Avoidance of Density Anomalies as a Structural Principle for

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Avoidance of Density Anomalies as a Structural Principle for Semicrystalline Polymers: The Importance of Chain Ends and Chain Tilt Keith J. Fritzsching,† Kanmi Mao,‡ and Klaus Schmidt-Rohr*,† †

Department of Chemistry, Brandeis University, 415 South St., Waltham, Massachusetts 02453, United States ExxonMobil Research and Engineering, 1545 Route 22 East, Annandale, New Jersey 08801, United States

‡

S Supporting Information *

ABSTRACT: On the basis of sufficiently realistic chain models and simulations, it is concluded that the commonly used models of the lamellar structure of melt-crystallized semicrystalline polymers unintentionally but inevitably contain layers with a higher density than in the crystallites (density anomalies). The density excess would be particularly pronounced in polymers with planar zigzag conformations in the crystallites, such as polyethylene (PE). To avoid density anomalies, the structural models must be modified, with chain ends at the crystal surface and/or chain tilt in the crystallites. NMR and X-ray evidence for these structural features in PE is presented. Termination of chains at the crystal surface keeps dangling chain ends out of the crowded interfacial layer, reducing the density at the interface by about 17% for Mn = 15 kg/mol, a common value in commercial high-density polyethylenes. NMR of PEs shows that most CH3 end groups are indeed in all-trans chains in a nearly solid-like environment. When the ends of polydisperse polymers are trapped at the crystal surface, many chain folds cannot be tight, in agreement with NMR showing fast trans−gauche isomerization even for solution crystals of PE. We propose that for polydisperse PE, interfacial chain ends are required for the formation of regular stacks of flat melt-crystallized lamellae without extreme chain tilt or density anomalies. Chain tilt in the crystallites, which decreases the area density of chains emerging from the crystal surface, is another indispensable structural adjustment in PE that reduces excess noncrystalline density; it has occasionally been reported but was usually considered as incidental. For instance, in X-ray analyses of PEs, chain tilt was ignored, and differential broadening of the (hk0) Bragg peaks was mistakenly attributed to mosaicity, in extreme cases resulting in the assumption of overly thick, rod-shaped rather than lamellar crystallites. For various oriented PE samples, including blown films and annealed fibers, published scattering patterns exhibit clear evidence of macroscopically aligned lamellar stacks with pronounced chain tilt. Literature data for PE also show examples of the increasing importance of chain tilt at high molecular weights, where the density reduction by chain ends is minor and evidence of extreme chain tilts of up to 60° has been shown. Alternatively, when chain-end concentrations are very low, the assumption of wide regularly stacked lamellae may have to be given up in favor of ribbon-shaped crystallites. The interplay of the density effects of chain ends and chain tilt can explain many molecular-weight-dependent structural features in polyethylene. On the basis of the combined effects of chain ends at the crystal surface, chain tilt in the crystallites, and adjacent reentry of ∼1/3 of chains, we can construct a lamellar model of PE without density anomalies. Chain tilt and chain ends at the crystal surface are required to “make space” for short loops, in conjunction with noncrystalline chain segments emerging from the crystal roughly along the surface normal. These four effects together enable a structure without density anomalies. In solution crystals, the absence of tie molecules and long loops, which produce a more extensive density increase than short loops, reduces the crowding problem and allows for lamellar crystals (with strongly tilted chains) even without chain ends, e.g., at ultrahigh molecular weights. In poly(ethylene oxide) and other polymers with helical conformations in the crystallites, the higher density along the chain axis associated with greater bond tilt angles in the crystal reduces the problem of amorphous excess density relative to Flory’s prediction and thus the need for chain tilt in the crystallites.

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The chains run parallel to the short dimension LC of the nanocrystals, as depicted in Figure 1a, in the vast majority of structural models.1−13 Here we show that these models

INTRODUCTION

Rigid crystalline lamellae and softer noncrystalline layers, >100 nm wide and alternating with an ∼25 nm periodicity, are the mesoscale building blocks of most semicrystalline polymers and account for many of the macroscopic properties of these materials, such as a favorable combination of stiffness and toughness. © XXXX American Chemical Society

Received: September 13, 2016 Revised: January 9, 2017

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DOI: 10.1021/acs.macromol.6b02000 Macromolecules XXXX, XXX, XXX−XXX

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a decades-old debate,3,4,6,8,14−24 and must be taken into account in the analysis of polymer crystallization and deformation. In the textbook models of melt-crystallized polymers,7,10,12,13,25 shown schematically in Figure 1, chains pass through a crystalline lamella along the shortest path and emerge from the crystal surface with high area density. However, as was shown by Flory, Frank, and others,1,6,15 if all chains continued along random directions (see Figure 1a), the density would be about twice higher than in the crystallites. In the Appendix, we show that the excess density is larger for chains with planar all-trans crystalline conformations such as PE than for helical polymers such as poly(ethylene oxide) (PEO), since the former have a larger spacing of backbone atoms along the chain axis (0.125 nm for PE vs 0.095 nm for PEO), resulting in a lower density along the chain axis in the crystallites. In the following, we will focus on PE, where the density problems are particularly pronounced. Experiments had shown very early on that the amorphous density is usually not larger, but rather smaller (by ∼15% in PE), than in the crystallites.12,26−28 To solve the excess-density or chain-flux problem, it was proposed that more than half of the chains need to reenter the crystal, forming more or less tight folds or loops.1,6,15 It is shown here that this does not avoid layers with excess density. For instance, simple counting shows that the iconic1,3,29 loose-fold structure in Figure 1b contains (at least) 1.3 times more repeat units (dots) than does the corresponding volume in the crystallite. Similarly, a simple model of PE (right column of Figure 1) with a realistic 1.25:5 ratio of the spacings of backbone atoms along the chain and of chain stems in the crystallites shows, again by counting, that loops, and even an extremely tight fold18 (Figure 1c), place many backbone atoms between the crystalline chain stems; as a result, excess density (see atom numbers highlighted in red) is unavoidable. Nonadjacent reentry (Figure 1b) produces a larger density anomaly than tight reentrant folds (Figure 1c), and loose loops produce a particularly pronounced excess density; in effect, they are chains of a Flory−Frank1,15 amorphous region, with nearly doubled density. The “leapfrog” or “wicket” model30−32 (Figure 1d) does not resolve the issue either, since it contains long loops that run nearly parallel to the crystal surface, which results in a particularly high projected density ρ(z) (see Appendix). Note that in the widely used cubic-lattice models with unrealistically large, 0.45 nm, spacings between “segments”,17 most of the folds or loops in the right column of Figure 1 would not have shown any excess density. The predicted density anomaly near the crystal surface is indeed seen in atomistic simulations of polyethylene,20,31 and we have confirmed it in exhaustive searches of self-avoiding loops of a specified length on a realistic diamond lattice (see Figure 2 and Figure S1). For loops consisting of eight bonds, corresponding to the shortest fold in orthorhombic PE,18 the excess density is around 20%. The simulations underline that long loops, which are unavoidable with nonadjacent reentry (Figure 2e), produce particularly pronounced density anomalies. Density is mass per volume, and therefore the density profile depends on the size of the volume in which the mass distribution is sampled. With a pm3 sampling volume, every atom or chain would produce a spike in the distribution, and excess density relative to the crystal density would be difficult to recognize. On the other hand, with a μm3 sampling volume, none of the features of the density distribution across a crystalline−amorphous repeat unit would be resolved. An intermediate size of the sampling volume of about 0.6 nm3 is suitable to yield a constant density (within ±2%) along any direction in the crystallite (see Figure 2f),

Figure 1. Excess density due to random chain orientations, loops, and chain folds near the crystal surface in the standard textbook models of the lamellar semicrystalline structure. The models shown on the right are views of chains on a diamond lattice, with a realistic ratio of the spacings between backbone atoms and between chain stems, which is the crucial parameter for density analysis. (a) Chains emerging from the crystal surface and going off in random directions have an approximately twice higher density than in the crystal (see Appendix).1 (b) Iconic model (after Flory,1 Fischer,2 and Keller3) of chains emerging from the crystal surface and folding back with random reentry. The ratio of noncrystalline:crystalline “repeat units” (i.e., of red:green dots) is 154:120 in equal volumes, corresponding to a 1.3 times higher density in the noncrystalline regions. While the spacing of the repeat units is unrealistically large here relative to the chain spacing, it is shown correctly in the loop on the right. (c) Adjacent reentry with very tight fold (left) and moderately long loop (right of either panel). The number of carbon atoms, which is proportional to the density, is given for each “layer” in the models on the right; excess is highlighted in red. (d) “Leapfrog”30 or “wicket”31,32 loop model, with significant excess density.

unavoidably result in regions with excess density, higher than in the crystallites, in particular for polymers with planar all-trans conformation in the crystallites such as polyethylene (PE). We will present clear experimental evidence from nuclear magnetic resonance (NMR) and wide-angle X-ray diffraction (WAXD) of PEs that the structure adjusts significantly to avoid such density anomalies, by chains terminating at the crystal surface, chain tilt in the crystallites, and, when necessary, small lateral width of the crystallites. We will show that the interplay between these parameters accounts for the pronounced but previously unexplained molecular weight dependence of several structural features in semicrystalline polyethylenes,8 including the crystallite thickness in ethylene copolymers. At high molecular weights, chain-end concentrations are low and pronounced chain tilt is predicted for lamellar PE crystallites. The avoidance of density anomalies also imposes new, stringent constraints on chain folding at the crystalline−amorphous interface, the topic of B


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Figure 2. Density anomalies due to chain folding. Density profiles were obtained from the exhaustive search of all possible conformations of selfavoiding loops of a given length connecting two chain stems separated by 5 or 10 Å on a diamond lattice. The crystal stem locations mimic loops in the [010] direction on the (001) surface of orthorhombic polyethylene. (a) 3D view of the density of 12-bond loops, with one loop outlined in red. (b−e) Density profiles for loops of increasing length connecting adjacent stems separated by 5 Å, with Gaussian sampling of 0.6 nm width giving the smooth red profiles. Dash-dotted lines: density profiles in the amorphous region decreasing from ρ = 0.85 g/cm3 due to the excluded volume around the fold. (b) Folds or short loops of 8-bond length, with g+g− conformations allowed.18 (c) Adjacent reentry loops of 14-bond length, without g+g− conformations due to a 3.6 Å radius selfexcluding volume around the chain. (d) Adjacent reentry loops of 20-bond length, with 3.6 Å self-excluding volume, documenting a pronounced density anomaly. (e) Nonadjacent reentry with loops of 14-bond length connecting stems separated by 10 Å, with 3.6 Å selfexcluding volume. (f) Density profile perpendicular to the chains in the crystal. With Gaussian sampling of 0.6 nm width, the density (red horizontal line) in the crystal is constant within ±2%.

Figure 3. Structural adjustments to relieve interfacial chain crowding. (a) For reference, typical textbook model of semicrystalline polymers with chains perpendicular to the crystal surface and dangling chain ends in the noncrystalline regions. Excess density near the crystal surface is unavoidable (see Figures 1 and 2). (b−f) Reduction of chain crowding at the interface by (b) chains ending at the crystal surface, which keeps these chain segments from entering the crowded interfacial layer; (c) chain tilt, which results in the stem length LLAM exceeding the crystal thickness Lc and (d) increases the distance at which the chains emerge from the crystal surface. (e) Dependence of the relative area per chain at the crystal surface on the tilt angle. (f) Finite lateral width of crystals, which allows for spill-out of chains away from the crowded interfacial region. shown in Figures S4 and S5. The solution NMR signal of the chain ends at 14 ppm indicates ∼4.6 CH3/1000 carbons, and the concentration of methyl branches resonating at ∼20 ppm was determined to be ∼0.5 CH3/1000 carbons. 1H NMR showed only 0.09 vinyl end groups per 1000 CH2. Low-molecular-weight polyethylene (Mw ≈ 3 kg/mol; PDI = 1.1; “Pwax3k”) manufactured by Baker Hughes (POLYWAX 3000 polyethylene) was purchased from Baker Hughes. The manufacturer reports a density of 0.98 g/cm3, a melting point of 129 °C, and a heat of fusion of 260 J/g. The chain length is consistent with the chains being once folded.33,34 The solid-state 13C NMR spectrum of this material is shown in Figure S6. An oil shale of Permian age (Glen Davis Torbanite) from the Sydney Basin, New South Wales, Australia, kindly provided by Dr. J. Birdwell, USGS, was used as a reference material with amorphous (CH2)nCH3 chains of limited mobility. The organic matter is type I kerogen, with very low heteroatom content (58.7 wt % organic C), derived from chlorophyceae (Botryococcus braunii), colonial cellular green algae,35 while the mineral matter is mostly quartz with 4%. These inconsistencies can be removed by taking into account chain tilt: The WAXD pattern simulated using the Debye formula for 4.5−5 nm thick lamellar crystals with ∼39° chain tilt reproduces the salient features of the experimental data of Heink et al.9 (see Figure 14). In thicker crystals, the Bragg peak broadening due to paracrystallinity7,91 competes with the finite-size mechanism, and the two effects have to be separated through careful analysis.91 In solution crystals of PE, the thickness perpendicular to the (200) planes has been determined as D200 = 27 ± 2 nm,91 in agreement

Figure 14. Chain tilt in PE crystallites is reflected in differential broadening of Bragg peaks. (a): Data from Heink et al., 19919 for a PE fraction with 4.4 ethyl branches per 100 carbons (reprinted with permission from Springer). The fit used the peak widths for 10.9-nm wide and 9.7-nm thick rod-shaped crystallites, inconsistent with Lc = 4.3 nm from SAXS, and treated the peak intensities as free parameters. (b): Simulation for 4.5-5 nm thick lamellar crystals with ∼39° chain tilt, obtained using the Debye formula; Gaussian bell-curves attributed to the scattering from noncrystalline regions (dashed) were also included. The tilt was not exactly in the a−c plane, consistent with the orientation of the shape ellipsoid.9

with the prediction from chain tilt, 12 nm/sin(30°) = 24 ± 4 nm. The experimental value was increased only slightly, from 25 to 29 nm, by annealing at 125 °C, while a significant increase would be predicted by the mosaicity model. The ∼63 nm correlation length perpendicular to the (120) planes91 may be the result of bending or corrugation of the thin crystalline lamellae, which would be exacerbated by solvent removal.64 Chain Tilt in Aligned PE Lamellae. If our analysis is correct, chain tilt must also occur in macroscopically aligned PE lamellae. In the following, we will show that scattering patterns characteristic of chain tilt in aligned lamellar stacks, as sketched in Figure 15, have been reported repeatedly in the literature7,66,85,87,89,92 but were often not recognized as such.7,85−89 In drawn LDPE sheets annealed close to the melting point, Hay, Keller, and co-workers66,67,92 documented pronounced chain tilt, by up to 50°, in oriented, well-stacked crystalline lamellae, with pole figures as shown schematically in Figure 15d.67,86,92 SAXS proves that the lamellar normal is aligned with the draw direction,66,67which in turn makes an angle of 35°−50° with the chain axis.66,67,92 The chain tilt is uniform, with the lamellar normal in the a−c plane. The origin of this structure was “not fully comprehended”,86 and when Vonk rediscovered it,7,87 its origin remained “still a source of speculation”.7 Apparently, the relaxed structure with parallel lamellae made up of tilted chains is energetically favored in the absence of stress; during (re)drawing,67 a lamella is prone to be rotated by the torque exerted by amorphous tie chains on a crystalline chain stem since the two ends of the stem are “laterally displaced” relative to each other by the chain tilt. A second instance of this type of structure can be recognized in the data of Rastogi et al. for a solution-crystallized UHMWPE film.89 TEM clearly shows lamellae aligned as sketched in Figure 15a, parallel to the film surface.89 The chain tilt angle can be estimated from the ratio of the crystalline stem length to the crystal thickness and corroborated by the WAXD pattern. The well-defined Raman LAM corresponds to a stem length of LLAM = 11.9 nm,89 and the crystallinity is 80 mass % from NMR93 and vc = 77 vol %, while the long period is Lp = 12 nm from sharp J


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Because of the macroscropic alignment of the lamellae, these materials, whose crystalline structure we can finally explain and understand, are ideally suited for detailed future scattering or NMR studies of the chain trajectories in the noncrystalline layers of both melt- and solution-crystallized polymers. They offer an excellent framework for distinguishing between different kinds of chain folds and possibly identifying the long-elusive tie molecules connecting neighboring crystallites. Interplay of Chain Ends, Chain Tilt, and Crystallinity. Our analysis predicts that with decreasing concentration of chain ends, i.e., increasing molecular weight, the chain tilt should become more pronounced or the lamellae will start to fragment. Indeed, Voigt-Martin et al.70 concluded from the increasing curvature of lamellae that “the tilt angle ... increases with molecular weight”. In the same vein, chain crowding at the interface provides a rationale for the empirical observation by Mandelkern that “with increasing molecular weight the lamellae are more curved and segmented, and the lateral dimensions are reduced.”8 A particular compelling example of the interplay between chain tilt and chain-end concentration is shown in Figure 16, based

Figure 15. Characteristic scattering patterns of macroscopically aligned PE lamellae with chain tilt by an angle ϕ. (a) Structural cartoon of mats of carefully deposited solution crystals of UHMWPE,89 or of parallel lamellae in drawn LDPE annealed close to the melting point,7,66,87,92 or in blown HDPE films.85 (b) Simple “row structure” with planar lamellae produced by high-stress film blowing according to Keller and Machin,86 but with chain tilt. As usual, the b-axis is the lamellar growth direction and perpendicular to the machine direction “MD”. (c) Two-point SAXS pattern proving alignment of lamellae with their surface normals along the vertical direction.85,87 (d) Resulting pole figures of the (200), (020), and (002) directions in PE.86,92 The distributions of the a-axis ((200) pole) and b-axis ((020) pole, lamellar growth direction) are determined easily, and the main features of the distribution of the c-axis ((002) pole, chain direction, tilted relative to the surface normal) are obtained based on its orthogonality to the a- and b-axes. The pair of characteristic circular ridges of the (200) pole at 90° − ϕ from the normal of the lamellae was observed in blown HDPE films85 and contracted to spots in biaxially oriented LDPE.66,92 The corresponding patterns for chains aligned with the MD are shown in Figure S9.

Figure 16. Dependence of chain tilt angles in ethyl-branched PE on molecular weight Mn, obtained from crystalline stem length LLAM and crystal thickness Lc data from Alamo et al.95 according to ϕ = cos−1(Lc/ LLAM) (see Figure 3c).

SAXS peaks and ≤9 nm from a clear TEM image. The crystal thickness is then Lc = Lp × vc = (10.5 ± 1.5) nm × 0.77 = 8 ± 1.2 nm, which with cos ϕ = Lc/LLAM = 8/11.9 ± 0.1 corresponds to a tilt angle of ϕ = 48 ± 7°. Chain tilt is proved by the WAXD pattern of the well-stacked lamellae (“viewed edgeon”)89 in the solution-crystallized UHMWPE film, where the (200) reflection shows two broad arcs, each apparently with two maxima separated by nearly 90°, similar to the pattern in Figure 6(z) of ref 92, which corresponds to a chain tilt angle close to 40°. If instead the chains were aligned along the lamellar normal, these reflections would be concentrated on the equator.92 Finally, we recognize the hallmarks of such macroscopically aligned lamellae with chain tilt in scattering data from uniaxial or slightly biaxial blown HDPE films produced at high stress.85 This corresponds to the Keller−Machin high-stress morphology,86 but with chain tilt (see Figure 15b). While Pazur and Prud’homme88 concluded that the high-stress Keller−Machin morphology, without chain tilt, had hardly ever been observed, we propose that the characteristic features of this simple and intuitively appealing structure, with chain tilt, are clearly seen in the classic HDPE blown-film data of Choi et al.85 Figure 15d shows cartoons of the observed (200), (020), and (002) pole figures, i.e., the distribution of the a-, b-, and c-axes, respectively,86 of the crystalline unit cell in the blown film. While the equatorial distribution of the b-axis, the common lamellar radial growth direction, is a shared feature of several structural models,85,86,88,94 the pair of pronounced nonequatorial circular ridges in the (200) pole figure observed by Choi et al.85 arises naturally from chain tilt, with no meaningful alternative explanation.85 89

on beautiful data measured by Alamo et al.95 on model polyethylenes (hydrogenated polybutadienes) of well-defined molecular weights. Above about 15 kg/mol, the crystal thickness measured by SAXS and corroborated by TEM95 decreases significantly, to as little as 3 nm, while the stem length in the crystal, LLAM, remains constant. While the authors of the study concluded that “The reason that the two methods do not agree is not clear”95 and suspected the LAM data to be in error, our analysis provides an easy explanation: The crystallizable stem length LLAM is constant due the fixed branch fraction. At higher Mn, fewer interfacial chain ends are available to reduce interfacial crowding; to compensate, the chain tilt increases, up to 60° (see Figure 16). Regularly Stacked Lamellar Crystallites: Not an Intrinsic Polymer Property? Our analysis indicates that chain ends are essential for avoiding density anomalies at the crystallite surface of polymers with crystalline all-trans conformation. Avoidance of excess density by extreme chain tilt usually results in curved crystals that stack poorly.63,70 This suggests that wide, flat lamellar crystallites with good stacking order (with a lamellar aspect ratio exceeding 20:20:1) are not an intrinsic property of melt-crystallized polydisperse polymers with crystalline all-trans conformation but require chain ends. This is indeed reflected in the micrographs of Voigt-Martin and Mandelkern, which show nicely stacked, flat lamellae in meltcrystallized PE only if Mn ≤ 20 kg/mol,63 unless the crystallites K


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space for about 34% of all chains to form short loops, mostly with adjacent reentry. To keep crowding moderate at the interface, the other chains that continue into the amorphous core layer do not go off in random directions but emerge relatively straight and perpendicular to the crystal surface; this change in chain direction at the crystal surface is consistent with atomistic simulations of PE, where the orientational order parameter relative to the crystal-surface normal showed a maximum in the noncrystalline regions near the crystal surface when the chains in the crystal were tilted.20 The second step can be viewed as similar to the “leapfrog” or “wicket” models;30−32 however, without the first density drop at the crystal surface due to chain ends and tilt, those models produce density anomalies. As a result of the 15% chain ends and 34% chain folds, only 51% of crystalline chain stems continue into the core of the amorphous layer. The resulting density, with Flory’s density doubling (see Appendix), in the amorphous core layer is 0.51 × 2 g/cm3 cos(35°) = 0.84 g/cm3. This model, which appears to be consistent with all the experimental data and resembles the atomistic simulation of a low-molecular-weight polyethylene,20 demonstrates the pronounced constraints that the principle of the avoidance of density anomalies imposes on structural models. It would be highly desirable to confirm these considerations in realistic atomistic simulations of HDPE of Mn = 20 kg/mol with chain tilt and interfacial chain ends, and to extend them to higher Mn to observe how the system responds to the interfacial crowding resulting from the lack of chain ends. Solution Crystals. The model with two-step density reduction described in the preceding section can also be adapted for polymer single crystals. Above we have shown evidence for loose, mobile loops and chain ends at the surface of PE solution crystals. In the model of Figure 17b, the density effect of the chain ends is moderate, ∼−10%, due to the smaller “long period” of ∼13 nm, and so is that of chain tilt, cos(30°) or −14%. The resulting smaller (−22%) reduction in interfacial crowding is compensated by the relatively small excess density (ca. +30%, based on Figure 2c) of the loosest loops that take the role of the amorphous core layer in the model. This allows for ∼56% of chains to connect in loose loops. With increasing molecular weight, the chain-end concentration decreases, which should be compensated by increased chain tilt. Data from solutioncrystallized films of UHMWPE89 confirm this prediction with a tilt angle of ≥40°, as discussed above. A crucial difference from melt crystals is the absence of tie molecules in solution-crystallized polymers. A tie molecule increases the total amorphous density more strongly than a corresponding (half) loop near the crystal surface, since it nearly doubles the density (see Appendix) throughout the whole amorphous layer. A near-surface loop, by contrast, while locally increasing the density, does not contribute at all to the density in the lamellar core layer and therefore reduces that density. Because of tie molecules and long loops, the requirement for a density reduction at the crystal surface, by chain ends and chain tilt, is more stringent in melt-crystallized polymers. A Nonlamellar Model for Ultrahigh Molecular Weights. For PE of large Mn and moderate crystallite thickness, the low concentration of chain ends makes it difficult to produce a lamellar model without density anomalies. This is consistent with Mandelkern’s observation that “For low molecular weights, ... the lamellae have larger lateral dimensions and are geometrically well developed. However, with increasing molecular weight the lamellae are more curved and segmented, and the lateral dimensions are reduced. This effect is accented at very high

are very thick due to days of annealing at low undercooling; for thick crystallites and resulting large long periods, the density reduction by chain ends is proportionally increased, as shown above. At ultrahigh molecular weights, well-stacked flat lamellae do not form easily,59,60,63 and when lamellae are present, they probably consist of strongly tilted chains (ϕ ≥ 40°), as documented above (under “Chain Tilt in Aligned PE Lamellae”) for sedimented UHMWPE single crystals from solution. The need for chain ends is even greater in melt-crystallized PE since, unlike folds or short loops, tie molecules and long loops enhance the density throughout the noncrystalline layer. Accordingly, it has been shown96 that high crystallinity can be achieved by blending PE with low-Mn oligoethylenes, which provide many chain ends (in addition to decreasing the viscosity and speeding up chain transport, which reduces the effect of entanglements). Along the same lines, we propose that the lowmolecular-weight component in the typical bimodal molecular weight distribution of modern industrial HDPEs serves to provide the chain ends that reduce interfacial crowding and therefore favor a high volume fraction of well-stacked crystalline lamellae, which in turn increases the stiffness of the materials. A Lamellar Model without Density Anomalies. Interfacial chain ends for HDPE of Mn = 20 kg/mol in a 20 nm repeat unit reduce the chain crowding by about −15%, and a similar reduction is achieved by a 35° chain tilt in the crystallites. The combined density reduction of −30% does not reach the value of ∼−50% required to avoid crowding in the amorphous core layer. Nevertheless, the necessary additional density reduction can be achieved by including some adjacent chain reentry (see Figure 17a) in a lamellar model in which density

Figure 17. Sketch of lamellar structural models of HDPE of Mn = 20 kg/mol without density anomalies, with a chain tilt angle of 35°. (a) Melt crystallized material with fully amorphous structure in the core layer of the noncrystalline regions. With 15% chain ends (red dots) and 34% short adjacent loops (in blue), only 51% of chains enter the core amorphous layer. The excess density due to the short loops is compensated by the density drop at the crystal surface due to the chains terminating at the crystal surface and chain tilt in in the crystal, in combination with interfacial noncrystalline chain segments (in green) orienting preferentially perpendicular to the crystal surface. The density profiles from terminating (red), folded (blue), and remaining (purple, green, and black) chains are schematically shown on the right, together with the total density profile. (b) A corresponding model for a solution crystal, with a 30° tilt angle. Because of the smaller long period, the density reduction due to chain ends is less pronounced, but this is acceptable since crowding beyond the folds is similarly reduced.

anomalies are avoided in two stages: a first one for the interfacial layer and a second for the amorphous core layer. In the interfacial layer, the reduction by −30% from chain ends and tilt makes L


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Macromolecules molecular weights (M ≥ 106).”8 However, these problems have been ignored in most of the literature, where it is usually assumed implicitly that the presence of crystalline material implies the presence of wide, stacked crystalline lamellae with chains perpendicular to the lamellar surface, even for ultrahigh molecular weights.55,59,97 The effects of chain ends are not necessarily negligible for some commercial UHMWPEs, since their polydispersity index (PDI = Mw/Mn) is usually quite large;59 Mw = 3 000 000 g/mol and PDI = 10 results in Mn = Mw/PDI = 300 000 g/mol. Furthermore, these materials may have crystal thicknesses of 20−40 nm and long periods of 45−100 nm,58,61 which are ∼3 times larger than in HDPE. This reduces the total crystal surface area, so proportionally fewer chain ends are needed to reduce interfacial crowding (see above). We can also assume a larger chain tilt in UHMWPE, resulting in a 1.3 times larger crystalline stem length. Combining these factors of 300 000/ (3 × 1.3), the chain ends in UHMWPE are comparable to those in HDPE with Mn = 75 000 g/mol, and the surface density reduction by chain ends is still about −4%. Combined with a chain tilt by 50°, a density reduction by −30% at the interface would still be achievable. While lamellae are observed in some commercial UHMWPEs,61,98 they are not stacked but rather randomly oriented, with spacings that much exceed the lamellar thickness.98 Thus, these lamellae do not form a “regularly stacked lamellar structure” and are too sparse to generate the full crystallinity of ≥50%. On the other hand, TEM micrographs of some UHMWPEs do not show lamellae at all.59,60,63 For instance, only short, poorly defined crystallites are seen in TEM of nascent UHMWPE reactor powders.59,60 Several studies59,97 assumed that UHMWPEs form a lamellar structure without chain folding, i.e., with many tie molecules, which would result in an amorphous layer with a density of >1.3 g/cm3,6 incompatible with experiment. In fact, the data for this UHMWPE and other high-molecular-weight or cyclic99,100 melt-crystallized polymers with planar all-trans conformation show no evidence for a regularly stacked lamellar morphology and usually have relatively low crystallinities around 50%. We propose that such materials will contain crystallites of limited width, consistent with TEM.63 For the nascent-chain UHMWPE carefully characterized by Egorov et al.,59 the bulging-ribbon

Figure 19. Effect of crystalline chain conformation on the segment density along the chain axis, shown in terms of projected segment positions (vertical row of filled circles); the conformation and therefore the density in the noncrystalline regions is the same in (a) and (b). For simplicity, no chain tilt is shown in the crystalline regions. (a) Planar alltrans chain; (b) helical chain with low pitch, resulting in a larger bond tilt angle and higher segment density. The excess density in the noncrystalline region is higher for the planar crystalline structure in (a) than for a helical structure with large bond tilt angles as in (b).

crystal thickness, and the Raman LAM spectrum documents a wide chain-stem length distribution. The finite lateral width and curved surfaces in the model reduce chain crowding at the crystal surface. It is interesting to note that the disordered morphology of this material cannot be attributed to entanglement problems since the nascent chains are barely entangled. Helical Polymers. Our discussion so far has focused on polyethylene and would apply similarly to other polymers with planar crystalline chain structures, including nylons and various polyesters. By contrast, density anomalies would be less pronounced in polymers with helical conformation in the crystallites: For instance, the backbone-atom number density along the c-axis is lower for PE [1/(1.25 Å)] than for helical polymers such as poly(ethylene oxide), PEO [1/(0.95 Å)], or isotactic polypropylene, iPP [2/(2.04 Å)]. This is visualized schematically in Figure 19. The larger the average backbone bond tilt angle in the crystallites, the higher the crystalline density per chain along the lamellar normal, and the need for chain tilt is correspondingly reduced. Thus, the amorphous density is relatively larger in PE than in the helical polymers (see Figure 19) as long as the amorphous chain conformation is the same. This is confirmed by our density calculations in the Appendix (eqs A21−A23). For instance, in PEO with its large crystalline bond tilt angle of ∼52°, the excess density of isotropic tie molecules is more than twice smaller than in PE. Accordingly, there are few if any reports of chain tilt in PEO crystals. Cyclic PEO, having no chain ends to reduce interfacial crowding, is the best candidate for at least moderate chain tilt in PEO. While the SAXS peak positions of cyclic PEO have been interpreted in terms of the traditional model without chain tilt,101,102 the scattering data actually show subtle evidence of a tilt angle of up to 40°. Contrary to the prediction in the traditional model without chain tilt, the WAXD data do not exhibit a broadening of the (032) peak as the crystallite thickness is reduced from ∼11 nm in the linear to 30%, so an amorphous layer must contribute to the long period.102 Furthermore, WAXD data showed an amorphous halo even for a relative short (2000 g/mol) cyclic PEO and the crystallinity from DSC was around 67%,102 which gives a crystal thickness of Lc = 4 nm. With a maximum collapsed ring length of 6 nm, the data are consistent with a tilt angle of up to ϕ = arccos(4 nm/6 nm) = 48°. For PEO of Mn = 5000 g/mol, SAXS and DSC analysis showed a crystal thickness dc = 12 nm for a cycle length of 2 × 16 nm,102 consistent with a tilt angle of up to ϕ = arccos(12 nm/ 16 nm) = 41°. Whether such significant chain tilt really occurs in cyclic PEO would need to be verified by further studies. Avoidance of Density Anomalies and Energetics. The analyses in this paper have been based on the assumption that semicrystalline polymers cannot contain regions of density higher than in the crystallites. While such a principle of avoidance of density anomalies (in short, “no-density-anomaly principle”) is intuitively appealing and has led to several correct, nontrivial predictions such as chain ends at the crystal surface and pervasive chain tilt in PE crystals, we have not provided proof that regions of excess density can never occur. While a sigmoidal density profile without excess density at the interface has often been used for fitting SAXS data,27,28 a moderate density anomaly near the interface may currently be difficult to exclude by experiment for poorly stacked and twisted lamellae. The avoidance of density anomalies should be considered as an energetic effect, based on the free energy penalty of unfavorably dense chain packing. It has to compete with other unfavorable free energy terms, such as the free energy cost of extreme chain tilt, the lateral-surface energy of small crystallites, or the entropic penalty of amorphous layers with liquid-crystalline order. It is also possible that density anomalies do exist in certain PE samples with few chain ends, specifically for monodisperse chains with Mn between 80 and 500 kg/mol, which show poorly stacked, stressed lamellae.58,61,63 The energetically unfavorable interfacial crowding without enough chain ends in these materials is reflected in the large curvature and small width of the lamellae as well as the appearance of defects in the crystallites, such as dark lines indicating that parts of the crystal are accessible to the staining agent.62,63 While energetics may underlie the no-density-anomaly principle, our density-based criterion is valuable precisely because it can be applied without energy calculations. Excess density is easy to identify in a model, by counting repeat units or measuring the length of chain segments (see Figure 1), and it could be quantified using the equations derived in the Appendix. Therefore, the no-density-anomaly principle is a valuable tool to rule out impossible models and provide a criterion for likely structures.

Figure 20. Visualization of the concept of the area per chain for a semicrystalline polymer. (a) A column of cross-sectional area Ac,z and height Ztot can be assigned to each chain. The structure is shown without chain tilt in the crystal. (b) The analysis focuses on the portion Z of the column in the noncrystalline regions. For a treatment of density per chain, each chain needs to be conceptually restricted to its own column of area Ac,z, which can be achieved by periodic boundary conditions in x/y; this does not affect the density ρ(z) projected onto the z-axis. (c) With chain tilt by an angle ϕ in the crystal, the cross-sectional area Ac,z at constant z is increased by a factor 1/cos ϕ.

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CONCLUSIONS AND OUTLOOK We have shown that the standard models of melt-crystallized semicrystalline polymers presented in textbooks and used in many structural analyses9,49,55,59,86,89,90,97,103−105 inevitably contain layers of density higher than in the crystallites. To prevent this problem, we have proposed a “no-density-anomaly” N


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Figure 21. Excess density of chain segments and loops in the noncrystalline regions of semicrystalline polymers. (a) Relation between corresponding infinitesimal lengths along the chain and along the z- and x-axes, which results in an increase in ρ(z) with chain tilt. (b) Representation of the contour length L and z-displacement Z of a noncrystalline chain segment. (c) Representation of the sequence of z-displacements of an up-and-down meandering noncrystalline chain segment that need to be evaluated separately. (d) Contour length L of a semicircular loop, which exceeds that of the chain stems in a corresponding layer of thickness r in the crystal; the corresponding density profile is shown on the right. (e) Geometric relations for the derivation of the excess density due to variously oriented bonds in the noncrystalline regions. An arrow highlights a rare bond orientation giving a lower local density than in the crystal with its moderately tilted bonds.

crystallites reduces the need for chain tilt and for chain ends at the crystal surface. As the next step, state-of-the-art atomistic models should be constructed for a range of molecular weights and different crystalline chain conformations, taking into account all experimental facts and avoiding significant density anomalies. These could then serve as the basis for a new generation of accurate textbook representations of the chain trajectories in the semicrystalline morphology. In the future, it will be interesting to use scattering and spectroscopic methods to explore structural details such as the density profile near the crystalline−amorphous interface, the correlation length of chain tilt within a crystallite and between lamellae, the effect of αc mobility106 on chain-end location, and the influence of long- and short-chain branching. These properties could be studied particularly well in samples consisting of macroscopically aligned lamellae with tilted chains, which can be produced through a variety of methods66,85,87,89,92 that we have briefly reviewed.

Figure 22. Model with randomly linked segments of fixed helical or (as shown) planar zigzag conformation, with (a) actual repeat units and bond orientations (some pointing down, as highlighted by the red arrow) and (b) corresponding effective chain. In (a), the angles β, θ, and θb between the z-axis (crystal-surface normal), local chain axis, and the bond direction are shown. The angle α of rotation of the atoms and bonds around the local chain axis will be randomly distributed.

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weight (TEM); and the tilted-lamellar structure of many semicrystalline polymers after drawing (four-point SAXS patterns). Molecular weight (specifically Mn) is seen to play an important role beyond entanglements, since it is inversely proportional to the chain-end concentration and thus strongly affects the interfacial crowding. Chain tilt in the crystallites is recognized as a structural requirement for PE. Our analysis suggests, in agreement with experimental observations, that regularly stacked flat lamellar crystallites are not an intrinsic property of entangled polydisperse polyethylene but require chain ends. For as-polymerized or melt-crystallized ultrahigh molecular weight PEs, where chain ends are negligible, ribbon-like crystallites may explain the experimental data better than do stacked lamellae. We have presented models without density anomalies that demonstrate the fairly stringent constraints imposed by the new principle. For polymers with helical conformation in the crystallites, we have shown that bond tilt in the

APPENDIX. CALCULATION OF EXCESS DENSITY IN THE AMORPHOUS REGIONS Flory1 has provided the best-known analysis of the twice-higher density in isotropic tie molecules between crystals, but the derivation was mostly verbal and did not spell out a general formula. This omission may have contributed to a limited understanding and appreciation of the random-tilt effect in the polymer community. Flory did not write down an equation for mass density; he in fact assumed a constant density in the noncrystalline regions, and his “density Na of chain intersections” decreased when crowding increased. Therefore, even some specialists believe that there is only a “chain flux”, but not an excess-density problem for a tie-molecule amorphous region.55,59,97 In deriving explicit formulas for density, we will also show that Flory overlooked two factors that can lead to O


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This equation can also be obtained from ρave = m/V with the mass m = ρ1DL in the volume V = Ac,zZ (see Figure 20). It needs to be noted that this eq A7b applies only in a range where the chain goes only up or only down, i.e., where neither dz/dx nor dz/dy changes sign. For a chain meandering up and down (see Figure 21c), it has to be applied piecewise, and the resulting densities ρn(z) of the pieces must be added up.

density enhancements somewhat smaller or larger than the famous factor of 2. We consider the density profile ρ(z) along the crystal-surface normal, which is chosen as the z-axis. The analysis is performed per chain stem in the crystallites. The whole volume is partitioned into columns of cross-sectional area Ac,z and length Ztot, one per chain stem in the crystal (see Figure 20). Because of chain connectivity across the interface, each chain can be assigned a column of cross-sectional area Ac,z in the noncrystalline regions, too. Each chain needs to be conceptually restricted to its own column, which can be achieved by imposing periodic boundary conditions in x and y (see Figure 20b). The volume of each column is Ac,zZtot, where Ac,z is the area per chain stem in the crystal, evaluated at constant z. The density profile ρ(z) is the projection of the regular mass-density distribution ρ(x, y, z) onto the z-axis: ρ(z) = 1/Ac, z

∫ ∫Ac,z ρ(x , y , z) dx dy

Density Profiles of Wormlike Chains

For a chain parametrized as z(x), with dz = |dz/dx| dx and dl = √[dx2 + dz2] = √[1 + (dz/dx)2] dx, eq A6 can be evaluated as ρ(z)Ac, z = ρ1D dl /dz = ρ1D √[1 + (dz /dx)2 ]/|dz /dx| = ρ1D √ [1 + 1/(dz /dx)2 ] = ρ1D √ [1 + (dx /dz)2 ] (A8)

For instance, for a semicircular loop, z = √[r − x ], |dz/dx| = |x/z|, and 2

(A1)

The shifts along x and y due to the periodic boundary conditions do not change this integral and can therefore be ignored. The total mass m of the chain in the column is m=

Z tot

∫0 ∫ ∫Ac,z ρ(x , y , z) dx dy dz = ∫0

Z tot

ρ(z)Ac, z = ρ1D √ [1 + (z /x)2 ] = ρ1D 1/x√ [x 2 + z 2] = ρ1D r /x = ρ1D r /√[r 2 − z 2] (A9)

ρ(z)Ac, z dz

This function with its integrable singularity at z = r is shown on the right in Figure 21d. Given that the length of the quarter circle is L = rπ/2 and the height Z = r (see Figure 21d), according to eq A7b the average density is

(A2a)

so dm /dz = ρ(z)Ac, z

(A2b)

Ac,z depends on the tilt angle ϕ of the chain axis relative to the crystallite surface normal according to

ρave,circ Ac, z = ρ1D L /Z = ρ1D rπ /2/r = ρ1D π /2 = 151% ρ1D (A10)

Ac, z = Ac ⊥ /cos ϕ

(A3)

which can be confirmed by direct integration of eq A9 with x = r sin θ and z = r cos θ:

(see Figure 20c), where Ac⊥ is the “regular” area per chain, evaluated in a plane perpendicular to the chain axis. For polyethylene, Ac⊥ = ab/2 = 0.182 nm2, where a and b are dimensions of the orthorhombic unit cell containing two chains. The chain is treated as having a constant one-dimensional density ρ1D along its contour (with the contour-length variable l): ρ1D = dm /dl = const

ρave,circ Ac, z = 1/Z = ρ1D /r

ρ(z)Ac, z dz = dm = ρ1D dl

ρ1D 1 Ac, z Z

L

=

∫0

ρ(z) dz = 1/(ZAc, z )

dl =

ρ1D L Ac, z Z

π /2

∫0

Z

dz =

1/sin θ r sin θ dθ = ρ1D

Z

1/(r sin θ)r d(r cos θ)

∫0

π /2

dθ = ρ1D π /2

∫0

N

L

(dz /dl) dl =

∑ (dz/dl)j Δl j=1

N

=

(A6)

∑ cos θj Δl = N ⟨cos θ⟩Δl (A12)

j=1

The average density of a chain segment of contour length L and projected length Z onto the z-axis (see Figure 21b) can be calculated by integrating eq A6: Z

∫0

Next we consider a chain that keeps moving up and consists of N straight segments each of length Δl (see Figure 21e). Segment j makes an angle θj with the z-axis, and (dz/dl)j = Δzj/Δl = cos θj. Z=

dl

∫0

ρ1D r /x dz = ρ1D /Z

Average Density from Segment-Orientation Distribution

or ρ(z) = ρ1D dl /dz Ac, z −1

Z

The compensatory effect of excluded volume is discussed in Figure S10.

(A5)

ρ 1 = 1D dz Ac, z

∫0

∫0

(A11)

(A4)

For PE with bonds of 0.154 nm length connecting CH2 units of 14 g/mol molar mass each, we have ρ1D = 14 × 1.66 × 10−24 g/ 0.154 nm = 1.51 × 10−22 g/nm. As shown in Figure 21a, for a given short chain segment of length dl and mass dm, ρ(z) is linked to ρ1D based on eqs A2b and A4:

ρave = 1/Z

2

∫0

since ⟨cos θ⟩ = gives

1/N∑Nj=1

cos θj. Inserting eq A12 into eq A7b

L

ρ1D dl

ρave = ρ1D L /(Ac, z Z) = ρ1D

(A7a)

= ρ1D

(A7b) P

N Δl Ac, z N ⟨Δz⟩

Δl = ρ1D /⟨cos θ ⟩Ac, z −1 Ac , z ⟨Δl cos θ ⟩

(A13)


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Macromolecules Overlooked Effects of Chain Conformation

This central formula applies for tie molecules that do not turn back on themselves,1,6 for the crystal, and for any noncrystalline interfacial layer close enough to the crystal surface that chains have not folded back. It will give a stringent lower limit for the excess density due to isotropic tie molecules. The average ⟨cos θ⟩ in eq A13 can be calculated from an segmental orientation distribution P(θ) according to ⟨cos θ ⟩ =

∫0

The derivation so far, which is equivalent to Flory’s analysis, has disregarded two effects of chain conformation (fortunately, these result in opposing changes and therefore partially cancel): (i) The derivation implicitly assumed that the chain does not turn back downward; this has already been pointed by DiMarzio and Guttman.6 A chain meandering up and down in the noncrystalline regions (see Figure 21c) could increase the density beyond Flory’s prediction.1,15 This means that eqs A20−A25 provide lower limits for the density of isotropic tie molecules. (ii) When we combine eqs A17 and A20 to express the amorphous density in terms of the crystalline density, we find that θb, the bond tilt angle in the crystallites, affects the amorphous density in this model

π /2

cos θ P(θ) dθ

(A14)

The orientation distribution for the semicircular chain fold is constant, Pcirc(θ) = 2/π, which gives ⟨cos θ⟩ = 2/π, so eq A13 reproduces ρave,circ = ρ1Dπ/2Ac,z−1 of eq A10. For a fixed bond tilt angle θb, e.g., θb = 35° in crystalline PE chains without chain tilt, P(θ) = δ(θ − θb), and ⟨cos θ⟩θb = cos θb, so ρ = ρ1D /cos θ b Ac, z −1

ρave,rand = 2 cos θ b ρxtal cos ϕ

(A15)

This reflects that even when the chain axis is vertical (ϕ = 0), the bonds in the crystalline phase are tilted. With cos θb = cos(35°) = 0.82 for PE, we find

Note that one cannot obtain the density from an equivalent chain with “virtual” segments; such a fictitious chain may yield the right contour length and end-to-end distance but does not distribute the mass in the same locations as in the real chain. If the chain axis is tilted by an angle ϕ, half the bonds are at an angle θb + ϕ and the others at θb − ϕ, relative to the z-axis, so ⟨cos θ ⟩ϕ =

ρave,rand,PE = 1.64 ρxtal cos ϕ

(A16)

Since for a tilted chain in a crystal, the area per chain in the crystal at constant z, Ac,z, also increases with 1/cos ϕ (see eq A3), the density in the crystal calculated from eqs A15 and A16 is independent of chain tilt: ρxtal = ρ1D /(cos θ b cos ϕ Ac ⊥ /cos ϕ) = ρ1D /cos θ b Ac ⊥−1 (A17)

Here, ρ1D/cos θb is the density projected onto the chain axis. For orthorhombic polyethylene, ρ1D/cos θb = 14 × 1.66 × 10−24 g/ 0.125 nm and

ρave,rand,PEO = 1.23 ρxtal cos ϕ

ρxtal = ρ1D /(cos(35°)ab /2) = 1.51 × 10−22 g/nm/(0.82 × 0.182 nm 2)

(A18)

ρave,rand,iPP = 1.23 ρxtal cos ϕ

For randomly oriented segments, Prand(θ) = sin θ and therefore

=

∫0

∫0

π /2

cos θ sin θ dθ =

∫0

cos θ d cos θ (A19)

Helical-Segment Model

If the tie-chain conformation in the noncrystalline regions can be described as consisting of linked and isotropically oriented, upward-moving segments of the same helical or planar all-trans chain structure as in the crystallites (see Figure 22a), a simpler effective chain model can be used (see Figure 22b). For the local chain axis in the crystallite, θb = 0 (see Figure 22b), and therefore from eq A21

so eq A13 gives an average density of randomly oriented tie molecules of ρave,rand =

(A24)

These results confirm quantitatively that the density problem is less pronounced, and chain tilt less required, in polymers with lowpitch helical conformations in the crystal than for those with planar zigzag structures, as concluded qualitatively from Figure 19.

1

1

z dz = 1/2

(A23)

For isotactic polypropylene, the average cos θb value to be used in eq A21 is [cos(0) + cos(70°)]/2 = 0.67 and

= 1.0 × 10−21 g/nm 3 = 1.0 g/cm 3

⟨cos θ ⟩rand =

(A22)

Considered in terms of the actual mass distribution (concentrated in the atoms, rather than homogeneously distributed along the bonds), this reflects that when the chain in the amorphous regions happens to have the carbon−carbon backbone bond pointing along the z-direction the atoms are spaced more widely (see Figure 19), and the density is locally lower than in the crystallites (see an example highlighted in Figure 21e). Interestingly, eq A21 does not contain the area per chain (except for the tilt factor): A larger area per chain decreases the amorphous density (see eq A20), but the crystalline density (eq A17) is reduced accordingly. The result of eq A22 applies to nylons, aliphatic polyesters, and PET with their planar zigzag structures in the crystallites. However, for polymers with helical conformations in the crystallites, the bond tilt is generally larger. For instance, in PEO with its 72 helical structure, the bond tilt angle calculated from the spacing, 0.95 Å, of backbone heavy atoms projected onto the chain axis is 52° and thus

1 1 cos(θ b + ϕ) + cos(θ b − ϕ) 2 2

= cos θ b cos ϕ

(A21)

ρ1D

ρ1D 1 = = 2ρ1D cos ϕ Ac ⊥−1 Ac, z ⟨cos θ ⟩rand 1/2Ac, z (A20)

In the last step, eq A3 has been used. The factor of 2 in eq A20 is equivalent to Flory’s famous conclusion about the chain flux,1 but the result is written directly in terms of density and with chain tilt in the crystal included explicitly.

ρave,rand,helix = 2ρxtal cos ϕ

(A25)

This is a higher density value than in eqs A22−A24. The earlier equations were obtained with a model in which noncrystalline Q


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Macromolecules bonds do not “point down”, while the model for eq A25 does contain bonds that “point down” (red arrow in Figure 22a), when the local chain axis is approximately horizontal. To derive eq A25, consider that when the local chain axis makes an angle β with the z-axis and the bond is rotated by an angle α around the chain axis (see Figure 22a), the “three-axes relation”107 gives cos θ = cos α sin β sin θb + cos β cos θb. Averaged over a random distribution of α and β, with ⟨cos α⟩rand = 0 and ⟨cos β⟩rand = 1/2, this results in ⟨cos θ ⟩rand,helix = ⟨cos α⟩rand ⟨sin β⟩rand sin θ b 1 + ⟨cos β⟩rand cos θ b = cos θ b 2

(13) Sperling, L. H. Introduction of Physical Polymer Science, 4th ed.; Wiley: New York, 2006. (14) Bassett, D. C.; Keller, A. Some new habit features in crystals of long chain compounds. Part II. Polymers. Philos. Mag. 1961, 6, 345− 358. (15) Frank, F. C. Organization of Macromolecules in the Condensed Phase - General Introduction. Faraday Discuss. Chem. Soc. 1979, 68, 7− 13. (16) Stamm, M.; Fischer, E. W.; Dettenmaier, M. Chain Conformation in the Crystalline State by Manes of Neutron Scattering Methods. Faraday Discuss. Chem. Soc. 1979, 68, 263−278. (17) Flory, P. J.; Yoon, D. Y.; Dill, K. A. The Interphase in Lamellar Semicrystalline Polymers. Macromolecules 1984, 17, 862−868. (18) Chum, S. P.; Knight, G. W.; Ruiz, J. M.; Phillips, P. J. Computer Modeling of {110} Adjacent Reentry of Polyethylene Molecules. Macromolecules 1994, 27, 656−659. (19) Strobl, G. The Physics of Polymers; Springer: Berlin, 1996. (20) Gautam, S.; Balijepalli, S.; Rutledge, G. C. Molecular Simulations of the Interlamellar Phase in Polymers: Effect of Chain Tilt. Macromolecules 2000, 33, 9136−9145. (21) Hütter, M.; in ’t Veld, P. J.; Rutledge, G. C. Polyethylene {201} crystal surface: interface stresses and thermodynamics. Polymer 2006, 47, 5494−5504. (22) Miyoshi, T.; Hu, W.; Hagihara, H. Local Packing Disorders in a Polymer Crystal by Two Dimensional Solid-State NMR. Macromolecules 2007, 40, 6789−6792. (23) Hong, Y. L.; Chen, W.; Yuan, S. C.; Kang, J.; Miyoshi, T. Chain Trajectory of Semicrystalline Polymers As Revealed by Solid - State NMR Spectroscopy. ACS Macro Lett. 2016, 5, 355−358. (24) Savage, R. C.; Mullin, N.; Hobbs, J. K. Molecular Conformation at the Crystal Amorphous Interface in Polyethylene. Macromolecules 2015, 48, 6160−6165. (25) Gedde, U. W. Polymer Physics; Chapman & Hall: London, 1995. (26) Fischer, E. W.; Schmidt, G. F. Long Periods in Drawn Polyethylene. Angew. Chem., Int. Ed. Engl. 1962, 1, 488−499. (27) Koberstein, J. T.; Morra, B.; Stein, R. S. The determination of diffuse-boundary thicknesses of polymers by small-angle X-ray scattering. J. Appl. Crystallogr. 1980, 13, 34−45. (28) Stribeck, N.; Alamo, R. G.; Mandelkern, L.; Zachmann, H. G. Study of the Phase-Structure of Linear Polyethylene by Means of SmallAngle X-Ray-Scattering and Raman-Spectroscopy. Macromolecules 1995, 28, 5029−5036. (29) Fischer, E. W.; Goddar, H.; Schmidt, G. F. A remark on the surface structure of polyethylene single crystals. J. Polym. Sci., Part B: Polym. Lett. 1967, 5, 619−624. (30) Sadler, D. M. Organization of Macromolecules in the Condensed Phase - General Discusion. Faraday Discuss. Chem. Soc. 1979, 68, 106. (31) Guttman, C. M.; Hoffman, J. D.; DiMarzio, E. A. Monte Carlo calculation of SANS for various models of semicrystalline polyethylene. Faraday Discuss. Chem. Soc. 1979, 68, 297. (32) DiMarzio, E. A.; Guttman, C. M.; Hoffman, J. D. Study of Amorphous-Crystal Interfaces in Polymers Using the Wicket Model Estimates of Bounds on Degree of Adjacent Reentry. Polymer 1980, 21, 1379−1384. (33) Ungar, G.; Stejny, J.; Keller, A.; Bidd, I.; Whiting, M. C. The crystallization of ultralong normal paraffins: the onset of chain folding. Science 1985, 229, 386−389. (34) Urabe, Y.; Tanaka, S.; Tsuru, S.; Fujinaga, M.; Yamamoto, H.; Takamizawa, K. Synthesis of Ultra Pure Long Normal Alkanes to Hexacohectane, Their Crystallization and Thermal Behavior. Polym. J. 1997, 29, 534−539. (35) Audino, M.; Grice, K.; Alexander, R.; Kagi, R. I.; Boreham, C. J. Unusual distribution of monomethylalkanes in Botryococcus brauniirich samples: origin and significance. Geochim. Cosmochim. Acta 2001, 65, 1995−2006. (36) Fritzsching, K. J.; Kim, J.; Holland, G. P. Probing lipid-cholesterol interactions in DOPC/eSM/Chol and DOPC/DPPC/Chol model lipid rafts with DSC and (13)C solid-state NMR. Biochim. Biophys. Acta, Biomembr. 2013, 1828, 1889−1898.

(A26)

which inserted into eq A13 and using eq A17 confirms eq A25.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b02000. Figures S1−S10 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (K.S.-R.). ORCID

Klaus Schmidt-Rohr: 0000-0002-3188-4828 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Dr. August W. Bosse for helpful discussions, Douglas J. Harris for the synthesis of the 13C-enriched polyethylene, and Dr. Justin Birdwell and the USGS for making the oil shale sample available.

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REFERENCES

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