Juan J Alonso

B Allés

V Russier

Phase diagram for ensembles of random close packed Ising-like dipoles as a function of texturation

We study random close packed systems of magnetic spheres by Monte Carlo simulations in order to estimate their phase diagram. The uniaxial anisotropy of the spheres makes each of them behave as a single Ising dipole along a fixed easy axis. We explore the phase diagram in terms of the temperature and the degree of alignment (or texturation) among the easy axes of all spheres. This degree of alignment ranges from the textured case (all easy axes pointing along a common direction) to the non-textured case (randomly distributed easy axes). In the former case we find long-range ferromagnetic order at low temperature but, as the degree of alignment is diminished below a certain threshold, the ferromagnetic phase gives way to a spin-glass phase. This spin-glass phase is similar to the one previously found in other dipolar systems with strong frozen disorder. The transition between ferromagnetism and spin-glass passes through a narrow intermediate phase with quasi-long-range ferromagnetic order.

I. INTRODUCTION

The study of ensembles of magnetic nanoparticles (NP) is an active field of research due to its potential application in areas as disparate as biomedicine, data storage or nanofluids. 1,2 Present technology allows to synthesize NPs with a wide variability of sizes and shapes, in addition to coating them with non-magnetic layers. Moreover they can be produced in nearly monodisperse ensembles so as to enjoy a good control on their spatial distribution.

3 This know-how opens the possibility to realize densely packed ensembles of NPs that behave as systems of interacting dipoles. It is the magnetic order of such structures that stirs a renewed interest in their use in technological applications.

4,5

NPs with diameters d p up to a few tens of nanometers have a single domain (typical values are 15 nm for Fe, 35 nm for Co, 30 nm for maghemite γ-F e 2 O 3 ) that behaves as a magnetic dipole. 6 Even when they are spherical, such NPs can have anisotropies that oblige the dipole to lie along a local easy axis and to surmount an anisotropy energy barrier E a whenever the magnetic moment is inverted, resulting in a blocking temperature T b ≃ E a /30k B . 2,4 When the NPs are closely packed, their dipolar interaction energies E dd are not negligible but typically larger than E a /10, leading to E dd /k B T b ≳ 3. Consequently, low-temperature signatures of collective order induced by the dipolar interaction can be (and have indeed been) observed experimentally.

7 This is to be compared with the super-paramagnetism observed in very diluted systems for which E dd /k B T b ≪ 1.

2,8

Dilute dispersions of NPs gather into highly ordered 3D super-crystals on account of their ability to selfassemble after the evaporation of the solvent. 9,10 Such crystals exhibit dipolar super-ferromagnetism in FCC, BCC of I-tetragonal lattices. This behavior was predicted to exist in such lattices by Luttinger and Tisza. 11 Less ordered (non-crystalline) dense packings may be obtained by pressing powders to obtain a granular solid, 12 or in concentrated colloidal suspensions by freezing the carrier fluid. 13 The frozen disorder on the positions of the NPs and on the orientation of the anisotropy axes in those systems may induce frustration resulting in super spin-glass (SG) behavior. 14,15 This behavior, originated by dipolar interactions, has been observed experimentally in random close packed (RCP) samples of dipolar spheres 7 with volume fractions φ about 64%. 16 An equilibrium SG phase for non-textured RCP ensembles of dipolar spheres has recently been found by numerical simulations.

Nevertheless, the role of positional and orientational disorder in non-crystalline ensembles is far from being completely understood. Numerical simulations have shown that frozen amorphous densely packed systems with volume fractions as high as φ = 0.42 order ferromagnetically provided they are textured. 18,19 This texturation shows up in colloidal suspensions by freezing the solution in the presence of large magnetic fields h.

20

Even when h = 0, ensembles of dipolar spheres moving in a non-frozen fluid with volume fractions as low as 42% tend spontaneously to become textured by aligning their axes, exhibiting nematic order (i.e. with no positional long range order).

21,22

The picture that emerges is that the ordering of dense non-crystalline systems may change from ferromagnetic (FM) to SG as the anisotropy-axes alignment dwindles from textured (i.e., parallel axes dipoles or PAD) to nontextured (random oriented axes dipoles or RAD).

The purpose of the present work is to depict the phase diagram of non-crystalline dense packings of Ising dipoles with different degrees of texturation by employing Monte Carlo (MC) simulations (see Fig. 2). In this effort, spe-cial attention will be paid to (i) examine whether a SG phase exists comparable to the one previously found for very diluted as well as RAD systems of Ising dipoles, and (ii) explore the transition between FM and SG in order to look for possible intermediate phases. We will pursue this investigation on ensembles of Ising dipoles placed at the center of RCP spheres that occupy a 64% fraction of the entire volume. Given that here we do not focus on time-dependent properties, we concede to the Ising dipoles (i.e. dipoles with large anisotropy energies) all the necessary time to flip up and down along their easy axes and reach equilibrium, which is tantamount to say that we choose T b = 0. Such a model may be relevant for experimental situations in which one expects E a ∼ 10E dd . 7 In order to investigate the effect of the easy axes alignment we will introduce a parameter σ that interpolates from the textured to the completely random axes cases. The nature of the low temperature phases are investigated by measuring the spontaneous magnetization, the SG overlap parameter, and the associated fluctuations and probability distributions.

The paper is organized as follows. In Sec. II we carefully define the model, give the details of the MC algorithm, and introduce the observables that will be measured. The results are presented in Sec. III and some concluding remarks in Sec. IV.

II. MODEL, METHOD, AND OBSERVABLES

A. Model

We study RCP systems of N identical NPs that behave as single magnetic Ising dipoles. The NPs are labelled with i = 1, . . . , N . We will regard each NP as a sphere of diameter d carrying a permanent pointlike magnetic moment ⃗ µ i = µs i âi at its center, where the unit vector âi is the local easy-axis and s i = ±1.

The Hamiltonian governing the interaction is

H = <i,j> ε d d r ij 3 âi ⋅ âj - 3(â i ⋅ ⃗ r ij )(â j ⋅ ⃗ r ij ) r 2 ij s i s j , (1)
where

ε d = µ 0 µ 2 /(4πd 3
) is an energy and µ 0 the magnetic permeability in vacuum. ⃗ r ij is the vector position of dipole j viewed from dipole i, and r ij = ∥⃗ r ij ∥. The summation runs over all pairs of dipoles i and j, with i / = j. The particles' positions as well as their easy axes âi remain fixed during the simulations.

The spheres are placed in frozen RCP configurations in a cube of edge L assuming periodic boundary conditions. As in previous work, 17 these configurations are obtained by using the Lubachevsky-Stillinger algorithm, 23,24 in which the spheres, that are initially very small, are allowed to move and collide while growing in size at a sufficiently high rate until the sample gets eventually stuck in a non-crystalline state with volume fraction φ = 0.64.

16,24

We shall specify the size of the system by the number N of spheres inside it, or, equivalently by the lateral size of the cube they fill to capacity,

L = N π 6φ 1/3 d . ( 2
)
where d is the final diameter attained by the spheres after they ended growing.

To investigate the effect of texturation, we consider that the alignment of the vectors âi with the direction ẑ follows a Gaussian-like distribution

p(θ i ) ∝ {e -θ 2 i /2σ 2 + e -(θ i -π) 2 /2σ 2 } sin θ i , (3)
where θ i is the polar angle of the i-th dipole while each azimuthal angle is chosen at random. The variance σ controls the degree of texturation, intended as the amount of alignment of the easy axes along the Cartesian axis ẑ. σ ranges from σ = 0 for textured systems (PAD) to σ = ∞ for non-textured samples with axes completely oriented at random (RAD). We let each Ising dipole flip up and down along its easy axis âi , assuming that the dipoles are able to overcome the local anisotropy barriers. In what follows, distances and temperatures will be given in units of d and ε d /k B respectively, where k B is the Boltzmann's constant.

B. Samples

We define a sample J as a given, arbitrary realization of disorder which, for the systems under study, comes from two sources: from the randomness of the positions of the spheres and from the degree of texturation or of alignment of their easy axes âi . This disorder does not participate in the dynamics but remains frozen during MC simulations. Only the signs s i evolve during a simulation.

As a consequence of the above definitions, we shall call configuration any set of N signs {s i } i=1,...,N . In Figs. 1(a,b) two statistically independent configurations obtained from a given sample by MC simulation are depicted. Dark blue (red) colored spheres in the figures stand for dipoles pointing up (down) along axes âi nearly parallel to ẑ, while light greyish spheres stand for those whose axes deviate significantly from ẑ.

Results susceptible to be compared with empirical data require an average over N s independent samples. The need of this average is crucial at large σ due to the sizeable sample-to-sample fluctuations that appear in this regime, where SG order is expected. Moreover, because of the lack of self-averaging associated with SG order, we have not made N s smaller with increasing N . However, for large systems (the largest ones contain N = 1728 dipoles) we could employ no more than 3000 samples because of computer time limitations. The number of samples N s is listed in Table I for the values of N and σ explored in the simulations.

C. Method

Since by decreasing the degree of texturation, the system could end up in a SG phase, we have performed parallel simulations with the tempered Monte Carlo (TMC) algorithm as this algorithm has proved to be satisfactorily efficient in beating slowing down.

25 Indeed, the TMC method allow replicas to overcome energy barriers within which the system could sink and remain confined at low temperatures. These potential wells are minima of the rough energy landscapes that characterize glassy phases. Concretely, for each sample J , we run in parallel n + 1 identical replicas at temperatures T = T min + k∆ where k = 0, 1, 2, ..., n. We have found useful to choose the highest temperature, T max = T min + n∆, larger than twice the transition temperature from the paramagnetic (PM) phase to the ordered one. The TMC algorithm involves two steps. In the first one, 10 Metropolis sweeps are applied separately to all n + 1 replicas, in order to make them evolve independently from each other. Dipolar fields are updated whenever a sign s j flip is accepted.

After that step, we give to any pair of replicas evolving at neighboring temperatures (T, T ± ∆) a chance to be exchanged, according to tempering rules that satisfy detailed balance. 25 We choose ∆ such that at least 30% of all attempted exchanges are accepted. Due to limitations in computer time we simulate systems containing up to N = 12 3 = 1728 dipoles and choose T min larger than half the transition temperature.

We have imposed periodic boundary conditions in the simulations. That means that each dipole i is allowed to interact with all dipoles within an L × L × L box centered at i, see (2). Due to the long-range nature of the dipolardipolar interaction, we need to take into account contributions from beyond this box by using Ewald's sums.

27

Details on the use of Ewald's sums for dipolar systems are given in Ref.

28 . In these sums, the use of neutralizing Gaussian distributions with standard deviation α/2 allows to split the computation of the dipolar fields into two rapidly convergent sums: a first sum in real space with a cutoff r c = L/2, and a second sum in reciprocal space with a cutoff k c . We have used k c = 10, and α = 7.9/L as a good compromise between accuracy and computational speed.

28 More importantly, given that textured systems in our model are expected to exhibit spontaneous magnetization at low temperatures, we have chosen the socalled conducting external conditions using surrounding permeability µ ′ = ∞, in order to eliminate shape dependent depolarizing effects.

21,29

The thermal equilibration times t 0 are assessed by the same procedure of Ref.

17 . The overlap q(t) of configurations created from two replicas of the same sample J are obtained by evolving the replicas independently after having started from random configurations. Then t 0 is the average over samples of the value of t at which q(t) attains a plateau q 0 for each sample. In order to test the value thus obtained for t 0 , we observed that a second overlap q(t 0 , t 0 + t) calculated for pairs of configurations of a single replica taken at times t 0 and t 0 + t remains stuck to q 0 as t increases. [START_REF] Alonso | [END_REF] It is found that the less textured the system is, the longer the equilibration time appears. This is due to the large roughness of the free-energy landscapes for non-textured systems. For these hard-to-equilibrate systems, the overlap distributions p J (q) exhibit numerous spikes associated with the existence of several pure states. 31 In the simulations we have examined the ±q symmetry of the overlap distributions p J (q) as an additional indication that all samples are well thermalized.

17

A double average, the thermal one for each sample J and the above-mentioned average over the N s samples, is needed to achieve physical results. The first average is taken within the time interval [t 0 , 2t 0 ]. Given an observable u, the result of both averages will be symbolized by ⟨u⟩. For simplicity, ⟨|u| p ⟩ will often be denoted by u p .

The values of all the simulation parameters are listed in Table I.

D. Observables

The observables that have been measured in the course of the work are the following:

(i) the specific heat c from the fluctuations of the energy e ≡ ⟨H⟩/N ;

(ii) the m z component of the magnetization vector

⃗ m ≡ 1 N i âi s i , (4)
as a way to characterize the FM behavior. Note that for a given sample, ⃗ m does not rotate during the MC simulation. Rather, it aligns along the nematic director 21,29 λJ that, for the model under study, is the eigenvector corresponding to the largest eigenvalue of the tensor

Q Q Q J ≡ 1 2N ∑ i (3â i ⊗ âi -I I I). Since Q J is constant in time, λJ remain frozen during the simulation.
We find that, for the values of σ considered here, λJ practically coincides with ẑ. Then, it makes sense using m z as the FM order parameter instead of ∥ ⃗ m∥. In fact, we have also computed ∥ ⃗ m∥ and their related quantities and found that they provide the same qualitative results that m z .

(iii) The moments m p = ⟨|m z | p ⟩ for p = 1, 2, 4, that prove useful to calculate the magnetic susceptibility

χ m ≡ N k B T (m 2 -m 2 1 ), (5)
and the dimensionless Binder cumulant

B m ≡ 1 2 (3 - m 4 m 2 2
).

(iv) As an useful tool to look for SG behavior, we calculate the overlap parameter,

32 q ≡ 1 N i s (1) i s (2) i , (7)
given a sample J . s

(1) j and s

(2) j in this expression are the signs at site j of two replicas of the given sample, denoted (1) and (2), that evolve independently in time at the same temperature. Similarly as it has been done for m z , we also measure q p ≡ ⟨|q| p ⟩ for integer p, and the corresponding

Binder parameter B q ≡ 1 2 (3 -q 4 q 2 2
).

(v) Finally, for each sample J we compute the probability distributions p J (m) and p J (q), as well as their average over samples, which will be denoted by p(m) and p(q). Errors for all quantities are obtained from the mean squared deviations of the sample-to-sample fluctuations.

III. RESULTS

A. The FM Phase

The main result of the paper is the phase diagram on the plane temperature-degree of texturation shown in Fig. 2. It displays regions with FM, PM and SG phases. The FM order arises at low temperatures in the range 0 ≤ σ ≲ 0.53. A thermally driven second order transition takes place at the phase boundary between the PM and FM phases. Next we give the numerical evidence that supports this interpretation.

FM phases are defined by the presence of a nonvanishing magnetization. In Fig. 3(a) we show the behavior of the moment m 2 with the temperature for σ = 0.3 in a number of system sizes. We obtain similar results for the magnetization for all values of σ below 0.53. This is a first piece of evidence of the existence of the FM phase. Fig. 4(a) shows plots of the specific heat c vs T . The sharp variation of c near T = 2.5 suggests the presence of a singularity as N increases, as it is expected for a second order PM-FM phase transition. The same happens with the plots of the magnetic susceptibility χ m vs T shown in Fig. 4(b). The data are consistent with a logarithmic divergence of c, and with an approximate power-law divergence of χ m with N p (up to logarithmic corrections ln N ) where p ∼ 2/3.

Next we examine the dependence of m 2 on the number decay. The dot-dashed line shows the N -1 decay expected for paramagnets in the thermodynamic limit.

N of dipoles. Fig. 3(b) shows log-log plots of m 2 vs N for several temperatures. The data at T below T c = 2.55 (5) reflect that m 2 does not vanish in the N → ∞ limit. On the contrary the plot of m 2 vs N for T > T c shows a faster than a power-law decay with a T -dependent exponent, and consequently the slope of the curves is steeper for increasing T and approaches a 1/N trend, which is the expected trend in PM phases. The dashed line in Fig. 3 allow a precise determination of T c , we have followed this criterion as a first rough approach for establishing the boundary of the FM phase.

The Binder parameter B m grants a more precise determination of the transition temperature. It follows from its definition in ( 6) that B m → 1 as N → ∞ in the FM phase. On the other hand, from the law of large numbers it follows that, in the PM phase, with short-range FM order, B m → 0 as N increases. Finally, at a critical point, B m becomes size independent, as it must occur to every scale-free observable (recall that B m is dimensionless). The latter is also true in the case of a marginal phase with quasi-long-range magnetic order. Then, curves of B m vs T for various values of N should cross at T c if it (T /T c -1) + v (log L) -1/2 for σ = 0.3 using T C = 2.57( 2) and v = -1.83 (8). ▵, ⋄, □ and • stand for systems with N = 216, 512, 1000 and 1728 respectively. is a second order transition. Note however that when a marginal phase exists these curves should colapse rather than cross for all the critical region.

1.6 2 2.

33

The plots of B m vs T are shown in Fig. 5(a) for different values of N at σ = 0.3. It is apparent that all curves intersect at a precise temperature, allowing to extract the Curie temperature T c (σ), and permitting to establish a clear-cut boundary between the PM and FM phases. The relatively modest system sizes that we have used (a limitation due to the long-range nature of the dipolar interaction) does not allow the precise determination of the critical exponents.

However, from finite size scaling relevant for dipolar Ising models we get acceptable data-collapse plots of B m vs L 3/2 log 1/6 L (T /T c -1)+v (log L) -1/2 , that provide a more reliable determination of T c , (see Fig. 6). This finite size scaling behavior corresponds to the mean field one and agrees with the fact that the upper critical dimension of the dipolar Ising model be d u = 3. 34,35 For σ = 0.3, we get T c = 2.57 (2). Likewise, precise determinations of T c (σ) can be obtained for σ ≤ 0.53, the overall result being shown in Fig. 2.

For σ = 0.55 and 0.57 the curves B m vs T merge rather than cross at low temperatures, giving a less precise determination of T c . We will return to this point in subsection III C. Given that for our model ⃗ m does not rotate, m z and the overlap q are expected to give similar information in the FM phase. Thus, crossing points in the plots of B q vs T like the ones shown in Fig. 5(b), may in principle provide an additional way for obtaining T c . This is true for σ ≥ 0.53 for which clean crossing points are obtained. For smaller values of σ, see Fig. 5(b), a characteristic dip near the transition temperature makes it difficult to accurately locate the critical point.

36

B. The SG phase

This subsection is devoted to the study of small texturations, which quantitatively entails large values of σ. As σ grows, we observe large sample-to-sample fluctuations which obliges us to increase the number of samples up to roughly ten thousand (see Table I) in order to attain trustworthy averages. Also large relaxation times are observed, a typical feature of SG behavior. Indeed, we are going to report numerical data that evidence the absence of magnetic order and the existence of an equilibrium SG phase for systems with σ ≥ 0.6. With the aim of exploring this low-temperature ordered phase within a reasonable amount of computer time, we have performed the TMC simulations at temperatures no less than T = 0.55 and system sizes no larger than N = 1728, to the detriment of the accuracy.

Plots of the moment m 2 vs T are shown in Fig. 7(a) at σ = 0.6. m 2 decreases as N increases at all temperatures. In the inset of the figure, we show the plots of the specific heat c/k B vs T . They display a gentle variation and no signature of any possible singularity is seen. Similar TAD on RCP σ=0.6 . graphs follow if the study is repeated at larger values of σ. These are the first pieces of evidence that point to the non-existence of FM order and of any PM-FM transition for σ ≥ 0.6.

In Fig. 7(b) we show log-log plots of m 2 vs N . They exhibit a decay faster than 1/N 1/2 for all available temperatures. At low temperatures T ≲ 1 the results are in principle consistent with quasi-long-range magnetic order. We will further discuss this point in the next subsection. For the PM phase (with short-range magnetic order), we expect to observe m 2 ∼ 1/N for large enough systems. For the available system sizes, we discern such a trend only for extremely large temperatures, (see for example the data at T = 2.55).

A definite signature of the presence of a SG phase is the divergence of the magnetic susceptibility at low temperatures. The plots of χ m vs T for σ = 0.6 showing an increase with N , see Fig. 8(a), are consistent with that scenario. Notice that this is in clear contrast with the behavior shown in Fig. 4(b) for σ = 0.3. Log-log plots of χ m vs N for low temperatures show a power-law increase χ m ∼ N p with an exponent p that changes slightly with T but that is never greater than p = 0.55 (see Fig. 8(b)). For T ≳ 1.25, the curves detach from an algebraic growth and bend downwards indicating a non-diverging χ m in the macroscopic limit, as expected for a PM phase.

The most convincing evidence for the absence of FM order at low temperatures for σ = 0.6 is given in Fig. 9(a). The B m vs T plots show that B m diminishes as N increases for all temperatures. As a consequence, curves for different system sizes do not cross, in contrast with the behavior found in Fig. 5(a). Recall that, in case of short-range FM order, B m should vanish in the thermodynamic limit. In the inset of Fig. 9(b), we have represented B m vs 1/L for T = 0.55, showing that that is indeed the case. We obtain a similar trend for all σ ≥ 0.6 and temperatures. This finding, consistent with shortrange FM order, seems to be in contradiction with the effective power-law decay of m 2 with N observed for low T for the system sizes we have used (see Fig. 7(b)). Some clues could be obtained by inspecting the two independent magnetic configurations displayed in Fig. 1. These are thermalized configurations at σ = 0.6, T = 0.55 in the largest system size considered in this work, N = 1728. The sample appears to be broken into large magnetic domains whose frontiers appear to be frozen. The large size of the domains explains the effective power-law decay found in the m 2 vs N plots in Fig. 7(b). In striking contrast, the overlap between the two configurations covers practically the whole system (see Fig. 1(c)), suggesting a diverging SG overlap correlation length.

Provided that the magnetic correlation length (i.e. the size of the magnetic domains) does not diverge, then m z would be expected to be normally distributed, as follows from the law of large numbers. In Fig. 10(a) we represent the distribution p(m r ) where m r ≡ m z /m 1 averaged over all samples for σ = 0.6 and the lowest temperature available, T = 0.55. Clearly, p(m r ) tends to (1/π) exp(-m 2 r /π) as N → ∞, in agreement with shortrange magnetic order. We obtain qualitatively similar results for all σ ≥ 0.6 and T , a fact that leads us to discard the existence of a critical FM phase with quasi-longrange order at low temperature. For this to be the case, we should have seen a non-Gaussian broad distribution p(m r ) that behaves as an scaling function that does not change with the system size. [START_REF]At criticality, the probability distribution of m z behaves as P (m z ) = L (1+η)/2 p(m z L (1+η)/2 ) being p a scale invariant function, and m 1 ∼ L -(1+η)/2[END_REF] It seems to be the case, within errors, for a bit larger texturation (σ = 0.55), as shown in Fig. 10(b) for T = 0.55. More details on this point will be discussed in the next subsection. Finally, we report numerical evidence in favor of the positive existence of a SG phase for σ ≳ 0.6 by studying the overlap parameter q 2 and B q . Plots of q 2 vs T are shown in Fig. 11(a) for σ = 0.6. It is worth comparing this figure with its counterpart for m 2 , Fig. 7(a), to appreciate the qualitative differences between the behavior of q 2 and m 2 at low temperature. Note however that q 2 also decreases appreciably as N increases for all temperatures. This fact raises the question on whether or not q 2 vanishes as L → ∞. To clarify this, we have prepared the log-log plots of q 2 vs N shown in Fig. 11(b). Data are consistent with q 2 ∼ 1/N p for low temperatures, and with a T -dependent exponent p. The N -1 trend, expected for PM phases, shows up only at large temperatures. All of this suggests the presence of a phase with quasi-long-range SG order. We draw additional evidence on this point from the behavior of B q . Recall that in the thermodynamic limit B q → 1 in case of strong long-range order, vanishes in the PM phase, and tends to some intermediate value at criticality. In Fig. 9(b), plots of B q versus T for σ = 0.6 show that curves of different system sizes cross at a precise temperature T sg that delimits the extend of the region with SG order. These crossings permit to obtain the points T sg (σ) of the PM-SG transition line in Fig. 2.

38 Note that T sg does not vary strongly with σ. The results agree well with the limiting value T sg = 0.8 found in previous work for the RAD case (σ = ∞).

17 It is important to stress that the fact that the B q curves cross at T sg does not imply the existence of strong longrange order for T < T sg .

39 Indeed, plots of B q vs 1/L for T ≤ T sg (σ) show that B q stays below 1 (see the inset in Fig. 9(b)). Then, the B q curves should collapse in the N → ∞ limit when T ≤ T sg (σ), which is consistent with the algebraic decay found for q 2 . In summary, the data for σ ≥ 0.6 point to the existence of a SG phase delimited by T sg (σ) for which quasi-longrange SG order occurs, like in the 2D XY model.

40,41

A similar SG phase has been previously found for other dipolar systems with strong frozen disorder, namely for systems of parallel Ising dipoles with strong dilution as well as in dense arrays, both crystalline of not, of nontextured systems of Ising dipoles with the axes oriented completely at random. 17,44 However, given the moderate range of system sizes considered here, our data cannot rule out completely the so-called replica symmetry breaking scenario in which q 2 does not vanish in the N → ∞ limit, but there are long range SG order fluctuations which provoke B q < 1.

42,43

C. The FM-SG transition.

From the previous sections, we expect to find a transition within the narrow region 0.53 < σ < 0.6. In order to identify it, we have carried out TMC simulations for several values of σ in the interval [0.45, 0.6] and a range of temperatures in the TMC between T max = 3.5 and T min = 0.55. The highest temperature has been chosen well into the PM phase in order to refresh configurations and ensure equilibrium results for T min = 0.55 which is, in turn, a temperature well deep into the low-temperature phase. This procedure facilitates the exploration of the FM boundary along several isothermal lines, allowing to investigate whether there is an intermediate phase between this boundary and the SG phase determined in the previous section. In addition, the slope of the FM boundary line may discern between a forward or a reentrant behavior.

The magnetization m 2 vs σ in Fig. 12(a) for a low temperature T = 0.55 shows that m 2 decreases with N for σ > 0.5. Log-log plots of m 2 vs N in Fig. 12 4, 0.45, 0.5, 0.53, 0.55, 0.57, 0.6, 0.7, and 0.8 respectively. Dotted lines are guides to the eye. The dashed line shows the N -1 decay expected for the paramagnetic phase.

magnetization. In contrast, for σ = 0.55 and 0.57 we find a power-law decay, giving some room for the existence of an intermediate region with quasi-long-range FM order. This decay is consistent with the behavior found for the p(m r ) distributions of Fig. 10(b) for σ = 0.55. All p(m r ) curves tend to collapse into a non-Gaussian broad distribution for large N , as expected when quasi-long-range order settles. We obtain the same qualitative results for σ = 0.57. Finally, curves for larger values of σ tend to the N -1 decay characteristic of short-range FM order, as discussed in the previous section.

The plots for q 2 are shown Fig. 13. Similarly as for m 2 , q 2 does not vanish for σ ≤ 0.53, as it is expected for a FM phase. For larger values of σ we find instead a 1/N p algebraic decay of q 2 . Note that the slope of the decay is small. For example, for σ ≥ 0.7 we find p ≈ 1/8, indicating that we are far from a PM phase (for which p = 1 is expected).

We next examine how the cumulants B m and B q vary with σ and N at low temperatures. For the FM phase, both quantities tend to 1 in the thermodynamic limit while for the SG phase B m should vanish as N → ∞, and B q should tend to a non-zero value. Then, if there is a transition line separating the FM and the SG phases, we expect the related B m vs σ curves to cross at the transition point σ c (T ). As for the B q vs σ curve, it should merge for σ ≥ σ c and splay out only for σ < σ c .

In Fig. 14 (Color online) (a) Plots of the squared overlap parameter q 2 vs σ for the lowest available temperature, T = 0.55. ▵, ⋄, □ and • stand for N = 216, 512, 1000 and 1728 respectively. Solid lines are guides to the eye. (b) Loglog plots of q 2 vs the number of dipoles N for T = 0.55. From top to bottom, ⋄, ▫, •, ▵, ⬩, •, ▪, ▾, and ◂ stand for σ = 0.4, 0.45, 0.5, 0.53, 0.55, 0.57, 0.6, 0.7, and ∞ respectively. Dotted lines are guides to the eye. The dashed line corresponds approximately to a N -1/8 decay.

values of σ, as shown in Fig. 14(b), we see that B m tends to values that are neither 1 nor 0, which is a trait of quasi-long-range order, only in this intermediate region.

Similar plots are given for a lower temperature, T = 0.55, in panels (c) and (d) of the same figure. We obtain the same qualitative picture found for T = 0.8, apart from the fact that finite size effects are larger within the intermediate region. However, 1/L extrapolations of B m for σ = 0.55 and 0.57 tend to non-vanishing values, which is consistent with marginal behavior. We have performed averages over thousands of samples in order to improve the statistics. However, the error bars of B m do not allow a precise determination of the FM boundary σ c (T ). The points along the FM boundary shown in Fig. 2, are just rough estimates obtained by taking the mean value of the crossing points of the pairs of curves B m vs σ for different sizes

(N 1 , N 2 ) = (8 3 , 10
3

) and ( 103 , 12

). We find a boundary line which is nearly vertical with a positive slope suggesting a slight reentrance near σ = 0.55. However, at least for the system sizes we have employed, plots of m 2 vs T for σ = 0.55 do not allow to discern any intermediate region with strong FM order separating the low temperature SG phase from the PM region (not shown).

More extensive simulations for larger systems and for ad- ditional values of σ within the interval (0.53, 0.6) would be needed to address this issue. In summary, the results point to the existence of a narrow intermediate region with quasi-long-range order between the FM boundary line and the SG phase, a phase which covers the lowtemperature region for all σ ≥ 0.6. For σ = 0.57 and all temperatures below the PM boundary, we obtain a non vanishing B m and an algebraic decay of m 2 with N , indicating that that region of the T -σ plane still stays in the quasi-long-range regime. The area shaded with grey color in Fig. 2 exhibits the extent of this intermediate phase.

Additional information could be gathered from comparison of plots in Fig. 14 with their counterparts for B q vs σ shown in Fig. 15. Note that, in contrast to B m , the curves of B q vs σ do not splay out for σ ≥ 0.6 but merge for large N . This is expected for the SG phase described in the previous section. On the other hand, for σ ≤ 0.53 we find that both B m and B q tend to 1 in the thermodynamic limit, indicating the existence of strong FM order. Finally, for σ = 0.55 and 0.57 (the only values we have simulated in the intermediate region), B q increases with the size of the system. 1/L extrapolations of B q for T = 0.55 point to values which are less than 1, suggesting that the intermediate phase includes quasilong-range FM and SG order contemporaneously. Note however that the data for T = 0.8 shown in Fig. 15(b) do not exclude the possibility of having strong SG order in this intermediate region. Simulations for larger systems far beyond our present CPU-time resources would be needed in order to address this point.

IV. CONCLUSIONS

We have studied by Monte Carlo simulations the effect of texturation on the collective behavior of disordered dense packings of identical magnetic nanospheres that behave as Ising dipoles along local easy axes. The local axes orientations follow a probability distribution parameterized by a single parameter σ. This allows to vary the amount of orientational disorder ranging from the complete textured case (σ = 0) with all axes pointing along a common direction, to the non-textured one with the axes oriented at random (σ = ∞).

We have obtained the phase diagram on the temperature-σ plane (see Fig. 2), from studying the magnetization, the spin-glass overlap parameter q, their fluctuations, as well as some other related observables, see II D. The region σ ≤ 0.53 contains a low-temperature ferromagnetic phase with strong order separated by a second order transition line from a paramagnetic hightemperature phase. For large orientational disorder (namely, for σ ≥ 0.6) the ferromagnetic order gives way to a spin-glass phase for temperatures below a nearly flat transition line T sg (σ) that extends up to T sg (∞) = 0.8. The spin-glass phase is similar to the one previously observed in systems of Ising dipoles with strong structural disorder, at σ = ∞. The Binder cumulants allow to estimate the position of the low-temperature boundary separating the ferromagnetic and spin-glass phases. It is located near σ = 0.55 and consistent with a small reen-trance. Moreover, a narrow intermediate region with quasi-long-range ferromagnetic order seems to lie between the ferromagnetic and the spin-glass phases.

Finally we comment on the applicability of our results to actual experimental situations. As stated in the introduction, the model corresponds to the limit T c /T b ≫ 1 where T b is the blocking temperature of the dispersed system and T c a dipolar ordering temperature. This is for instance the situation of the maghemite NP ensembles with diameters d p 6 nm < d p < 12 nm studied in Ref. 7 . In them, PM/SG freezing is observed for randomly distributed easy axes and a volume fraction φ ca. 0.67 at a ratio of temperatures 4 < T c /T b < 12. Moreover the aging phenomenon used to characterize the SG state is observable only at temperatures above T b . We can thus conclude that the present model applies at a qualitative level to the latter experimental situations whenever the SG region of the phase diagram is reached.

FIG. 1 .

 1 FIG. 1. (Color online) (a) and (b) show two statistically independent configurations of a sample with 1728 magnetic nanospheres with σ = 0.6 at the temperature T = 0.55. The position of the spheres and the orientation of their local easy axes are both frozen. The color of each sphere i stands for the value of the z component of the magnetic moment ⃗ µ i /µ = s i âi , where âi is the local easy-axis and s i = ±1. Picture (c) represents the overlap between the configurations (a) and (b). Black (white) color of spheres in (c) means s (a) i s (b) i = +1 (-1).

σ

  = 0 (T max = 4.5, T min = 1.55) = 3.5, T min = 0.55)

  FIG. 2. (Color online)Phase diagram on the temperaturedegree of texturation plane for the dipolar Ising model. Symbols ⬩ indicate the PM-FM transition and have been obtained from data of B m vs T . Symbols ○ stand for PM-FM and PM-SG transitions and were obtained from the B q vs T plots. Symbols ◇ represent the FM-SG transition and follow from the B m vs σ plots. The error bars for the data marked with ○ and ◇ are smaller than the size of these symbols. FM quasilong-range order cannot be discarded in the grey region.

1 FIG. 3 .

 13 FIG. 3. (Color online) (a) Plots of the squared magnetization m 2 vs temperature T for σ = 0.3. ▵, ⋄, □ and • stand for N = 216, 512, 1000 and 1728 dipoles respectively. Lines are guides to the eye. (b) Log-log plots of m 2 vs N for different temperatures at σ = 0.3. From top to bottom, •, ▪, ⬩, ▵, ⋄, □ and • stand for T = 1.6, 2.2, 2.4, 2.5, 2.6, 2.8 and 3.4 respectively. Dotted lines are guides to the eye. The dashed line separates two regimes and stand for a 1/N 0.35

FIG. 4 .

 4 FIG. 4. (Color online) (a) Plots of the specific heat versus T for σ = 0.3. ▵, ⋄, □ and • stand for systems with N = 216, 512, 1000 and 1728 dipoles respectively. (b) Plots of the magnetic susceptibility χ m vs T for σ = 0.3. Same symbols as in (a). Lines in both panels are guides to the eye.

FIG. 5 .

 5 FIG. 5. (Color online) (a) Plots of the Binder cumulant B m vs T for σ = 0.3. ▵, ⋄, □ and • stand for systems with N = 216, 512, 1000 and 1728 dipoles respectively. The dashed vertical line indicates the Curie temperature, at which curves cross. (b) Plots of the Binder cumulant for the overlap parameter B q vs T . Same symbols as in (a). Solid lines in both panels are guides to the eye.

FIG. 6 .

 6 FIG. 6. (Color online) Finite size scaling plots for B m vs L 3/2 (log L) 1/6

FIG. 7 .

 7 FIG. 7. (Color online) (a) Plots of the squared magnetization m 2 vs T for σ = 0.6. In the inset, the specific heat vs T . ▵, ⋄, □ and • stand for N = 216, 512, 1000, and 1728 dipoles respectively. All lines in this panel are guides to the eye. (b) Log-log plots of m 2 vs N for σ = 0.6. From top to bottom, •, •, ⋄, ▵, and ▪ stand for temperatures T = 0.55, 0.85, 1.25, 1.55, and 2.55 respectively. The arrow marks the onset of the PM phase. Dotted lines are guides to the eye. The dashed line shows the N -1 decay expected for a paramagnet in the thermodynamic limit.

FIG. 8 .

 8 FIG. 8. (Color online) (a) Plots of the magnetic susceptibility χ m vs T for σ = 0.6. ▵, ⋄, □ and • stand for systems with N = 216, 512, 1000 and 1728 dipoles respectively. Solid lines are guides to the eye. (b) Log-log plots of χ m vs N for σ = 0.6.•, ⋄, and ▵ stand for temperatures T = 0.55, 1.25, and 1.55 respectively. As stressed by the dotted lines connecting the points, data ceases to grow linearly (the solid lines) at large temperatures. The arrow marks the onset of the PM phase.

FIG. 9 .FIG. 10

 910 FIG. 9. (Color online) (a) Plots of B m vs T for σ = 0.6. ▵, ⋄, □ and • mean N = 216, 512, 1000 and 1728 dipoles respectively. The solid lines are guides to the eye. (b) Plots of B q vs T . Same symbols as in (a). The curves cross at the SG transition temperature, marked in the figure with a vertical dashed line. The inset contains plots of B m and B q vs 1/L for the lowest temperature available, T = 0.55. • (⋄) stands for B m (B q ). The dashed lines in the inset are extrapolations.

1 FIG. 11 .

 111 FIG. 11. (Color online) (a) Plots of the squared overlap parameter q 2 vs T for σ = 0.6. ▵, ⋄, □ and • stand for N = 216, 512, 1000 and 1728 respectively. The dashed vertical line indicates the SG transition temperature Solid lines are guides to the eye. (b) Log-log plots of q 2 vs the number of dipoles N for σ = 0.6. From top to bottom, •, ▪, ⬩, ⋄, □, •, and ▵ stand for T = 0.55, 0.85, 1.0, 1.05, 1.25, 1.55, and 2.55 respectively. The arrow marks the onset of the PM phase. Dotted lines are guides to the eye. The dot-dashed line shows the N -1 decay expected for the PM phase.

  FIG. 12.(Color online) (a) Semilog plots of the squared magnetization m 2 vs σ for the lowest available temperature, T = 0.55. ▵, ⋄, □ and • stand for N = 216, 512, 1000 and 1728 respectively. Solid lines are guides to the eye. (b) Loglog plots of m 2 vs the number of dipoles N at T = 0.55. From top to bottom, ⋄, ▫, •, ▵, ⬩, •, ▪, ▾, and ◂ stand for σ = 0.4, 0.45, 0.5, 0.53, 0.55, 0.57, 0.6, 0.7, and 0.8 respectively. Dotted lines are guides to the eye. The dashed line shows the N -1 decay expected for the paramagnetic phase.

  FIG. 13.(Color online) (a) Plots of the squared overlap parameter q 2 vs σ for the lowest available temperature, T = 0.55. ▵, ⋄, □ and • stand for N = 216, 512, 1000 and 1728 respectively. Solid lines are guides to the eye. (b) Loglog plots of q 2 vs the number of dipoles N for T = 0.55. From top to bottom, ⋄, ▫, •, ▵, ⬩, •, ▪, ▾, and ◂ stand for σ = 0.4, 0.45, 0.5, 0.53, 0.55, 0.57, 0.6, 0.7, and ∞ respectively. Dotted lines are guides to the eye. The dashed line corresponds approximately to a N -1/8 decay.

  FIG. 14. (Color online) (a) Plots of B m vs σ for T = 0.8, and the values of N indicated in the panel. Solid lines are guides to the eye. (b) Plots of B m vs 1/L for T = 1 for various values of σ. From top to bottom, ▫, •, ▵, ⬩, •, ▪, and ▾ stand for σ = 0.45, 0.5, 0.53, 0.55, 0.57, 0.6, and 0.7 respectively. Dotted lines are guides to the eye. (c) Same as in (a) but for T = 0.55. (d) Same as in (b) but for T = 0.55.

  FIG. 15. (Color online) (a) Plots of B q vs σ for T = 0.8, and the values of N indicated in the panel. Solid lines are guides to the eye. (b) Plots of B m vs 1/L for T = 1 for various values of σ. From top to bottom, ▫, •, ▵, ⬩, •, ▪, and ▾ stand for σ = 0.45, 0.5, 0.53, 0.55, 0.57, 0.6, and 0.7 respectively. Dotted lines are guides to the eye. (c) Same as in (a) but for T = 0.55. (d) Same as in (b) but for T = 0.55.

TABLE I .

 I The values taken by the parameters utilized in the TMC simulations. σ is the degree of texturation, N the number of dipoles, N s the number of samples with different realizations of disorder, and T max and T min the highest and lowest and temperatures respectively. ∆ = 0.05 is the temper-

ature step in all simulations. The number of MC sweeps for equilibration is t 0 = 10 6 in all cases. Measurements are taken during the MC sweeps comprised in the interval [t 0 , 2t 0 ].

ACKNOWLEDGEMENTS

We thank the Centro de Supercomputación y Bioinformática at University of Málaga, Institute Carlos I at University of Granada and Cineca for their generous allocations of computer time in clusters Picasso, and Proteus. We thank also access to the HPC resources of CINES under the allocation 2018-A0040906180 made by GENCI, CINES, France. Work performed under grants FIS2017-84256-P (FEDER funds) from the Spanish Ministry and the Agencia Española de Investigación (AEI), SOMM17/6105/UGR from Consejería de Conocimiento, Investigación y Universidad, Junta de Andalucía and European Regional Development Fund (ERDF), and ANR-CE08-007 from the ANR French Agency. J.J.A. also thanks the Italian "Fondo FAI" for financial support.

Each author also thanks the warm hospitality received during his stays in the other authors' institutes: ICMPE, the Pisa INFN section, and the University of Málaga.


Aly Shamseddin

C Eline Crauste

Erwann Durand

Pierre Villeneuve

Gregor Dubois

Thierry Durand

Joseph Vercauteren
email: jvercauteren@univ-montp1.fr

Francisco Veas
email: francisco.veas@ird.fr

Resveratrol formulated with a natural deep eutectic solvent inhibits active matrix metalloprotease-9 in hormetic conditions

Keywords: Hormesis / MMP-9 / Metalloproteases / nADES (natural deep eutectic solvents) / Resveratrol active matrix metalloproteinase, BxPC3, human pancreatic adenocarcinoma cell line, hcMec/D3, human brain microvascular endothelial primary-derived cell-line, HUVEC, human umbilical vein endothelial cells, MDA MB-231, human breast adenocarcinoma cell-line, MMP, matrix metalloproteinase, NADES, natural deep eutectic solvent

Despite the promising anti-oxidant and anti-inflammatory effects of resveratrol (RES) on human health, pre-clinical and clinical studies are frequently disappointing, probably due to its low water-solubility and poor bioavailability. Even though a hormetic mode of action was clearly established for RES, the high doses commonly used to mitigate these issues, lead to adverse effects. Common hallmarks of multiple pathologies results from pathological-enhanced endothelial permeability due to both enhanced inflammation and matrix metalloprotease-9 (aMMP-9) activities. The main aim of this work was to optimize the RES capacity to inhibit aMMP-9 by using a new class of solvents, natural deep eutectic solvents (NADES) for a new RES formulation as compared with dimethyl-sulfoxide (DMSO). To obtain the appropriate NADES, 18 compounds combinations were prepared to select those exhibiting the optimized capacity to dissolve RES. The RES-NADES 1,2-propanediol:choline-chloride:water (PCW, 1:1:1 molar ratio) and compared with RES-DMSO for their aMMP-9-inhibitory capacities. Low concentrations of RES-NADES/PCW formulation exhibited both a biocompatible solubility and a strong increased aMMP-9-inhibitory activity, at least 10-fold, higher than RES-DMSO, reaching its hormetic mode of action. Following in vivo validations, some particular NADES could potentially be considered as the new generation of formulation for druggable compounds.

Practical applications: Formulation of resveratrol in Natural Deep Eutectic solvents (NADES) optimizes its capacity to inhibit active matrix metalloprotease-9. The Resveratrol-NADES 1,2-propanediol:choline-was the most efficient and low concentrations exhibited both a biocompatible solubility and an increased aMMP-9-inhibitory activity, at least 10-fold, higher than RES-DMSO. Consequently, the NADES/PCW formulation allowed resveratrol to reach its hormetic mode of action. Following in vivo validations, some particular NADES could potentially be considered as the new generation of formulation for druggable compounds.

Introduction

Numerous clinical studies have been performed with resveratrol (3,4 0 ,5-trihydroxystilbene, RES), a plant polyphenol from the stilbenoid series (Fig. 1), to assess its capacity to promote beneficial effects on human health [START_REF] Brasnyo | Resveratrol improves insulin sensitivity, reduces oxidative stress and activates the Akt pathway in type 2 diabetic patients[END_REF][START_REF] Magyar | Cardioprotection by resveratrol: A human clinical trial in patients with stable coronary artery disease[END_REF]. Indeed RES, is considered as a dietary anti-oxidant [START_REF] Leighton | Plasma polyphenols and antioxidants, oxidative DNA damage and endothelial function in a diet and wine intervention study in humans[END_REF], able to decrease inflammatory processes, including the inhibition of the gelatinolytic metalloproteases (gMMP)-2 and 9 activities [START_REF] Park | Gold-conjugated resveratrol nanoparticles attenuate the invasion and MMP-9 and COX-2 expression in breast cancer cells[END_REF]. The activity increase of gMMP is strongly related to pathological enhancement of endothelial permeability is a common characteristic of several infectious and systemic diseases, including HIV-1 [START_REF] Misse | HIV-1 glycoprotein 120 induces the MMP-9 cytopathogenic factor production that is abolished by inhibition of the p38 Figure 4. MMP-9-inhibitory effect of RES formulated in NADES/PCW or DMSO. The gelatinolytic activity of MMP-9 was revealed by a gel zymography assay. This assay evidenced that RES exhibit an inhibitory effect on the enzymatic activity of MMP-9 (A). In addition, RES dissolved in NADES/PCW exhibited a higher inhibitory activity of MMP-9 than the RES-DMSO formulation. Data from the pixel image analysis of individual zymography gel band of MMP-9 activity was plotted for each RES concentration (B). Data were normalized against negative control treated with 0.2% NADES/PCW or 0.2% DMSO in RPMI medium. Values are mean of three experiments AE standard deviation. Ã p 0.05, ÃÃ p 0.01, and ÃÃÃ p 0.001. mitogen-activated protein kinase signaling pathway[END_REF], dengue virus infection [START_REF] Luplertlop | Dengue-virus-infected dendritic cells trigger vascular leakage through metalloproteinase overproduction[END_REF], Hantavirus cardiopulmonary syndrome [START_REF] Marsac | Infection of human monocyte-derived dendritic cells by ANDES Hantavirus enhances pro-inflammatory state, the secretion of active MMP-9 and indirectly enhances endothelial permeability[END_REF], and cancer [START_REF] Scherer | Imaging matrix metalloproteinases in cancer[END_REF]. There are multiple MMP-9 upstream enhancers, such as pro-inflammatory cytokines (tumor necrosis factor-a, TNF-a), growth and transcriptional factors [START_REF] Zhang | Differential regulation of monocyte matrix metalloproteinase and TIMP-1 production by TNF-alpha, granulocytemacrophage CSF, and IL-1 beta through prostaglandindependent and -independent mechanisms[END_REF]. Despite the previous mentioned positive effects of RES in human health, its low-water solubility, mode of action, targets, route and dose of administration, as well as its low bioavailability, preclude its regular use as a therapeutic approach [START_REF] Bansal | Advanced drug delivery systems of curcumin for cancer chemoprevention[END_REF][START_REF] Ponzo | Resveratrol: A supplementation for men or for mice?[END_REF][START_REF] Pujara | Enhanced colloidal stability, solubility and rapid dissolution of resveratrol by nanocomplexation with soy protein isolate[END_REF][START_REF] Visioli | The resveratrol fiasco[END_REF]. In order to overcome the poor watersolubility of RES, and to be able to observe measurable effects, some groups, using elevated RES doses (1000 to 1500 mg/person/day), did not get the positive effects observed in humans at low concentrations (150 mg/day) as reported by others [START_REF] Cho | Differential effects of low-dose resveratrol on adiposity and hepatic steatosis in diet-induced obese mice[END_REF][START_REF] Timmers | Calorie restriction-like effects of 30 days of resveratrol supplementation on energy metabolism and metabolic profile in obese humans[END_REF] but, rather observed adverse effects [START_REF] Heeboll | Effect of resveratrol on experimental non-alcoholic steatohepatitis[END_REF][START_REF] Poulsen | High-dose resveratrol supplementation in obese men: an investigator-initiated, randomized, placebocontrolled clinical trial of substrate metabolism, insulin sensitivity, and body composition[END_REF]. For more than10 years, such a phenomenon has been assigned to hormetic effect of RES [START_REF] Calabrese | Converging concepts: Adaptive response, preconditioning, and the Yerkes-Dodson Law are manifestations of hormesis[END_REF][START_REF] Calabrese | Resveratrol: An assessment of its dose response an introduction[END_REF][START_REF] Calabrese | Dose response biology: The case of resveratrol[END_REF][START_REF] Mattson | Dietary factors, hormesis and health[END_REF][START_REF] Richer | Resveratrol based oral nutritional supplement produces long-term beneficial effects on structure and visual function in human patients[END_REF][START_REF] Scapagnini | Dose response biology of resveratrol in obesity[END_REF]. Some synonymous terms of hormesis have also been used, such as biphasic dose-response effect, adaptive response, or preconditioning effects [START_REF] Calabrese | Converging concepts: Adaptive response, preconditioning, and the Yerkes-Dodson Law are manifestations of hormesis[END_REF][START_REF] Mattson | Neurohormetic phytochemicals: Low-dose toxins that induce adaptive neuronal stress responses[END_REF]. All these data advocate for the use of the lowest effective RES doses, in preclinical and/or clinical studies. Different formulations have been set up to reach the goal to solve both the RES solubility and bioavailability issues: Micellar solutions with bile acids [START_REF] Atanackovic | Solubilization of resveratrol in micellar solutions of different bile acids[END_REF], carboxymethyl chitosan (CMCS) and RES-loaded CMCS nanoparticles (RES-CMCSNPs) [START_REF] Zu | Preparation and in vitro/in vivo evaluation of resveratrol-loaded carboxymethyl chitosan nanoparticles[END_REF], N-trimethyl chitosan (TMC) grafted by palmitic acid (TMC-g-PA) [START_REF] Ramalingam | Improved oral delivery of resveratrol from N-trimethyl chitosan-g-palmitic acid surface-modified solid lipid nanoparticles[END_REF], cyclodextrin, and co-encapsulation of cyclodextrin inclusions in liposomes [START_REF] Soo | Enhancing delivery and cytotoxicity of resveratrol through a dual nanoencapsulation approach[END_REF]. Some of these nanotechnology-based formulations were able to successfully "enhance the RES clinical potential" [START_REF] Summerlin | Resveratrol nanoformulations: Challenges and opportunities[END_REF].

For the first time in this field, this study is using a new class of solvents so called Natural Deep Eutectic Solvents (NADES), recently discovered by Choi et al. [START_REF] Choi | Are natural deep eutectic solvents the missing link in understanding cellular metabolism and physiology?[END_REF], to prepare new formulations of RES. NADES consist in a mixture, in specific molar ratios, of two or more plant-based metabolites, exhibiting strong capacity of self-association through noncovalent interactions, ionic interactions, hydrogen bonding, and van der Waals forces (lipophilic-lipophilic and p-p interactions), leading to a supramolecular assembly. The molecules forming such structures stay connected to each other through a well-organized three-dimensional system, acting as a new entity with properties mostly different from those of pure isolated components (melting point, solving power,. . .). One of the most important changes is the drastic lowering of melting point: NADES act like a pure compound (eutectic) and melt at very low temperature (deep). Unlike the very similar noncovalent derivatives such as ionic liquids (IL) and deep eutectic solvents (DES), the main components of NADES are all natural: Sugars, amino acids, and organic acids. NADES are very "good solvents" due to the strong intermolecular interactions between NADES components, They are used in broad-spectrum applications, including cosmetics, media for enzymatic reactions [START_REF] Durand | Evaluation of deep eutectic solvent-water binary mixtures for lipase-catalyzed lipophilization of phenolic acids[END_REF][START_REF] Weiz | Enzymatic deglycosylation of flavonoids in deep eutectic solventsaqueous mixtures: Paving the way for sustainable flavonoid chemistry[END_REF], enhancers for extraction of bioactive natural products, and nucleic acids [START_REF] Dai | Natural deep eutectic solvents as new potential media for green technology[END_REF][START_REF] Mamajanov | DNA and RNA in anhydrous media: Duplex, triplex, and G-quadruplex secondary structures in a deep eutectic solvent[END_REF]. In addition, NADES have been shown to be good solvents for non-water-soluble small molecules (polyphenols), or even, macromolecules, (starch, cellulose, nucleic acids [START_REF] Dai | Natural deep eutectic solvents as new potential media for green technology[END_REF]).

From 18 compound combinations were used to prepare different NADES, we selected those exhibiting the highest RES solubilizing power and were used to formulate RES (RES-NADES). The best combination used as a RES formulation was compared with a RES DMSO-formulation (RES-DMSO), which is the most current solvent of medium polar substrates for in vitro studies. For the first time here, we show that RES was not only easily dissolved in a particular NADES, but also that this formulation exhibited at least a 10-fold higher aMMP-9 inhibitory effect as compared with the RES-DMSO formulation, making possible to lower RES doses and thus able to reach the hormetic effects conditions.

2 Materials and methods

Chemicals and reagents

Dimethyl sulfoxide (DMSO) and MTT (3-[4,5-Dimethylthiazol-2-yl]-2,5-diphenyltetrazolium bromide; Thiazolyl blue), as well as individual NADES components were purchased from Sigma-Aldrich TM (Marnes-la-Coquette, France), Choline chloride (ChCl) was dried under vacuum at 608C over a silica gel for 72 h before use. TNF-a was purchased from PeproTech TM (Paris, France), and all other chemicals used in this study were highly purified molecular grade reagents (Sigma, Paris, France). RES was isolated from stalks of Vitis vinifera, Vitaceae, and highly-purified (98%) according to the process described by Delaunay et al. [START_REF] Delaunay | Preparative isolation of polyphenolic compounds from Vitis vinifera by centrifugal partition chromatography[END_REF].

Preparation of natural deep eutectic solvents

A set of 18 natural deep eutectic solvents were prepared according to the procedure modified from Abbott [START_REF] Abbott | Deep eutectic solvents formed between choline chloride and carboxylic acids: Versatile alternatives to ionic liquids[END_REF]. Briefly, these NADES were prepared as follows: Components 1, 2 and 3, from (Table 1), were directly weighed in a flask, avoiding any contact with the air moisture, to get a molar ratio as described in (Table 1). Finally, all mixtures to form a potential NADES were heated and stirred with an orbital shaker at 250 rpm and incubated at 608C until a clear solution was formed. Among these solvents, three solvents were able to solubilize RES (as visually assessed), NADES composed of: 1,2-propanediol: choline-chloride:water (1:1:1) (NADES/PCW), Glucose: ChCl:Water (1:2.5:2.5) (NADES/GCW), and Malic acid: ChCl (1:1) (NADES/MA). NADES/PCW was further selected based upon both its higher RES solubilizing capacity as well as its individual low cytotoxicity.

Determination of resveratrol solubility in NADES/PCW and DMSO

To assess the RES concentration by UPLC-UV analysis (Accela-Thermo scientific), a calibration curve was realized using standard samples of RES, diluted in water to respectively get final concentrations of 16, 8, 4, 2, and 1 mg/mL. The UPLC-UV analysis of RES was performed using an Xbridge BEH C18 2.5 mm column (2.1 Â 100 mm) and an Xbridge BEH C18 2.5 mm pre-column (2.1 Â 5 mm). Solvents A (H 2 O) and B (CH 3 CN) were used according to the timetable in the following gradient: 80/20 from 0 to 3 min, 80/20 to 50/50 from 3 to 10 min, 50/50 to 80/20 from 10 to 11 min, all of them at a flow rate of 0.3 mL/min. The RES detection was performed at 310 nm. In order to determine the experimental working solubility of RES in both solvents NADES and DMSO, 20 mg of RES powder were dissolved in either 1 mL of NADES/PCW or 1 mL of DMSO and heated overnight at 408C. Subsequently, these preparations were centrifuged for 5 min at 1500 rpm and supernatants were recovered. To prepare a theoretical RES concentration of 1 mg/mL, 10 mL of the respective recovered supernatant were diluted with 190 mL of methanol, from which a theoretical 10 mg/mL solution was realized in H 2 O (10 mL in 990 mL H 2 O). All these solutions were analyzed by the UPLC-UV method (using the calibration curve) allowed us to determine the actual concentration in each sample and to compare the corresponding RES solubility in both NADES/PCW and DMSO (supplementary Fig. S1).

Cell culture

A TNF-a activated human leukemia cell line (THP-1) was used as a source of active MMP-9 production. In the aim to assess the MMP-9-inhibitory activity of RES, RES was dissolved in two kinds of solvents, NADES/PCW and DMSO. The cytotoxic activity of different compounds was assessed using the MTT assay on different cell types, including primary human umbilical vein endothelial cells (HUVEC), human leukemia cell line (THP-1), the human brain micro-vascular endothelial cell line (hcMEC/D3) was derived from primary human temporal lobe micro-vessels (immortalized with a lentiviral vector transduction with the catalytic subunit of human telomerase (hTERT) and SV40 large T antigen [START_REF] Weksler | Blood-brain barrier-specific properties of a human adult brain endothelial cell line[END_REF] as well as other cell lines including the human pancreatic adenocarcinoma BxPC3, and the human breast adenocarcinoma MDA MB-231. Both THP-1 and BxPC3 were cultured in RPMI 1640, whereas MDA MB-231 was cultured in DMEM medium, and both media were supplemented with 10% heat-inactivated FBS, penicillin G 100 U/mL and streptomycin 100 mg/mL purchased from Fisher Scientific TM (Illkirch-Graffenstaden, France). HUVEC were maintained in EndoGro TM low-serum culture media kit purchased from Merck Millipore TM (Paris, France) and hcMEC/D3 were maintained in EBM-2 mL V TM culture media kit purchased from Lonza (Basel, Switzerland).

Inhibition of the MMP-9 gelatinolytic activity

In the presence of TNF-a 10 ng/mL, THP-1 monocytes were seeded at 3 Â 10 5 cell/mL in 24-well plate and were incubated with different concentrations of RES (1, 2, 4, 8, 10, 20, and 30 mM) dissolved either in DMSO or NADES/PCW. Different THP-1 controls were prepared by incubating cells with each solvent in RPMI cell culture medium at a final concentration of 0.2% DMSO or 0.2% NADES/PCW, in the presence or in the absence of TNF-a. After 24 h of cell culture in these conditions, 1 mL of supernatants were collected and centrifuged at 1 200 rpm at 258C for 5 min before their storage at À808C up to the zymogram analysis.

Zymography

The gelatinolytic activity of MMP-9 was assessed using a zymography approach as described by Pan et al. [START_REF] Pan | Electrophoretic transfer protein zymography[END_REF]. Briefly, supernatants were collected as mentioned above and thawed to be electrophoretically separated in a 10% SDS-polyacrylamide gel electrophoresis (PAGE) containing 1% gelatin in the absence of reducing agents. After the electrophoresis, gels were washed three times in a solution of 2.5% Triton X-100, followed by incubation with a gelatinase buffer (NaCl ) on an orbital shaker at 100 rpm at 378C for 24 h. Then, gels were stained using the Coomassie Blue-staining solution (0.025% Coomassie Blue, 40% methanol, and 10% acetic acid) for 1 h, followed by treatment of distaining solution (glacial acetic acid 10% and methanol 20%), until white bands are formed surrounded by the blue background. Gel bands were photographed and analyzed using GelAnalyzer 2010a TM software.

MTT cytotoxicity assay

The MTT cytotoxicity assay relies on the reading intensity of the converted formazan blue crystals from tetrazolium dye by living cell, the followed protocol was modified of the protocol recommended by Mosmann [START_REF] Mosmann | Rapid colorimetric assay for cellular growth and survival: Application to proliferation and cytotoxicity assays[END_REF]. THP-1, BxPC3, MDA-MB 231, and hcMEC/D3 cells were seeded at 5 Â 10 3 cells/well in 96-well plate for 24 h. While HUVEC were seeded at 1 Â 10 4 cells/well density in the same conditions. Then, supernatants were discarded and plates were either treated with different concentrations of NADES/PCW, NADES/GCW, NADES/ MA or DMSO (0.5, 1, and 2%), or with serial dilutions (10, 20, 40, and 80 mM) of each RES formulation (RES in NADES/ PCW or RES in DMSO). Samples were assayed in duplicates for each concentration, and normalized against negative control of each solvent diluted in the corresponding cell culture medium ie., 0.2% NADES (PCW, GCW, MA)treated or 0.2% DMSO-treated. Subsequently, plates were incubated for additional 72 h in a CO 2 humid chamber at 378C before replacing the medium by MTT in a serum-free medium and incubated for four additional hours. To dissolve formazan crystals, SDS-HCl 10% was added to each plate and incubated at 378C for 2 h, the optical density was measured at 570 nm (against the reference, 690 nm) using a TECAN TM plate reader (Paris, France).

Statistic analyses

Experimental data from each group were analyzed using GraphPad Prism TM v.5 software. To analyze the differences between NADES/PCW and DMSO treated groups, onepaired Student t-test was carried out. All experiments were done least three times with replicates equal or higher than two. The same test was also applied for the cytotoxicity assay between different groups. Values of p 0.05 were considered statistically significant.

Results

Toxicity assay of NADES (PCW, GCW, and MA) and DMSO

Some particular NADES including PCW, GCW, and MA, preselected for their capacity to solubilize RES (by visualization), were assessed for their cytotoxic effects on either THP-1 (Fig. 2A) monocytes or primary HUVEC (Fig. 2B), using the MTT assay. The results showed that at a 2% NADES/PCW concentration exhibited high deleterious impact on respective THP-1 and HUVEC viability of 71 and 85%, respectively. DMSO decreased the viability of THP-1 and HUVEC to 60 and 80%, respectively. When NADES/ PCW or DMSO was assayed at the lower concentration of 0.5%, no toxicity was observed neither against THP-1, nor HUVEC. NADES/GCW exhibited high cell toxicity at 2% that rapidly declined in function of a decreased concentration. In contrast, NADES/MA was extremely toxic at all the tested concentrations for both THP-1 and HUVEC. Since, NADES/PCW and DMSO exhibited the best solubility (supplementary material, Fig. S1) and the lowest cytotoxicity of NADES/PCW (Fig. 2C) and DMSO (Fig. 2D) was extended to other cell types, including BxPC3, hcMec/D3 and MDA MB-231. The highest cytotoxicity effect of NADES/PCW and DMSO was observed at 2% that remained relatively moderated, since the cell observed viability varied from 60 to 81% in these cells (Table 2). In contrast, the cytotoxicity was drastically reduced at the lowest assayed concentration of 0.5%.

Resveratrol cytotoxicity

To properly assess the cytotoxicity of RES, the final used concentration of solvents, either NADES/PCW or DMSO to prepare RES was 0.2%, at least fivefold reduced concentration as compared with their cytotoxic effect at a concentration of 1% (Fig. 2). These results show that high concentrations (40 and 80 mM) of RES have a similar high cytotoxicity level for both THP-1 monocytes (Fig. 3A) and primary HUVEC (Fig. 3B). In addition, as shown in Table 3, the 50% of lethal concentration (LC 50 ) of both RES formulations for THP-1 and HUVEC were relatively high, as established using GraphPad TM from Prism v.5 software. Indeed, both LC 50 values of RES-NADES/PCW and RES-DMSO tested with THP-1 monocytes were close to 40 mM. While the LC 50 of RES-NADES/PCW and RES-DMSO tested with primary HUVEC were similar, respectively 60 and 64 mM. In contrast, for the lower concentrations NADES/PCW and DMSO allowed high cell viability as compared to the negative control cells submitted to RES vehicles.

Resveratrol formulated with NADES/PCW or DMSO and the MMP-9 activity

The MMP-9-inhibitory activity of RES dissolved in 0.2% NADES/PCW or 0.2% DMSO was assessed on a culture of TNF-a-activated THP-1 monocytes, which shed large amounts of active MMP-9 into the supernatant. Zymography results (Fig. 4A) clearly showed a MMP-9-inhibitory activity of RES. The pixel density analyses of each gel band representing the MMP-9 activity confirmed a decrease of this activity in function of the inhibitory activity of RES. These data were plotted for each concentration of both RES formulations (Fig. 4B). We must stress that a significant increase of the RES-NADES/PCW-MMP-9 inhibitory activity was observed at low RES concentrations up to 1 mM that was more than 10-fold stronger than the limited inhibitory activity of RES-DMSO. Thus, evidencing that the NADES/PCW vehicle was able to confirm and enhance the hormetic mode of action of RES as compared with its limited dose-response activity when dissolved in DMSO.

Discussion

Our findings revealed that the selected natural deep eutectic solvent 1,2-Propanediol 1,2: ChCl:water or NADES/PCW (1:1:1) was able to dissolve stalks-extracted RES native pure powder. The UPLC-UV quantification analyses have evidenced that RES was completely soluble in NADES/PCW at a RES-saturated concentration of 20 g/L at 408C, and that this solubility was similar than the one measured in the RES-DMSO formulation (supporting information file, Fig. S1). Since the NADES/PCW has exhibited both the highest visual solubilizing power and lower cytotoxicity of RES as compared with the two other selected RES solvents NADES/GCW and NADES/MA (Fig. 2), latter were discarded and only NADES/PCW was used to assess the capacity of RES to inhibit the active form of MMP-9 produced from TNF-a-activated THP-1 monocytes. The individual toxicity of NADES/PCW, DMSO, or resveratrol was assessed using the MTT assay. Thus, we observed that both NADES/PCW and DMSO, in the absence of RES were both toxic at the concentration of 2%, as shown in (Table 2). In contrast, at a final concentration of 0.5% of either NADES/PCW or DMSO vehicle alone allowed a high viability rate, close to 100%, of tested cells. In contrast, RES-NADES/PCW and RES-DMSO formulations exhibited similar toxic effects when used at 40 and 80 mM on THP-1 monocytes and HUVEC. Moreover, lethal concentrations able to kill 50% (LD 50 ) of cultivated primary HUVEC were respectively reached at 60 and 64 mM of RES-NADES/PCW and RES-DMSO respectively as compared with LD 50 on THP-1 monocytes, which both were reached around 40 mM. These latter values showed that these formulations were less toxic for primary endothelial cells, which are the main physiological targets of the active MMP-9. In addition, independently of the RES formulation, below these concentrations, no toxic effects of RES were observed. These observations are extremely encouraging for the future in vivo experiments.

Since this RES-NADES/PCW was diluted in different cell culture media to proceed to test the biological activity of RES proposed in this work, we have observed that RES dissolved in the NADES/PCW vehicle keep its soluble form for longer durations in all these cultivation media in the absence or the presence of tested cell types as compared with DMSO. Therefore, the RES-NADES/PCW formulation, similarly to RES-DMSO formulation, is compatible with in vitro experiments to evaluate the RES capacity to inhibit active MMP-9 produced by activated monocytes. Using a range of decreasing RES concentrations prepared in both formulations, we have observed that the NADES/PCW as compared with DMSO, resulted in a considerable enhancement by a factor of at least 10-fold of the RES capacity to inhibit the MMP-9 activity. Thus, when using RES a concentration as low as 1 mM with the NADES/PCW formulation, RES was able to reduce MMP-9 activity up to 52% as compared with RES formulated with DMSO, which did not reduce this enzymatic activity at all, at this low concentration. These findings contribute to a considerable enhancement of the RES potential therapeutic effects in the context of NADES formulation, since they support that RES could exhibit its optimal effects under hormetic conditions. While, a similar reduction rate of MMP-9 activity induced by RES formulated with DMSO was only observed at concentrations of RES comprised between 10 and 20 mM.

Due to the poor bioavailability of RES, Sinclair et al. have suggested that a normal human weighing 75 kg would need a daily RES dose close to 100 mg/kg/day [START_REF] Baur | Therapeutic potential of resveratrol: The in vivo evidence[END_REF], which, obviously, is not applicable. Consequently, new derivatives and/or formulations of RES remain as unmet need that must be overcome to reach both acceptable bioavailability and activity of this compound.

This comparative study for the first time the possibility of getting improved drug's responses at low concentrations, as compared with traditional DMSO assays. We hypothesized that with the NADES formulation, these interesting RESbiological effects were probably due to the fact that NADES/ PCW could improve RES bioavailability, diffusion, transport, and cell uptake for specific cell compartment localization. In addition, taking into account the considerable number of possible NADES combinations, potentially it would be likely to optimize a particular NADES solvent for RES and any other drug compound for their clinical use at the lowest possible dosage. Since the limitation of the use of DMSO in animal studies is due to its toxicity [START_REF] Galvao | Unexpected low-dose toxicity of the universal solvent DMSO[END_REF] and although very encouraging results of our work, further investigations (mainly pharmacokinetic properties) are required to set up in vivo assays of RES administration with non-covalent NADES formulations.

Conclusions

Altogether these results show, for the first time, that without altering the cell viability, the formulation of RES dissolved in the particular NADES/PCW solution significantly enhances the RES inhibitory effects on MMP-9 activity of a more than 10-fold factor as compared with its formulation in DMSO. These data evidenced that NADES/PCW is able to enhance the hormetic mode of action of RES to decrease the MMP-9 activity. In addition, NADES are low-cost, simple to prepare, and biodegradable resulting in less toxic "green" compounds. Consequently, we propose that NADES could be use as formulation method that would be considered as a suitable "adjuvant" for druggable compounds, providing major advantages, including bio-compatibility, lowered toxicity, solubility, as well as the optimization of their active concentrations for an enhanced biological activity. Therefore, these findings could help to improve the pharmacological activity of RES for preclinical and clinical studies.

Figure 1 .

 1 Figure 1. Resveratrol structure and numbering of transresveratrol.

Figure 2 .

 2 Figure 2. In vitro cytotoxic effects of different RES solvents: NADES (PCW, GCW, and MA) and DMSO. Increasing concentrations 0.5, 1, and 2% of NADES/PCW, NADES/GCW, NADES/MA, or DMSO were incubated with either THP-1 monocytes (A) or primary HUVEC (B) and then submitted to MTT cytotoxicity assay. Subsequently, in the same conditions these data were extended to other three different cell types including BxPC3, hcMec/D3, and MDA MB-231 for solvent having previously shown the lowest cytotoxic effects NADES/PCW (C) and DMSO (D). Values are mean of three experiments AE standard deviation. Ã p 0.05, and ÃÃÃ p 0.001.

Figure 3 .

 3 Figure 3. In vitro cytotoxic effects of resveratrol formulated in NADES/PCW or DMSO. NADES/PCW or DMSO concentrated at 0.2% was used to dissolve RES. In these Res solubilization conditions, increasing concentrations of 10, 20, 40, and 80 mM RES were submitted to MTT assay to assess their cytotoxic effects on either THP-1 monocytes (A) or primary HUVEC (B). Data were normalized against 0.2% of DMSO or NADES/PCW alone as negative control and 1% SDS was used as positive control in cell culture medium. Values are mean of three experiments AE standard deviation. ÃÃÃ p 0.001.

Table 1 .

 1 Components of tested natural deep eutectic solvents

			Compound	Compound	Molar
	NADES Compound 1	2	3	ratio
	1	Glycerol	ChCl	-	2:1
	2	Lactic acid	Betaine	-	2:1
	3	Lactic acid	ChCl	-	10:1
	4	Malic acid	ChCl	-	1:1
	5	Malic acid	Glycine	-	1:1
	6	Malic acid	Proline	-	1:3
	7	Malonic acid	ChCl	-	1:1
	8	Urea	ChCl	-	2:1
	9	1,2-propanediol	ChCl	Water	1:1:1
	10	Glucose	ChCl	Water	1:2.5:2.5
	11	Glycerol	ChCl	Water	2:1:2
	12	Glycerol	Proline	Sucrose	9:4:1
	13	Lactic acid	Glucose	Water	5:1:1
	14	Lactic acid	Proline	Water	1:1:0.25
	15	Malic acid	Proline	Water	1:1:3
	16	Sucrose	ChCl	water	1:1:1
	17	Sucrose	ChCl	Water	1:4:4
	18	Urea	ChCl	Water	2:1:2

Table 2 .

 2 Cell viability assay of either NADES/PCW or DMSO in the absence of RES

	Cells viability (%)

Table 3 .

 3 LC50 effect of NADES/PCW-or DMSO-formulated RES on HUVEC and THP-1

		LC 50 (mM)
	Cells	NADES/PCW	DMSO
	HUVEC	60	65
	THP-1	40	40

Eur. J. Lipid Sci. Technol. 2017, 119, 1700171 ß 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ejlst.com

ß 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ejlst.com

Eur. J. Lipid Sci.Technol. 2017, 119, 1700171

This work has been funded by participation of the IRD, the CNRS, the CIRAD, France. AS has been a PhD fellow at IRD supported by Misr University for Science and Technology (MUST), Egypt and IRD.

The authors have declared no conflict of interest.