Bernard Lotz
email: bernard.lotz@ics-cnrs.unistra.fr

Brill transition in nylons: the structural scenario (#)

Keywords:

Pleated and rippled sheet crystal structures of even-even nylons that feature twisted amide groups-aliphatic parts links were introduced in the previous contribution (Macromolecules, xxxx, xx, xxxx). Pleated sheets are made of isochiral conformational isomeric stems, rippled sheets are racemic stereocomplexes of enantiomeric stem conformations. These models are used to reanalyze the various and puzzling manifestations of the Brill transition first reported in 1942. The transition is generally considered to result from the generation of gauche bonds in the aliphatic parts of the nylons alpha phase that travel to the junctions with the amide groups. Recognition of the pleated/rippled sheets structures suggests a different scenario that takes in account, but challenges this analysis. The transition takes place in pleated/rippled sheets with twisted chain conformations. It is a dynamic interconversion between the two possible conformers (mirror images) of the stems in the crystal lattice. The interconversion timescale is in the ≈100 picosecond range, as determined by NMR. During the interconversion, the aliphatic parts C-C-C plane flips by ≈120° (from + to -60°). The intra-sheet H bonds are preserved, but transitory inter-sheet H bonds may be formed, which accounts for the frequent, but not systematic pseudo-hexagonal cell of the Brill structure. This scenario is fully consistent (and more so than the earlier models) with the wide body of experimental data available on the Brill transition. The interconversion between conformers is generic for all types of nylons (even, even-even, etc.) and is therefore a valid scenario for their Brill transitions as well. On this basis, the Brill transition should be defined by a molecular process (the conformation interconversion) rather than by an experimental criterion that lacks general validity (the merging of equatorial reflections).

Introduction.

Since the discovery of nylons, their structural analysis has been a constant interest and concern.

The crystal structures of PA66, PA610 and PA6 were solved in the late 1940s -mid 1950s by Bunn and his group in England. [START_REF] Bunn | The crystal structures of two polyamides ("nylons[END_REF][START_REF] Holmes | The crystal structure of polycaproamide: Nylon 6[END_REF] However, back in 1940, Brill in the Oppauer Research Laboratory of the I.G. firm in Germany observed a phenomenon (published in 1942) that has become known as the Brill transition: when heated, the diffraction pattern of nylon 66 changes in a very specific way. [START_REF] Brill | Über das Verhalten von Polyamiden beim Erhitzen[END_REF] The X-ray powder diffraction pattern changes in a very specific way on heating. The smaller spacing (the inter-sheet distance) increases progressively and ends up merging with the larger spacing (the intra-sheet inter-stem distance) to form a single reflection near 160°C, which is maintained at higher temperatures (this temperature, known as the Brill temperature, depends on the nylons and crystallization conditions). The process is reversible: on cooling, but with some undercooling, the two reflections are restored.

In the eighty years since its discovery, numerous studies have been devoted to the Brill transition. It has been observed in a range of nylons with different chemical constitutions. Its understanding is still far from complete -for good reasons. The transition has very unique characteristics -if only for the unusually wide temperature range over which a given nylon has its Brill transition, depending on the way it was crystallized. Even the nature of the transition is debated. It is generally accepted that the Brill transition is a first order transition, but with hints of a "second-order-like transition character" [START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF] or that this "inherently first order process, comprises a second order element". 5 The generally accepted view (with exceptions) on the molecular process at the root of the transition is based on the thermally induced mobility of the aliphatic parts: at the Brill transition, these parts are way above the melting temperature of polyethylene. The heat-induced aliphatic parts dynamics generates gauche bonds that move and end up at the amide-aliphatic parts junctions. Also, the generally accepted view (again, with exceptions) is that, at the Brill transition and in the "Brill structure" (in the domain where this structure is stable) the pattern of hydrogen bonds that stabilizes the alpha phase-type nylons sheets is maintained.

Original crystal structures of even-even nylons made of pleated and rippled sheets have been introduced in the companion paper. [START_REF] Lotz | Original crystal structures of even-even nylons made of pleated and rippled sheets Macromolecules[END_REF] They are reminiscent of the hydrogen bonded b sheets of polypeptides and proteins first described by Pauling and Corey in the 1950s. [START_REF] Pauling | Configurations of polypeptide chains with favored orientations around single bonds: two new pleated sheets[END_REF][START_REF] Pauling | Two Rippled-Sheet Configurations of Polypeptide Chains, and a Note about the Pleated Sheets[END_REF] Their torsional angles at the amide links differ from 180°. Two conformational isomers, i.e. two mirror images of any one chain conformation can exist. The pleated sheets are made of one conformer only -the pleated sheets are chiral, as are the b sheets of proteins. The rippled sheets are made of an alternation of stems in a given chain conformation and of its mirror image. This possibility is forbidden to chiral proteins, but is offered to nylons due to the absence of any chiral atom in the molecule. Of relevance in the present context, for some nylons at least, the pleated and rippled sheets make up crystal structures that grow at temperatures very close to the melting temperature, that is, way above their usual Brill transition and Brill domain.

In the present contribution, a reanalysis of the Brill transition is attempted in the frame provided by the newly introduced pleated and rippled sheets of nylons. It considers possible molecular movements taking place at high temperature in these sheet structures. More specifically, it focuses on an interconversion between the two conformation enantiomers accessible to nylons. The approach is disruptive in two respects: (a) the Brill transition is not associated with the extended chain conformation, as has been assumed so far and (b) the usual emphasis on the aliphatic parts mobility issue is shifted to the amide groups. The novel features are compared with the experimental, mostly crystallographic and spectroscopic data.

Even though the present scenario does not settle all the issues linked with the Brill transition it provides a clearer perception than available so far and reflects better the underlying molecular processes involved in this transition.

Experimental.

All the experimental data used in the present analysis are from the published literature. Model building and crystal packing are as used in the companion paper. The packing energy minimization is performed with the "Crystal packer" module of the Cerius 2 program. [START_REF]Cerius modeling software[END_REF] No constraints are set in the minimization procedure. In particular, the symmetry used is systematically P1, which does not impose any symmetry element on the cell content and cell geometry. As tested with the known alpha phase of PA66 the approach yields reasonable approximations of the structures under consideration. [START_REF] Lotz | Original crystal structures of even-even nylons made of pleated and rippled sheets Macromolecules[END_REF] Results: A Reanalysis of the Brill Transition.

In the present section, structural information on the Brill transition in different even-even polyamides is described. On this basis, a dynamic scenario for the Brill transition is introduced, that develops the theme of the pleated/rippled polymorphs of even-even nylons. The validity of the model is tested by its compatibility with the wide body of available information: structural, spectroscopic (NMR, IR) and molecular simulations (MD). Since these techniques "see" (NMR, IR) or "image" (MD) molecular movements, they provide a challenging test for any model that assumes dynamic structural rearrangements.

I. The Brill transition: a combination of two different transitions.

The most characteristic manifestation of the Brill transition is the variation of the inter-chain and inter-sheet distances with temperature, on heating and cooling. Figure 1a illustrates their "archetypical" variation for PA66 during cooling, but the process is virtually identical on heating -with however a shift of the Brill temperature TB to higher temperature, as illustrated in Figure 1b. [START_REF] Ramesh | Studies on the crystallization and melting of nylon-6,6: 1. The dependence of the Brill transition on the crystallization temperature[END_REF] The 100 spacing decreases from ≈4.35-4.4Å to ≈4.2Å. The common 010/110 reflection associated with the inter-sheet distance and the inter-stem one increases significantly from its room temperature (RT) value of ≈3.7Å to 4.15-4.2Å. The process appears however to include two stages.

From RT to≈100°C, the increase is limited and regular. Above 100°C, it increases progressively and becomes a clear upswing in the higher temperature range. The point of confluence of these reflections, when the unit-cell becomes pseudo-hexagonal, is taken as the Brill transition temperature TB. The data for PA46 single crystals (Figure 2c) [START_REF] Jones | Temperature induced changes in of aliphatic polyamides[END_REF] and bulk crystallized material (Figure 2b) [START_REF] Ramesh | New Crystalline Transitions in Nylons[END_REF] take this conclusion one step further. In both cases, the room temperature PA46 is in the alpha phase.

On heating, it converts to a pleated/rippled sheet type structure with spacings at ≈4.1Å and 3.9Å.

On further heating, the variation of the spacings parallels the behavior observed in the PA26 case (part a). The low temperature transition TL increases with the crystallization temperature TC, as does TB (insert in Figure 2b). In the PA46 single crystals (Figure 2c), a similar transition is observed at ≈200°C, with spacings at ≈4.2Å and ≈4.0Å that remain nearly constant on further heating up to ≈230°C before a sharp upswing to TB at 245°C. Comparison of the transition temperatures is of interest. In the present case, the observed TBs are comparable (≈240-250°C) for the single crystals and the bulk material crystallized at the highest TC (280°C). The low transition temperatures are however well apart (120°C and ≈200°C). Widely different Brill temperatures are typically observed for solution and bulk-crystallized nylons (cf. below). For PA46, these differences are reflected rather in the low temperature a to pleated sheets structures transitions. These different behaviors observed well before reaching TB must reflect genuine structural differences between the solutionand bulk-crystallized materials, enhanced in the present case by the diversity of alpha structures accessible to the 2,2n+2 family of nylons (antiparallel with an amide group in the fold, or parallel with no such structural constraints). [START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF] Mats of single crystals of PA46 produced in superheated water illustrate crystal transitions that are quite abrupt (Figure 2d). [START_REF] Vinken | Role of Superheated Water in the Dissolution and Perturbation of Hydrogen Bonding in the Crystalline Lattice of Polyamide 4[END_REF][START_REF] Dijkstra-Vinken | Polyamides: Hydrogen bonding, the Brill transition, and superheated water[END_REF] On heating, the a phase reflections (at 50°C: 4.44Å and 3.76Å) tend smoothly towards each other, with possibly a change of regime around 120°C, as judged from a change in the slopes of both curves. The transition to the hexagonal phase, at ≈210°C takes place with only a minimal inter-sheet spacing increase (≈0.1Å). On cooling, the Brill structure remains stable down to ≈120°C (i.e. 165°C below TM), where an abrupt conversion takes place to a pleated/rippled sheet structure (as indicated by its spacings at 50°C: 4.22Å and 3.90Å). [START_REF] Vinken | Role of Superheated Water in the Dissolution and Perturbation of Hydrogen Bonding in the Crystalline Lattice of Polyamide 4[END_REF][START_REF] Dijkstra-Vinken | Polyamides: Hydrogen bonding, the Brill transition, and superheated water[END_REF] All in all, the above data show that the transition from the alpha phase to the Brill structure involves two different transitions with an intermediate pleated/rippled sheet structure. In the nylons 2,2n family one of the transitions is skipped since the starting crystal structure is not the alpha phase but already the intermediate pleated/rippled sheets one. For PA46, the transition is a straightforward two-stage process with a clearly identified intermediate stage, both on heating and on cooling. For most nylons, the two steps are (or appear to be) "merged", which may be interpreted, as in the conventional analysis of the Brill transition, as a single transition process.

Even for these apparent "single process" transitions, hints of the two steps do sometimes exist. For example, the heating curve of PA66 illustrated in Figure 1b is drawn as a smooth line, but the data points at 150°C -190°C suggest a levelling off as in Figure 2c -which itself was described as a "curious plateau" and could not be analyzed in any further detail. As explained later, using a slower heating rate might have expanded the T° range of this intermediate stage.

II. The Brill "structure": a dynamic interconversion between enantiomeric twisted stems.

All the analyses of the Brill transition developed up to now rest on structural and/or conformational modifications of the (extended chain) alpha phase of nylons and emphasize the structural disorder resulting from the heat induced mobility of the alkyl parts.

Polyamides with short alkyl segments and their response to rising/cooling temperatures tell a very different story. Their main message is a simple, but a novel one: in nylons, the Brill transition is not associated with the standard a phase but with the structure(s) made of pleated/rippled sheet(s). These materials help also partition the inter-sheet spacing increase between two different processes (cf. Figure 2d). The recorded inter-sheet distance increase is not the sole result of thermal agitation in the alkyl parts. This novel approach imposes to (re)consider the molecular dynamics that may take place in the pleated/rippled sheets of polyamides at high temperature, and thus may be associated with the Brill transition.

If the Brill transition takes place in a structure with twisted chain conformations, a novel structural element comes into play, that had not been considered so far. It is an interconversion between the two enantiomorphic conformers that are available for the twisted nylon stems. This process will be the central theme of this contribution. It is now considered at the stem and the crystal structure levels.

The stem conformation.

The conformational isomerism is conveniently illustrated with the help of a conformational energy map. As far as we are aware, no such 2D map exists for nylons, in which both torsional angles at the CO and NH bonds are varied. For the present illustrative purpose, a redrawn conformational energy map of polyglycine or nylon 2 must be used as a plausible approximation of the nylons one (Figure 3). [START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF][START_REF] Dickerson | The Structure and Action of Proteins[END_REF] These protein energy maps are known as the Ramachandran plots and the extended chain conformation is located at the upper left corner. Here the extended chain conformation is shifted to the center of the diagram to highlight the symmetry of the conformations accessible to the aliphatic parts-amide groups links in nylons.

Interconversion takes place in a very wide, accessible domain of the conformational energy map with fully allowed (in yellow) and allowed (in orange) domains (this domain is known in protein structure and nomenclature as the b plateau). A "direct" conversion path from the alpha to the pleated/rippled conformation and the interconversion via an alpha phase is illustrated in Figure 3.

More indirect pathways will be discussed later.

Stem interconversion and atomic movements.

Transformation of a stem in its mirror symmetry involves atomic movements. In the alpha phase, with near-planar chain conformation, these movements are limited. They are more important for the twisted chain conformations, and furthermore vary significantly for the various atoms of the repeat unit. The structural alterations associated with the interconversion of any one stem in its mirror image are illustrated in Figure 4 or even more clearly in Figure 5. In this model, the stem has ≈115° torsional angles at both amine and acid junctions. The major alterations are the symmetric tilt of the amide groups and the different orientation at +60° and at -60° to the H bonded sheet of the C-C-C planes in the aliphatic segments for both acid and amine parts. The different positions and amplitudes of the movements are illustrated for all the atoms of the repeat unit in

Consequences in terms of the crystal structures.

Interconversion of pleated sheets. Pleated sheets are conformationally chiral. The interconversion of all the stems to the mirror image generates a twinned but still chiral structure.

This scheme is illustrated in Figure 5 (and locally in Figure 4). The chain conformation is maintained, except for the mirror symmetry: the c axis repeat distance is not modified, contrary to many crystal-crystal transformations in polymers. The pattern of hydrogen bonds is maintained because the plane of the amide groups is only at a small angle to the mirror plane. The figure illustrates a dynamic interconversion process taking place at all the crystallographic sites but that does not change the location of the chains in the crystal lattice (the time scales of this dynamic process will be considered later). In c axis projection, the a axis is at 4 o'clock, the b axis is vertical.

The C-C-C plane of the aliphatic parts flips back and forth from positions oriented at ≈+60° and -60° to the H bonded sheet plane, from the bc plane to the (110) plane. As a result, in c axis projection, the apparent sweep angle is only 60°, but the magnitude of the sweep is ≈120°, as illustrated in Figure 4. Note also that in the process, the structure and roles of the (100) and (1-10) planes are exchanged, as best seen in the c axis projections of Figure 5. Interconversion of rippled sheets. The synchronous interconversion of all the stems of a rippled sheet structure has a more original and unexpected outcome: it generates exactly the same sheet and crystal structure. The stems appear to have been shifted along the a axis by one inter-stem distance, which of course is not the case (Figure 6). The dynamic aspects remain the same as for the pleated sheet: the conformational energy map (Figure 3) and the molecular movements (Figure 4) are valid for both structures. It remains that a dynamic interconversion taking place in the rippled sheet is possibly one of the mildest conceivable crystal-crystal transformations.

Figure 6. Conversion of conformational stereoisomers in a rippled sheet structure of nylon 66: parent and daughter crystal structures are identical. In the rippled sheet, made of alternating conformational isomers, interconversion of all the stems in their mirror images generates the same crystal structure but with an apparent shift along the a axis by one inter-stem distance.

For both the pleated and rippled crystal structures, the interconversion affects all the stem sites in the same manner. It is clear however that the above transformations are not synchronous. Packing energy minimization of the "dynamic" structure thus created can be performed by superposing the two extreme positions of the atoms in a one chain unit-cell. Of course, this approach does not have much physical meaning since in the actual structures and at any given time, each chain site is occupied by only one of the two conformational isomers. Also, possible adjustments and c axis shifts during the interconversion (depending on the local environments) are erased in this simplified approach. By design, the minimization overestimates the cell dimensions. Its outcome is nevertheless of interest: all reflections are now at ≈4.35Å. The cell geometry projection is clearly pseudo-hexagonal, which results in a single diffraction peak, as observed for the Brill structure.

Note that a similar approach was used by Tashiro and Yoshioka for nylon 1010. [START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF] These authors superposed several molecular dynamics trajectory files (from an initial alpha phase), which takes better in account the thermally induced disorder of the aliphatic segments, an aspect of the process that is neglected here.

Conformational isomerism of even-even nylons in the Brill structure: the main features.

Before evaluating the compatibility of the present interconversion model of the Brill transition with available experimental evidence, it is appropriate to recall its major structural and molecular features.

The proposed mechanism rests in the first place on the recognition that stable chain conformations can exist with twisted bonds at the aliphatic parts/amide groups links. The torsional angles are in the present models close to 115° -but this angle may vary and may be different at the acid and amine links.

The twisted chain conformations make it possible to consider conformational enantiomers and therefore also a dynamic conformational interconversion between them.

In a hydrogen bonded sheet, such an interconversion is a very mild process: the final (and possible intermediate, all trans) conformations are located on a wide so-called "b plateau" in the related Ramachandran plot of the nylon 2 (or polyglycine) conformational energy map.

The structure of the sheet is maintained and the inter-stems hydrogen bonding remains virtually undisturbed.

The interconversion has more bearing on the inter-sheet distance, since the azimuthal setting of the aliphatic C-C-C plane flips by plus and minus 50° to 60° away from the H bonded plane. The amplitude is determined by the torsion angles at the amide-aliphatic parts. The torsion angles and therefore the amplitude may progressively increase as the Brill transition is approached on heating.

The interconversion can take place in the two conceivable types of hydrogen-bonded nylon sheets: in the pleated sheet (chiral, since made of stems with the same conformation) or in the rippled sheet (with alternating mirror images stem conformations).

More generally, the interconversion can take place in sheets of all types of nylons since it is defined by a local conformational interconversion at the amide-aliphatic parts links allowed by the achiral character of the nylons.

The interconversion results in crystal-crystal transformations, but with very special characteristics. In pleated sheets, it generates the same, but twinned crystal. In rippled sheets, it generates exactly the same crystal structure, but apparently shifted by one inter-stem distance along the hydrogen bonded sheet -a virtual shift of ≈4.8 to 5Å.

This conformation interconversion appears to be a plausible and "mild" molecular process that can take place at high temperature, triggered by or combined with the thermally induced mobility of the aliphatic segments.

III. The dynamic interconversion and experimental data: an evaluation.

The above structural scheme rests on a dynamic interconversion between two enantiomeric conformations of the nylon crystalline stems. The dynamic aspects at the stem level -as opposed to the static, crystallographic approach developed in the preceding contribution -are therefore highly relevant. Are these dynamic aspects compatible with the available experimental results? For this purpose, global techniques (DSC, X-ray fiber patterns) are of course useful but NMR, IR and molecular dynamics modeling data that "see" the nature and the frequency of these molecular movements are mandatory. The development to follow relies heavily on the experimental contributions and in-depth analyses due to A. English (NMR), and to K. Tashiro et al. The latter team in particular has provided the most comprehensive work on the Brill transition by combining fiber X-ray diffraction, infra-red analysis, molecular modeling and molecular mechanics. Also, quite understandably, given the wealth of data available, the present evaluation cannot give credit to every contribution in the field -not to mention the works that have been missed in the literature search.

One approach in evaluating the compatibility of the model with experimental data would be to consider the various techniques in succession. It is more instructive to compare and evaluate the movements of the different parts of the molecule as derived from the structural model and how the different techniques perceive these movements.

The amide parts and hydrogen bonds.

Early analyses of the Brill transition considered 60° flips of the amide units (even if temporary)

and formation of inter-sheet H bonds in order to account for the Brill structure hexagonal symmetry. [START_REF] Schmidt | Gitterstrukturen mit räumlichen Wasserstoffbrückensystemn und Gitterumwandlungen bei Polyamiden[END_REF][START_REF] Olf | NMR Observations of Drawn Polymers. VII. Nylon 66 Fibers[END_REF] Various investigation techniques agree however in pointing out that the amide bonds are relatively "stable" in the Brill transition. The NMR data due to A. English et al. indicate that only a minor amount of motional averaging takes place up to the highest temperature examined (228°C). [START_REF] Hirschinger | Segmental dynamics in the crystalline phase of nylon 66: solid state deuterium[END_REF] It corresponds to a "small librational motion about the N-C axis" that preserves the hydrogen bonds, with mean correlation times 2ptc varying from 500 to 200 picoseconds over a 185°C to 228°C temperature range. A polarized µ-FTIR study on µm to mm-large single crystals of a model 3-oligoamide for PA66 due to Cooper et al. indicates that "a maximum of 5° rotation of the N-H bonds about the extended chain axis occurs prior to melting". [START_REF] Cooper | A polarized -FTIR study on a model system for nylon 6 6: implications for the nylon Brill structure[END_REF] The angle between the two NH orientations in Figure 4 is ≈8°.

The overall agreement of the interconversion scenario with the H bonds geometry is thus satisfactory. However, this analysis carries on the traditional view of different intra-sheet H bonds and inter-sheet (van der Waals) molecular interactions. As such, it does not explain the hexagonal unit-cell geometry of the Brill structure. This issue will be considered and developed later.

The aliphatic parts: observations and modeling.

The dynamic aspects of the aliphatic parts are one of the most investigated issue of the Brill transition. NMR, IR and molecular dynamics have provided a wealth of information on their movements and associated time scales.

Solid state 2 H NMR.

The seminal early contributions of Alan English et al. have provided the most detailed information available so far on the segmental dynamics in the crystalline phase of nylons. [START_REF] Hirschinger | Segmental dynamics in the crystalline phase of nylon 66: solid state deuterium[END_REF] Solid state 2 H NMR on selectively deuterated PA66 coupled with line shape analysis helped to determine the nature, amplitude and rate of the molecular motion of all the methylene units and the amide groups in the chemical repeat unit, as illustrated in Figure 8. The movements of the CH2s are quite complex. [START_REF] Hirschinger | Segmental dynamics in the crystalline phase of nylon 66: solid state deuterium[END_REF] The line shapes indicate a "large-amplitude uniaxial libration" occurring about the main chain bonds. The observed movements are "a fast jump motion between two equally populated states… with an additional libration motion in each site of the two-site jump". More precisely, the authors consider "an inhomogeneous distribution of amplitude of motions" that combines an inhomogeneous distribution of jump angles P(F) between two different conformation sites with librations at these sites P(Dq) (Figure 8d). These librations have a Gaussian azimuthal distribution of standard deviation Dq with a low angle cutoff (Figure 8 a and b). The variation of the angular range P(Dq) illustrates vividly how the libration amplitude increases during the upswing of the inter-sheet spacing below the Brill transition. Thereafter, in the Brill structure, "the amplitude of the rapid librational motion of all the methylene group C-D bonds is quite similar (Figure 8c). [START_REF] Hirschinger | Segmental dynamics in the crystalline phase of nylon 66: solid state deuterium[END_REF] ["all" is here emphasized, as it will feature in an analysis to be developed soon].

The broad line NMR analysis also indicated that the libration amplitude depends on the position of the methylenes relative to the amide groups. Specifically, the methylenes groups next to the amides have a much larger libration amplitude than the middle methylenes.

These studies provided the magnitude of the angular jumps and the associated ≈ tens of picoseconds time scales, but no clear-cut structural model could be developed to explain these data.

Two-site jumps were considered early on in the analysis of the Brill transition and structure. Itoh 23 mentions the possibility of two equilibrium positions and a "violent oscillation" between them at high temperature as well as the transition over a wide temperature interval from a low temperature one-state model. In his analysis, the two "equilibrium" positions have however two weaknesses: they correspond to unrealistic azimuthal settings of the stems that moreover are in the alpha phase. [START_REF] Itoh | Change with Temperature in Crystal Structures of Nylons 6, 66 and 610 Jap[END_REF] Other detailed observations remain to be understood. At low T° (i.e. below 160°C) a constant feature is the higher mobility of the CH2s in the amidoyl part than in the acid part -a "counterintuitive" feature in Hirschinger et al.'s words, [START_REF] Hirschinger | Segmental dynamics in the crystalline phase of nylon 66: solid state deuterium[END_REF] but also observed by IR. [START_REF] Yoshioka | Structural change in the Brill transition of nylon m/n (2) conformational disordering as viewed from the temperature-dependent infrared spectral measurements[END_REF] The different mobility may be due to different torsion angles in the acid and amine junctions at low temperature (cf. the models of Figure 5 in the companion paper). Rastogi et al. observe also that, in nylon 46, the NH-CH2 bond becomes twisted on heating at≈140°C before the CO-CH2 link, at≈180°C whereas the opposite is true for PA66. [START_REF] Vinken | Role of Superheated Water in the Dissolution and Perturbation of Hydrogen Bonding in the Crystalline Lattice of Polyamide 4[END_REF][START_REF] Dijkstra-Vinken | Polyamides: Hydrogen bonding, the Brill transition, and superheated water[END_REF] Beyond these more detailed issues, the two-site jump process determined by solid state 2 H NMR is fully consistent with an interconversion process between two mirror chain conformations. The nature of the movements is adequate. The movements of all the CH2s are confined in a plane normal to the chain axis. Below the Brill transition a progressive increase of the torsion angles at the amide links (Figure 8c) pushes the methylene groups away from the medium H bonded sheet plane, which accounts for the upswing of the inter-sheet distance on heating (Figure 1). Above the Brill transition the amplitude of the CH2s librations is quite large and constant. The F angle in Figure 8d has reached its maximum value, when the CH2-CH2-CH2 planes orientation sweeps a ≈120° angle between sheet structures that are "stable" with "equilibrium" torsion angles values at the amidealiphatic parts junctions They correspond to the two mirror images of the same energy minimized pleated/rippled sheets structures illustrated in Figures 456and to the two energy wells in Figure 8d.

Their identical energy accounts for the equal population of the two sites in the interconversion.

Note the distinction that can be made between the angular sweep and the magnitude of the libration.

In the present scenario, the angular sweep is identical above the Brill transition for all the methylenes of the chemical repeat unit. This stems from the assumption of a rigid aliphatic part (or an averaging of motions that results in a near-extended aliphatic part) and the use made in the models of equal torsion angles at the acid and amine links, imposed by the NMR data above the Brill transition (below the transition, these angles may well be different, cf. the companion article). [START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF] However, even though the angular sweep is identical, the magnitude of the sweep, the distance travelled by the different methylenes, varies with their position relative to the amide groups. As illustrated in Figure 4, the methylenes attached to the amide units travel over longer distances. They are the outer parts of a CH2-CO-NH-CH2 "peptide group", a long and rigid entity that swivels about its middle axis defined by the hydrogen bonds. By contrast, the sweep is much more limited for the central CH2s, where the aliphatic parts cross the middle plane of the H-bonded sheet.

The above analysis considers the pleated/rippled sheets "at rest" in their mirror symmetry structures (the wells bottoms in Figure 8d). The additional libration motion in each site of the two-site jump model appears to be related with thermal agitation and is therefore appropriately investigated by infra-red techniques and molecular dynamics methods.

Infrared data and vibrational decoupling of the CH2s alpha to the amide groups.

Extensive analyses of the spectroscopic information provided by various types of nylons and their model compounds are mostly due to Tashiro et al. The analysis of the IR progression bands of several PAs (PA66, 1010, 1012) is of particular relevance in the present context. [START_REF] Yoshioka | Structural change in the Brill transition of nylon m/n (2) conformational disordering as viewed from the temperature-dependent infrared spectral measurements[END_REF] Thanks to a novel and innovative analysis of these progression bands, Tashiro et al. could characterize the length (or number of carbons) included in an oscillating linear stretch of aliphatic carbon atoms. [START_REF] Yoshioka | New interpretation of progression bands observed in infrared spectra of nylon-m/n[END_REF] They observe at high temperature "an invasion of gauche bonds" that reduces the length of these linear stretches (to be considered soon). Most importantly, they also observe systematically a "motional inhibition" of the CH2s attached to the amides. To quote: "The inner methylene segments are vibrationally decoupled with the terminal methylene unit adjacent to the amide group". [START_REF] Yoshioka | Structural change in the Brill transition of nylon m/n (2) conformational disordering as viewed from the temperature-dependent infrared spectral measurements[END_REF] Related work by two-dimensional correlation infrared spectroscopy singles out also the methylene attached to the NH: it moves before (at lower T°) than the inner methylenes in the diamine segment of PA66. [START_REF] Luo | Mechanism in Brill transition of polyamide 66 studied by two-dimensional correlation infrared spectroscopy[END_REF] The "vibrational decoupling" of the sole CH2s adjacent to the amide groups revealed by the IR progression bands is at first sight inconsistent with the NMR data that indicate "essentially identical" angular sweeps for all the CH2s in the Brill structure. This inconsistency is only apparent.

Actually, two very different time scales apply for the thermally induced mobility in the aliphatic segments and for the oscillation of the amide groups. The IR bands are in the ≈800 to ≈1100 cm -1 range, which sets the frequency at ≈300-400 THz: the inner methylenes vibration time scale is in the ≈10 -15 sec range. The rate of oscillation of the amide units can be determined by the rate of libration of the attached CH2 groups since, in the present analysis, they are coupled (cf. Figure 4).

The rate of libration is in the tens of picoseconds over the entire temperature range examined by Infrared data and the role and impact of gauche bonds in the aliphatic parts.

All IR studies indicate specific bands corresponding to the formation of gauche bonds in the aliphatic parts, an aspect of the Brill transition that has been neglected so far in the present analysis.

In the thorough study performed by Yoshioka and Tashiro, PA1010 and three model systems were investigated: one sample with a middle amide group sandwiched between the amine and acid aliphatic parts and two samples with a central diamine or diacid segment end-capped with the aliphatic segments of the diacid or diamine part, respectively. [START_REF] Yoshioka | Structural change in the Brill transition of Nylon m/n (1) Nylon 10/10 and its model compounds[END_REF] For all three materials, evidence of twisted conformations at the amide linkages appears at low temperatures (<100°C), with as a rule, The rate of formation and the lifetime of the gauche bonds is difficult to evaluate. Molecular dynamics simulations (cf. below) suggest that they appear at intervals of several tens of picoseconds [START_REF] Wendoloski | Molecular dynamics in ordered structures: Computer simulation and experimental results for nylon 66 crystals[END_REF][START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF] and have lifetimes lasting less than or a few picoseconds (cf. Figure 6 in ref. 29)lifetimes that remain however long compared to the time scales explored by IR. During that short life, they are considered to "travel" towards the amide groups and induce the twisted conformations at the amide-aliphatic parts links. [START_REF] Wendoloski | Molecular dynamics in ordered structures: Computer simulation and experimental results for nylon 66 crystals[END_REF][START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF] It is possible (and tempting) to transfer this information gained on the alpha phase to the pleated/rippled sheets structures -which brings in yet another (very tentative) scenario. The gauche bonds generated in the aliphatic parts "traveling" to the amide junctions may have two different functions and fates. Since NH and CO bonds are already twisted, two possibilities may be considered: (a) the incoming twist (gauche or skew) is opposite to the existing twist at the amide link. It triggers the conformation "untwisting" that possibly goes on and ends up in a conformation reversal (b) the twist sense is the same as the amide link one. The gauche (or skew) conformation cannot "add up" to the existing twist of the CH2-amide bond. It is stopped at the previous CH2-CH2 link, which translates in a shortening of the extended aliphatic part by one link (or three links -the distinction between odd and even aliphatic bonds [START_REF] Hirschinger | Segmental dynamics in the crystalline phase of nylon 66: solid state deuterium[END_REF] would apply here also). In brief, in this scenario, the generation of shorter, odd numbered CH2 stretches results from the different, longer life times of the twisted conformations at the amide links compared with the segmental dynamics within the aliphatic parts. Note that these insights (if valid) could be gained only because polyamides with long enough aliphatic sequences are available in which the progression bands show up and were analyzed.

Molecular dynamics.

The molecular mobility and amplitude of movements below and above the Brill transition have been further tackled by molecular dynamics methods, both by English 28 and Tashiro [START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF] and associates. Historically, the origin of the Brill transition has been ascribed to a "conformational disordering" in both the methylene segments and CH2-amide links occurring in a wide temperature region (90°-170°C according to Luo et al. for bulk crystallized PA66) [START_REF] Luo | Mechanism in Brill transition of polyamide 66 studied by two-dimensional correlation infrared spectroscopy[END_REF] . The conformational disordering in the methylene segment is related strongly with (or induces?) the twisting motions about the amide-CH2 bonds. Also, this "wide temperature region" is not linked with the "polymeric" nature of the materials since it exists for the low molecular-weight model compounds investigated by Cooper et al. [START_REF] Cooper | A polarized -FTIR study on a model system for nylon 6 6: implications for the nylon Brill structure[END_REF] Molecular dynamics simulations performed on PA66 support the statement that first and foremost, the Brill transition is a "structural disordering process". [START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF] They reproduce the molecular movements in the aliphatic parts, up to the generation of gauche bonds. The frequency of appearance of the gauche bonds depends of course on the temperature, but as indicated above, the time scales are in the picosecond range. [START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF] On the structural side, they result in a significant departure from the trans conformation at the aliphatic part-amide links (mostly at the CO-CH2 links) [START_REF] Wendoloski | Molecular dynamics in ordered structures: Computer simulation and experimental results for nylon 66 crystals[END_REF][START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF] , maintain a "near-rigid restricted rotation of the entire diamine moiety" [START_REF] Wendoloski | Molecular dynamics in ordered structures: Computer simulation and experimental results for nylon 66 crystals[END_REF] and confirm the limited rotational motion of the amide groups below 230°C.

Molecular dynamics methods as performed by Wendoloski et al. reproduce very accurately the mean values of the librational distribution for all the methylenes of PA66 at room temperature. [START_REF] Wendoloski | Molecular dynamics in ordered structures: Computer simulation and experimental results for nylon 66 crystals[END_REF] On heating, the librational amplitude increases relatively smoothly. At high temperature however, the calculated amplitudes are very significantly lower than the experimental ones: their mean amplitudes ranges are 22-27° at 230°C and 30-32° at 240°C whereas the 2 H NMR experimental values are in the 55-62° bracket. [START_REF] Hirschinger | Segmental dynamics in the crystalline phase of nylon 66: solid state deuterium[END_REF] This disturbing mismatch might question the validity of the MD approach. It does not. The origin of the failure in the present case is linked with the starting model used: the MD simulations use the PA66 alpha phase in the whole temperature range investigated -up to 228°C 28 and 600K 29 in the two different investigations. In the interconversion scenario, as indicated above, the amplitude of the libration is no longer determined by the sole thermally induced libration. It includes also a major component linked with the conformation interconversion of the stems. This component is missing in the MD simulations as performed. The MD simulations explore in effect only the thermally induced libration, i.e. the half-width Dq. In the experimental Gaussian curve, this halfwidth is in the ≈20° range above TB and is indeed consistent with the MD modeling. and HT-2 (230°C), that is, as indicated, way above the Brill transition. [START_REF] Wendoloski | Molecular dynamics in ordered structures: Computer simulation and experimental results for nylon 66 crystals[END_REF] The various conformations reached are far from the pleated/rippled sheets ones. Their path away from the extended chain conformation helps reach further aspects of the transition from extended to pleated/rippled conformations i.e. processes taking place below the Brill transition. Since the MD approach used does not generate pleated chain conformations, the high temperature MD helps visualize the (presumably maximum) "violence" of molecular movements that the aliphatic segments of the alpha phase can sustain. Remarkably, the values of the torsion angles and their distribution determined for the HT-1 case (nominally at 240°C) nearly match the experimental NMR data obtained at 162.7°C. The level of "violence" of the molecular motions modeled by MD at higher T° appears to be reached already at lower T°, actually at the transition stage when the alpha phase structure is destabilized and switches to pleated/rippled sheet-type conformations (cf.

Figure 8a). [START_REF] Hirschinger | Segmental dynamics in the crystalline phase of nylon 66: solid state deuterium[END_REF] In addition, the MD simulations also agree with the different thermal response of the CO and NHaliphatic parts connections. In figure 9, the F angle (NH-CH2 bond) variation is much less than the y angle of the CO-CH2 bond. Similar differences in variations are actually observed by NMR (low T° part of Figure 8c, when the structure is still in the alpha phase).

The above approach combining two powerful investigation tools helps analyze details and causes of molecular processes. It would be of interest to repeat these simulations for example (a) for PA46, where contrary to PA66, NMR data indicate that the NH-CH2 torsion angle changes at a lower temperature than the CO-CH2 one (Is this due to a different alpha phase structure with antiparallel stems?) [START_REF] Vinken | Role of Superheated Water in the Dissolution and Perturbation of Hydrogen Bonding in the Crystalline Lattice of Polyamide 4[END_REF][START_REF] Dijkstra-Vinken | Polyamides: Hydrogen bonding, the Brill transition, and superheated water[END_REF] (b) for the pleated/rippled sheets, in order to evaluate more precisely the contribution of the dynamic sub-molecular movements that are an important ingredient in the Brill transition and structure.

IV. The Brill transition scenario: structural aspects linked with the interconversion, thermally induced molecular movements and reorganization.

The scenario of a Brill transition implying pleated/rippled sheets is consistent with NMR, IR and (presumably, after correction of the starting model used) molecular dynamics analyses. This scenario offers a new frame to approach unsolved and/or controversial issues of the Brill transition.

In the following, some of these topics are touched upon: (a) the exact characteristics of the Brill structure unit cell (b) the partition of the inter-sheet distance increase between the interconversion and mobility or "melting" of the aliphatic parts (c) the prevailing, but not exclusive role of the hexagonal unit-cell of the Brill structure and the possibility of inter-sheet hydrogen bonding (d) the structural root(s) of the very wide temperature range over which the Brill transition takes place for any given polymer.

The Brill structure: variability of the cell geometry.

The Brill structure is frequently characterized with bulk crystallized materials and powder patterns, which makes it difficult to define a clear three-dimensional unit-cell geometry. The cell is described as triclinic with a pseudo-hexagonal projection that does not differ much from the RT alpha phase unit-cell. However, the experimental data are frequently limited. [START_REF] Hirschinger | Segmental dynamics in the crystalline phase of nylon 66: solid state deuterium[END_REF] They are also relatively old: no recent redetermination of the Brill structure unit-cell using modern equipment and data processing facilities has been attempted or reported.

The cell geometry of the Brill structure raises actually several issues. Two of them -an observation and a puzzle -are briefly outlined in the following. The biaxially oriented sample used by Itoh illustrates the simplest transformation mode between the RT and the Brill structure (but with its own problems, cf. below) (Figure 10). [START_REF] Itoh | Change with Temperature in Crystal Structures of Nylons 6, 66 and 610 Jap[END_REF] The RT tilt (≈38°) of the 00l reflections of the alpha phase is maintained in the Brill structure, at 160°C. In These tilts are reminiscent of the PA48 ones and tend to suggest that slightly different torsion angles at the NH and CO bonds may be involved. [START_REF] Lotz | Original crystal structures of even-even nylons made of pleated and rippled sheets Macromolecules[END_REF] The data of Tashiro and Yoshioka on PA1010 tell a more complex story. [START_REF] Tashiro | Conformational disorder in the Brill transition of uniaxiallyoriented nylon 10/10 sample investigated through the temperature-dependent measurement of X-ray fiber diagram[END_REF] The 00l reflections are at some 22° to the fiber axis (Figure 11,part d). This tilt is observed on first heating, with a "clean" 4.2Å equatorial reflection and it is maintained after two excursions at 190°. After these excursions however, the equator reflections are more spread out, with a clear indication of two reflections confirmed by a densitometric scan. They indicate a (partial?) conversion to a form of, probably, pleated/rippled sheet crystal structure.

The various tilts of the 00l reflections suggest that the Brill structure was reached via different structures and that it keeps a memory of these parent structures at least as far as the shear of the layers is concerned. A progressive increase of the 00l spacings with temperature (meaning progressively lower shear between layers) has been documented in lamellar crystals of even-even nylons. The conclusion of the simple 00l tilt test stands however: even for a single, "homogeneous" family such as the even-even nylons, the "Brill structure" cannot be defined by a "standard", single unit-cell geometry and, by implication, by a detailed crystal structure. Presence of two reflections only may reflect either a structural disorder or the fact that the interactions involved in two of the spacings become similar. The structure of the doubly oriented films investigated by Itoh appears to support the first option. [START_REF] Olf | NMR Observations of Drawn Polymers. VII. Nylon 66 Fibers[END_REF] Only two reflections are seen in Figure 10, in the room temperature pattern taken end-on (along the PA66 chain axis of the film). The sheets are parallel to the film, the relative orientations of the 010 intersheet and 100 reflections are consistent with an alpha phase PA66 structure -but the 110 reflections are missing. Their expected positions and spacing are indicated by the question marks shown in the inset of Figure 10.

Their absence may be due to twinning. However, in single crystals twinning preserves the 110 reflections. In these mechanically oriented films, the density of twin planes is probably much higher, resulting in a form of polysynthetic twinning that leaves the inter-sheet 010 reflection unaffected but spreads and reduces the intensity of the 100 and 110 reflections, up to the extinction of the latter, initially weaker one.

Their absence may be associated with the interconversion process. Above the Brill transition, the interconversion does indeed "symmetrize" the roles and the structure of the (100) and (1-10) planes (cf. Figure 5) and these planes remain at 60° to one another. After the double orientation processing and subsequent annealing at 230°C for one day, this more symmetric structure may have been memorized and "frozen in" during the cooling to room temperature.

At any rate, this puzzle must be settled in future investigations, in particular by investigating the impact of deep quenching of thin samples in the Brill structure.

The Brill structure: the various contributions to the inter-sheet increase.

The MD simulations indicate that the segmental motion in the aliphatic parts cannot account for the whole inter-sheet spacing increase observed in the Brill transition. The partition between the different contributions of heat induced motions and crystal transformation is nevertheless accessible from an analysis of the models and experimental data.

The interconversion is expected to have a large impact on the inter-sheet distance and thermal expansion. As seen in Figure 4, the distances between the two extreme positions are ≈1.2Å and ≈1.9Å apart for the aCH2s in the acid and amine segments, respectively. The distances are more limited for the middle CH2s since the aliphatic segment crosses the medium H bonded sheet.

For crystal structures at rest, the interdigitation of the pleated sheets limits however the intersheet expansion compared to the alpha phase. Figure 2d illustrates this for PA46. Both alpha and pleated sheet structures exist at 50°C. The inter-sheet increase is only 0.14Å whereas the 100 reflection decrease is larger: 0.22Å. Most of the inter-sheet increase is therefore associated with dynamic aspects of the molecular interconversion.

On heating, the libration sweeps a progressively larger angular range. The impact on the position of the methylene units (i.e. the distance away from the medium H-bonded plane) varies like the hexagonal unit-cell bears similarities with PGII, the hexagonal crystal form of polyglycine or nylon 2. The nylons cell symmetry and spacings are the same, which strongly suggests a structure with three different orientations of the hydrogen bonds as exist in PGII or, at least, with formation of partial and/or temporary inter-sheet hydrogen bonds.

Numerous studies, especially by polarized µ-FTIR on single crystals of low Mw models of PA66 [START_REF] Cooper | A polarized -FTIR study on a model system for nylon 6 6: implications for the nylon Brill structure[END_REF] indicate however that the room temperature hydrogen bonds orientation is maintained above the Brill transition -thus the pre-eminence of intra-sheet bonds. The hexagonal symmetry is therefore explained by the conformational/structural disorder modeled by molecular dynamics that produces a hexagonal packing of aliphatic parts "cylinders" in a rotator phase. These cylinders would have a cross section of ≈20Å 2 , comparable to the smectic phase of PE. As just mentioned, the pleated/rippled sheets would be in line with a preservation of hydrogen bonds: the seesaw oscillation of the amide groups increases the inter-sheet spacing but maintains the orientation of the hydrogen bonds (cf. Figures 5 and6).

Another, geometric argument against the formation of inter-sheet hydrogen bonds has frequently been given: based on the then available unit-cell determinations, the shift of layers in the Brill structure is too large, the CO and NH groups are too far apart to allow inter-sheet hydrogen bonding -a situation that parallels the alpha phase case also.

This view of exclusive intra-sheet hydrogen bonds has however been challenged. Ted Atkins has strongly advocated the impact of a "symmetrization" process above the Brill transition, with formation of inter-sheet hydrogen bonds -even if only intermittent and with different populations in the three trigonal directions. [START_REF] Jones | Comparison of structures and behavior on heating of solution-grown, chain-folded lamellar crystals of 31 even-even Nylons[END_REF][START_REF] Jones | Investigation of Solution-Grown, Chain Folded Lamellar Crystals of the Even-Even Nylons[END_REF][START_REF] Atkins | Structural and morphological investigations of nylon 8 chain-folded lamellar crystals[END_REF][START_REF] Cooper | Temperature-Induced Changes in Lamellar Crystals of Monodisperse Nylon 6 and Nylon 6 6 Oligoamides[END_REF] This is the only reasonable way to explain the cohesion of the layered structure when thermal agitation is high. It also justifies the symmetric, uniform thermal expansion up to TM, sometimes over more than 100°C in a supposedly asymmetric crystal structure (cf. Figure 2d). The translation of these assumptions in molecular terms was however difficult. The models use an extended chain conformation and must assume that a proportion of stems are shifted along the chain axis to allow formation of the H bonds in three directions.

The recognition of pleated and rippled sheets makes it possible (a) to better analyze the experimental data (b) to provide an alternative model (or hypothesis) for the establishment of intersheet hydrogen bonds (c) on this basis, and to concur with many authors, to explain why the hexagonal unit-cell symmetry is (rightly) no longer considered as a mandatory marker of the Brill transition.

Experimental data: thermal expansion and inter-sheet spacings.

The X-ray diffraction data of nylons with short aliphatic segments (e.g. PA46) illustrate quite vividly the thermal responses for the various crystal structures accessible to aliphatic nylons.

According to the data illustrated in Figure 2d, the (linear) thermal expansion of the Brill structure spacings in PA46 is 9.5x10 -4 Å/°C in a 130°C interval, from 120 to 250°C (in bulk [START_REF] Ramesh | New Crystalline Transitions in Nylons[END_REF] and in single crystals [START_REF] Vinken | Role of Superheated Water in the Dissolution and Perturbation of Hydrogen Bonding in the Crystalline Lattice of Polyamide 4[END_REF] ). This very same constant value (9.72x10 -4 Å/°C) can be determined from the data published by Slichter on nylon 57 for a 200°C interval (RT to 225°C). [START_REF] Slichter | Molecular motion in Polyamides[END_REF] These odd-odd nylons with hydrogen bonding in three directions provide therefore reliable thermal expansion coefficients for "isotropic" hydrogen bonded nylon structures. These nylons also experience thermally induced mobility of the aliphatic segments, but the potential impact of this mobility is annihilated by stronger inter-stem and inter-sheet interactions -which points at a hydrogen bonds hexagonal-type network as the most likely explanation.

Inter-sheet hydrogen bonding: an alternative model.

In this unsettled controversy context, it is of interest to examine what insights may be suggested by the interconversion process developed above -and to develop an alternative structural model for the occurrence of inter-sheet hydrogen bonding.

The revised "polyglycine" conformational energy map is again useful for this purpose (Figure 12).

The two enantiomeric conformations are symmetrically positioned relative to the extended chain conformation. A "direct" interconversion would be via the extended chain conformation. The weakness of this trajectory is that it crosses the extended chain conformation, which would result in an accordion-type "breathing" of the stem c axis repeat distance. The energy barrier at the CH2-NH bond is much lower than for the CH2-CO bond (0.9 kcal/mol versus 3.2 kacl/mol in Tashiro's analysis). [START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF] More sinuous paths may be preferred that avoid or limit this breathing. Possible or likely paths may cross the conformations that exist in the polyglycine II crystal structure, indicated as PGII on the map (Note that this PGII position may also be reached in "aborted", incomplete interconversions). Here again, a path that crosses the extended conformation (illustrated as a thinner line) may not be ideal. A path that amounts to travel in a "round about" around the central energy may be preferable (thick blue line). For such a conformational pathway, the H bonds orientation is deviated and can allow formation of inter-sheet hydrogen bonds as they exist in the polyglycine II crystal structure. The lifetimes of these transitory H-bonds in nylons would be very short but repeated, since the interconversion frequency is in the 10 10 to 10 11 Herz range. Even if H bonds are not made at every interconversion, their frequency is sufficient to translate in a permanent structural feature (the hexagonal symmetry of the unit-cell) but the lifetime of these transient H bonds is too short to impact the global orientation of the H bonds as determined by IR.

Breaking and reorganization of hydrogen bonds is a well-known process, especially in the temperature range of the Brill transition. In the nylons case, H bonds break and reorganize when the lamellar thickness doubles as a result of annealing. [START_REF] Burmester | On the annealing of polyamide crystal mats[END_REF] They also break and reorient in the hightemperature morphologies (spherulitic aggregates and triangular crystals). [START_REF] Lotz | Original crystal structures of even-even nylons made of pleated and rippled sheets Macromolecules[END_REF] In nylons with short

Conversely, no such reorganization can take place in the doubly oriented nylon 66 investigated by Itoh. The large 38° tilt of the 001 reflections indicates a prohibitive shear for the formation of intersheet hydrogen bonding. In their absence, the three reflections do not merge, in both the PA66 and PA610 doubly oriented samples. [START_REF] Olf | NMR Observations of Drawn Polymers. VII. Nylon 66 Fibers[END_REF] The geometric rule that imposes sufficient proximity of the amide groups has actually general validity: nylons with a chemical constitution modified in different ways, or for that matter nylon 6 [START_REF] Murthy | Premelting crystalline relaxations and phase transitions in nylon 6 and 6[END_REF] experience the Brill transition without reaching the hexagonal symmetry.

A summary of the Brill transition characteristics.

The above scenario is consistent with the present consensus on the Brill structure and transition.

The transition is not defined by the meeting point of the hk0 reflections, i.e. by the pseudohexagonal symmetry of the unit-cell. Thermally induced motions as observed in the first stages of the inter-sheet upswing are random ones and are not specific enough to characterize the Brill transition. In the present model, the transition results specifically from the formation of twisted chain conformations and their interconversion allowed by the achiral character of these aliphatic nylons. This conformation isomerization takes place between two stable (or metastable) conformations.

In nylons with short aliphatic segments, the twisted chain conformations exist at low T° in their "still" versions, in the form of a different pleated/rippled sheets crystal structure. For these crystal structures, the Brill transition appears when the conformation interconversion sets in, at higher temperature. The interconversion process accounts for the major part of the molecular movements:

the angular sweep is ≈120°, which leaves only a limited margin for the additional and unavoidable thermal agitation to contribute to the inter-sheet distance increase.

In nylons with longer aliphatic segments -typically nylon 66 -the thermal agitation of the aliphatic parts induces a crystal-crystal transformation to the pleated/rippled sheets structures, followed by the dynamic interconversion that defines the Brill transition. The two processes may be merged, which has been a major difficulty in defining the nature of the transition. For these same materials and for even longer aliphatic segments, the pleated/rippled sheet structures may be formed at high temperature either after self-seeding or, more insidiously, during annealing. Since annealing is routinely used to stabilize or increase the crystal perfection, this little-known complication has led to many reports of puzzling observations -and definitions of the Brill transition. The Brill transition is not some ill-defined form of "packing change" but is defined by the attainment (or the appearance, for nylons with short aliphatic parts) of a stable, dynamic regime of interconversion between stereo-conformers, which is a highly specific intra-sheet form of structural disordering.

As an obvious aside, no reverse Brill transition is expected on cooling e.g. nylon 57 with its hexagonal unit-cell and three H bonds orientations. The Brill transition is specific to sheet-type structures and the possibility to generate mirror image conformations of these sheets without H bonds breaking and/or (major?) reorganization.

nylons samples were "cold crystallized" by annealing at Tc. The TBs vary from 110°C to 190°C

for TCs between 120°C and 175°C: the variation of TB is more important than TC. Extrapolation to the melting temperature (≈ 202°C) would correspond to long periods of ≈100Å. Lamellar thicknesses in the ≈130Å range formed on cooling after self-seeding would correspond to TB ≈240°C, well above TM.

The spread of TB is usually explained by differences in crystal perfection, but in ways that remain to be understood. Is only the state of the crystal involved (crystal perfection, lamellar thickness, etc.)? Is the inter-crystalline, folds and amorphous material and its connection with the crystal core also involved? If the process is mostly an intra-sheet one and the sheet as a whole is the relevant structural element, does the connection between sheets play a role? These impacts need to be investigated, or earlier analyses may need to be recast in the present frame of different processes associated with the Brill transition. For the present, a few simple observations must be recalled.

(a) TB varies considerably with the lamellar thickness. The ultimate illustration of this trend is provided by pleated and rippled sheet structures formed at high Tc (sometimes referred to as "high temperature a phases"). These sheets are "stable" yet they crystallize in temperature domains well above the conventional Brill transition and their aliphatic parts experience the same thermal conditions as in the Brill structure -well above the onset of TB.

(b) The "mobility" of the aliphatic parts revealed by NMR, IR and e.g. quasi-elastic neutron scattering 45 led Wunderlich to consider that the aliphatic parts "loose the(ir) solid nature through a glass transition below the melting point, best measured by the change of CP". [START_REF] Xenopoulos | Conformational motion and disorder in aliphatic nylons. The case of nylon 6.6[END_REF] (c) The amorphous phase appears to play an important role. Feldman et al. observe a 15°C increase of TB in trans-crystalline PA66 compared with the parent PA66 spherulites. They invoke the role of "reciprocal interactions between the crystalline and amorphous phase". 5,[START_REF] Wolanov | Amorphous and crystalline phase interactions during the Brill transition in nylon 66 eXPRESS[END_REF] This same line was followed earlier on by Murthy et al. from an analysis of SAXS data: "a change in the packing of crystalline segments… provoke a proportionate response in the packing of the amorphous chain segments in the interlamellar regions". [START_REF] Murthy | Interactions between Crystalline and Amorphous Domains in Semi-crystalline Polymers: Small-Angle X-ray Scattering Studies of the Brill Transition in Nylon 6[END_REF] (d) The role of the amorphous phase is suggested by different, but little considered experimental observations. Brill already mentioned that some residual solvent left in the material decreases the transition temperature, which questioned early on the role of crystal perfection as the sole factor determining TB. [START_REF] Brill | Über das Verhalten von Polyamiden beim Erhitzen[END_REF] Similar observations deal with the impact of water. [START_REF] Brill | Über das Verhalten von Polyamiden beim Erhitzen[END_REF][START_REF] Vinken | Role of Superheated Water in the Dissolution and Perturbation of Hydrogen Bonding in the Crystalline Lattice of Polyamide 4[END_REF][START_REF] Dijkstra-Vinken | Polyamides: Hydrogen bonding, the Brill transition, and superheated water[END_REF] The mobility/structure of the amorphous part must be involved.

(e) The role and impact of the different components of the amorphous parts must be elucidated.

Following the approach used by Wunderlich et al., the folds, rigid amorphous and amorphous phases are expected to influence the Brill transition in different (but still poorly understood) ways.

As a development of the pleated/rippled sheets involvement in the Brill transition, the possible nature of the "reciprocal interactions" must be addressed. It is here focused on the existence and role of the "rigid amorphous" component and more specifically on inter-sheet folds. At the same time, it must be stressed that the Brill transition takes place in short model compounds with no folds. [START_REF] Cooper | A polarized -FTIR study on a model system for nylon 6 6: implications for the nylon Brill structure[END_REF][START_REF] Cooper | Temperature-Induced Changes in Lamellar Crystals of Monodisperse Nylon 6 and Nylon 6 6 Oligoamides[END_REF][START_REF] Yoshioka | Structural change in the Brill transition of Nylon m/n (1) Nylon 10/10 and its model compounds[END_REF] Folds are not necessary to induce the transition but they influence the location of TB.

The Brill transition is characterized by an unusual expansion of the inter-sheet spacing. Nylons are the only, or among the few polymers for which thermal expansion takes place in one crystallographic direction only. On a molecular scale, the magnitude of the expansion, and stresses on folds, is important. It is comparable to or higher than the surface stresses (in that case, unbalanced) leading to twisting or scrolling of polymer lamellae (cf. S2). [START_REF] Lotz | A critical assessment of unbalanced surface stresses as the mechanical origin of twisting and scrolling of polymer crystals[END_REF][START_REF] Lotz | Structure of Negative Spherulites of Even-Even Polyamides. Introducing a Complex Multicomponent Spherulite Architecture Macromolecules[END_REF] by heat-induced segmental mobility in the crystal lattice that expand the inter-sheet distance, as established by MD simulations. This expansion affects the amorphous component. In that amorphous part, the inter-sheet folds (a part of the "constrained amorphous phase" 51 ) play a specific, dual role. They are strained as a result of the crystal thermal expansion, and because they are strained they destabilize the crystalline stems (and thus sheets) they connect. Since the density of inter-sheet folds depends on crystallization conditions, it may range over large limits -and in particular may approach zero at high TC or in single crystals. The "structure" (orientation) of this amorphous phase element, together with the crystalline core perfection and lamellar thickness contribute to determine the Brill transition and are at least in part at the root of its unusually wide temperature range.

II. Possible additional insights and further investigations.

Developing what appears to be a reasonable structural scenario for the Brill transition is not the end of the road. As just illustrated, a number of additional insights must be reached. The most critical ones deal with the transposition of these local process to the whole material. How and how fast does the interconversion spread within (along) any one stem, to neighbor stems within the hydrogen bonded sheets and to neighbor sheets? How cooperative is the Brill transition? Various, and sometimes scattered data and approaches must be considered in this context: (a) the contributions of the DSC (b) the additional insights provided by polymers that differ from the nylons but that also experience the Brill transition (c) the analogy between the Brill transition and the amine inversion.

II.1. Differential scanning calorimetry data.

DSC is a priori a powerful technique to detect and analyze transitions of all kinds. Extensive DSC information on the Brill transition in nylons has been collected and should have been included

in the above analysis. It turns out that in the Brill transition case, DSC illustrates the uncommon features of this transition and the difficulty to reach any clear-cut conclusion about its nature. An endothermic peak, that would be expected for a first order transition, is not usually observed in bulk crystallized samples. A broad peak appears only in solution grown single crystals. [START_REF] Starkweather | Crystalline transitions in powders of nylon 66 crystallized from solution[END_REF] The present recognition of two different processes needed to complete the Brill transition does not invalidate the reported data, but certainly questions the possible (frequently tentative) interpretations provided.

The enthalpies determined for the Brill transition are quite scattered. A latent heat of 4.26 kJ/mol.

was reported for the transition by Olf and Peterlin. [START_REF] Olf | NMR Observations of Drawn Polymers. VII. Nylon 66 Fibers[END_REF] Starkweather and Jones note that the broad calorimetric response of the Brill transition DHB is only a fraction of the melting enthalpy DHM: -18.8 versus -125.6 J/g, i.e. 15%. [START_REF] Starkweather | Crystalline transitions in powders of nylon 66 crystallized from solution[END_REF] Wunderlich et al. mention a latent heat for the (Brill only?) transition of 2.44 kJ/mol. For a 44% crystallinity material, the enthalpy of melting DHM and of the Brill transition DHB are indicated as 27.7 and 0.9kJ/mol., respectively. [START_REF] Qiu | The phase structures of nylon 6.6 as studied by temperature modulated calorimetry and their link to X-ray structure and molecular motion[END_REF] Vinken et al. observe an exotherm with a DH of 20±2J/g on the first heating of several water-crystallized even-even PAs (46, 66, 610, 612) (the heats of fusion are 100±10 J/g). [START_REF] Dijkstra-Vinken | Polyamides: Hydrogen bonding, the Brill transition, and superheated water[END_REF] This peak corresponds nearly with the recorded TBs, but is not observed in a second heating, which suggests that it corresponds to the removal of water included in the crystal. As already indicated, these evaluations and assignments need to be reassessed in the two different transitions scheme. The task will not be easy, due to the breadth of the transition linked with the structural heterogeneity of crystalline polymers.

Thermally modulated DSC (TMDSC) indicates an original variation of the heat capacity CP of nylons. For a crystallized PA66 sample (with ≈30 to 35% crystallinity) CP exceeds that of the liquid above the Brill transition, and drops to the liquid value on melting (cf. Figure S3). [START_REF] Qiu | The phase structures of nylon 6.6 as studied by temperature modulated calorimetry and their link to X-ray structure and molecular motion[END_REF] In the present analysis, this observation highlights the important contribution of molecular mobility in the crystalline part of the material. Within the crystal, the librations resulting from the interconversion take place in an adapted and "dedicated" crystallographic space and environment: the stems are indeed confined but can librate. In addition, these librations involve the full crystalline stem length.

This contribution to the entropy by the interconversion-driven "conformational motion" 53 is therefore much larger than expected from the "vibrational limit" [START_REF] Qiu | The phase structures of nylon 6.6 as studied by temperature modulated calorimetry and their link to X-ray structure and molecular motion[END_REF] of the same parts in the melt, where the movements are hindered by the more random hydrogen bonding. In short, above the Brill transition, the crystal lattice (still stabilized by H-bonds) shelters two different molecular movements contributing to the overall entropy: thermal agitation (10 -15 sec time scale) and the much ampler interconversion (10 -10 sec time scale). Note that these molecular movements can hardly be considered as indicating a "glass transition of the crystalline phase", to refer to the elegant and provoking oxymoron introduced by Wunderlich to describe the Brill transition. [START_REF] Wunderlich | The glass transition of polymer crystals[END_REF] The above evaluations refer to global features. A noteworthy and up to now only (!) attempt at reaching a more local insight of the Brill transition is due to Starkweather. [START_REF] Starkweather | Deconvolution of the Excess Heat-Capacity of the Brill Transition in Nylon 66[END_REF] This author deconvolutes the diffuse endotherm extending from 130°C to 230°C in solution-grown PA66 single crystals with a two-state model also applied for transitions in globular proteins. He concludes that "the Brill transition appears to be characterized by two energy states and to occur in a space corresponding to about 25 chemical repeat units" (which corresponds to ≈ 5 stems of the crystal).

Since the homogeneous crystal domains are much larger, "the Brill transition occurs point by point involving small groups of adjacent stems". 52

II.2. The Brill transition in aliphatic polymers with different constitution.

The mobility of the aliphatic parts observed for nylons is known, to quote Wunderlich "for polyesters, polyurethanes, and polymers of other functional groups separated by longer CH2 sequences". [START_REF] Wunderlich | The glass transition of polymer crystals[END_REF] As such, these polymers may be useful in deciphering further details of Brill (or Brill-like in this case) transitions.

Clear examples of such "Brill transition"-type processes are to be found in poly(hexamethylene- The conformational isomerism described above for the Brill transition in nylons is not established yet for any of these systems. If Brill transition there is (as opposed to the sole onset of thermal agitation) some form of twisted chain conformation at the aliphatic parts-functional group junctions must be involved -which calls for further structural investigations on these systems.

II.3 A parallel between the Brill transition and the amine inversion process.

The close analogy between the conformation inversion described here for the Brill transition and the much-studied ammonia and amine inversion must be underlined.

The ammonia and amine inversion stem from the tri-coordination of the nitrogen atom, which generates a pyramidal molecule. [START_REF] Lehn | Nitrogen Inversion: Experiment and Theory in : Dynamic Stereochemistry[END_REF] This pyramid may be inverted, much like an umbrella, typically through a planar intermediate stage.

The nylon interconversion and the amine inversion both generate identical but mirror-symmetric molecules. More interestingly, the similarity of the rates of interconversion in the nylons and As an echo to the questions raised in the introduction about the nature of the Brill transition, it is appropriate to recall the difficulty of the task, that has led at times to bold or abrupt statements (e.g. the Brill transition is recognized only for even-even nylons of the type 6Y (with Y>8) and the socalled Brill transition in melt-crystallized nylons "is not a phase transition, but an abrupt thermal expansion" [START_REF] Ishikawa | Phase Transition in Even-Even Nylon Crystals[END_REF] or "a packing change in the crystal" [START_REF] Xenopoulos | Conformational motion and disorder in aliphatic nylons. The case of nylon 6.6[END_REF] ). These statements are for the most part right, if one accepts the criteria used to define the transition or within the frame set by the experimental techniques used. The commonly accepted view emphasizes a first order transition, seasoned however with second order elements or character. [START_REF] Tashiro | Molecular dynamics simulation of the structural and mechanical property changes in the Brill transition of nylon 10/10 crystal[END_REF]5 The most ambiguous manifestations, that also led to approximate or inappropriate analyses of the Brill transition stem from the progressivity of the transition, as illustrated by diffraction and thermal analysis data. Most of these approximate analyses result from the exceedingly specific criteria that have been used to define the Brill transition, i.e. the merging of reflections and pseudohexagonal symmetry of the Brill structure unit-cell, or the near-absence of a calorimetric indication.

The present scenario suggests that the Brill transition should not be defined by a structural (crystallographic) criterion but rather by the underlying molecular process, namely the conformation interconversion between the stereo-conformers accessible to the aliphatic nylons.

This criterion avoids the ambiguities linked with the known variability of the consequences of the transition, which do not necessarily sum up in the merging of the hk0 reflections. Rather, this criterion focusses on the cause. The cause, the conformation inversion is common to all types of nylons (even-even, even, etc.) submitted to all types of processing conditions and thermal histories.

3

 3

Figure 1 . 15 Figure 2 .

 1152 Figure 1. (a) The variation of d 100 and the combined d 010/110 spacings with temperature on cooling to room temperature a PA66 bulk material crystallized at Tc = 260°C (open circles), 232°C (filled circles), 196°C (filled squares). The corresponding Brill transition temperatures are indicated (box). (b) Heating and cooling curves for a sample crystallized at 196°C. Note that the data points for the heating curve at 150-180°C (but not the curve that was drawn) could indicate an intermediate phase, following the scheme illustrated in Figure 2c for nylon 46. Very similar data have been obtained by Brill back in 1942 (cf. Supplementary Information S1). Reprinted and adapted from Ramesh et al. (Reference 10), with permission from Elsevier.

Figure 3 .

 3 Figure 3. Conformational energy map of polyglycine modified and adapted for nylons. The

Figure 4 .

 4 Figure 4. The sweeps during the interconversion of the main chain (C or N) and the larger sweeps

Figure 4 .

 4 Figure 4. The two orientations of the amide units (bottom) and of all the CH2s in the PA66 repeat unit (top) generated by the interconversion between the two possible, mirror symmetry conformations of any one stem above the Brill transition. The angular sweeps are illustrated for ≈115° torsional angles at the amide-CH2 links. Labelling of the C and N atoms (the bottom right figure) is borrowed from Wendoloski et al. (Reference 28) and reprinted with permission from AAAS.

Figure 5 .

 5 Figure 5. Interconversion between mirror image chain conformations in a PA66 crystal structure

Figure 7 .

 7 Figure 7. Illustration of the impact of a dynamic interconversion between the mirror chain

Figure 8 .

 8 Figure 8. A sampling of the solid state 2 H NMR results and analyses due to A. English at al. (a)

A

  . English. (Note that Yoshioka et al. already mentioned a difference in rate of motion and dfferent, larger librational amplitude of the terminal methylenes among possible reasons for the vibrational decoupling.[START_REF] Yoshioka | New interpretation of progression bands observed in infrared spectra of nylon-m/n[END_REF] ) Comparison of the time scales involved -femtoseconds for the IR progression bands and tens of picoseconds for the amide groups oscillation -indicates that above the Brill transition the CH2s alpha to the amide groups only appear as immobile to the IR technique even though they reorient over large angles at a much lower frequency as a result of the interconversion. In the present structural scenario, the behavior of these alpha CH2s is thus very telling about their dual status: being part of the CH2-CO-NH-CH2 "peptide" group (a) they give away part of their "aliphatic character" and do not vibrate as the other methylenes (b) yet they remain parts of the aliphatic segments as attested by the "essentially identical", large angular swing of all the aliphatic parts during the interconversion. As such, they indicate that the global molecular mobility of the Brill structure includes two processes (conformational interconversion and thermal agitation) with widely different manifestations (large azimuthal jumps versus smaller scale librations: F versus Dq in Figure8d)) and with their own, different time scales (picoseconds versus femtoseconds).

  more twist for the CO links. An "enhancement of torsional motion" around both acid and amine links with the aliphatic parts was established, based on the half-width of the n[C-C(O)] band and the intensity reduction of the n(C-N) band. Frequency shifts and decrease of infrared band intensities indicate thermally induced conformational fluctuations of the aliphatic parts. The fate of progression bands may be more telling: the bands corresponding to straight, full length aliphatic parts are observed to decrease progressively as the temperature approaches the Brill transition, at ≈ 140°C. Additional bands characteristic of shorter linear stretches of the aliphatic parts set in, indicating the impact of gauche bonds: for PA1010, C8 stretches are replaced by C7 and C5 stretches.

Figure 9

 9 Figure 9 illustrates, as plotted on a polyglycine conformational energy map, the different

Figure 9 .

 9 Figure 9. The different conformations at the amide-CH2 links determined by molecular dynamics

  (a) The merging of the equatorial reflections, which identifies the "Brill structure" in powder patterns does not by itself indicate a uniqueness of its cell geometry. Fiber patterns of various nylons that all display the Brill structure (with the single ≈4.2Å reflection on the equator) do actually differ by the tilts of the 00l reflections to the fiber axis: they range from ≈20° to ≈40° (Figures10 and 11).

Figure 10 .

 10 Figure 10. The fiber pattern (top) and (bottom, in c axis projection) the equatorial plane of a

  itself, this result is surprising since the inter-sheet spacing has increased from 3.7Å to 4.2Å. It is however in line with a later observation made by Jones et al. on heating PA8 crystals in the l phase.[START_REF] Jones | Discovery in nylon-8 chain-folded lamellar crystals of new structure--structure-with progressive intersheet shear along the chain axis[END_REF] Jones et al. established that the inter-sheet distance increases in two different, successive processes. First, at low temperature, the layers move apart along the b axis of the unit-cell, which maintains the orientation of the (001) planes. At higher T, a combination of b and c axes translations takes place. In the doubly oriented specimen investigated by Itoh, the Brill transition appears to involve the first of these processes only with a lengthening of the b axis compared to the RT unitcell characteristics. 30 **** The situation with other samples is more diverse or even more contrasted. Tilts of ≈30° are frequently observed, as illustrated for PA610 in Figure 11, parts b and c by Slichter 31 and Itoh 23 .

Figure 11 .

 11 Figure 11. Fiber patterns of nylons in the Brill structure due to Itoh (a and c), Slichter (b) and

Figure 12 .

 12 Figure 12. The conformational map of polyglycine used here to illustrate possible conformation

  ) [(CH2)6-S-CO-S]n[START_REF] Zhao | Sharp and strong "Brill Transition[END_REF] and poly(octamethylene carbonate)[(CH2)8-O-CO-O]n 56 .The former case is probably closest to the examples set by the nylon family. The material displays "sharp and strong" transitions. The low temperature a phase has reflections at 4.65Å and 3.7Å, the high temperature b phase at 4.2Å, with clear transitions in both X-ray and DSC data (TB and TM at 69 and 92°C, respectively). In that specific case, the enthalpies of the Brill transition and of melting are in a 2 to 3 ratio, most probably because of a lower melting enthalpy.

Acknowledgments. Many colleagues and friends helped me understand their data, shape and frequently correct the views expressed in this paper: Andy Lovinger, Freddy Khoury, Alan English, Sanjay Rastogi, Chris Li, Ben Hsiao, Stephen Cheng as well as my colleagues at the Institute Charles Sadron, Martin Brinkmann, Philippe Mésini and Gilbert Weill.

sine of the libration angle and is therefore important in the early stages of the heating process.

When the C-C-C planes have reached their "stable" orientation at 60° to the H bonded plane and stay there for ≈10 -10 second the additional libration due to thermal agitation (time scale 10 -15 second) contributes little to the expansion -basically from sin 60° to sin 90°, i.e. from 0.87 to 1.

At the same time, the existence of two "stable" positions of the C-C-C plane 120° away from each other combined with this additional libration generates a dynamic chain axis projection very similar to the polyethylene "rotator phase". The PE "rotator phase" is frequently offered as a potential model for the nylon Brill structure and its characteristic 4.15Å spacing. When applied to an alpha phase-type nylon structure, the rotator phase analogy suffers from the fact that the H bonds "pin" one direction, which does not offer much leeway for the aliphatic parts to "rotate" unless an excessive concentration of gauche conformations is involved. The conformation interconversion of pleated chains does also maintain the H-bonds "pinning". It offers however a possible structural answer/remedy that takes care of this difficulty. The thermally induced rotations remain limited, but they take place at two "base camps" positions of the aliphatic segments that are symmetrically located away from the medium H-bonded sheet.

The Brill structure and its very frequent (although not universal) hexagonal unit-cell may however reflect more complex intermediate stages during the interconversion process than considered up to now.

The Brill structure: why and when a hexagonal unit-cell?

The issue.

The Brill transition temperature has been associated for years with a structural marker: the point of convergence of all three equatorial spacings at a common value of ≈4.2Å. The resulting (pseudo) aliphatic segments these dynamic H-bonds movements set in on heating and stop on cooling at ≈200-250°C for bulk crystallized materials (Figure 2b). The single crystals of nylon 46 produced in superheated water display cleaner transitions (Figure 2d). On heating, the hexagonal cell appears at ≈200°C when the 010/110 spacing is already very close to 4.2Å. On cooling the interconversion process stops at ≈120°C with immediate reversal from the dynamic Brill structure to the "still" pleated sheet structure. In a broader context, in proteins and polypeptides, transitions with hydrogen bonds reorganization take place in the 130-200°C range. [START_REF] Lotz | A Family of Double Helices of Alternating Poly(g-Benzyl-D-L-Glutamate), a Stereochemical Model for Gramicidin A[END_REF] By contrast, a type of "Brill transition" takes place in poly(alkylcarbonates) (C6 and C8) at much lower temperature (≈80°C), which reflects the structural similarities but also the weaker dipolar interactions in these remote structural cousins of the aliphatic even-even nylons. [START_REF] Zhao | Brill Transition" Shown by Green Material Poly(octamethylene carbonate)[END_REF] The above scenario emphasizes the dynamic aspects of the Brill structure. In line with Ted Atkins' views, the hexagonal symmetry would result from transitory inter-sheet H-bonds. This process requires however that certain structural criteria be satisfied.

The transient H-bonds can be formed only when the CO and NH groups located in different sheets are, when they reorient, within reach to establish a hydrogen bond. It is valid only for the pleated/rippled sheet structures with relatively limited shear, which translates in tilts of 00l reflections probably less than ≈ 30°. The pleated/rippled sheets are also characterized by a tilt of the amide groups to the ac plane that limits the amide groups separation, even for sheared sheets (cf. Figure 5). Alternatively, the inter-sheet shear may also decrease as part of the annealing process. Such a variation has been observed by Tashiro for fibers of nylon 66 maintained at high temperature: the lamellae become oriented normal to the fiber axis, which would imply that the stems become normal to the lamellar surface and by implication that the inter-sheets shear is reduced. [START_REF] Tashiro | Structural Evolution in Isothermal Crystallization and Phase Transition Processes of Crystalline Polymers as studied by the Simultaneous and Timeresolved FTIR/WAXD/SAXS Measurements[END_REF]

Discussion.

The above analyses of the Brill transition and Brill structure remain essentially structural and emphasize a dynamic interconversion of twisted chain conformations. This insight is novel, but does not solve many questions about the Brill transition. This is clearly a task for future work. Only a few recurrent themes are briefly discussed in the following, without attempting to arrange them in order of importance.

I. Variability of the Brill transition temperature. Role of crystal perfection.

The Brill temperature is manifested in very diverse ways depending on the polymer chemical constitution and physical condition. For short amine and/or acid parts the transitions from the pleated structures to the Brill structure take place at low temperature, the Brill structure is stable over several tens of degrees. For intermediate lengths (including PA66), it is preceded by a wide temperature range in which the original alpha phase crystal structure rearranges or, to say the least, is modified -but the resulting spread is considerable: the reported TBs for PA66 range from 139 to 230°C, as reviewed by Feldman et al. 5 For long aliphatic parts, the preparation conditions may generate considerable gaps or discrepancies in the reported TBs. For example, TB for PA1010 and PA1012 TBs are indicated in single crystals at 200°C and 193°C (also equal to TM,) [START_REF] Jones | Comparison of structures and behavior on heating of solution-grown, chain-folded lamellar crystals of 31 even-even Nylons[END_REF] and reported at ≈135°C 42 and ≈120°C [START_REF] Yan | X Brill transition in Nylon 10 12 investigated by variable temperature XRD and real time FT-IR[END_REF] after bulk crystallization, respectively. In a more systematic study, the TBs of nylon 1010 have been determined as a function of TC and crystal size by Yang et al. [START_REF] Yang | Dependence of the Brill Transition on the Crystal Size of Nylon 10 10[END_REF] The initially quenched, glassy

In solution grown single crystals, the folds are mostly parallel to the H bonds, as established by the polymer decoration technique. [START_REF] Jones | Investigation of Solution-Grown, Chain Folded Lamellar Crystals of the Even-Even Nylons[END_REF][START_REF] Wittmann | Polymer decoration: the orientation of polymer folds as revealed by the crystallization of polymer vapors[END_REF] When neighbor H-bonded sheets are not connected by folds, their structural integrity and stability is maintained during the thermally induced inter-layer expansion. In addition, these solution-grown nylon crystals are thin (≈6 to ≈10nm) yet their TBs are very high. Atkins et al. use specifically such single crystals to reach for the highest possible TBs on the basis of their "crystal perfection" (meaning implicitly "crystal core perfection" and, implicitly also, of the alpha phase). [START_REF] Jones | Comparison of structures and behavior on heating of solution-grown, chain-folded lamellar crystals of 31 even-even Nylons[END_REF] This "perfection" does not include the lamellar thickness component, which remains quite small in solution grown crystals. The paradox is that thin, chain folded crystals for nylons with long aliphatic segments have TB≈TM, whereas the same bulkcrystallized polymers with thicker lamellae have TB<<TM although the thermal expansion of the crystalline parts remains similar.

The missing element defining the "perfection" of crystals appears to be the presence or absence of inter-sheet folds. Higher/lower TBs must be influenced by the absence/presence of connections (meaning folds) and their number/density between H-bonded layers. If folds link different layers, thermal expansion of the crystal lattice induce surface stresses and strain the folds. These strained folds will, in return, destabilize the crystalline stems that they link. Of course, these stresses may be "damped" for long aliphatic segments, a feature that can contribute to the high TBs of these nylons. If there are no inter-sheet folds, the transition remains a pure intra-sheet interconversion process, with less or no destabilization induced by strained inter-sheet connections (and thus higher TBs).

This analysis develops the insights gained by Murthy et al. and by Feldman et al. on the "reciprocal interactions" [START_REF] Murthy | Interactions between Crystalline and Amorphous Domains in Semi-crystalline Polymers: Small-Angle X-ray Scattering Studies of the Brill Transition in Nylon 6[END_REF] or "proportionate responses" 5 between crystalline and amorphous parts.

In this scheme, the Brill transition is a first order transition -but with qualifications. It is triggered inversion in the amines is of interest. At room temperature, this rate is high for ammonia: it is ≈2

x 10 10 Herz. It decreases (but remains high) for substituted, chiral diamines R1R2NH or trisamines R1R2R3N, which makes it usually impossible to separate their two conformation isomers (it becomes possible by hindering the inversion with cyclic substituents). The NH3 inversion is therefore in the same ≈10 10 range as is the rate of interconversion of the present nylons, at higher T°, as established by NMR. On a more "mechanistic" side, the flat ammonia sp 2 transition stage between the inverted initial and final sp 3 states would correspond to the "direct" interconversion process illustrated here in Figure 3. However, stereo-inversion pathways with multistep, intermediate stages have been suggested/modeled in more complex substituted amines and their salts that are prone to chiral inversion in the presence of water. [START_REF] Kaur | Vikas From nitrogen inversion in amines to stereoinversion in aminium salts: role of a single water molecule[END_REF] These multistep processes would correspond to the more "contorted" route for the nylon interconversion suggested in Figure 12.

The rates of amine inversion and nylon interconversion depend on the energy barriers and temperature. For the "simple" amines and ammonia molecules, these values are available. In NH3, this barrier is 25KJ/mol. or 6Kcal/mol., which is only two times larger than the rotation about a single C-C bond. [START_REF] Kaur | Vikas From nitrogen inversion in amines to stereoinversion in aminium salts: role of a single water molecule[END_REF] The corresponding values for the Brill transition are more difficult to identify since the separation between the a to pleated rippled transition and the interconversion process itself have not been clearly identified and thus discriminated. This is clearly a topic for future investigations.

III. What further work? What criteria define best the Brill transition?

III.1 What further work?

A large number of earlier observations are compatible with or are validated by the present scenario of the Brill transition, but further work is still ahead. These experiments and analyses should tackle the features that characterize and make the Brill transition so singular. Staying on the structural side, the most obvious undertaking is to evaluate the complex (and different, depending on the nylons) relationship and reciprocal impact of the two transitions uncovered in the present analysis. In this approach, the issue of orientation should be a constant concern. Orientation, and even better double orientation is an important criterion in all aspects of the Brill transition and of the reversed transition. Preservation (or not) of the initial H bonds and unit-ell orientation after an excursion in a more symmetric packing mode is an important element of the underlying structural processes. Single crystals as investigated by Atkins and his group [START_REF] Jones | Temperature induced changes in of aliphatic polyamides[END_REF][START_REF] Hill | Morphology and Structure of Nylon 6 8 Single Crystals[END_REF] and Li et al. widely different behaviors on heating, thus providing a set of data that allows meaningful comparisons (cf. Figure S4). [START_REF] Atkins | Structural and morphological investigations of nylon 8 chain-folded lamellar crystals[END_REF] Fast quenching of solution-grown single crystals in the Brill structure helps preserve this structure at room temperature and offers means to follow the transformation process. The innovative hydrogen bonds decoration should be a major help in visualizing initial and (sometimes, after transformation) mixed patterns of H bonds orientations. [START_REF] Li | On the crystal structure and morphology of nylon 6,6 from single crystals to spherulites[END_REF] Epitaxial crystallization generates double oriented samples similar to the mechanically produced material but with characteristics (lamellar thickness, etc.) closer to the bulk ones -with the additional benefit of double orientation. [START_REF] Wittmann | Epitaxial crystallization of polymers onto benzoic acid: polyethylene and paraffins, aliphatic polyesters and polyamides[END_REF][START_REF] Wittmann | Epitaxial crystallization of polymers on organic and polymeric substrates[END_REF] Following the example set by Tashiro et al. [START_REF] Tashiro | Conformational disorder in the Brill transition of uniaxiallyoriented nylon 10/10 sample investigated through the temperature-dependent measurement of X-ray fiber diagram[END_REF] elaborate, better targeted thermal treatments (given the additional insights gained) should help to reach further insights on the nature, location and interplay of these crystal-crystal transformations.

III.2 What criteria define best the Brill transition?

Conclusion.

The recognition of pleated and rippled sheets of nylons with twisted chain conformations 6 made possible the present novel structural approach and analysis of the Brill transition.

The Brill transition is not the result of the sole thermal agitation of the aliphatic parts in the alpha phase. It is the sum of two different processes: (a) generation of pleated and/or rippled sheet structure triggered by thermal agitation (b) dynamic conformational interconversion of the twisted stems that build up these pleated/rippled sheets, which is an intra-sheet process (c) under favorable conditions, formation of temporary inter-sheet hydrogen bonds that result in the characteristic 4.2Å reflection.

The two first processes may be separate. In even-even nylons with short aliphatic parts, the pleated/rippled sheet structures exist or may be generated as "clean" crystal phases that experience the Brill transition on heating (which allows to evaluate the relative contributions of the two processes to e.g. the overall inter-sheet distance increase). More frequently, as in PA66 in the transition regime near 160°C, the crystal transformation and interconversion are more or less simultaneous and cannot be discriminated.

The dynamic interconversion between conformers (frequency: ≈10 10 Herz) is allowed by the achiral character of the nylon structure. A seesaw flip of the amide part is associated with large amplitude movements and reorientations (+60° to -60°) of the aliphatic parts planes relative to the medium plane of the hydrogen bonded sheet. The thermal agitation adds an extra molecular vibration of lower amplitude but higher frequency (≈10 15 Herz).

The formation of inter-sheet hydrogen bonds appears not to be the result of thermal agitation as such but rather is linked with transitory stages during the interconversion process and are thus formed and broken at a very high frequency. The standard geometrical constraints do apply: H bonds are formed only when the CO and NH units in different sheets are "within reach" to establish these bonds, which rules out e.g. excessive shear of layers, nylons with excessive inter-sheet separation (with e.g. bulky chemical modifications), etc.

The merging of equatorial reflections spacings was (and still is) frequently used as the experimental criterion to define the Brill transition. In line with earlier observations and analyses, [START_REF] Yoshioka | Structural change in the Brill transition of Nylon m/n (1) Nylon 10/10 and its model compounds[END_REF] this criterion is not satisfactory: nylon 6 or modified nylons also display a Brill transition but without merging of the reflections. This approximate criterion will probably stand, if only for convenience. The dynamic interconversion between mirror image conformers would provide a more pertinent means to define the Brill transition since it refers to a molecular and structural process that applies in all cases and for all types of nylons that experience this transition.

Author information:

Bernard Lotz: bernard.lotz@ics-cnrs.unistra.fr