a long heritage and renown that reveal their high quality [START_REF] Beverland | The 'real thing': Branding authenticity in the luxury wine trade[END_REF][START_REF] De Barnier | Do consumers perceive three levels of luxury? A comparison of accessible, intermediate and inaccessible luxury brands[END_REF] : Hermès (10,6 million followers) and Dior (35,7 million followers). The feature product category is common (fashion) to reflect a realistic situation and because these brands are very well-known in France (our country of data collection). Both rap singers are Kaaris and Eddy de Pretto differ in terms of popularity6 , they are perceived as incongruent by respondents (Minc Kaaris = 3,36 and Minc De Pretto = 2,88, F = 3,290, p = 0,072) and this difference is significant. So, we can compare two levels of incongruency between brands and celebrity. The association between Dior and Kaaris is considered as less incongruent than Hermes and De Pretto. Congruency was measured with a single item (Do you think that this association between this brand and this endorser is relevant?, 7 points Likert Scale). Using real advertisements from Dior and Hermes, the authors added a picture of each selected celebrity (Appendices A1&2). All the created pairings were fictitious but the use of real brands and celebrities in realistic ads increased the ecological validity of the study. Brand personality was measured using the 15 most salient adjective traits from [START_REF] Aaker | Dimensions of Brand Personality[END_REF] scale, in line with prior research [START_REF] Eisend | Brand personality: A meta-analytic review of antecedents and consequences[END_REF] on five dimensions (excitement, sincerity, competence, sophistication and ruggedness).

Results

We conduct an ANOVA using a repeated measure design with incongruency (neutral versus incongruent) as a within-subject variable. In the model, we find a main significant effect of celebrity incongruency on brand personality dimensions. Results shown in Table 1 (Appendix 3) confirm a significant effect of celebrity/brand incongruency on three dimensions of brand personality for brand 1 (DIOR), on excitement (F = 3.28, p = 0.077) supporting H1a but with reverse effect, on competence (F = 13.111, p = 0.001) supporting H1c and on Sophistication (F = 49.518, p = 0.000) supporting H1d. Incongruent endorser leds to more excitement for Brand 1 (DIOR, M = 4.224) compared to neutral endorser. On the reverse, incongruent endorser conveys less competence (Mincongruent = 4.187 vs Mneutral = 4.609) and sophistication (Mincongruent = 4.242 vs Mneutral = 5.527) than neutral endorser. This means that incongruency can increase some dimension of brand personality as excitement and decrease others as competence and sophistication. No effects were found for sincerity (F = 1,081, p = 0,302) and Ruggedness (F = 4,188, p = 0,275), so H1b and H1e are not supported. Results of the replication of the study for another pair celebrity/brand shown in Table 2 (Appendix 4) confirm a significant effect of celebrity/brand incongruency on four dimensions of brand personality for brand 2 (HERMES), on sincerity (F = 3.553, p = 0.064) supporting H1b, on competence (F = 44.489, p = 0.000) supporting H1c, on sophistication (F = 82.668, p = 0.000) supporting H1d and on ruggedness (F = 13.732, p = 0.000) supporting H1e. Regarding sincerity, the mean rating is higher for neutral endorser (Mcongruent = 4,51) than incongruent endorser (Mincongruent = 4,184). Moreover, the incongruent endorser conveys less competence (Mincongruent = 4.10 vs Mneutral = 5,257), less sophistication (Mincongruent = 3.661 vs Mneutral = 5,665) and less ruggedness (Mincongruent = 3.546 vs Mneutral = 4,092). The results suggest that incongruency has no effect on excitement (F = 0,005, p = 0,943) for brand 2, so H1a is not supported.

Discussion

The goal of this study was to compare the effectiveness of brand/endorser incongruency on brand personality perception. We have investigated our main hypothesis in two experiment, using different celebrity (neutral versus incongruent) across two luxury brands. Our results show that incongruency play an important role through brand personality perception by decreasing four dimensions. Competence and sophistication decrease in both cases (brand 1 and 2) when the endorser is incongruent with the luxury brands. Moreover, Brand 2 (Hermès) seems to be less sincere and rugged when there is no fit between endorser and brand. In contrast to expectations, incongruency seems to be able to convey better evaluation of excitement for the luxury brand DIOR. This result is of particular interest because it means that the luxury brand personality can be expanded when the rap artist has a personality moderately incongruent to the luxury's brand. As an association between a luxury brand and a rap singer seems to convey less sophistication for the brand, it appears to lead a positive effect on excitement. Although participants are more likely to consider the brand as less sophisticated when it is endorsed by an incongruent celebrity, they could consider the brand as more exciting as well. These results have many implications. In line with previous research, the results improve current understanding of celebrity endorsement by focusing on the antecedents of endorsement formation but suggest that incongruent endorsement alliances can be beneficial [START_REF] Zamudio | Matching with the stars: How brand personality determines celebrity endorsement contract formation[END_REF]. They, also, show both negative (dilution of the original luxury brand personality) and positive (expansion of personality) effects of brands alliances with rap artists as [START_REF] Kim | Using brand alliances with artists to expand retail brand personality[END_REF] demonstrated between artist and retail brand personality. The findings have several implications for marketers. The integration of rap artists into luxury brands is a growing trend and the risk of a such alliance demand a better understanding of the mechanisms that govern how consumers react to such alliances. Luxury brands often need to broaden their personality dimensions to increase the number of consumers and to catch new young consumers but this strategy is critical because it exists a risk of dilution of the original brand personality and a risk of loss of the existing consumers.

https://www.statista.com/topics/4149/luxury-advertising-and-marketing/

https://www.gq-magazine.co.uk/grooming/article/robert-pattinson-dior-interview

https://www.businessoffashion.com/articles/luxury/the-new-kings-and-queens-of-fashion-kanye-west-asaprocky-cardi-b

https://www.wsj.com/articles/these-are-the-fashion-brands-rappers-namedrop-the-most-11555943601

https://www.businessinsider.fr/porsche-gucci-rolex-les-10-marques-de-luxe-les-mieux-valorisees-au-monde-

3,2 million followers on Instagram and 1 047 818 114 views for Kaaris and 220 000 followers on Instagram and 96 000 000 view on Youtube for De Pretto