Pier Luigi Silvestrelli

Evelyne Martin

Mauro Boero

Assil Bouzid

Guido Ori

Carlo Massobrio
email: carlo.massobrio@ipcms.unistra.fr

The atomic structure of glassy GeTe 4 as a playground to assess the performances of density functional schemes accounting for dispersion forces

The atomic structure of glassy GeTe 4 is obtained in the framework of first-principles molecular dynamics by considering five different approaches for the description of the electronic structure within density functional theory. Among these schemes, one is not corrected by the account of the dispersion forces and it is based on the BLYP exchange correlation (XC) functional, while all the others are intended to account for the dispersion forces according to different theoretical strategies. In particular, by maintaining the BLYP expression for the 1 XC functional, two of them (BLYP-D2 and BLYP-D3) exploit the Grimme expressions for the dispersion forces, while the fourth scheme is based on the Maximally Localized Wannier functions (MLWF). Finally, we also considered the rVV10 functional constructed to include seamlessly the dispersion part. Our results point out the better performances of the BLYP-D3 and MLWF in terms of comparison with experimental data for the total pair correlation functions, BLYP-D2 and rVV10 being closer to the uncorrected BLYP data. The implications of such findings are discussed by considering the overall limited impact of dispersion forces on the atomic structure of glassy GeTe 4 .

Introduction

Two main reasons are at origin of the widespread interest in disordered network forming materials containing Ge-Te bonded structural units. The first is deeply rooted into the role they play as representative systems in phase change materials (PCMs) or in devices prone to applications for optics and far infrared transmission [START_REF] Huang | Deposition and characterization of germanium sulphide glass planar waveguides[END_REF][START_REF] Kato | GeS 2 /metal thin film bilayered structures as write-once-type optical recording materials[END_REF] . These latter exhibit enhanced glass stability as an essential prerequisite while PCMs are typically poor glass formers due to their inherent structural instability [START_REF] Wuttig | Phase-Change Materials for Rewriteable Data Storage[END_REF] .

The second reason is more fundamental, since chalcogenide disordered networks are an excellent playground for the assessment of bonding properties escaping any clear-cut classification in terms of standard ionic or covalent bonding. This limitation has been somewhat overcome by invoking the notion of iono-covalent bonding to describe localized distribution of charges around the atomic sites coexisting with highly directional bonding [START_REF] Bouzid | Atomic-scale structure of the glassy Ge 2 Sb 2 Te 5 phase change material: A quantitative assessment via first-principles molecular dynamics[END_REF][START_REF] Bouzid | First-Principles Study of Amorphous Ga 4 Sb 6 Te 3 Phase-Change Alloys[END_REF][START_REF] Bouzid | Role of the van Der Waals Interactions and Impact of the Exchange-Correlation Functional in Determining the Structure of Glassy GeTe 4[END_REF] . However, the search of a quantitative theoretical description remains challenging. The case of Ge-Te glasses is even more elusive. These systems are not only highly sensitive, within density functional theory (DFT), to the choice of the exchange-correlation (XC) functional but also, for specific concentrations, to the neglect or the account of dispersion forces. In a paper devoted to glassy GeTe 4 , we pointed out that its structure depends on the combined choice of the XC functional and of dispersion (van der Waals, vdW) corrections [START_REF] Bouzid | Role of the van Der Waals Interactions and Impact of the Exchange-Correlation Functional in Determining the Structure of Glassy GeTe 4[END_REF] . This work was carried out by relying on the first-principles molecular dynamics (FPMD) plane waves DFT scheme. In particular, it was demonstrated that the impact of vdW forces is not as crucial as the optimal choice of the XC functional (BLYP [START_REF] Becke | Density-functional exchange-energy approximation with correct asymptotic behavior[END_REF][START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF] turning out to be the preferred one) but it cannot be neglected a priori as it occurs in the original Kohn-Sham DFT expression.

Such conclusion was reached via the application of the vdW scheme due to Grimme (vdW G ), in the version named D2 [START_REF] Grimme | Semiempirical GGA-type density functional constructed with a long-range dispersion correction[END_REF] . A further set of calculations on glassy GeTe 4 was motivated by two considerations [START_REF] Massobrio | Sensitivity to Dispersion Forces in First-Principles Modeling of Disordered Chalcogenides[END_REF] . First, it appeared that a single dispersion scheme cannot be taken as a universal recipe equally effective for any system, this calling for the account of at least one alternative choice to compare with. Second, there is some concern on the neglect, in D2-Grimme, of updates in the dispersion coefficients resulting from the evolution in time of the electronic structure, closer in spirit to the combined dynamical evolution of ionic and electronic degrees of freedom occurring in first-principles molecular dynamics (FPMD) [START_REF] Car | Unified approach for molecular dynamics and density-functional theory[END_REF] . Accordingly, in Ref. 10, we focused on the comparison between distinct sets of BLYP data, namely the two already available (not including dispersion forces, named NovdW and adopting the D2-Grimme scheme, named vdW G ) and a third one, making use of the maximally localized Wannier functions (MLWF) framework (vdW W hereafter), known for exploiting the evolution of the electronic structure when updating the dispersion coefficients [START_REF] Wannier | The structure of electronic excitation levels in insulating crystals[END_REF][START_REF] Marzari | Maximally-localized generalized Wannier functions for composite energy bands[END_REF][START_REF] Silvestrelli | Maximally-localized Wannier functions for disordered systems: Application to amorphous silicon[END_REF][START_REF] Ikeda | Role of van der Waals corrections in first principles simulations of alkali metal ions in aqueous solutions[END_REF] . Taken altogether, the results obtained did not provide unambiguous indications.

In fact, the vdW W strategy produced pair correlation functions agreeing well with the NovdW set of data but differing from those obtained via vdW G . In turn, vdW G data were in better agreement with the experimental counterpart. These pieces of evidence call for a more complete analysis, extended to other vdW schemes, in order to achieve the following goals: a) a more robust conclusion on the impact of the dispersion forces, b) the assessment of the actual improvements induced by dispersion coefficients adapting to changes in the electronic structure with respect to more empirical ones and, c) some hints on the overall capability of the correction schemes not to modify artificially the structural data for bonding situations not crucially demanding their explicit account. This latter point is likely to be quite subtle since one would like to estimate correctly dispersion forces regardless of their being a priori of limited impact. For instance, in the case of glassy GeTe 4 , plausible physico-chemical arguments (i.e. the presence of predominant iono-covalent interactions) indicate that they should not affect substantially structural properties.

In this paper atomic-scale modelling of glassy GeTe 4 is carried out within DFT-FPMD by performing extensive calculations with five different approaches for the description of the dispersion forces. This new insight considerably improves upon the preliminary report of Ref. 10. We have accounted not only for two additional vdW schemes but we have also produced new time trajectories to reach a total number of 11 independent ones. As a result, we have gathered more compelling information on the changes induced to the network topology by a variety of different treatments of the long range van des Waals interactions. This paper is organized as follows. The introduction is followed by section 2 describing our methods of calculation, comprising the theoretical framework and the technical details of the FPMD simulations as well as the different schemes employed to account for the dispersion forces. Section 3 contains in two different subsections results for the total and the partial pair correlation functions. Conclusive remarks are collected in section 4.

Theoretical Methods

In order to fully appreciate the insight provided by the present investigation and the underlying methodology, we recall the computational framework employed in Ref. 10. Glassy GeTe 4 was made of 215 atoms (43 Ge, 172 Te) in a supercell of cubic shape with edge 19.24 Å. Such system was studied within the Car-Parrinello 11 molecular dynamics (CPMD) scheme as implemented in the CPMD code [START_REF] Hutter | Computer code CPMD[END_REF] , by adopting the exchange functional by Becke [START_REF] Becke | Density-functional exchange-energy approximation with correct asymptotic behavior[END_REF] and the correlation functional by Lee, Yang and Parr 8 (BLYP). An energy cutoff of 40 Ry was employed to account for valence electrons within a plane wave basis set, by restricting the Brillouin zone integration to the Γ point [START_REF] Bouzid | Role of the van Der Waals Interactions and Impact of the Exchange-Correlation Functional in Determining the Structure of Glassy GeTe 4[END_REF][START_REF] Bouzid | Origin of structural analogies and differences between the atomic structures of GeSe 4 and GeS 4 glasses: A first principles study[END_REF][START_REF] Lampin | Impact of dispersion forces on the atomic structure of a prototypical network-forming disordered system: The case of liquid GeSe 2[END_REF] . Other relevant features were the control of the temperatures of the ions and of the fictitious electronic degrees of freedom via Nosé-Hoover thermostats [START_REF] Nosé | A unified formulation of the constant temperature molecular dynamics methods[END_REF][START_REF] Hoover | Canonical dynamics: equilibrium phase-space distributions[END_REF][START_REF] Blöchl | Adiabaticity in first-principles molecular dynamics[END_REF] and the choice of an integration step of 5 a.u. (0.121 fs). The Nosé-Hoover chain consisted of two thermostats, characterized by an oscillation frequency of 6 and 60 THz, respectively.

In regard to the choice of the BLYP exchange-correlation functional, this is rooted into the better performances exhibited by this recipe for disordered Ge-based chalcogenides [START_REF] Micoulaut | Improved modeling of liquid GeSe 2 : Impact of the exchange-correlation functional[END_REF][START_REF] Giacomazzi | Vibrational properties of vitreous GeSe2with the Becke-Lee-Yang-Parr density functional[END_REF] . In particular, one obtains an improved description of the short range Ge environment corresponding to an enhanced localized behavior of the electron density. Such effect is predominant over electronic delocalization that favors the metallic character, a drawback inherent in generalized gradient approximation approaches based on the uniform electron gas as reference system (as the one due to the Perdew and Wang [START_REF] Perdew | Accurate and simple analytic representation of the electron-gas correlation energy[END_REF] .

Dispersion forces were included by using either the DFT-D2 formula devised by Grimme [START_REF] Grimme | Semiempirical GGA-type density functional constructed with a long-range dispersion correction[END_REF] or by resorting to a recipe (refered to hereafter as vdW W ) based on the concept of the maximally localized Wannier functions (MLWFs) [START_REF] Wannier | The structure of electronic excitation levels in insulating crystals[END_REF][START_REF] Marzari | Maximally-localized generalized Wannier functions for composite energy bands[END_REF][START_REF] Silvestrelli | Maximally-localized Wannier functions for disordered systems: Application to amorphous silicon[END_REF][START_REF] Ikeda | Role of van der Waals corrections in first principles simulations of alkali metal ions in aqueous solutions[END_REF] . Briefly, the vdW W scheme can be described by relying on the following relationships, the first containing the C 6,nl coefficients 25

E vdW = -∑ n<l f (r nl ) C 6,nl r 6 nl , (1)
(in which the subscripts n and l run over all atoms or fragments of the system and f (r nl ) is a damping function avoiding unphysical divergence and double counting of correlations at short range) and the second expressing them in terms of the MLWFs w n (r) as follows

C 6,nl = 3 32π 3/2 r≤r cut r ≤r cut w n (r) • w l (r ) w n (r) + w l (r ) d 3 r d 3 r. (2)
The MLWFs approach allows for an intuitive interpretation of the bonding properties of condensedmatter systems and are at the heart of the modern theory of polarization [START_REF] Resta | Macroscopic Electric Polarization as a Geometric Quantum Phase[END_REF][START_REF] Vanderbilt | Electric polarization as a bulk quantity and its relation to surface charge[END_REF] . In principle, this schema is very well suited to provide additional information on the electronic structure characterizing the different realizations of glassy GeTe 4 . However, such an extension is outside the scope of the present study focused on the correlation between structural properties and different vdW correction to the Kohn-Sham DFT energy functional. Van der Waals-corrected DFT approaches based on MLWFs proved to be successful in describing many systems: small molecules, water clusters, graphite and graphene, water layers interacting with graphite, interfacial water on semiconducting substrates, hydrogenated carbon nanotubes, molecular solids, the interaction of rare gases and small molecules with metal surfaces,... (see, for instance, Ref. 28 and references therein). The ML-WFs approach performs well since it combines the simplicity of the semiempirical formalism with the accuracy of the first principles approaches. In fact, being based on localized electronic orbitals, they are well suited to describe small electronic effects following configuration changes, thereby largely increasing the transferability of the approach (for instance, charge polarization effects are naturally included).

In the present study, the set of methods to account for dispersion forces is completed by considering two additional theoretical schemes. We also resorted to a computational package other than CPMD. More explicitly, FPMD calculations have been performed in the framework of the Quantum Espresso package [START_REF] Giannozzi | QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials[END_REF][START_REF] Giannozzi | Advanced capabilities for materials modelling with QUANTUM ESPRESSO[END_REF][START_REF] Giannozzi | Quantum ESPRESSO toward the exascale[END_REF] by selecting either the Car Parrinello 11 or the Born-Oppenheimer (BO)

scheme. This latter amounts to a search of the electronic ground state for each ionic configuration and is characterized by an integration step of 50 a.u. (1.21 fs).

We have increased the statistical sampling of our analysis by taking advantage of new trajectories produced for the same system at room temperature, thereby allowing for averages to be taken by considering previous and new results. As in Ref. 10, trajectories were produced for the BLYP (NovdW), BLYP-D2 (vdW D2 herafter) and MLWF (vdW W ) cases. In addition, we also considered the Grimme D3 expression (vdW D3 herafter) expected to ensure improved accuracy and range of applicability with respect to D2 [START_REF] Grimme | A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu[END_REF] . In view of its character of exchange-correlation functional constructed to include seamlessly the dispersion part, our study has been also extended to FPMD trajectories produced by using the rVV10 functional (vdW VV ) [START_REF] Stohr | Theory and practice of modeling van der Waals interactions in electronic-structure calculations[END_REF][START_REF] Sabatini | Nonlocal van der Waals density functional made simple and efficient[END_REF] . In fact, unlike the Grimme and MLWF methods, rVV10 (as its pristine counterpart VV10 [START_REF] Vydrov | Nonlocal van der Waals density functional: The simpler the better[END_REF] ) does not feature any separation between the DFT Kohn-Sham functional and the dispersion part. This makes its use particularly appealing in the context of a thorough analysis intended to assess the impact of vdW forces on a prototypical disordered chalcogenide. Overall, the theoretical tools employed in Ref. A summary of the different combinations of DFT schemes together with the way FPMD has been implemented (CP or BO) is given in Table 1.

(Grimme-D2) vdW D2 (Grimme-D2) vdW D2 (Grimme-D2) vdW W (BLYP-MLWF) vdW VV (rVV10) vdW D3 (Grimme-D3) vdW W (BLYP-MLWF) vdW VV (rVV10)
The first column refers to Ref. 10. First of all, the same heating and cooling process are used to produce a second MLWF(vdW W ) simulation run. Concerning the production of the other trajectories, one configuration at T = 300 K among those exploited in Ref. 10 is taken as the initial one to initiate two kinds of thermal cycles. In the first (CPMD method) heating has occurred up to a temperature T = 900 K over 15 ps to allow for significant diffusion, followed by a rapid quench (in 4 ps) to T = 300 K with statistical averages calculated over 6 ps at this same temperature. In the second (BO method) one of the previous sets of coordinates at T = 900 K is selected to start a cooling process lasting 50 ps down to T = 300 K, followed by data collection over a further trajectory of 20 ps at the same temperature. BLYP-D3 results are available within the FPMD-BO calculations only.

The availability of a distinct number of equilibrium trajectories (three, two or a single one)

among the different choices for the dispersion forces does not affect the essence of our results.

Indeed, for a specific case, deviations from the mean found when several trajectories are available are smaller than the differences occurring among the different schemes.

3 Results and Discussion

Total pair correlation functions.

In what follows, we concentrate on the total pair correlation function g T (r) obtained as a weighted linear combination of the three partial pair correlation functions g αβ (r), where α and β denote the chemical species considered (Ge, Te). The focus will be on a comparative analysis of the pair correlation functions obtained in the absence (NovdW, BLYP data) and in the presence of dispersion forces, these latter expressed in four different forms (vdW D2 , BLYP-D2, vdW D3 , BLYP-D3, vdW W , BLYP-MLWF and vdW VV , rVV10).

As a prerequisite, it is instructive to define under which conditions the performances of the different vdW schemes will be examined. Having in mind the impact of the dispersion forces on the structural properties, we choose not to account for short range contributions corresponding to distances up to the first minimum of the pair correlation function, say smaller than 3 Å. For this reason, there is little that can be added to the fact that the depth of the first minimum observed experimentally is not entirely reproduced by any of schemes we implemented (see Fig. 1). Overall, quantitative comparisons and conclusions drawn thereof will not depend on the degree of variability found in the height and shape of the first peaks in the pair correlation functions, clearly not directly sensitive to the action of the dispersion forces. Variations of this specific feature can be mostly ascribed to statistical fluctuations affecting, in the first place, Ge-Ge correlations, the number of Ge atoms being limited to 43 in our simulations. This is why our set of comparisons are based most exclusively on the behaviors recorded for distances larger than 3 Å.

For g T (r) we provide a direct comparison among all cases and also the function ∆(exp,tot) that results from the difference between the various g T (r) and the experimental data. Analogously, ∆(NovdW, tot) is the difference between all calculated g T (r) and the NovdW counterpart. Finally, the ∆ functions are exploited to calculate the integrals of their absolute values on the range [3 -9] Å such quantities being inversely proportional to the agreement between two given curves. The case of g T (r) (Fig. 1) is indicative of small differences between the five different schemes we considered.

One notes that vdW D3 improves considerably upon vdW D2 , both of them performing better than the NovdW scheme. Interestingly, vdW W is also closer to experiments than NovdW while vdW VV is the farthest from the experimental data. A quantitative summary of this analysis is presented in Table 2 with further evidence given in Fig. 2

(∆(exp,tot) function).

Table 2: Integral of the absolute value of the differences between a given set of data of g T (r) and the experimental counterpart on the range [3 -9] Å. This analysis is based on the assumption that the experimental curve is the one to be employed when assessing the reliability of the different strategies to account for vdW contributions. Globally, it underscores the legitimacy of schemes correcting Kohn-Sham recipes via the use of dispersion formulas (vdW D3 , vdW W ) either highly refined (the first) as a posteriori addition to the total energy or (the second) adapting to the dynamical evolution of the electronic structure.

An additional method to analyze our results consists in taking as reference data those obtained within the BLYP scheme with no account of dispersion forces. The situation is summarized in Table 3 again by taking advantage of the integrals of the absolute value of ∆(NovdW, tot), this function being visualized in Fig. 3.

Results obtained via vdW VV , rVV10 are by far the closest to NovdW, followed by vdW D2 .

However, vdW W behaves in a way not too dissimilar. This proves that this scheme is capable of ensuring a twofold effect. On the one hand, it improves upon BLYP approaching the experimental results. On the other hand, this occurs without departing drastically from the same BLYP data in a way that could be interpreted as an artificial effect overestimating the actual correction due to dispersion forces. This is exactly the pitfall invoked in our previous study [START_REF] Massobrio | Sensitivity to Dispersion Forces in First-Principles Modeling of Disordered Chalcogenides[END_REF] when comparing the performance of schemes depending or not depending explicitly on changes of the dispersion coefficients during the time evolution.

Therefore, it appears that the schemes employed can be grossly classified in two categories, i.e.

those reducing the gap between the experimental data and the NovdW solution(vdW D3 , vdW W ) and those closer to NovdW than to the experimental data (vdW VV and to a smaller extent vdW D2 ).

However, vdW W stands as the best compromise between these two tendencies, under the assump-tion that for this specific system one expects both corrections with respect to the mere BLYP NovdW data and a better agreement with experiments.

Partial pair correlation function

In the absence of experimental data for the partial pair correlation functions it is instructive to employ them to complete the analysis relative to the performances of the different schemes when compared to BLYP NovdW (Fig. 4, 6 and 5). In the case of g GeTe (r) the five different profiles shown in Fig. 4 are quite similar, with vdW D3 mostly departing from NovdW for r > 3 Å. This is confirmed by the data on the integrals of the absolute value of ∆(NovdW, GeTe)(Table 4). The pair correlation function g TeTe (r) (Fig. 5) is at very origin of the close behavior of NovdW and vdW VV mentioned when considering g T (r), keeping in mind that this partial pair correlation function bears the largest contribution in the weighted sum of partials leading to g T (r). For vdW VV the departure from NovdW is very limited 4. The case of g GeGe (r) is the one featuring the largest discrepancies between BLYP NovdW and the other schemes (see Fig. 6) and yet is the one with the smallest impact on g T (r). Regardless of the behavior in the region of the first peak, one notices that the height of the second peak is overestimated (vdW VV , vdW D2 ) or underestimated (vdW W ,vdW D3 ), while for r > 4 Å once again vdW VV and vdW D2 are those closer to NovdW. We conclude that the decomposition in the three partial structure factors helps confirming that there are sizeable differences among the four different vdW schemes employed in this work to include the dispersion forces. Of particular interest is the close behavior of NovdW and vdW VV data for Te-Te correlations, keeping in mind that vdW VV is the only scheme employed including the dispersion contribution directly in the energy functional.

Conclusion

Assessing the performances of theoretical recipes for the dispersion forces in a specific case is a valuable objective in the search of quantitative atomic-scale descriptions. This has been stressed by recent contribution in the area of DFT methodologies applied to the treatment of long range van der Waals interactions [START_REF] Raty | Aging mechanisms in amorphous phase-change materials[END_REF][START_REF] Lee | Higher-accuracy van der Waals density functional[END_REF][START_REF] Tkatchenko | Accurate molecular van der Waals interactions from groundstate electron density and free-atom reference data[END_REF] .

Our selection of glassy GeTe 4 proved worthwhile since this system is highly sensitive to the kind of vdW schemes employed although none of them brings crucial improvements upon the original DFT-BLYP approach devoid of dispersion contributions. The present work is the third of a series devoted to the structure of glassy GeTe 4 and to the relationship between the structural features, expressed via the pair correlation functions, and the account of dispersion forces. In the first paper, we came to the conclusion that the atomic structure does depend on the choice of the exchangecorrelation functional and, to a minor extent, to the account of dispersion forces [START_REF] Bouzid | Role of the van Der Waals Interactions and Impact of the Exchange-Correlation Functional in Determining the Structure of Glassy GeTe 4[END_REF] . In the second, we considered a scheme accounting explicitly for changes in time of the van der Waals coefficient (Maximally Localized Wannier Functions method) [START_REF] Massobrio | Sensitivity to Dispersion Forces in First-Principles Modeling of Disordered Chalcogenides[END_REF] . Here we aimed at a more complete analysis of the effect of dispersion forces by relying on four different methods and on large set of equilibrium FPMD trajectories. Overall, we were able to demonstrate that the best agreement with experiments in terms of total pair correlation function is obtained by using the D3 or the MLWF schemes. This is the most straightforward conclusion one can come up with in terms of an objective analysis of our data. The good performance of the D3 scheme is related to its less empirical character. In fact, in the D3 method the specific molecular environment is considered by using atom-pair specific C 6 coefficients which include local information in the form of geometrymotivated, fractional coordinated numbers. For this reason, D3 is certainly closer in spirit to the MLWF method than its precursor D2.

In view of the expected non negligible and yet limited impact of the dispersion forces on our specific system, we have also paid attention to the performances of the different recipes in terms of comparison with the bare BLYP results. In this case, the best result is given by the rVV10 scheme, i.e. a functional including dispersion forces via a non-local construction in which there is no analytical separation from the energy functional. Consideration of both criteria (better agreement with experiments and moderate departure from the original BLYP data with no dispersion) led us to conclude that the MLWF scheme is the best compromise for situations in which expected changes due to dispersion forces are deemed to be of limited extent. These considerations are confirmed by the analysis of the partial pair correlation functions. In summary, this work clearly points out that the assessment of schemes for the account of the dispersion forces is not necessarily univocal, although the search of a better comparison with experiments might always appear as a primary legitimate objective. Our extended analysis and the use of different approaches contribute to a better understanding of a very subtle and controversial issue fostering a longstanding debate on the role and the impact of dispersion forces. The results obtained will prove especially important to better control bonding situations where the weight of the dispersion forces is smaller in comparison to other dominant bonding contributions.

Acknowledgment

Calculations were performed by using resources from GENCI (Grand Equipement National de Calcul Intensif) under Grants No. DARI-A0080906092, DARI-0910296 and DARI-095071, and

Pôle HPC Equipex Equip@Meso project (Programme Investissements d'Avenir) and the CPER Alsacalcul/Big Data of the University of Strasbourg, and clustphy2 at the IEMN.

  10 and in the present work can be grouped into three categories: those accounting for dispersion forces via coefficients reflecting (MLWF) (a) or not reflecting (BLYP-D2 and BLYP-D3) (b) changes in the electronic structure during the motion and (c) the rVV10 method built as a functional including dispersion forces via a non-local construction with no analytical separation from the energy functional.

Figure 1 :

 1 Figure 1: g T (r). Comparing the five different schemes employed to account for dispersion forces with experiments (from Ref. 36) in the case of the total pair correlation function.

FunctionalFigure 2 :

 2 Figure 2: g T (r). Difference functions obtained by subtracting the experimental points from each one of the calculated total pair correlation function. Experiments from Ref. 36.

Figure 3 :

 3 Figure 3: g T (r). Difference functions obtained by subtracting the BLYP (NovdW) points from each one of the other available (calculated) total pair correlation functions

Figure 4 :

 4 Figure 4: GeTe partial pair correlation functions.

Figure 5 :

 5 Figure 5: TeTe partial pair correlation functions.

Figure 6 :

 6 Figure 6: GeGe partial pair correlation functions.

Table 1 :

 1 DFT schemes and FPMD methodologies employed in this work and in Ref. 10. CP stands for Car-Parrinello and BO for Born-Oppenheimer first principles molecular dynamics

	CP 10	CP	BO
	NovdW (BLYP)	NovdW(BLYP)	NovdW(BLYP)
	vdW D2

Table 4 :

 4 Integral of the absolute value of the differences between a given set of data of partial pair correlation functions and the BLYP (NovdW) data on the range [3 -9] Å. TeTe) taking the smallest value among the twelve available values in Table

	Functional	I |∆(NovdW,GeTe)| (a. u.) I |∆(NovdW,TeTe)| (a. u.) I |∆(NovdW,GeGe)| (a. u.)
	BLYP-D2 (vdW D2 )	0.474	0.617	1.784
	BLYP-D3 (vdW D3 )	0.914	1.107	3.599
	BLYP-MLWF (vdW W )	0.673	0.993	2.210
	rVV10 (vdW VV )	0.429	0.227	1.751
	in the range considered, with the integral of the absolute value of the corresponding difference
	function ∆(NovdW,

Supplementary material

The data that support the findings of this study are available from the corresponding author upon request.


Yingjie Liao

Kangping Liu

Wen-Chang Chen

Redouane Borsali
email: redouane.borsali@univ-grenoble-alpes.fr

Robust Sub-10 nm Pattern of Standing Sugar Cylinders via Rapid 'Microwave Cooking'

Keywords: carbohydrate, microwave heating, self-assembly, nanostructured thin films, grazingincidence small angle X-ray scattering

Block copolymer self-assembly is a widely used technique for obtaining many interesting nanostructures. Development of efficient and rapid processes for driving block copolymer selfassembly has remained a challenge. Microwave heating has attracted much attention in bioenergy production and food industry due to the advantages associated with dielectric heating effects. We report here a simple method (like microwave cooking) of microwave-heatinginduced microphase separation of a carbohydrate-based block copolymer, maltoheptaose-block-polystyrene (MH-b-PS). Highly ordered and perpendicularly orientated MH cylinders 7 nm in diameter were formed in the PS matrix on a bare silicon wafer after annealed at 333 K for one second. The whole process is performed at a relatively low temperature under one atmosphere, and no solvent is needed during the process. The mechanism of this method is investigated by using grazing-incidence small angle X-ray scatting technique to make a detailed comparison with conventional thermal annealing process. The MH-b-PS cannot achieve self-assembly by conventional thermal annealing. In contrast, microwave energy can transfer directly to the polar MH blocks in the MH-b-PS thin films, which helps to make MH-b-PS self-assemble quickly.

This study circumvents a major barrier to using carbohydrate-based block copolymer materials.

Introduction

Block copolymer (BCP) materials have attracted a lot of research interest since they can selfassemble into many useful nanostructures like hexagonal cylinders, cubic spheres, alternating lamellae, and bicontinuous gyroids. [START_REF] Albert | Self-assembly of block copolymer thin films[END_REF][START_REF] Matsen | Effect of Architecture on the Phase Behavior of AB-Type Block Copolymer Melts[END_REF] These nanostructures have exhibited a wide variety of applications in nanolithography, [START_REF] Mansky | Monolayer films of diblock copolymer microdomains for nanolithographic applications[END_REF][START_REF] Cushen | Oligosaccharide/Silicon-Containing Block Copolymers with 5 nm Features for Lithographic Applications[END_REF] electronic and energy devices, [START_REF] Hung | Conception of stretchable resistive memory devices based on nanostructure-controlled carbohydrate-block-polyisoprene block copolymers[END_REF][START_REF] Yoo | Performance enhancement of electronic and energy devices via block copolymer self-assembly[END_REF][START_REF] Chiu | High-performance nonvolatile transistor memories of pentacence using the green electrets of sugar-based block copolymers and their supramolecules[END_REF][START_REF] Darling | Block copolymers for photovoltaics[END_REF][START_REF] Wang | Top-down meets bottom-up: organized donor-acceptor heterojunctions for organic solar cells[END_REF] nanoscale templating and patterning, [START_REF] Fasolka | Block copolymer thin films: Physics and applications[END_REF][START_REF] Aissou | Thin films organized in nanodomains on the basis of copolymers having polysaccharide blocks for applications in nanotechnology[END_REF] which have led to important developments in the semiconductor and many other industries. To fully realize the enormous potential of BCP materials in these applications, however, many problems need to be addressed.

A main challenge is that the size of self-assembled BCP features is required to be extremely small in many applications. [START_REF] Fasolka | Block copolymer thin films: Physics and applications[END_REF][START_REF] Sinturel | High chi-Low N Block Polymers: How Far Can We Go?[END_REF][START_REF] Kim | Functional nanomaterials based on block copolymer self-assembly[END_REF][START_REF] Tseng | Block Copolymer Nanostructures for Technology[END_REF] Developing BCPs with high Flory-Huggins interaction parameters (χ) and low degrees of polymerization (N) is a strategy used by many people for obtaining a small feature size in BCP self-assembly. [START_REF] Sinturel | High chi-Low N Block Polymers: How Far Can We Go?[END_REF] Carbohydrates have high incompatibility with synthetic polymers, so a BCP will have a high χ parameter if it is comprised of carbohydrates and synthetic polymers. Many carbohydrates-based BCPs have been synthesized successfully via click chemistry in our group [START_REF] Cushen | Oligosaccharide/Silicon-Containing Block Copolymers with 5 nm Features for Lithographic Applications[END_REF][START_REF] Aissou | Thin films organized in nanodomains on the basis of copolymers having polysaccharide blocks for applications in nanotechnology[END_REF][START_REF] Aissou | Nano-Organization of Amylose-b-Polystyrene Block Copolymer Films Doped with Bipyridine[END_REF][START_REF] Otsuka | Control of 10 nm scale cylinder orientation in self-organized sugar-based block copolymer thin films[END_REF] . By decreasing the degree of polymerization, the feature diameters of carbohydrate-based BCPs can go down to sub-10 nm. [START_REF] Cushen | Oligosaccharide/Silicon-Containing Block Copolymers with 5 nm Features for Lithographic Applications[END_REF][START_REF] Otsuka | Control of 10 nm scale cylinder orientation in self-organized sugar-based block copolymer thin films[END_REF] Maltoheptaose-block-polystyrene (MH-b-PS) is one of the carbohydrates-based BCPs which are capable of forming sub-10 nm scale cylindrical morphologies after solvent vapor annealing (SVA). [START_REF] Otsuka | Control of 10 nm scale cylinder orientation in self-organized sugar-based block copolymer thin films[END_REF] A chemical structure of the MH-b-PS is shown in Figure 1. MH-b-PS can be synthesized through copper-catalyzed azide-alkyne cycloaddition of propargyl-functionalized maltoheptaose with azido end-functionalized polystyrene in dimethylformamide [START_REF] Aissou | Thin films organized in nanodomains on the basis of copolymers having polysaccharide blocks for applications in nanotechnology[END_REF][START_REF] Aissou | Nano-Organization of Amylose-b-Polystyrene Block Copolymer Films Doped with Bipyridine[END_REF][START_REF] Otsuka | Control of 10 nm scale cylinder orientation in self-organized sugar-based block copolymer thin films[END_REF] . It has been demonstrated that MH-b-PS can be applied to non-volatile transistor devices, [START_REF] Chiu | High-performance nonvolatile transistor memories of pentacence using the green electrets of sugar-based block copolymers and their supramolecules[END_REF] directed selfassembly and nanolithography. [START_REF] Otsuka | Carbohydrate-based block copolymer systems: directed self-assembly for nanolithography applications[END_REF] Although these carbohydrates-based BCP materials are promising for many applications [START_REF] Chiu | High-performance nonvolatile transistor memories of pentacence using the green electrets of sugar-based block copolymers and their supramolecules[END_REF][START_REF] Otsuka | Control of 10 nm scale cylinder orientation in self-organized sugar-based block copolymer thin films[END_REF] , it is difficult for them, e.g. MH-b-PS, to self-assemble with thermal annealing (TA) which is the method preferred by industry. And their SVA process takes a long time (about 24 h for MH-b-PS) [START_REF] Otsuka | Control of 10 nm scale cylinder orientation in self-organized sugar-based block copolymer thin films[END_REF] . Developing versatile annealing methods for achieving a fast self-assembly process with long-range ordering currently is an outstanding challenge for many BCP materials. [START_REF] Majewski | Rapid ordering of block copolymer thin films[END_REF] As a consequence there is much research in this area. Many rapid ordering techniques, e.g. laser zone annealing, [START_REF] Majewski | Millisecond Ordering of Block Copolymer Films via Photothermal Gradients[END_REF] laser-spike, [START_REF] Jiang | Ultrafast Self-Assembly of Sub-10 nm Block Copolymer Nanostructures by Solvent-Free High-Temperature Laser Annealing[END_REF] rapid thermal treatment, [START_REF] Perego | Ordering dynamics in symmetric PS-b-PMMA diblock copolymer thin films during rapid thermal processing[END_REF] microwave annealing, [START_REF] Zhang | Fast assembly of ordered block copolymer nanostructures through microwave annealing[END_REF][START_REF] Borah | Swift nanopattern formation of PS-b-PMMA and PS-b-PDMS block copolymer films using a microwave assisted technique[END_REF][START_REF] Jin | Deconvoluting the mechanism of microwave annealing of block copolymer thin films[END_REF][START_REF] Liao | Carbohydrate-Based block copolymer thin films: ultrafast nano-organization with 7 nm resolution using microwave energy[END_REF][START_REF] Mokarian-Tabari | Study of the Kinetics and Mechanism of Rapid Self-Assembly in Block Copolymer Thin Films during Solvo-Microwave Annealing[END_REF][START_REF] Toolan | Selective molecular annealing: in situ small angle X-ray scattering study of microwave-assisted annealing of block copolymers[END_REF][START_REF] Qiang | Evolution in surface morphology during rapid microwave annealing of PS-b-PMMA thin films[END_REF] [START_REF] Zhang | Fast assembly of ordered block copolymer nanostructures through microwave annealing[END_REF] Morris's group reported a microwave assisted annealing method for obtaining self-assembled PS-b-PMMA and polystyrene-b-polydimethylsiloxane (PS-b-PDMS) thin films in short times. [START_REF] Borah | Swift nanopattern formation of PS-b-PMMA and PS-b-PDMS block copolymer films using a microwave assisted technique[END_REF] They suggested that the polar molecules (polymethylmethacrylate and polydimethylsiloxane) could interact with the oscillating field resulting in dipole rotation which drives microphase separation of the PS-b-PMMA and the PS-b-PDMS. However, Buriak et al. later determined that it is not the polar molecules but the microwave absorber which plays a dominate role in the fast self-assembly with the microwave annealing process. They also demonstrated that the annealing time can be reduced to 1 minute by placing a piece of high resistivity silicon wafer (a microwave absorber) in contact with the sample. [START_REF] Jin | Deconvoluting the mechanism of microwave annealing of block copolymer thin films[END_REF] Jonathan's group also reported that the block copolymer polystyrene-blockpoly(ethylene-co-butylene)-block-polystyrene is a poor microwave absorber, resulting in no change in the block copolymer morphology upon application of microwave energy. [START_REF] Toolan | Selective molecular annealing: in situ small angle X-ray scattering study of microwave-assisted annealing of block copolymers[END_REF] Our group has reported a fast ordering method termed high temperature solvent vapor annealing (HTSVA). [START_REF] Liao | Carbohydrate-Based block copolymer thin films: ultrafast nano-organization with 7 nm resolution using microwave energy[END_REF] The HTSVA provided an ideal binary solvent vapor environment at a high temperature using microwave energy that achieved self-assembly of the MH-b-PS in one second.

In summary, researchers have tried to find an effective way to heat the medium (e.g. a rapid thermal processing system, a germanium layer, high resistivity silicon wafers or a high temperature solvent vapor environment) contacted with the target BCP materials. In these ways, energy could transfer easily to the target BCP materials through the medium and thus the BCP self-assembly process could finish in a short time.

The concept "microwave annealing" is found in many publications, but in these publications the microwave energy is used in different ways. There is no clear definition of "microwave annealing" until now. Here we would like to clarify the microwave approach used in this manuscript. It does not use a combination of solvent and heat like most publications, but use heat only. The microwave energy was utilized to heat the carbohydrate block maltoheptaose, residual moisture and the doped silicon substrates. While in our prior work [START_REF] Liao | Carbohydrate-Based block copolymer thin films: ultrafast nano-organization with 7 nm resolution using microwave energy[END_REF] , a combination of solvent and heat was used for annealing. The microwave energy in our prior work was used to heat the annealing solvent only. Here we would like to probe how to transfer microwave energy directly and efficiently to carbohydrate-based BCP materials and make them self-assemble rapidly. We expect microwave heating can be used for heating carbohydrate-based BCP materials, because it is widely used in carbohydrate chemistry [START_REF] Das | Application of Microwave Irradiation in the Synthesis of Carbohydrates[END_REF] , food preparation [START_REF] Icier | Dielectrical properties of food materials -1: Factors affecting and industrial uses[END_REF] , bioenergy [START_REF] Kostas | The application of microwave heating in bioenergy: A review on the microwave pre-treatment and upgrading technologies for biomass[END_REF] , organic synthesis [START_REF] Kappe | Controlled Microwave Heating in Modern Organic Synthesis[END_REF] and so on. This unique heating mechanism provides many advantages, including a higher energy transfer efficiency, a shorter heating time to a set temperature, etc. that could benefit tremendously the self-assembly of carbohydrate-based BCP materials. Microwave heating is the reverse of conventional heating. Conventional heating (e.g. oven heating) transfers heat from an external heat source first to the surface of a material and then to the cooler interior regions by thermal conduction. Microwave heating, however, is a form of energy conversion rather than a form of heating, because electromagnetic energy is converted into heat directly in microwave heating [START_REF] Motasemi | A review on the microwave-assisted pyrolysis technique[END_REF] . In addition, the effect of microwave heating is almost instantaneous, which is different from conventional heating methods. This characteristic is of great interest for many practical applications since no time is spent waiting for the source to heat up or cool down [START_REF] Kappe | Controlled Microwave Heating in Modern Organic Synthesis[END_REF] . Microwave heating displays a penetration depth (with penetration depth depending on dielectric constant and wavelength). Most likely this penetration depth is in the mm or higher range, so much thicker than the film thickness. What is more, microwave heating enables better control of the heating process itself and eliminates the risk of the sample surface overheating and possibly degrading (whilst the inner part has not yet been heated). This could be quite useful in improving the degree and range of ordering of whole BCP thin films.

Results and Discussion

In this study, atomic force microscopy (AFM) and grazing-incidence small-angle X-ray scattering (GISAXS) techniques have been used for investigating the microphase separation of MH-b-PS thin films in the real and reciprocal space. 2a exhibits disordered spherical morphology. Its GISAXS profile in Figure 3a shows two symmetrical Bragg rods at around -0.7 and 0.7 nm -1 linked by a so-called Debye-Scherrer ring, indicating that disordered spheres were formed in the as cast MH-b-PS thin film. This is consistent with what is observed in its AFM phase image. After microwave heating at 328 K for 1 s, the AFM phase image (Figure 2b) of the MH-b-PS thin film does not show sensitive changes in terms of degree of ordering. Its GISAXS pattern in Figure 3b is also similar to that of as cast sample in Figure 3a, except an increase in the intensity of the Bragg rods. This suggest that disordered spherical MH-b-PS nanostructures do not self-assemble into periodic nanostructure under this condition. When the set temperature is increased to 333 K, the GISAXS pattern changes dramatically. As shown in Figure 3c, the Debye-Scherrer rings observed in Figure 3a and 3b disappear and four symmetrical Bragg rods appear along the Yoneda band. The positions of the Bragg rods are at ±0.61 and ±1.06 nm -1 , which are consistent with characteristic pattern of a hexagonally packed array of cylinders normal to the surface. The AFM phase image in Figure 2c confirms the cylindrical morphology with a feature size about 8 nm. The domain spacing is around 10.3 nm, which is calculated from q*=0.61nm -1 by Bragg's equation. The two main Bragg rods at ±0.61 nm -1 in Figure 3c can be assigned to the (10) reflection of the 2D hexagonal lattice formed by the MH cylinders. The other two rods at ±1.06 nm -1 are just located at the scattering vectors of √3q 10 and are thus assigned to (11) reflections. Intensity profiles along the q y direction are shown in Figure 3g. A center-to-center distance of 11.8 nm can be calculated from the scattering vector of q 10 . The intensity of both (10) and (11) rods are increased when extending the microwave heating time from 1 s to 20 s at 333 K, see Figure 3d. The corresponding AFM phase image (Figure 2d) also exhibits an improvement in the degree of ordering. The trend towards a higher degree of ordering, however, changes when extending the microwave heating time at 333 K. The AFM phase image of the MH-b-PS thin film in Figure 2e seems to some extent degrade after 40 s microwave irradiation. The intensity of diffractions in its GISAXS pattern in Figure 3e decreases as compared to Figure 3d. It is worth noting that the phase transition from spherical domains (as cast) to perpendicularly oriented cylindrical domains was fast. As the set temperature is increased to 338 K, the GISAXS pattern in Figure 3f does not show sensitive changes as compared to Figure 3c at 333 K, except that the (11) reflections become rather weak. This could be explained by the existence of a minimum in the form factor at smell of burnt sugar during the experiment. It is believed that the MH segment of the MH-b-PS assembly of the MH-b-PS is at around 333 K for a few seconds. This may due to a high efficient heating for the small volume of the MH-b-PS in the thin films, and probably could be solved by reducing the frequency of the microwaves. The full width at half maximum (FWHM) of the first order peak of as cast sample is 0.15 nm -1 . After microwave heating at 328 K for 1 s, the FWHM increases to about 0.27 nm -1 , which implies the grain size of disordered nanostructures decreases under this condition. When the set temperature increases to 333 K and keep annealing for 1 s, the FWHM decrease to 0.04 nm -1 . It means cylindrical structures with a correlation length of 25 nm are formed in the MH-b-PS thin film. The FWHW at 333 K increases to about 0.06 nm -1 when the annealing time is extended to 20 s and even 40 s, which means that the correlation length decreases at the beginning and then keep constant during microwave heating. When the set temperature goes up to 338 K, the FWHM is still 0.06 nm -1 . This suggests the correlation length dose not increase once the cylindrical structures are self-assembled in the thin films during microwave heating. Here the set temperature of microwave heating is around 333 K, which is lower than the glass transition temperature of the MH-b-PS (T g = 365 K, see Figure 4a). It implies that the mechanism of microwave heating induced self-assembly of the MH-b-PS is different from that of conventional heating induced self-assembly. Although the silicon wafer substrates used in this study have a relatively small size and low sheet resistance, they could also be heated by the microwaves and contribute to the self-assembly of the MH-b-PS through thermal annealing.

Morris's group performed a study on the actual temperature of silicon substrates in a microwave heating system by attaching an independent thermometer to the silicon substrate [START_REF] Mokarian-Tabari | Study of the Kinetics and Mechanism of Rapid Self-Assembly in Block Copolymer Thin Films during Solvo-Microwave Annealing[END_REF] . They found the actual temperature of the silicon substrate did not exceed the set temperature and is even lower than the set temperature. This means the infrared sensor in the microwave heating system can provide a correct temperature measurement during microwave heating. In order to understand microwave heating induced self-assembly of MH-b-PS, we compare it with conventional heating by performing an in situ GISAXS measurement of MH-b-PS thin films on silicon wafers from 323 K to 473 K. Figure 5 shows q y scans of the GISAXS patterns from 323 K to 473 K and 4 representative GISAXS patterns at 323, 403, 433 and 473 K, respectively. As presented in Figure 5a, all the GISAXS patterns exhibit only one pair of Bragg rods at around q y =-0.72 and 0.72 nm -1 during the whole heating process. Neither the position nor the shape of the Bragg rods change from 323 K to 373 K, which suggests MH-b-PS chains do not get enough mobility necessary to form any nanostructure in this temperature range. This can also be explained by the T g of MH-b-PS which is around 365 K (see Figure 4a). After the heating temperature rises to 373 K, we find some effect of temperature on the GISAXS patterns. The position of the Bragg rods shifts towards q y =-0.80 and 0.80 nm -1 from 383 K to 403 K, which indicates the domain space of MH-b-PS nanostructure decreases and the MH-b-PS thin film has a trend towards self-assembly. However, no higher order Bragg rods occur and the Bragg rods become broader. That means the MH-b-PS thin films do not achieve self-assembly between 383 K and 403 K. As the temperature continues to increase, the position of the Bragg rods firstly goes back to q y =-0.72 and 0.72 nm -1 at 433 K and then shifts to q y =-0.66 and 0.66 nm -1 at 473 K. The GISAXS patterns at 403, 433 and 473 K (see Figure 5b,c,d,e) are quite similar to the one at 323 K in terms of density distribution. All of this evidence suggests that the MH-b-PS thin film does not self-assemble in this heating process even when the temperature is increased to 473 K. We thus conclude that it is quite difficult to make MH-b-PS self-assemble by conventional heating. In comparison, MH-b-PS thin films can self-assemble at 333 K into well-ordered cylindrical structures under microwave irradiation as shown before. This indicates that the main mechanism of microwave heating induced ultra-fast self-assembly of MH-b-PS is possibly not due to the heating of the silicon substrate. To eliminate the effect of silicon substrates in the microwave heating, we replaced them with microwave transparent fused silica substrates. These fused silica substrates cannot be heated by microwaves, so microwaves will work only on the MH-b-PS thin films. These MH-b-PS samples were annealed with the same microwave heating parameters for comparison and then characterized by GISAXS. As presented in Figure 6a, the as cast MH-b-PS on fused silica substrate has a similar GISAXS pattern to the as cast MH-b-PS on silicon wafer. The positions of Bragg rods in both GISAXS patterns are around -0.7 and 0.7 nm -1 . So the effect of different surfaces of the two kind of substrates on the self-assembly of MH-b-PS before annealing could be ignored in this comparison. After microwave heating at 328 K for 1s, the GISAXS pattern of the MH-b-PS thin film on fused silica substrate does not show sensitive differences in the density distribution. The GISAXS pattern does not change at 333 K for 1 s. When the microwave heating time was extended to 20 s at 333 K, we found weak secondary order scatting peaks at around -1.1 and 1.1 nm -1 from the q y scan of the GISAXS pattern in Figure 6d. The two secondary order scattering peaks enhance in Figure 6e it is transparent to microwaves. But the maltoheptaose in the present study is obtained from a saturated sugar β-cyclodextrin which has a high dielectric constant [START_REF] Sakurai | A molecular orbital study of cyclodextrin (cyclomalto-oligosaccharide) inclusion complexes. III, dipole moments of cyclodextrins in various types of inclusion complex[END_REF] . As shown in Figure 1, there are 7 units of glucose with 22 polar oxygen-hydrogen bonds in the maltoheptaose. These hydroxyl groups are in an asymmetric geometry, so maltoheptaose is overall highly polar. The hydroxyl groups in saccharides like maltoheptaose could create a high shear environment by attempting to continuously reorient in the electromagnetic radiation's oscillating field, which is similar to the behavior of the water dipole. Dependent on the frequency, the dipole may move in time with the field, lag behind it or remain apparently unaffected [START_REF] English | Molecular dynamics simulations of microwave heating of water[END_REF] . When the dipole lags behind the field then interactions between the dipole and the field lead to an energy loss by heating, the extent of which is dependent on the phase difference of these fields. The dipole-dipole interactions could also exist in the maltoheptaose domains during microwave heating and contribute to the self-assembly of MH-b-PS. In addition, hydrogen bonds and hydroxyl-water interactions also play a significant role in high sugar, maltodextrin, starch hydrolysate, and lactose-like disaccharide-based food formulations [START_REF] Roebuck | Dielectric Properties of Carbohydrate-Water Mixtures at Microwave Frequencies[END_REF] . Since microwaves act only on the MH segment, the heat produced in the MH domain transfers to the PS segments as microwave heating time is extended. The MH segment is a minority component of the MH-b-PS, so it is difficult for MH-b-PS to have enough chain mobility to achieve well-ordered self-assembly without the heating of silicon substrates.

Residual solvent and moisture inside the MH-b-PS thin films could be heated together with the MH-b-PS by microwaves during the annealing process. Although both of them could be removed after reaching a specific temperature, they may contribute to an increase in the temperature of MH-b-PS thin films and influence the self-assembly of MH-b-PS before their removal. The solvent used for fabricating MH-b-PS thin films is anisole. Because anisole is a nonpolar solvent and does not absorb energy from microwaves, the effect of the residual anisole inside MH-b-PS thin films is negligible. On the contrary, depending on the hydrophilic property and relative humidity of the surrounding environment, saccharide polymers, like maltoheptaose, usually contain several percent moisture which can be heated by microwaves. In order to estimate the amount of moisture inside MH-b-PS thin films and investigate physical and chemical processes of the MH-b-PS as the temperature increases, we performed thermogravimetric analysis (TGA).

Usually, the moisture content of a material is eliminated when the temperature goes up to 373 K.

As shown in the TGA curves (see Figure 4b), the weight of the MH-b-PS decreases just 1% from room temperature to 373 K. This TGA curve stays flat up to 473 K, which proves that there is only a very small amount of moisture (about 1.5 wt %) inside the MH-b-PS. Since the PS segment of the MH-b-PS is hydrophobic, the moisture could only adhere to the hydrophilic segment of the MH-b-PS (the MH segment). This assumption is proved from the TGA curve of the MH. As compared to the TGA curve of MH-b-PS, the TGA curve of MH shows the same characteristic from room temperature to 473 K. This means that the moisture lost from the MHb-PS is originally involved in the MH segment as the temperature is elevated from room temperature to 473 K. It should be noted that this small amount of moisture inside the MH segment is not fully eliminated at the annealing temperatures (328 K, 333 K and 338 K). Some moisture (less than 1%) is still trapped inside the MH segment until 373K. We think the moisture as a dielectric material could contribute to an increase in the temperature of the MH-b-PS during the annealing process, but it does not play a main role in driving the self-assembly of the MH-b-PS due to its very small amount.

Conclusion

It is demonstrated here that microwave energy can be absorbed directly by thin films of the carbohydrate-based BCP material MH-b-PS and then drives the MH-b-PS self-assembly in a few seconds without need of any solvents. Perpendicularly orientated cylinder-forming MH-b-PS nanopatterns with 7 nm feature size and long-range ordering across the whole thin film were obtained in one second at 333 K using this microwave heating method. The rotation of the MH dipole under microwave irradiation is the main reason for achieving ultra-fast self-assembly of the MH-b-PS. The small amount of moisture inside the MH-b-PS is also helpful for the effective microwave heating of MH-b-PS. In addition, the silicon substrate is indispensable in the microwave heating induced self-assembly of the MH-b-PS, because it can help enhance the flexibility of the PS chain of the MH-b-PS. In total, the microwave heating method is much more efficient than both conventional heating and solvent vapor annealing in driving the microphase separation of MH-b-PS. It is expected that this microwave heating method can be applied to other carbohydrate-based materials, morphologies, and directed self-assembly strategies and would be useful in microelectronic industries.

Experimental Section

Thin film preparation: The Si wafers (p-type, B-doped, orientation=100 ± 0.5 °, ρ=1-50 Ω cm) were purchased from Siltronix and then diced into approximately 1.0 cm × 0.5 cm pieces. In order to avoid the effect of difference in the resistance of silicon substrates, all the silicon substrates used are from the same piece of silicon wafer. The silicon pieces were degreased in an ultrasonic bath of acetone for 20 min, rinsed in ethanol, and then dried by a nitrogen stream. A radio-frequency oxygen plasma (ION3 KHZ plasma system, PVA TePla) treatment at 0. Grazing-incidence small-angle X-ray scattering (GISAXS): GISAXS measurements were performed on the CRG-BM02 beamline at the European Synchrotron Radiation Facility in Grenoble. GISAXS experiments used photo energies of 11 and 16 keV. The samples were positioned on a 6 circle-goniometer, allowing an accurate alignment of the patterned area with respect to the incident beam direction. The GISAXS patterns were recorded after 10 s irradiation on a XPAD 2D pixel detector placed at a distance of 1880 mm from the sample. The intense incident and reflected beams were stopped by a 2 mm wide Ta strip. A photomultiplier with a removable kapton foil was placed in front of the beam stop chamber to adjust sample position and measure rough reflectivity curves to check the Si critical angle. The incident angle was chosen between the critical angles of the polymer (0.13°) and the Si substrate (0.18°). With the incident beam propagating along the x direction, the GISAXS images were recorded in the (q y , q z ) plane where q y and q z are the components of the scattering vector, related to the in-plane angle 2θ f and out-plane angle α f . The in situ GISAXS measurement was performed with a DHS 900 hot stage and TCU 150 temperature control unit. All the GISAXS patterns in the study were normalized to the incident beam intensity monitored by a front photomultiplier.

Thermogravimetric analysis (TGA) and Differential scanning calorimetry (DSC): TGA was conducted on a thermogravimetric analyzer (Setaram Instruments) using a ramp rate of 283 K per minute up to 773 K under nitrogen gas. DSC experiments were performed on a Q200 (TA instruments) under a nitrogen purge and at a heating rate of 293 K per minute. Data are reported on the second heat.

Corresponding Author

*Prof. Redouane BORSALI,

Table of Contents

Figure 1 .

 1 Figure 1. Chemical structure of the MH-b-PS used in this study.

  etc., have been developed to replace conventional TA and SVA methods. Majewski et al. designed a laser heating method by putting a light-absorbing layer of germanium beneath poly(styrene)-b-poly(methyl methacrylate) (PS-b-PMMA) thin films. 19MH 1.2k -b-PS 3.8k Highly localized laser heating produced photo thermal gradients which induced a thermophoretic force on morphological defects, and thus increased local ordering kinetics by at least 3 orders of magnitude compared to conventional oven annealing. Perego et al. used a so-called rapid thermal processing (RTP) technique to take advantage of the amount of solvent naturally trapped within the film during the spinning process and so obtained highly ordered lamellar grains of PS-b-PMMA in a few seconds 21 . Buriak et al. used microwaves to heat the sample and annealing solvent simultaneously and obtained horizontally orientated cylinder-forming PS-b-PMMA and poly(styrene)-b-poly(2-vinylpyridine) (PS-b-P2VP) patterns in few minutes.

Figure 2 .

 2 Figure 2. Atomic force microscopy phase images of the MH-b-PS thin films (a) before and (b-f)

Figure 3 .

 3 Figure 3. Grazing-incidence small-angle X-ray scattering (GISAXS) results of the MH-b-PS thin

Figure 2 shows

 2 Figure 2 shows AFM phase images of MH-b-PS thin films under different microwave heating

Figure 4 .

 4 Figure 4. (a) Differential scanning calorimetry (DSC) result of the MH-b-PS and (b)

Figure 5 .

 5 Figure 5. In situ GISAXS measurements of MH-b-PS thin films on silicon wafers using

Figure 6 .

 6 Figure 6. GISAXS patterns of MH-b-PS thin films on fused silica substrates (a) before and after

  when the microwave heating time is increased to 40 s at 333 K, but they are not as strong as the case of the MH-b-PS thin film on the silicon wafer. These results indicate that microwave energy can be absorbed by the MH-b-PS thin film and used for driving the self-assembly of MH-b-PS. However, the degree of ordering of the selfassembled MH-b-PS thin film on fused silica substrates is lower than the MH-b-PS thin film on silicon substrate. One possible explanation is that microwave-heated MH block is minor which has not enough ability to drive completely the whole MH-b-PS molecule to self-assembly. While the chain mobility of the PS matrix can be increased with the help of microwave-heated silicon substrate. MH-b-PS can self-assemble easier on silicon substrates than on fused silica substrates during microwave heating. It has been demonstrated above that the MH-b-PS cannot selfassemble with only conventional heating. So we think that the heating effect on both the MH-b-PS thin film and the silicon substrate is critical for achieving ultra-fast self-assembly of the MHb-PS by microwave heating. It is important to get insight on how the MH-b-PS thin film absorbs microwave energy. There are at least three components in the MH-b-PS thin films: MH-b-PS, residual solvent and moisture. The two segments of the MH-b-PS, polystyrene and maltoheptaose, have different dielectric properties. Polystyrene is a nonpolar molecule since it has a symmetrical structure. So

  77 torr, 41.3 cc/min, 40 W of power for 10 min was performed to render hydrophilic surfaces for the silicon substrates. Preparation of MH-b-PS thin films was performed immediately after the plasma treatment for the silicon substrates. The fused silica substrates were ordered from Edmund optics. The MH-b-PS thin films on fused silica substrates were fabricated using the same parameters as the MH-b-PS thin films on Si wafers. Anisole (purity > 99%) was acquired from Sigma-Aldrich and used without further purification. The diblock copolymer MH-b-PS was weighed and dissolved in anisole at room temperature by stirring overnight. The concentration of the MH-b-PS solutions is 20 mg/ml unless otherwise specified. A volume of 40 µl of the MH-b-PS solution was dropped onto a plasma-treated silicon substrate and spin coated (spin coater model: SPS Spin 150) at 2000 RPM for 1 min under ambient conditions at room temperature. The film thickness is around 40 nm. The spin-coated MH-b-PS films were transferred for microwave heating after drying in air. 'Microwave cooking': A commercial microwave synthesizer (Initiator 2.5, Biotage) was used as a microwave heating source for the annealing process. The Biotage reactor measures the temperature of the samples by an infrared sensor (accuracy: 5 K) at the bottom of the microwave cavity. Since the vials used in our experiments are transparent for microwaves, the temperature detected by the infrared sensor is coming from the sample. The measurement of the temperature is an in-situ measurement. The MH-b-PS sample was put into a reaction vial and then heated with 2.45 GHz microwave energy to reach the target temperature. The target temperature was kept for a given annealing time with the help of the temperature feedback-control system of the microwave synthesizer. The vial was then cooled down to 318 K with air flow. The whole heating and cooling process took approximately one minute, which depends on the target temperature. The output microwave power, the temperature of the vial and the pressure inside of the vial were recorded by the microwave synthesizer. Atomic force microscopy measurements: Phase images were collected by an atomic force microscope (Agilent 5500, Agilent Technologies) in tapping mode. Silicon cantilevers were purchased from Budget Sensors with a resonant frequency of 150 kHz and a force constant of 7.4 N/m.

CERMAV, BP 53, 38041 Grenoble cedex 9, France E-mail: borsali@cermav.cnrs.fr Tel: +33 (0)4 76 03 76 27 -fax : +33 (0)4 76 03 76 29 ACKNOWLEDGMENT This work was partially supported by the CNRS, the European project "GreeNanoFilms" which has received funding from the European Union Seventh Framework Program (FP7/2007-2013) under grant agreement No. 603519, and partially supported by the ANR project "Sweetmemory" (ANR-14-CE08-0021) and Carnot PolyNat Institut. This work is also supported by the program of Youth Eastern Scholar from Shanghai Municipal Education Commission. The authors are grateful to Prof. Robert Pecora (Stanford University, CA, USA) for helpful discussions. The authors thank ESRF for beam time allocation and thank D2AM staff Nathalie Boudet for her help in GISAXS measurement.

the corresponding scattering vector. When the set temperature goes up to 343 K, we found a caramelized under this condition. The working window for microwave irradiation induced self-

Supporting Information

Figures S1-S7.


Anthony Herrel
email: anthony.herrel@mnhn.fr

Dominique Joly
email: dominique.joly@egce.cnrs-gif.fr

Etienne G J Danchin
email: etienne.danchin@univ-tlse3.fr

Epigenetics in ecology and evolution -a primer

Keywords: ecology, epigenetics, evolution

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Over the past two decades, the study of epigenetics has emerged as a key stone discipline for the understanding of biological evolution (for reviews, see for instance [START_REF] Richards | Inherited epigenetic variation -revisiting soft inheritance[END_REF][START_REF] Rando | Timescales of genetic and epigenetic inheritance[END_REF][START_REF] Bossdorf | Epigenetics for ecologists[END_REF][START_REF] Jablonka | Transgenerational Epigenetic Inheritance: Prevalence, Mechanisms, and Implications for the Study of Heredity and Evolution[END_REF][START_REF] Skinner | Epigenetic transgenerational actions of environmental factors in disease etiology[END_REF][START_REF] Danchin | Beyond DNA: Integrating Inclusive Inheritance into an Extended Theory of Evolution[END_REF][START_REF] Wang | Lamarck rises from his grave: parental environment-induced epigenetic inheritance in model organisms and humans[END_REF][START_REF] Danchin | Epigenetically-Facilitated Mutational Assimilation: Epigenetics as a Hub within the Inclusive Evolutionary Synthesis[END_REF]. From the turn of the 3 rd millennium onwards, major advances have been generated by exceptional technological advances in high-throughput sequencing that nowadays allow the simultaneous analysis of genomes, epigenomes, and transcriptomes in all their complexity. These approaches, initially developed by molecular biologists, are becoming more and more accessible to researchers in other sub-disciplines of biology working on both model and non-model species.

Consequently, these approaches have provided a new impetus to studies of the intricate relationships between genes, environment and phenotypes in an eco-evolutionary context. A particularly surprising output of the use of recent high throughput molecular technologies was the discovery that the information encoded into the DNA nucleotide sequence by itself is often insufficient to explain biological variation in all its complexity [START_REF] Maher | Personal genomes: The case of the missing heritability[END_REF]. This resulted in the debate about missing heritability and revealed the existence of other forms of information transmission across generations [START_REF] Danchin | Avatars of Information: Towards an Inclusive Evolutionary Synthesis[END_REF].

The discovery of epigenetics (any modification other than changes in DNA sequences affecting gene expression, whether those modifications have been shown to be stable or not) profoundly changed our understanding of the functioning of biological organisms and generated considerable interest in its role as a 'missing link' between environmental and phenotypic variation. In particular, recent discoveries have fostered a full revision of the mechanisms of phenotypic plasticity [START_REF] Bonduriansky | Extended Heredity: A New Understanding of Inheritance and Evolution[END_REF][START_REF] Herman | How stable 'should' epigenetic modifications be? Insights from adaptive plasticity and bet hedging[END_REF][START_REF] Jablonka | Epigenetic inheritance and plasticity: the responsive germline[END_REF][START_REF] Pigliucci | Phenotypic plasticity and evolution by genetic assimilation[END_REF][START_REF] Reed | Phenotypic plasticity and population viability: the importance of environmental predictability[END_REF][START_REF] Sentis | Evolution without standing genetic variation: change in transgenerational plastic response under persistent predation pressure[END_REF][START_REF] Sultan | Evolutionary Implications of Individual Plasticity[END_REF][START_REF] Zhang | Epigenetic variation creates potential for evolution of plant phenotypic plasticity[END_REF], inheritance [START_REF] Bonduriansky | Rethinking heredity, again[END_REF][START_REF] Bonduriansky | Extended Heredity: A New Understanding of Inheritance and Evolution[END_REF][START_REF] Danchin | Avatars of Information: Towards an Inclusive Evolutionary Synthesis[END_REF][START_REF] Day | Epigenetic mechanisms in cognition[END_REF] and adaptation [START_REF] Danchin | Beyond DNA: Integrating Inclusive Inheritance into an Extended Theory of Evolution[END_REF][START_REF] Danchin | Cultural flies: conformist social learning in fruit flies predicts long-lasting matechoice traditions[END_REF][START_REF] Danchin | Epigenetically-Facilitated Mutational Assimilation: Epigenetics as a Hub within the Inclusive Evolutionary Synthesis[END_REF][START_REF] Pocheville | Physiology and Evolution at the Crossroads of Genetics and Epigenetics[END_REF], 2017). These three major lines of research in evolutionary biology are at the crossroads between individual-centred approaches (that mostly study proximate mechanisms) and supra-individual approaches (that mostly study ultimate processes) and are all impacted by our understanding of epigenetics [START_REF] Danchin | Inheritance Is Where Physiology Meets Evolution[END_REF][START_REF] Jablonka | Evolution in four dimensions. Genetic, Epigenetic, Behavioural, and Symbolic Variation in the history of life[END_REF][START_REF] Pocheville | Physiology and Evolution at the Crossroads of Genetics and Epigenetics[END_REF]. Epigenetics has emerged as a key discipline in the study of ecological and evolutionary dynamics, and this special feature contributes to highlighting the importance of this discipline.

As most disciplines of molecular biology, the study of epigenetics initially benefited considerably from the use of model organisms. Today, however, the time is ripe to transfer discoveries from the study of model organisms to non-model organisms, rendering these approaches more ecologically relevant. The transfer of knowledge to non-model organisms and more complex naturalistic settings may then allow us to study the molecular basis of the complex interactions among organisms within communities and ecosystems. As such, the study of epigenetic inheritance may provide novel insights into previously unexplained aspects of complex ecological interactions. With this special feature on the impact of epigenetics in the field of ecology, and particularly that of functional ecology, we want to stimulate ecologists to embrace epigenetics as a major source of information allowing the study of the interactions among organisms and their biotic and abiotic environment with the hope of fostering a better understanding of adaptation and evolution.

The potential role of epigenetics in answering important ecological and evolutionary questions is illustrated in this special feature. First, Pimpinelli and Piacentina (2020) Biological invasions are a global scourge and a major issue in the current context of globalisation and the world-wide and rapid transport of organisms outside of their native range. However, for an invader to be successful it needs to be able to rapidly respond to and cope with different environmental conditions. Marin and collaborators examine the possible link between stress and epigenetic changes and transposable element activity in generating phenotypic and genetic variation in invasive populations. They argue that these mechanisms can contribute to the success of biological invasions by facilitating plasticity as well as rapid adaptive evolution. Indeed, whereas epigenetic variation and transposable elements are generally well regulated and thus 'hidden' in natural populations, these mechanisms may be released from their regulation in new and stressful environments. As such, Marin and collaborators argue that the epigenome needs to be studied in greater detail if we want to understand the mechanisms by which populations successfully colonize and adapt to new environments.

Given the ability of epigenetic mechanisms to allow organisms to respond to rapid changes in their environment, they may also play an important role in conservation. Indeed, the paper by Rey et al (2020) argues strongly that epigenetic variation and more particularly DNA methylation represent important components of biodiversity linking genome to environment. They highlight the importance of DNA methylation in providing biomarkers for past and present environmental stress, the ecological structuring of wild populations, improving translocations and for studying landscape connectivity. As such, epigenetic mechanisms appear as a promising and important tool in conservation biology that may allow conservation biologists to overcome essential challenges. Together these papers not only show the versatility of epigenetic mechanisms in generating phenotypic variability but more importantly show the importance of including epigenetic thinking in addressing questions in ecology and evolution. Especially in our current ever more rapidly changing landscape, understanding how organisms may respond to change appears a crucial biological endeavour for which epigenetic approaches may provide a major contribution. Different aspects would need further theoretical and experimental investigation, including (a) the quantification of epigenetic reversion rate at the individual and transgenerational schales under different environmental scenario; (b) the characterization of the epimutation pattern (random, directed -as for phenotypic plasticity, or dependent or not on environmental conditions), here again at the different biological levels; (c) the study of the genetic control of epimutations in relation to the possible genetic transmission across generations; and (d) the causal relationship between epigenetics, phenotypes and fitnss in order to establish the adaptive value of epigenetics. Among these four pints, the latter is probably the most crucial to demonstrate the extent of epigenetics in evolution as well as its impact on the heritability and the stability of the genomic compartment.

We hope this special feature will stimulate readers to think about the role of epigenetics in ecology and evolution and that it will spark new studies trying to understand variation in nature and its role in allowing populations to respond to variation in biotic and abiotic factors.

  discuss how transposons may contribute to translating phenotypic plasticity into genetic variability through their impact on epigenetic mechanisms. Especially in the context of rapid environmental change, it may be important to be able to translate phenotypic plasticity into variation that is encoded in the genome, and thus under direct natural selection. Pimpinelli and Piacentina argue that modifications induced by transposons would not only impact epigenetic mechanisms but also be stably inherited. Indeed, transposable elements are environment-responsive molecular elements that can rapidly produce phenotypic and genotypic variants in response to environmental perturbations by causing regulatory changes in the transcription of the genome. As such, transposons may play an important role on the relevant ecological time-scales within which organisms need to respond to the current and ongoing rapid changes in their environment. At the end of their review, Pimpinelli and Piacentina discuss a few model systems where transposons are likely to have played a role in the rapid phenotypic diversification observed and suggest promising avenues for future research.The characterization of parental transmission of epigenetic variation to offspring is of major importance to understand how rapid individual phenotypic changes are passed on through generations and stay heritable over generations. A number of studies have been devoted to demonstrating how maternal processes shape the physical and social context of offspring development through behavioral and physiological mechanisms[START_REF] Adkins-Regan | Maternal effects in quail and zebra finches: Behavior and hormones[END_REF][START_REF] Galloway | Maternal effects provide phenotypic adaptation to local environmental conditions[END_REF]. Yet, very little is known on how the paternal transmission occurs. Champagne (2020) describes the current state of knowledge regarding paternal epigenetic effects, the interplay between maternal and paternal influences, and the importance of considering the complex nature of reproduction when predicting the transmission of phenotypes across generations. The paper reviews experimental evidence on how sperm can drive the paternal transmission of epigenetic marks by a diversity of environmental effects, such as nutritional or toxicological effects, and may shape offspring characteristics. The review also emphasizes how the environmental exposure history of males may alter female mate preference through sexual selection processes. It hence argues for a better integration of behavior in the transgenerational transmission of epigenetic variation.

Acknowledgements

The authors would like to thank the CNRS for funding the RTP and GDR 3E networks that have allowed to stimulate research on epigenetics and evolution in France. We would also like to thank all the members of the scientific committee of the RTP and GDR 3E networks for fruitful discussions. This work was supported by the "Laboratoires d'Excellences (LABEX)" TULIP (ANR-10-LABX-41) and Laboratoires d'Excellences (LABEX) BCDiv (ANR-10-LABX-03).