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

Original crystal structures of even-even polyamides made of pleated and rippled sheets (*)

Keywords: Crystal structure, Nylon 66, Pleated sheets, Rippled sheets, Original single crystals

Original crystal structures of even-even polyamides are introduced. They are based on twisted conformations at the amide-aliphatic parts links and are reminiscent of the pleated and rippled protein sheet structures introduced by Pauling and Corey in the early 1950s. Pleated sheets are chiral, rippled sheets are made of two conformational isomers. They have specific X-ray diffraction "profiles".

Publications on even-even polyamides describe such profiles, indicating that both pleated and rippled sheet variants exist. The pleated sheet structure is the stable crystal form of the whole family of nylons 2,2n. The rippled sheet applies for a high temperature "alpha phase variant" of nylons with long aliphatic parts and more generally appears to be (one of?) the so-called "high temperature a phase(s)" of even-even nylons. Pleated and rippled sheet crystal structures appear to be the entropically favored, high temperature crystal form(s) of nylons. This new family of crystal structures and their possible variants with different sequences of twisted amide-aliphatic parts junctions open a possible way for the (re)analysis of rare and unusual crystal morphologies produced at high Tc after partial melting/self-seeding.

Introduction.

The crystal structure of nylons is a standard in the crystallography of polymers. The structures of nylon 66 and nylon 6 were established long ago by Bunn et al. [START_REF] Bunn | The crystal structures of two polyamides ("nylons[END_REF][START_REF] Holmes | The crystal structure of polycaproamide: Nylon 6[END_REF] The triclinic structure of PA66 with its parallel stacking of hydrogen bonded sheets has a characteristic "fingerprint" that is valid for the whole family of even-even nylons: the powder patterns display two prominent hk0 reflections at 3.7Å and 4.4Å, and the ab plane of the cell is at an important ≈42° tilt to the chain axis.

A salient feature of these aliphatic nylons structures is the quasi-coplanarity of the amide groups and the C-C-C planes of the aliphatic parts. Twisted conformations at the amide-aliphatic parts are known to exist, but they are observed only in the metastable g crystal phase of even nylons (e.g. nylon 6) [START_REF] Itoh | Thermal properties of α-and γ-forms of Nylon 6[END_REF] or when they are needed to complete hydrogen bonding, for example in even-odd or odd-odd nylons. [START_REF] Navarro | Nylon 65 Has a Unique Crystal Structure with Two Directions of Hydrogen Bonds[END_REF] The present contribution considers crystal structures of even-even nylons that by contrast feature twisted amide groups-aliphatic parts connections. Such junctions generate "corrugated", pleated chain conformations that can hydrogen-bond to neighbor chains and generate a different type of nylon sheet.

The approach derives actually from the various sheet structures of proteins introduced in the early 1950's by Pauling and Corey. [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] Pleated and rippled sheets are considered. Pleated sheets are "chiral" as they are made of a single chain conformation, which mimics the situation found in proteins. Rippled sheets are built of alternating conformational isomers, and are therefore specific to the achiral nylons.

The models developed have characteristic X-ray powder fingerprints with well-defined sets of hk0 spacings. Comparison of these patterns with published but unexplained diffraction patterns indicates that such crystal structures do actually exist -sometimes, admittedly, only for specific nylons and/or under specific crystallization conditions.

In this contribution, we recall first some unsolved issues in the crystallography of even-even nylons. The various sheet structures of proteins due to Pauling and Corey are briefly described. [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] The pleated and rippled sheets of PA66 are then introduced. Their structural characteristics and X-ray diffraction patterns are valid for all even-even nylons. These patterns are then compared with different experimental results of the literature. Various possible crystallization patterns and transitions are tentatively considered.

In a companion contribution, [START_REF] Lotz | The Brill transition: A structural scenario Macromolecules[END_REF] a novel structural scenario including pleated/rippled sheet structures is introduced to analyze the Brill transition in nylons, a transition first observed in 1942. [START_REF] Brill | Über das verhalten von polyamiden beim erhitzen[END_REF] Experimental.

The present contribution rests almost exclusively on the wide body of experimental evidence accumulated over the years on the structure of polyamides, frequently in connection with the Brill transition. Although some experiments of our own have been performed, they did not provide any new experimental finding worth reporting. Extensive use is made of the Cerius 2 molecular modeling software developed by Accelrys -Molecular simulations, [START_REF]Cerius modeling software[END_REF] now owned by Biovia. Most useful modules are the Crystal Builder, calculation of powder and diffraction patterns (a part of the Cerius 2 software that, unfortunately, has not been migrated to the more recent "Molecular Simulations" package derived from the initial Cerius 2 software).

Minimization of structures is made with the "Crystal packer" module that maintains the chain axis of the polymer parallel to the c axis of the unit-cell.

In our minimizations, we had no access to the "Compass" set of potentials [START_REF] Sun | COMPASS: An ab Initio Force-Field Optimized for Condensed-Phase Applications -Overview with details on Alkane and Benzene Compounds[END_REF] that is superior to the Burchart 10.1-Universal 1.02 one that has been used. The latter yields consistently slightly larger unit-cells than the actual ones, most prominently along the H bonds direction. This limitation does not invalidate the logics of the present analysis and the major conclusions reached. This weakness could not be overcome (cf. Supplementary material SI1). The calculated interplanar distances of the models must be considered as "approximate" -typically by a fraction of an angstrom (<≈0.3Å) -which happens also to be in the range of the experimental data scatter. Even though the models are meant to illustrate high temperature phases, they are developed as if at room temperature. The aliphatic segments are left in the all trans conformation, which amounts to neglect thermal agitation. NMR indicates that the torsion angles in the aliphatic segments are in near-trans conformation, which justifies this simplification. [START_REF] Hirschinger | Segmental dynamics in the crystalline phase of nylon 66: solid state deuterium[END_REF] All models are built and minimized in the P1 symmetry, i.e. without imposing any a priori symmetry element in the unit-cell, as imposed by the "Crystal Packer" module of the Cerius 2 program. The energy minimizations frequently yield unit-cell angles that are close or very close to 90° or 120°, which suggests that the actual packing may have a higher symmetry. Although tempting, such higher symmetries have not been imposed in the additional minimizations.

The most relevant variables considered in the analysis of the crystal polymorphism are the torsion angles at the aliphatic parts-amide links and different packing modes (i.e. c axis shear of the H-bonded layers).

The resulting models have calculated diffraction patterns that are compared with published data. This comparison, this "measure of success" is by design not a quantitative one. It indicates that the model building and packing energy minimization procedures yield structures that, although not "final", are good representations of the actual ones.

The models are illustrated in a standard manner with three projections along the principal axes (c, b, a), a powder pattern displayed in the critical 15-30° range and a fiber pattern calculated in the precession mode to include also the critical 00l reflections. Figure SI2 shows the minimized structure of PA66 derived from, and very similar to, the Bunn and Garner alpha phase.

Development of original structural models for even-even nylons.

Known and unsolved crystal structures of even-even polyamides.

The alpha phases of nylon 66 and 610 solved in 1947 by Bunn and Garner 1 are well-established and are the prototypes of most (but not all, as will be seen) even-even nylons. The unit-cell is triclinic. The stems are linked by hydrogen bonds to form sheets that are stacked with a significant shift along the c axis of the unit-cell. As illustrated in Figure SI2, the powder pattern displays two strong reflections at 4.4Å and 3.7Å. The 3.7Å 010 reflection (blended with the 110 reflection) corresponds to the inter-sheet spacing, the 100 4.4Å reflection is linked with the intra-sheet inter-stem distance (but it is not the actual ≈4.8Å inter-stem distance). These values are the trademarks, the fingerprints of the alpha phase, irrespective of the length of the aliphatic parts. Since the aliphatic parts are parallel to the c axis, their length does not impact the spacings of hk0 reflections.

The other known structures are intrinsically or dynamically more disordered. The smectic phase formed by deep quenching is a disordered phase. The Brill structure formed at high temperature is described as a pseudo-hexagonal phase with generation of gauche bonds in the aliphatic parts. Both structures are characterized by a ≈4.2Å reflection.

The structural characterization of nylons is however far from complete, even after decades of investigations with progressively more elaborate experimental techniques. The literature, in particular on even-even nylons, mentions many "other" crystal structures that are not yet analyzed, as seen now.

The 2, 2n family of nylons. The structure and thermal properties of 31 even-even nylons (i.e. of the 2m, 2n type) have been investigated by Ted Atkins and his group and associates. 12 Their studies cover all combinations of diamine and diacid parts lengths: they range from 2 to 12 and from 4 to 12 carbons, respectively. The data of 30 of these polyamides (nylon 618 has been omitted) are condensed in a Table reproduced in Figure 1. For all polymers, the Table includes the crystal phase (a or b), the corresponding two major reflections in the X-ray powder pattern (in nm), the melt (TM) and Brill transition (TB) temperatures. The grey shading of the original table indicates polymers for which TM=TB (with TB either measured or extrapolated). All the polymers of the five right columns display the 4.4Å and 3.7Å reflections that characterize the "Bunn and Garner"-type triclinic a phase crystal structure. For all the members of the first, left column (highlighted in light yellow) the two major reflections are at 3.9Å and 4.2Å. The different crystal structure could not be determined, which led the authors to consider that these polymers have structures "between the room-temperature structure of even-even nylons… and the high-temperature pseudo-hexagonal structures" (i.e. between the alpha phase and the Brill phase). 12 The same unusual diffraction pattern was observed for PA216 [START_REF] Li | Crystal Structure of Novel Polyamides with Long Diacid Segment[END_REF] and for a much longer diacid segment, Nylon 2 34, a polymer that was considered as one of several PA2n34 possible alternatives to polyethylene to produce ultra-high modulus fibers. 14 Nylon 46. The usual crystal form of nylon 46 is the alpha phase. A different pattern, with the two reflections at 3.9Å and 4.2Å has been observed in connection with work on the Brill transition by Vinken and Rastogi (Figure 2). [START_REF] Dijkstra-Vinken | Polyamides: Hydrogen bonding, the Brill transition, and superheated water[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] Vinken at al. observe this structure under highly specific conditions, in mats of single crystals of nylon 46 produced in superheated water. On heating, the initial alpha phase transforms abruptly at ≈200°C to the Brill structure. On subsequent cooling from 250°C, the (pseudo)hexagonal Brill structure is maintained down to ≈110°C at which temperature it transforms to this different phase that remains stable down to 50°C (reflections at 4.2 and 3.9Å). Also, the ≈6nm low angle signal remains nearly constant throughout these complex transitions. [START_REF] Dijkstra-Vinken | Polyamides: Hydrogen bonding, the Brill transition, and superheated water[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] Somewhat similar results have been reported by Ramesh et al., this time on bulk-crystallized nylon 46. [START_REF] Ramesh | New Crystalline Transitions in Nylon 4,6, 6,10 and 6,12 Using High Temperature X-ray Diffraction Studies[END_REF] The room temperature alpha phase transforms on heating to a different form before reaching the Brill structure. On cooling, the reverse process takes place, at nearly the same transition temperatures as on heating. The nylon 4,6 crystallized at high Tc in the hexagonal form converts first to the "high temperature" alpha phase and on further cooling to the standard alpha phase. [START_REF] Ramesh | New Crystalline Transitions in Nylon 4,6, 6,10 and 6,12 Using High Temperature X-ray Diffraction Studies[END_REF] High temperature (metastable?) crystal phases. Y. Li and D. Yan have investigated a range of even-even nylons with long and/or highly unbalanced alkyl segments lengths at the amine and acid parts. [START_REF] Li | Crystal Structure of Novel Polyamides with Long Diacid Segment[END_REF][START_REF] Li | Preparation, characterization and crystalline transition behaviors of polyamide 4 14[END_REF][START_REF] Li | Investigation on Brill transition of nylons 6/16[END_REF][START_REF] Yan | Brill transition in Nylon 10 12 investigated by variable temperature XRD and real time FT-IR[END_REF][START_REF] Li | Crystalline transition in Nylon 10 10[END_REF] The structural investigation of PA414 revealed a most standard behavior -with however one original feature. [START_REF] Li | Preparation, characterization and crystalline transition behaviors of polyamide 4 14[END_REF] At room temperature, PA414 is in the a phase. On heating, the 4.4Å and 3.7Å reflections merge at 170°C (Brill transition). However, when further heated "above the melting temperature" and cooled, the crystals formed at 215°C display diffraction peaks with unusual spacings, at 4.36Å and 4.23Å. The crystalline form is "a kind of a phase… but much different from the room temperature a phase because the two d spacings are very close". [START_REF] Li | Preparation, characterization and crystalline transition behaviors of polyamide 4 14[END_REF] On cooling, "the two peaks are preserved but start to move away from each other" and "finally… reach 0.37 and 0.44 nm, respectively, which are the typical reflections of the a phase structure of nylons". This behavior was observed for other even-even nylons of different alkyl lengths. 20- 22 The extensive study by Ramesh on three different nylons revealed also crystal forms and transitions taking place in nylon 6,10 and 6,12. 17 A "high temperature alpha phase" transforms on cooling (at ≈200°C) to the alpha phase without any transition via an intermediate pseudohexagonal Brill structure.

Existence of "different" crystal structures formed at high temperature is also known through unusual crystal morphologies and growth patterns when crystallization takes place after annealing/self-seeding in the melting range. Triangular crystals, lancelets, scrolled crystals, spherulitic aggregates and negative spherulites are produced in solution and/or in bulk crystallization. Most of these morphologies remain unexplained. Possible features of the underlying structures will be suggested later.

Pleated and rippled sheets of fibrous proteins.

The novel crystal structures of nylons to be introduced soon derive from the four b sheet structures of proteins established by Pauling and Corey in the early 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] The major structural characteristics of these sheet structures are briefly recalled, as also illustrated in Figure SI3.

Proteins are substituted nylon 2, with different side groups R attached to the carbon atom: [HC*(R)-CO-NH-]. Like even nylons, they have therefore a chemical "sense" which in sheets may be parallel or antiparallel. In the models for the chiral poly(L-proteins and peptides) introduced in 1953, these sheets are "pleated" when seen along the H bonds direction, with one pleat ≈ 7Å long (cf. Figure 3 andSI3). [START_REF] Pauling | Configurations of polypeptide chains with favored orientations around single bonds: two new pleated sheets[END_REF] Two years later, Pauling and Corey also introduced the so-called "rippled" sheets that are formed when every other stem in the pleated sheet is replaced by its mirror image in the plane of the sheet. [START_REF] Pauling | Two Rippled-Sheet Configurations of Polypeptide Chains, and a Note about the Pleated Sheets[END_REF] As a result of this mirror symmetry, the pleats of the different stems appear to be shifted by half a c axis repeat distance. These sheets appear to have shorter "ripples" ≈3.5Å long (cf. again Figure 3 andSI3). In these rippled sheets, alternating stems are chemically poly(L-peptides) and poly(D-peptides). They are onedimensional stereo-complexes valid only for equimolecular blends of polypeptides or proteins stereoisomers, as opposed to e.g. two-dimensional stereo-complexes of the chiral poly(L-lactide) and poly(D-lactide). [START_REF] Ikada | Stereocomplex Formation between Enantiomeric Poly(lactides) Macromolecules[END_REF][START_REF] Cartier | Epitaxial crystallization and crystalline polymorphism of polylactides Polymer[END_REF] Twenty years later, in 1974, it was shown that the sheet-like structure of polyglycine (Polyglycine I, or PGI) is not the conventional pleated sheet of chiral proteins but is a rippled sheet structure. [START_REF] Lotz | Crystal structure of polyglycine I[END_REF][START_REF] Colonna-Cesari | Structure of polyglycine I: a comparison of the antiparallel pleated and antiparallel rippled sheets[END_REF] Polyglycine has a hydrogen atom as "side chain". Chemically, polyglycine is nylon 2: (CH2-CO-NH)n.

Since its carbon atom bears two hydrogens (rather than H and R), polyglycine is achiral and its conformational energy map is symmetrical (Figure 3, left). The rippled sheet explains the unusual monoclinic unit-cell geometry of PGI. Its "nylon-type" packing is more logical and more effective than the pleated sheet of chiral proteins such as poly(L-alanine). [START_REF] Marsh | An investigation of the structure of silk fibroin Biochim[END_REF][START_REF] Lotz | Structure of Periodic Copolypeptides of Alanine and Glycine. Their Relevance to the Structure of Silks[END_REF] The inter-sheet distances (calculated by packing energy minimization) are 3.35Å for the rippled sheet and 3.55Å for the pleated sheet (experimental: 3.40Å). [START_REF] Colonna-Cesari | Structure of polyglycine I: a comparison of the antiparallel pleated and antiparallel rippled sheets[END_REF] The unit-cell includes a center of symmetry, allowed by the achiral character of the structural unit. The rippled sheet structure model of polyglycine/nylon2 derived with the set of potentials used in the present work is compared with the early model 25 developed in 1974 in Figure SI4.

(Note that a different, helical conformation exists also in the polyglycine II hexagonal crystal form with a 4.15Å reflection and H-bonding in three directions). 29

RESULTS.

I. Structural models of nylons with twisted amide-CH 2 links.

The above experimental evidence indicates that nylon crystal phases do exist that are reminiscent of, but different from, the "standard" alpha phase. Parenthood with the alpha phase implies considering variants of sheet structures. Differences with the alpha phase, as illustrated by the different hk0 reflections can only result from departure(s) from the all trans conformation of the chain, i.e. indicate variations of torsional angles at the amide-CH2 links, much in the spirit of the proteins sheet structures.

Twisted conformations located at the amide-aliphatic parts links are usually considered to be temporary in nylons above the Brill transition and to result from the migration of gauche bonds generated by thermal agitation in the aliphatic segments. In the present analysis by contrast the twisted bonds at the amide linkages are treated as ("quasi"?) permanent and are first developed for PA66.

Parallel pleated sheet structure.

In a search for possible twisted chain conformations, and as is the case for the PA66 alpha phase, only "parallel" b sheets are formed due to the uneven length of the diamine and diacid aliphatic parts (6 and4 CH2s, respectively). As for the alpha phase, it results in a simple, one chain unit-cell.

Guided by the diffraction patterns of the PA2,2n and PA46, the twisted chains must yield spacings in the vicinity of 4.2Å and 3.9Å for the 100 and 010 reflections, respectively. These spacings can be reached with a simple, "generic" model, illustrated in Figure 4. For simplicity, equal torsion angles at the amine and acid links of the amide groups are considered. An obvious first choice is to use the torsion angles values determined for the gamma phase of nylon 6, i.e. 130°. [START_REF] Itoh | Thermal properties of α-and γ-forms of Nylon 6[END_REF] This chain conformation does not however expand enough the inter-sheet spacing, as illustrated in Figure SI3. Sets of progressively more important torsion angles must be considered.

The chain conformation of the pleated PA66 structure illustrated in Figure 4 has OC-CH2 and HN-CH2 torsional angles close to 115°. Are these angles reasonable? As far as we are aware, no complete, 2D conformational energy maps as exist for proteins are available for nylons. A comparison with the protein sheets is therefore of interest. The protein parallel pleated sheet structures have twist angles near 120° i.e. very similar to the present twist angles, which may be considered as a support for the present analysis.

In the proteins antiparallel sheets the chains are consistently more extended than in the parallel ones. The torsion angles in (antiparallel) polyglycine I are for example ≈140° (±3°) [START_REF] Lotz | Crystal structure of polyglycine I[END_REF][START_REF] Colonna-Cesari | Structure of polyglycine I: a comparison of the antiparallel pleated and antiparallel rippled sheets[END_REF] (to be compared with ≈135° for gPA6). [START_REF] Itoh | Thermal properties of α-and γ-forms of Nylon 6[END_REF] The PA66 model illustrated in Figure 4 differs in several respects from its alpha phase. The planes of the amide groups remain parallel to the a axis, but are tilted by about 20 to 25° to the H-bonded plane and the chain axis, which generates a "corrugated" sheet topography, best seen along the a axis projection. The aliphatic parts are oriented nearly parallel to the c axis (as in aPA66), but the plane defined by their carbon atoms is now some 60° away from the hydrogen bonded sheet plane: they are nearly in the (1-10) plane.

The unit cell is triclinic, with a near-hexagonal or more exactly, with a lozenge-shaped c axis projection. The powder and fiber diffraction patterns computed for this form differ significantly from the aPA66 ones, with their peak positions at ≈4.20Å and ≈3.9Å (4.4Å and 3.7Å for aPA66). Note that in the aPA66 pattern, the 4.4Å reflection is associated with the sole H-bonded sheets inter-stem distance; the 3.7Å peak combines the inter-sheet spacing 010 and the 110 reflection. In the present model, two reflections contribute to the larger spacing 4.2Å peak (110 and 1-10) whereas the 3.9Å peak corresponds to the sole 010 inter-sheet spacing.

The corrugated sheet topography would allow virtually no c axis shear in the stacking of layers (but this restriction will be removed in a later development of the model). The situation is very different from aPA66 in which the stagger of layers is important (the a angle is 48°) and the 00l reflections are significant tilted to the fiber axis (cf. Figure SI2).

In the model, the c axis repeat distance is reduced from the aPA66 one (17.2Å or 17.4Å, depending on the authors) to ≈16.9Å. This c axis shortening results only from the tilt of the amide groups since the aliphatic parts are parallel to the c axis of the unit-cell (due to the equal torsion angles at the CO and NH junctions). In the present modeling, this c axis repeat distance is not a crucial criterion since it depends on small adjustments of e.g. the dihedral angles of the aliphatic parts and may be significantly affected when considering different torsion angles at the CO and NH links (cf. later) or introduction of gauche bonds at high temperature.

To conclude, it must be reminded that the pleated sheet structure just described is chiral: all chains have the same conformation. The mirror image is a reflection twin in which the twin plane is the ac plane parallel to the hydrogen bonded sheet. The parent and mirror structures are illustrated together in Figure SI5. Structurally, the a axis and hydrogen bond orientations are unchanged (except for the symmetrical ≈20° amide units tilts) but the 60° orientations of aliphatic parts planes relative to the plane of the sheet are symmetrical -thus 120° apart. This twinning will be an essential ingredient of the reanalysis of the Brill transition. 7

A sheared parallel pleated sheet structure.

In the model of Figure 4, interlocking of corrugated sheets hinders the c axis shear between layers and the unit-cell is very nearly orthorhombic. Results to be developed later indicate that in the 2,2n nylons at least, neighbor layers are significantly sheared along the c axis. To some extent, the problem parallels the situation in aPA66, in which different shears are also conceivable. As illustrated in Figure SI2b, a smaller shear than the actual one in the aPA66 crystal structure yields a better packing energy and does not affect significantly the hk0 spacings, but is not observed.

Crystal structures with sheared, pleated layers are as a rule less dense. As illustrated in Figure 5, the interlayer distance increases by ≈0.15 to 0.2Å when keeping the above chain conformation. By introducing different torsion angles at the amide NH and CO links this inter-sheet distance may be modified (cf. SI 7). Adjustments would be possible by further slight modifications of torsional angles, but this line has not been pursued given the lack of guidelines about which features of the conformation are involved.

The "parallel rippled" sheet structure of even-even polyamides.

The pleated sheets described so far are reminiscent of the pleated sheet structures of (chiral) polypeptides.

In the PA66 parallel rippled sheet considered now, and in line with the analysis that led to the polyglycine I crystal structure, [START_REF] Lotz | Crystal structure of polyglycine I[END_REF] stems that are mirror images alternate along the a axis. The pattern of hydrogen bonds is maintained but the a axis length is doubled. The resulting crystal structure after packing energy minimization is illustrated in Figure 7.

The packing/interdigitation of the methylene segments with their different orientations is less favorable in the rippled sheets than in the pleated sheets, as judged by the increased unit-cell dimensions. The intersheet distance is now 4.25Å. Two shifts of layers must be considered in this structure: (a) neighbor layers are sheared along the c axis, as best illustrated in the a axis projection of Figure 6. As for the pleated sheet structure with shifted layers illustrated in Figure 5, this shift may be maintained in one direction only, which generates a mostly monoclinic unit-cell. If the shift sense alternates regularly as illustrated in Figure 6, it generates the "rippled sheet" version of the b phase of PA66 introduced by Bunn and Garner. 1 (b) In c axis projection, the C-C-C planes of the two stems that build up the sheet have different azimuthal orientations. Successive sheets stacked along the b axis may have the same position relative to the a axis, or may be shifted by ≈4.8Å along the a axis (the latter case is illustrated in Figure 6). This is a well-known problem in the antiparallel b structure of fibrous proteins and notably of the wild (Tussah) silk and the silk fibroin of Bombyx mori L. (the a parameter at≈9.8Å is observed in their fiber patterns). 29 To summarize the above analyses dealing with PA66, the SI7 section collects the experimental unit-cell parameters of aPA66 and of various minimized crystal structures made of rippled and pleated sheets.

II. Stable and metastable pleated/rippled sheets of even-even polyamides. The high temperature phase(s).

The X-ray powder patterns of the above pleated and rippled sheet models have already been observed in past works. As developed in the following, these patterns help establish the stable crystal structure of the whole family of 2,2n nylons and of a metastable phase of nylon 46. In addition, the pleated/rippled sheets are thermally stable. They are formed as seeds in annealing/self-seeding protocols and may on subsequent crystallization at lower temperatures generate unusual nylon crystal morphologies.

The pleated sheet structures. (a) The 2,2n family.

The extensive structural studies of Ted Atkins et al. single out the polymers with a short diamine aliphatic part (CH2-CH2). They have diffraction patterns with two major reflections at 3.9Å and 4.2Å, as opposed to the standard a phase 3.7Å and 4.4Å ones (cf. Figure 1). 12 The short CH2-CH2 sequence of the diamine part imposes this specific structure to the whole family, for lengths of the diacid part ranging from 4 to 34 carbon atoms. [12][START_REF] Li | Crystal Structure of Novel Polyamides with Long Diacid Segment[END_REF][START_REF] Ehrenstein | Structures of X 34-Nylons in Chain-Folded Lamellae and Gel-Spun Fibers[END_REF][START_REF] Li | Morphology and structure of nylon-2,14 single crystals from dilute solution[END_REF] At first sight, the similarity of chemical structures (the PA2,2n polymers of the first column are members of the even-even nylons family) and the perfect match of the two equatorial reflections at 3.9Å and 4.2Å with the computed models for PA66 suggest that the crystal phases of these 2,2n nylons are based on the "simple" pleated sheet structure illustrated in Figure 4. This is not so. As shown by Jones et al. [START_REF] Jones | Temperature induced changes in chain-folded lamellar crystals of aliphatic polyamides[END_REF] sedimented mats of single crystals of several PA2,2n (PA26, 28, 210, 212) provide a crucial information for the present investigation. The 3.9Å reflection (presumably arising from planes parallel to the H-bonded sheets) is oriented at some 30° to the lamellar normal, as illustrated in Figure 7a. (The same conclusion was reached also by Li and Yan by tilting single crystals examined by electron diffraction). [START_REF] Li | Crystal Structure of Novel Polyamides with Long Diacid Segment[END_REF] Assuming that the fold surface corresponds to the (001) plane (which ensures that all possible hydrogen bonds are made within the lamellar crystal core), the position of this reflection indicates that neighbor hydrogen bonded sheets are significantly sheared along the c axis of the unit-cell, with a 3.9Å inter-sheet spacing.

A model with torsion angles of 105° at the NH and 125° at the CO links satisfies both the 3.9Å inter-sheet spacing and the sheared layers requirements. It is illustrated in Figure SI6 for PA28, the material for which the best diffraction data are available. [START_REF] Jones | Temperature induced changes in chain-folded lamellar crystals of aliphatic polyamides[END_REF] Figure 7 compares the experimental diffraction pattern of mats of single crystals (in part a, where its indexing assumes an alpha phase) and the calculated one (in part b, using the modified pleated sheet structure). Note that even though the calculated patterns are similar, the indexing of the reflections is different: with the new cell, the reflection at ≈30° to the lamellar normal corresponds to the sole (010) planes. A more detailed analysis of the above structure derivation is provided in the supporting information (SI8).

b. Nylon 46, the 2n,2n+2 and 2n,2m nylons.

Nylon 46. As illustrated in Figure 2, powder diffraction patterns with the tale-telling 3.9Å and 4.2Å reflections have been recorded by Vinken et al. on cooling mats of PA46 crystals grown from superheated water and heated to 250°C. [START_REF] Ehrenstein | Structures of X 34-Nylons in Chain-Folded Lamellae and Gel-Spun Fibers[END_REF] Murthy et al. observe a similar transition on both heating and cooling of bulk crystallized PA46. [START_REF] Murthy | Premelting crystalline relaxations and phase transitions in Nylon 6 and 6,6[END_REF] PA46 therefore can exist in a metastable, occasionally only transient phase that, as indicated by the diffraction evidence, must be some form of pleated/rippled sheet structure.

PA46 is however a very specific case example. It is an even-even polyamide. It is also a member of the more specific 2n,2n+2 family in which the amine and acid aliphatic parts have equal lengths and hydrogen bonding can be completed between parallel and antiparallel stems. All the structural models illustrated in Figure 3 (parallel and antiparallel, pleated and rippled) may exist for PA46. This holds true for both the alpha phase and the twisted chain conformations considered in this work.

As for the alpha phase, some of these differences have already been illustrated by Atkins et al. in an investigation of aPA46 single crystals. [START_REF] Atkins | Lamellar Structure and Morphology of Nylon 46 Crystals. A New Chain Folding Mechanism for Nylons[END_REF] In particular, the authors describe a crystal structure with a monoclinic unit-cell and a g angle of 115°. It was explained by the stacking of pleated sheets, "closely analogous to nylon 6 fibers and the b-sheet proteins". [START_REF] Atkins | Lamellar Structure and Morphology of Nylon 46 Crystals. A New Chain Folding Mechanism for Nylons[END_REF] [Note however that in their analysis Atkins et al.

have omitted the possibility of rippled sheets of aPA46. The data might be better explained with a structure of aPA46 similar to the polyglycine I structure].

A similar variety of PA46 structures may potentially exist for twisted chain conformations. The characteristics of the various energy minimized structures made of pleated and rippled sheets are summarized in Table SI8. In line with the above structural analysis of PA28, crystal structures with nonshifted and with shifted layers are included. Several models do not reproduce the experimental pattern.

Defining which variant in this family may be formed, and under which specific crystallization conditions is clearly a topic for future work. The structures of PA46 and the thermal histories associated with them have also a bearing in the analysis of the Brill transition. 7 2m,2n nylons. Nylon 46 is also a cornerstone in the present analysis of original nylon structures. Nylon 46 avoids the possible specificity linked with the short sequence of the 2, 2n family of nylons. It indicates that a 2n,2m nylon can exist, besides the standard alpha phase, in a crystal phase with the ≈3.9Å and ≈4.2Å spacings fingerprint that calls for twisted chains conformations. For the 2m,2n family, parallel stems only must be considered. In this family, as illustrated in the SI9 Table, twisted chain conformations should generate crystal structures that have as a rule two major reflections in the 3.9-4.1Å and 4.2-4.3Å range.

No experimental data as clear as for PA46 are available for the PA2n,2m nylons with longer aliphatic segments. However, the (sometimes transient) formation of pleated and/or rippled sheets in various high temperature crystallization processes may well explain a number of observations reported in the literature. Since these phases are mostly produced at high temperature they are regrouped in the next two sections.

Pleated and rippled sheet structures: models for the "high temperature alpha phase".

Pleated and rippled sheet structures appear to be present in phases produced when crystallization takes place at high temperature, after an excursion in the melting range.

The "kind of a phase" reported by I. Li and D. Yan has an X-ray profile strikingly similar to the rippled sheet one illustrated in Figure 6: 4.36Å and 4.23Å observed, 4.40Å and 4.25Å for the model. [START_REF] Ramesh | New Crystalline Transitions in Nylon 4,6, 6,10 and 6,12 Using High Temperature X-ray Diffraction Studies[END_REF][START_REF] Hill | Morphology and Structure of Nylon 68 Single Crystals[END_REF] In addition, the reflections have comparable characteristics. The lower angle peak (at 4.36Å) is sharp. The model indicates that it is indexed 200, which corresponds to the lattice planes linked by hydrogen bonds and is sharp. The reflection at 4.23Å is significantly broader. This reflection blends the model inter-sheet 010 and the 2-20 reflections that are broad. It is therefore tempting to consider that these high temperature "kind of a phases" are made of rippled sheets and apply for other even-even nylons. [START_REF] Ramesh | New Crystalline Transitions in Nylon 4,6, 6,10 and 6,12 Using High Temperature X-ray Diffraction Studies[END_REF][START_REF] Hill | Morphology and Structure of Nylon 68 Single Crystals[END_REF][START_REF] Murthy | Premelting crystalline relaxations and phase transitions in Nylon 6 and 6,6[END_REF] X-ray powder diffraction data reported by Tashiro et al. on the early stages of the isothermal crystallization of PA1010 at 176°C after melting at 210°C suggest also that a pleated (rather than a rippled) sheet structure has been produced. [START_REF] Tashiro | Structural evolution in the isothermal crystallization process of the molten nylon 10/10 traced by time-resolved infrared spectral measurements and synchrotron SAXS/WAXD measurements[END_REF] The detailed analysis is developed in SI9. Note for future reference that comparable but not identical thermal treatments (annealing and/or self-seeding and recrystallization at high T°) yield apparently nearly "pure" PA1010 rippled sheet (Yan and Li [START_REF] Ramesh | New Crystalline Transitions in Nylon 4,6, 6,10 and 6,12 Using High Temperature X-ray Diffraction Studies[END_REF][START_REF] Hill | Morphology and Structure of Nylon 68 Single Crystals[END_REF] ) or nearly "pure" PA1010 pleated sheet structures (Tashiro et al. [START_REF] Tashiro | Structural evolution in the isothermal crystallization process of the molten nylon 10/10 traced by time-resolved infrared spectral measurements and synchrotron SAXS/WAXD measurements[END_REF] ) which underlines the close parenthood of the two structures but also the sensitivity of the structural outcome to the specific thermal treatments used.

Further structural studies may ultimately help precise the finer details of the even-even nylons crystal polymorphism. As examined now, the morphology of nylons crystallized at high temperature is also of help, although it carries its own load of interpretation challenges, linked with transformation processes both at Tc and on cooling.

High temperature phases of even-even nylons and crystal-crystal transitions.

Thermal treatments with annealing/melting "above the melting temperature" (implicitly: of the alpha phase) followed by crystallization at temperatures slightly below the melting range (or dissolution range in case of solutions) have been used repeatedly in the past. They yield different unusual morphologies, besides the expected and familiar alpha phase spherulites or single crystals. In the following, several hypotheses are presented that attempt to connect the present "new" phases of even-even nylons with earlier data and observations. The different crystal structures formed after self-seeding at high temperature are not established, even though the diffraction patterns (powder and/or fiber) of some of them are known. When examined at room temperature, the high temperature phases are frequently transformed to the alpha phase. This recurrent issue in the nylon literature sets interesting interpretive challenges. [START_REF] Barriault | Formation of spherulite structure in Polyhexamethylene Adipamide (66 Nylon). I. Structure and optical properties of spherulites at room temperature[END_REF] High temperature phases reveal their existence (possibly only transient) by original morphologies or growth features: spherulites with unusual negative birefringence, [START_REF] Khoury | The Formation of Negatively Birefringent Spherulites in Polyhexamethylene Adipamide (Nylon 66)[END_REF][START_REF] Lovinger | Crystallographic Factors affecting the structure of polymeric spherulites. I. Morphology of directionally solidified spherulites[END_REF][START_REF] Lotz | Structure of Negative Spherulites of Even-Even Polyamides. Introducing a Complex Multicomponent Spherulite Architecture[END_REF] so-called spherulitic aggregates, [START_REF] Lovinger | Crystallographic Factors affecting the structure of polymeric spherulites. I. Morphology of directionally solidified spherulites[END_REF][START_REF] Li | On the crystal structure and morphology of nylon 6,6 from single crystals to spherulites[END_REF] scrolled crystals [START_REF] Li | Nylon 6,6 scroll/tubular single crystal[END_REF] and (admittedly rare) single crystals with unexpected shapes (e.g. triangular crystals, elongated "lancelets"). [START_REF] Li | On the crystal structure and morphology of nylon 6,6 from single crystals to spherulites[END_REF][START_REF] Khoury | personal communication 44. Arimoto, H. ag Transition in nylon 6[END_REF] The link between negative spherulites and scrolled crystals has been established recently. [START_REF] Lotz | Structure of Negative Spherulites of Even-Even Polyamides. Introducing a Complex Multicomponent Spherulite Architecture[END_REF] In the present context, hypothetical assignments are attempted for spherulitic aggregates, lancelets and triangular single crystals.

a. Spherulitic aggregates and lancelets.

The spherulitic aggregates formed in the bulk or in concentrated solutions (cf. Figure 8) are elongated crystals, reminiscent of, but different from the standard aPA crystals that grow under the same conditions. Their growth rate is faster than that of the alpha phase. A most logical assignment would thus be to associate these morphologies with the pleated and/or rippled crystal phases just described: these phases are reminiscent of, but different from, the standard aPA phase. Spherulitic aggregates would be the morphological manifestation of the "high temperature" alpha phase(s). The assignment conveys further consequences. The most interesting one deals with the conversion process(es) from the pleated/rippled phase to the standard alpha phase. As mentioned by Yan and Li, [START_REF] Ramesh | New Crystalline Transitions in Nylon 4,6, 6,10 and 6,12 Using High Temperature X-ray Diffraction Studies[END_REF][START_REF] Hill | Morphology and Structure of Nylon 68 Single Crystals[END_REF] the high temperature phase converts slowly at room temperature to the alpha phase. The transformation process is unknown. However, the spherulitic aggregates, when examined at room temperature in their aPA crystal structure, have a very unusual unit-cell orientation, which may provide clues about the transformation process. Their apparent fast growth direction is unexpected: it is the aPA b axis (cf. later, Figure 10). [START_REF] Lovinger | Crystallographic Factors affecting the structure of polymeric spherulites. I. Morphology of directionally solidified spherulites[END_REF] This strange orientation can only result from a crystal transformation. Figure 9 illustrates two conceivable transformation mechanisms of a pleated sheet structure to the alpha phase. Generating an aPA with coplanar aliphatic parts/amide groups requires a ≈60° rotation of either the aliphatic parts or the amide groups. Rotation of the aliphatic parts maintains the original H bonds orientation. Rotation of the amide parts -as already introduced by Arimoto for the alpha to gamma transition in nylon 6 44 -is in principle a milder process when the aliphatic parts are long. It brings the b axis of the unit-cell parallel to the initial H-bonds orientation and moves the amide groups 60° away from that orientation. On this basis, the curious b axis orientation observed at room temperature would indicate that spherulitic aggregates grow at high Tc in the pleated/rippled sheet form (rather than the alpha phase) and that they transform to the alpha phase via reorientation of their amide groups. Pushing the model one step further, the crystal transformation just considered may well be operative also at the crystallization temperature. Several unusual features of the growth and structure of these aggregates suggest such transitions/transformations: preferred direction of fibrillar growth oblique to the spherulitic radius, variable and interrupted growth rate, apparent coarseness of texture, extinction at 45° to the crossed polars and formation of apparently two types of spherulitic aggregates, one more "ordered", with a growth more "directional" than the other (This important point will be developed soon). [START_REF] Lovinger | Crystallographic Factors affecting the structure of polymeric spherulites. I. Morphology of directionally solidified spherulites[END_REF] These features can only be explained by a crystal transformation process during growth at Tc. This conversion may be located at the growth front, or reach it during growth. In this scheme, the converted aPA phase can continue to grow also on its own with its a axis (growth axis) at 60° to the initial radial growth direction, but its growth rate is slower than the pleated/rippled sheets one. [START_REF] Lovinger | Crystallographic Factors affecting the structure of polymeric spherulites. I. Morphology of directionally solidified spherulites[END_REF] Crystal transformation during growth and lamellar branching are well known phenomena in polymer crystallization. Crystal growth transitions are documented in polydioxolan. [START_REF] Fraschini | Polymorphism and cross-nucleation in poly(1,3-dioxolan)[END_REF] and, in the nylons field, in e.g. nylon 65. [START_REF] Morales-Gamez | Study of the Brill transition and melt crystallization of nylon 65: a polymer able to adopt a structure with two hydrogen bonding directions[END_REF] Lamellar branching is well known also in the alpha phase of isotactic polypropylene (aiPP). [START_REF] Lotz | Structure and morphology of poly(propylenes): a molecular analysis Polymer[END_REF] The closest analogy with the present branching would be the composite structure of spherulites of stereo-defective iPP (typically produced with metallocene catalysts). These materials produce much gamma phase (giPP). In their spherulites, the alpha phase is produced first and creates a frame of radial lamellae with the aiPP a axis radial. Nucleation of branching lamellae of giPP takes place by epitaxy on the lateral faces of these radial aiPP lamellae. [START_REF] Lotz | Structure and morphology of poly(propylenes): a molecular analysis Polymer[END_REF] Growth along the fast growth rate axis (c axis) of these giPP branches is oriented at right angle to the elongated, radial aiPP lamellae. As a result, the unit-cell orientation within the composite, mostly giPP spherulite is not "logical": its c axis, the fastest growth direction in giPP should be radial. It is tangential.

Coming back to the nylons structure, and to summarize, the spherulitic aggregates appear to result from the combination of two unusual processes: (a) they crystallize initially in two related crystal structures (that may coexist) with twisted stems (pleated and rippled, i.e. chiral and racemic) [START_REF] Lovinger | Crystallographic Factors affecting the structure of polymeric spherulites. I. Morphology of directionally solidified spherulites[END_REF] (b) these crystal structures transform into the alpha phase, which results in their unconventional radial b axis.

At this point, it is of interest to introduce a recent development on the two different growth patterns of the nylon spherulitic aggregates mentioned earlier. The directionality of growth of one of these variants is much more "controlled", more "linear" than that of the other form, as indicated by the puzzling, very different arcing of the reflections in their respective X-ray diffraction patterns (Figure 10). [START_REF] Lovinger | Crystallographic Factors affecting the structure of polymeric spherulites. I. Morphology of directionally solidified spherulites[END_REF] Interestingly, these growth patterns have a connection with ongoing work on possible causes and remedies of neurodegenerative illnesses, including Alzheimer and Parkinson diseases. In the present context, potential candidate structures for the two types of spherulitic aggregates are the "high temperature alpha phase(s)" in their pleated and rippled sheets forms. The rippled and pleated sheets are structural "brothers", which accounts for the observed similarities. They are however different. The rippled sheet structure is symmetric relative to the H bond direction, which could result in a symmetric, and therefore more "linear" growth. The pleated sheet is asymmetric relative to the H bonds, thus the tendency to generate "bent" crystals, or twisted crystals. The 3D twist of polymer crystals (linked with various types of structural asymmetries 48 ) frequently "degenerates" in bending when growth is confined in a plane. The pleated sheets would correspond to the so-called "lancelets" observed and analyzed by F. Khoury [START_REF] Khoury | personal communication 44. Arimoto, H. ag Transition in nylon 6[END_REF] one example of which is illustrated in the top right corner of the micrograph of Figure 11. The (co)existence of pleated and rippled sheets structures would thus account for both the similarities and the differences of the two growth patterns observed for spherulitic aggregates. This hypothesis has received an unexpected and very early support -actually before it was devised recently. The relevant work is performed in a very different field -namely in research on neurodegenerative illnesses, such as Alzheimer or Parkinson diseases, or prion encephalopathies. These illnesses are "protein misfolding pathologies" that result from "unregulated self-assembly of peptides and proteins into amyloid aggregates". [START_REF] Dutta | Suppression of Oligomer Formation and Formation of Non-Toxic Fibrils upon Addition of Mirror-Image Ab42 to the Natural L-Enantiomer Angew[END_REF] It is safer to refer the interested readers to the original literature for the details and the philosophy of the approach. In brief however, cleavage of a transmembrane protein produces oligopeptides ≈ 40 amino-acids long. These so-called amyloid-b parts display short sequences of hydrophobic amino-acids and have a high tendency to fold (by a protein-type b -turn) and aggregate as fibrils made of hydrogen-bonded b sheets. The neurotoxic elements are in fact even shorter intermediates in this cleavage and aggregation process. Since these toxic intermediates co-crystallize with the fibrils, one possible means to limit their impact is to accelerate the fibril formation and aggregation. Such an acceleration may be achieved via racemic blends of the L-Amyloid b and its D-enantiomer. Racemates are frequently less soluble than the enantiopure analogs. Having shorter induction times and faster crystallization kinetics, they may act as scavengers for undesired chiral species. [START_REF] Dutta | New insights into differential aggregation of enantiomerically pure and racemic A 40 systems[END_REF] The racemic blends of amyloids and of biologically relevant peptide analogs form rippled b sheets of the Pauling-Corey type, cf. Figure 3). They have been observed and characterized by a variety of techniques (cf. SI12). [START_REF] Dutta | New insights into differential aggregation of enantiomerically pure and racemic A 40 systems[END_REF] Their efficiency in reducing the toxicity of the intermediates has been established. [START_REF] Dutta | Suppression of Oligomer Formation and Formation of Non-Toxic Fibrils upon Addition of Mirror-Image Ab42 to the Natural L-Enantiomer Angew[END_REF][START_REF] Chung | Enantioselective Molecular Recognition between -Sheets[END_REF] The crystal morphologies of the L-, the D-amyloid b and the racemates establish the connection with the present nylons morphology issues. As seen in Figure 10 (bottom) and Figure SI12 the chiral molecules produce twisted fibers (and, expectedly, the twists are opposite for the enantiomers). [START_REF] Lotz | Twisted single crystals of Bombyx mori silk fibroin and related model polypeptides with structure. A correlation with the twist of the sheets in globular proteins[END_REF] The racemate crystals are bigger and, more relevant in the present context, they form straight lamellae.

The connection between crystal morphology and configurational chirality is well documented. [START_REF] Flack | Pasteur's discovery of molecular chirality and spontaneous resolution in 1848, together with a complete review of his crystallographic and chemical work[END_REF] The even-even nylons aggregates appear to offer a subtler illustration of a morphology identification of different conformations of the same molecules. In summary, the existence of both pleated and rippled sheet structures of even-even nylons is supported by diffraction and morphology indicators. It also provides a unique answer to several, apparently unconnected and unsolved issues.

b. Triangular crystals.

Crystallization of self-seeded solutions of even-even nylons (PA66 in particular) yields rare single crystals with a triangular shape, as illustrated in Figure 11. [START_REF] Khoury | personal communication 44. Arimoto, H. ag Transition in nylon 6[END_REF][START_REF] Li | On the crystal structure and morphology of nylon 6,6 from single crystals to spherulites[END_REF] They suggest that even-even nylons may -at least occasionally -adopt a crystal structure that departs from the pleated/rippled or aPA sheets ones.

The structural models considered so far are all based on mirror conformations at successive amide groups (e.g. NH: 105° CO: -125°, CO: 125°, NH: -105°). The chain conformation has a center of symmetry and sheet structures are formed. Other sequences of twist angles may be considered, leading to crystal structures with two or even three hydrogen bonds directions. A simple model of this type is illustrated in Structures with two hydrogen bonds orientations have already been described for different nylons. These nylons blend frequently even and odd numbered aliphatic parts. Nylon 65 is one of these polymers. [START_REF] Navarro | Nylon 65 Has a Unique Crystal Structure with Two Directions of Hydrogen Bonds[END_REF][START_REF] Morales-Gamez | Study of the Brill transition and melt crystallization of nylon 65: a polymer able to adopt a structure with two hydrogen bonding directions[END_REF] Solution grown single crystals of nylon 65 are however elongated and symmetrical: they are very reminiscent of aPA66 ones. This different morphology (elongated versus triangular) highlights a major structural/conformational difference. In nylon 65, the hydrogen bonding is symmetrical: CO and NH groups are found on both sides of the chain. In the PA66 structure of Figure 11, the hydrogen bonding is polar: the CO groups are located on one side only of the stem, the NH units are located on the other side. This structural polarity generates different natures of the growth faces -less and more polar -which translate in Figure 11 in a morphology symmetry (left/right) and a morphology asymmetry (top/bottom).

The crystal has also transformed, as judged from the complex, three orientations of its H bonds decoration: the remnants of the initial cell structure (the model of Figure 11) generates the oblique decorating crystals

Discussion.

The model building/packing energy minimization approach used in the present investigation to analyze novel crystal structures of even-even nylons is clearly a risky one given the frequent paucity of experimental data. The models are clearly open to further improvements. The existence of both pleated and rippled sheet structures of even-even nylons is supported however by both diffraction and morphology indicators and the correctness of the general features is validated by their analogy with pleated crystal structures of proteins. The models provide also a simple and coherent structural frame that answers several unsolved and apparently unconnected issues.

The major novelty is the recognition of a family of stable or metastable crystal structures of even-even polyamides with torsion angles at the amide bonds that depart from the standard all trans ones. In this respect, and referring to the early work on PA6, they are the "b phases" of even-even nylons. These crystal structures "fill a gap" in our understanding of the nylons structures. More precisely, they fill two gaps.

They are valid for low temperature crystal structures -below the Brill transition for both nylons 2, 2n and nylon 46 -and for the very higher end of the crystal thermal stability range. As such, they establish a link between reported structures that were observed in faraway, apparently unconnected temperature domains and were conceived and described as different entities.

What feature(s) of the pleated and rippled structures can explain their high temperature crystallization range and associated thermal stability? Why are "self-seeds" of the nylons with longer aliphatic parts stable whereas the alpha phase experiences the Brill transition at much lower temperatures? At this stage, several remarks can be made, pending further experimental results.

The cross section of pleated and rippled sheet structures is intermediate between the alpha phase and the Brill structure. This is best evidenced by the progressive increase of the inter-sheet spacing in these structures. The distances range from 3.7Å (a phase) to 3.9Å-4Å (pleated/rippled -but up to 4.2-4.3Å for rippled)) to 4.2Å (Brill structure). As such, the sequence observed on heating nylon 48 is very illustrative.

Thermal expansion would thus appear as the major factor in the setup of this structural polymorphism. This analysis is however incomplete. The stability of self-seeds and the crystal growth of pleated and rippled sheets at high temperature indicates that the dynamic disorder associated with the Brill transition (cf. the companion paper) [START_REF] Lotz | The Brill transition: A structural scenario Macromolecules[END_REF] is not operative in the seeds and during growth of the crystals. At this stage, it is tempting to consider that a stabilization linked with the size of the crystals is involved. The stability of the nylon crystals, as assessed by the temperature shift of the Brill transition, is highly dependent on the "crystal perfection" (deliberately loosely defined in the present context). [START_REF] Yang | Dependence of the Brill Transition on the Crystal Size of Nylon 10 10[END_REF] Excursions at high temperatures result in lamellar thickening, doubling or even quadrupling of the initial, "standard" ≈6nm lamellar thickness frequently reported for the alpha phase. [START_REF] Mitomo | Lamellar thickening behavior of nylon-6,6 crystal by annealing[END_REF][START_REF] Burmester | On the annealing of polyamide crystal mats[END_REF] Considerable shifts, not only of this Brill transition, but also of the domain of stability of the pleated/rippled sheets are therefore likely.

The impact of the lamellar thickness or long period has been evaluated for PA1010 crystallized from the glassy state. [START_REF] Yang | Dependence of the Brill Transition on the Crystal Size of Nylon 10 10[END_REF] The variation is linear, in the form: TB= 2.894 L -89.18. For a long period of ≈130Å, [START_REF] Tashiro | Structural evolution in the isothermal crystallization process of the molten nylon 10/10 traced by time-resolved infrared spectral measurements and synchrotron SAXS/WAXD measurements[END_REF] the extrapolated Brill transition would be 287°C. In other words, the domain of existence of the pleated/rippled sheet structures is always located below the Brill transition corresponding to its lamellar thickness even though they may be formed at very high temperature, near the melting temperature of the same material with "conventional" lamellar thickness. These aspects will be further developed in the companion contribution.

Conclusion.

Novel insights on the structure of even-even nylons have been reached.

-Even-even nylons crystal structures with conformations featuring stable twisted torsion angles at the amide CO and NH bonds have been characterized. The resulting pleated chain conformation provides the basic structural element for original nylon structures.

-Different crystal structures of even-even nylons with rippled and pleated sheets and with different shifts between successive layers form a whole family of different but related crystal phases.

-The pleated sheets are the (only) crystal structure for all the PA2,2n nylons, irrespective of the length of the diacid part (with up to 32 CH2 units). They are actually metastable intermediate stages observed on heating, on the way to the Brill structure (cf. also the companion paper). [START_REF] Lotz | The Brill transition: A structural scenario Macromolecules[END_REF] -For even-even nylons with longer aliphatic parts (e.g. PA1010, but also shorter ones, e.g. PA610 (the exact length range remains to be explored) pleated and/or rippled sheet structures are formed next to the melting temperature but, again, below their (unreachable, extrapolated) Brill transition temperature. They provide models for the hitherto enigmatic "high temperature alpha phases".

They may be considered as the entropically favored, high temperature evolution of the alpha phase. The original features and manifestations of these high temperature structures were known, but no definite model could account for the similarities, but also the differences with the standard alpha phase.

-These structures are formed during crystallization histories that involve a self-seeding step. They are probably also formed during the heating/melting process: a clear and sharp exotherm is frequently observed just prior to melting. The nature of the underlying process is not established: recrystallization in thicker lamellae and/or in a different crystal modification? -Original morphologies of single crystals and growth patterns (e.g. spherulitic aggregates) suggest that these "high temperature phases" may take multiple forms. The variability is linked with the availability of different, nearly isoenergetic sheared or non-sheared packing modes of pleated and rippled sheets, and the possibility of slightly different torsion angles at the CO and NH links.

-The models developed in this contribution are based on sequences of two symmetrical conformations. Triangular crystals appear to depart from this simple scheme. More complex sequences of torsion angles could generate for example H-bonded sheets with crankshaft-shaped stems.

-Much work on these structures remains ahead, notably on their more precise characterization. Specific means of orientation, of handling and investigation methods will be needed since these structures are unstable and likely to transform, especially on cooling. Several structural hypotheses offered in the present contribution may thus be supported (or refuted!) on the basis of cleaner data.

-Extension of the present concepts to nylons with different chemical structure may provide new means to analyze for example the complex crystal polymorphism of nylon 6. [START_REF] Ramesh | High-Temperature X-ray Diffraction Studies on the Crystalline Transitions in the a-and g -Forms of Nylon 6[END_REF] -Elucidation of these new structures provides also an opportunity to reexamine the Brill transition of nylons, first reported in 1942. The fundamental importance of this transition in the history of nylons justifies a specific development, which is the topic of the companion contribution.

Figure 1 .

 1 Figure 1. Part of the table due to Jones et al. that colleects experimental data on single crystals of 30 nylons of the type 2m2n. Adapted from Jones, N. A. J.; Atkins, E. D. T.; Hill, M. J. Macromolecules 2000, 33, 2642-2650.

  Hill and Atkins observe a similar intermediate stage on heating single crystals of PA46.

18

 18

Figure 2 .

 2 Figure 2. The WAXD (left) and SAXS (right) d spacings recorded from a mat of PA46 single crystals formed in superheated water. On cooling, a structure different from the initial alpha phase is produced at ≈ 110°C. Reproduced from the PhD thesis of Vinken, E. et al. Langmuir, 2008, 24 (12), 6313-6326, with permission

Figure 3 .

 3 Figure 3. Left: Conformational energy map of polyglycine. Allowed to forbidden conformations are dark to white. The conformations of polyglycine I (PGI) are located on the so-called b -"plateau" of the map. Right: The Pauling and Corey four b sheet structures seen along the chain axis (top) and the hydrogen bonds directions (bottom).

Figure 4 .

 4 Figure 4. The structure of the pleated sheet of PA66. Unit cell parameters: a=5.0Å, b=4.63Å, c=16.93Å, a =83.76°, b =92.41°, g =122.89°. Energy (kcal/mol): Total: -30.35 (vdW: -24.45, H bonds: -5.90)

Figure 5 .

 5 Figure 5. Crystal structure of PA66 with the pleated layers shifted along the c axis. Cell dimensions: a=5.03Å; b= 5.06Å; c= 16.93Å; a = 71.6°; b = 85.83°; g = 118.4°, Energy: -27.34 kcal/mol (vdWaals : -21.8, H bonds : -5.5).

Figure 6 .

 6 Figure 6. The structure of the parallel rippled sheet of PA66 (a=10.07Å, b=10.05Å, c=16.93Å, a =88.4°, b =94.7°, g=122.2° E: -99.9Kcal/mol (vdWaals: -81.1, H bonds: -18.9).

Figure 7 .

 7 Figure 7. Analysis of the diffraction pattern of a sedimented mat of PA28 single crystals. (a) Experimental pattern as analyzed and indexed with the PA28 alpha phase unit-cell (b) Calculated diffraction pattern with indexing assuming an alternative crystal structure with staggered H-bonded pleated sheets as illustrated in part (c). Part (a) adapted from Jones, N.A. et al. J. Polym. Sci. Polym. Phys. Ed. 1997, 35, 675-688, with permission of John Wiley & Sons, ed.

Figure 8 .

 8 Figure 8. SEM images of (on the left) a complex assembly of a PA66 crystals and (on the right) spherulitic aggregates obtained in a concentrated solution in glycerin after self-seeding. Reproduced from Li, C.Y. et al. Polym. Mater. Sci. Eng., 2004, 91, 188-189, with permission.

Figure 9 .

 9 Figure 9. Two possible processes for a pleated sheet (on the bottom) to the a phase (on the top) transformation by reorientation of the aliphatic parts or the amide groups.

Figure 10 .

 10 Figure 10. The two types of "spherulitic aggregates" as seen in polarized optical microscopy (top, growth axis vertical) and their X-ray diffraction patterns (middle). Bottom: b -amyloid crystals made of L-amino-acids only (left) and of an equimolecular mixture of the L-and D-residues (right). Top parts reproduced from Lovinger, A.J., J. Appl. Phys. 1978, 49 (10), 5014-5028. Bottom part reproduced from Urban, J. M. et al. Molecules (Basel, Switzerland), 24(10), 1983, with permission.

Figure 11 (

 11 Figure 11 (cf. also Figure SI13 for a more detailed analysis). The sequence is here NH: 114° CO: 115°, CO: -115°, NH: 114°. It would be consistent with a triangular shaped crystal in which the hydrogen bonds orientations (presumably after transformation) are revealed by their additional, decorating growth.

Figure 11 .

 11 Figure 11. A possible structural model based on twisted conformations of the amide-aliphatic parts links with two H bonds orientations. This model is compatible with the triangular shape of, here, PA66 single crystals. The decoration of the oblique edges indicates a partial crystal transformation with alignment of the amide groups parallel to the aliphatic parts. The original (oblique) and transformed, horizontal H bonds orientations (the latter parallel to the C-C-C plane of the model) show up in the orientations of the decorating crystals (arrows). Micrograph adapted from Li et al. Polym. Mater. Sci. Eng. with permission.

(

  at 10 and 2 o'clock); in the transformed parts, the H bonds are reoriented parallel to the C-C-C planes of the initial model, at 3 and 9 o'clock.

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 and Gilbert Weill. Many thanks also to Prof. Jevgenij Raskatov (University of California at Santa Cruz) who made me aware of the link between my old work on the structure of polyglycine I and the recent research on rippled -amyloid sheet structures and neurodegenerative diseases.

Author information:

Bernard Lotz: bernard.lotz@ics-cnrs.unistra.fr ORCID: 0000-0001-8091-9014

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