Primary nucleation of polyethylene: Embryogenesis from a semidilute solution

Accepted Manuscript Primary Nucleation of Polyethylene: Embryogenesis from a Semidilute Solution Yi-Kang Lan , An-Chung Su PII:

S0032-3861(14)00391-7

DOI:

10.1016/j.polymer.2014.05.014

Reference:

JPOL 16967

To appear in:

Polymer

Received Date: 21 October 2013 Revised Date:

2 May 2014

Accepted Date: 5 May 2014

Please cite this article as: Lan Y-K, Su A-C, Primary Nucleation of Polyethylene: Embryogenesis from a Semidilute Solution, Polymer (2014), doi: 10.1016/j.polymer.2014.05.014. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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ACCEPTED MANUSCRIPT Primary Nucleation of Polyethylene: Embryogenesis from a Semidilute Solution Yi-Kang Lan and An-Chung Su*

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Using “realistic” molecular dynamics simulation extended up to 100 ns, we have investigated the evolution of cluster size, intrachain vs. interchain potential energies and pair correlations of polyethylene (PE) in a semidilute (ca. 28 wt%) 1,2,4-trichlorobenzene solution at 300 K. Results indicate that the embryonic development begins with the aggregation of trans-rich sequences of characteristic length lo ≈ 2 nm, forming clusters of short stems. This is immediately followed by reorganization/thickening via intracluster axial translation and reeling-in of segments from the surrounding matrix in dynamic competition with neighbouring embryos. Up to this stage, the embryonic clusters are loosely packed, retaining largely the conformer populations in the solution state but with gauche conformers enriched in the loose fold loops. After reaching a critical size with l* ≈ 4 nm, the intracluster order starts to significantly improve via a “solidification” process with sigmoidal decreases of valence and nonbonding energies, while axial diffusion dramatically slows down and intracluster torsions become fully adjusted to trans conformation by annihilation of gauche conformers. In these “solidified” embryos, although molecular packing remains deviated from the orthorhombic structure (as reflected in significant differences in pair correlations) while reminiscent of the mesomorphic “rotator” or hexagonal phase, the decrease in potential energy is already significant (corresponding to about half of the heat of crystallization) as the intrachain valence contribution is fully realized.

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Department of Chemical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan E-mail: [email protected]

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Introduction

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Primary nucleation of polymer crystals is poorly understood due to difficulties in studying embryos of limited size and transient lifetime. Previous studies of primary nucleation focused mainly on incipient crystals, with traditional wisdom extended from atomic crystals that the starting nuclei are natively similar to bulk crystals, only being much smaller in size and hence with significant surface effects [1]. More recently, with supporting evidences from small- and wide-angle X-ray scattering (SAXS/WAXS), Fourier-transform infrared spectroscopy (FTIR), differentials scanning calorimetry (DSC), electron microscopy, and molecular dynamics (MD) simulation, gradually emerged is an alternative picture of primary nucleation through bundling of extended chain segments (where the molecular packing within the bundle remains mesomorphic) into nanometer-sized embryos [2−8]. This lies much in line with the “bundle” theory proposed by Allegra et al. [9,10], in which the mesomorphic structure is assumed to comprise hexagonal arrays of extended stems. It is also reminiscent of Strobl’s “block” picture, where crystal growth is proposed to involve repeated scenarios of the secondary nucleation of mesomorphic blocks at the growth front that sequentially reorganize into matured/coalesced blocks of crystalline order [11]. Detailed evolution of chain conformation and molecular packing in the mesomorphic phase, however, remains to be clarified. Qualitative differences in chain conformation between the mesomorphic embryos and the amorphous matrix upon cold crystallization of poly(dimethyl siloxane) have been indicated by Lund et al. [12] from a combination of DSC, broadband dielectric spectroscopy, and wide/small-angle neutron scattering results.

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This is consistent with the observation by Konishi et al. [13] on the presence of a relatively narrow “mesophase” contribution superimposed on the broad amorphous halo in the WAXS profile of ice-quenched isotactic polypropylene, which suggests that the distribution of molecular spacing in the mesomorphic state is different from that in the amorphous matrix. On the basis of parallel WAXS and FTIR results for a racemic blend of poly(Llactide) and poly(D-lactide), we have further identified the correspondence between the mesomorphic peak in WAXS and the presence of long-lived precursors comprising loosely paired 32/31 helices prior to the emergence of βc crystals upon slow cooling from equilibrated melt state [14]. More quantitatively, Hikosaka et al. [15] showed that the primary nucleation rate I of polyethylene (PE) crystals under fixed supercooling scales with number-average molecular mass (Mn) according to I ~ Mn −η where the exponent η = 2.3 for foldedchain orthorhombic crystals and η = 1.0 for the extended-chain hexagonal (or “rotator”) phase. As the two values of η coincide with diffusion exponents for entangled vs. unentangled chains [16], Hikosaka et al. [15] conjectured that the nucleation process is kinetically controlled by the axial diffusion of polymer chains within the embryos. For linear alkanes, this rotator phase has been experimentally identified during the nucleation stage even for alkanes not exhibiting an equilibrium rotator phase [17,18]. Through theoretically treating crystal-melt interface as grafted brushes of loops in a self-consistent pressure field, Milner [19] showed recently that the rotator phase should indeed be favored in primary nucleation of PE crystals. Via a combination of analytical/numerical calculations and all-atom MD simulations for movement of soliton-like defects, Milner and Wenzel [20] further demonstrated low-barrier characteristics for axial motion in the rotator phase as compared to the orthorhombic phase of

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System and Method

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Results and Discussion 80

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With periodic boundary conditions, the semi-dilute (ca. 28 wt %) PE/TCB solution system comprised a 1000-mer PE chain and 400 TCB molecules. For initial system generation and MD simulation, the DREIDING force field with a hydrogen-implicit parameters (in the C-32 form of the DREIDING force field) was adopted for each methylene group and the hydrogen atoms in PE chain were removed for decreased atom number to speed up the simulation while keeping the backbone torsion behaviour unaltered, as the latter plays a key role in chain folding. The partial charge of TCB atoms was calculated by Gasteiger method whereas the partial charge in the PE chain was set to 0. The cutoff distance was set to 1.25 nm for non-bonding interactions. To build the initial state, the random PE chain was stochastically generated from the rotational isomeric state (RIS) model in a segment-by-segment manner in the 3D simulation box, taking into account interactions with all atoms already positioned. The solvent molecules were then randomly placed. The initial system density was The initial system was set to a relative large box, i.e., 0.600 g mL−1 in density, for easier setup of the system. The density was stepwise increased to 1.000 g mL−1 by gradually decreasing box size. To eliminate strains and stresses during system generation, geometry optimization followed by an MD pre-run of 500 ps (in time steps of 2 fs) at 300 K in the isobaric-isothermal (NPT) ensemble were applied, yielding converged density of ca. 1.252 g mL−1 with <1% fluctuations in temperature. Nose and Berendsen methods were adopted for thermo- and baro-stat purposes, respectively. With this NPT ensemble, a specific long-term MD run in time steps of 2 fs at 300 K was performed up to t = 100 ns. Commercial package Accelrys® Materials Studio was used for initial system

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Cell) and MD simulation engine (Forcite). In the semidilute condition, the initial polymer chain may randomly pass through the boundary several times and result in several periodic images within the simulation box. In view of the large ratio between the box size and the Kuhn length of PE, these periodic images were considered as effectively different chains. Further details are given in the Electronic Supplementary Information (ESI) where the capability of the united-atom DREIDING force field to sustain orthorhombic packing of PE chains was specifically checked via an MD test run up to 60 ns (cf. section S2 in ESI). To check for chain-length effects, MD simulation at 300 K for three shorter chains (each 333-mer in length) in 400 TCB molecules (i.e., 28 wt%) was also made. Results (cf. section S3) indicate embryonic evolution behaviour very similar to the single long-chain (1000-mer) case. A further MD simulation run using the Andersen barostat was also performed to check effects from the choice of barostat; results (cf. section S1) indicated generally consistent potential energy values but the extension stage was significantly prolonged, without entering the solidification stage up to t = 130 ns.

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Embryogenesis. Given in Fig. 1 are evolution profiles of the selected potential energy components and the embryonic size for the 28 wt% PE/TCB system at 300 K. There are 3 stages according to the evolution of the non-bonding energy between PE segments and that of chain-solvent interactions (cf. the lower two curves in Fig. 1). In the first stage (up to t ≈ 17 ns), intersegment nonbonding interactions decreased quickly (at the sacrifice of polymer-solvent interactions) upon aggregation of some extended stems to form embryos of characteristic axial length lo ≈ 2.2 nm. (As an operational definition of “stem”, we considered only extended segmental sequences with torsion angles within the range of 180˚ ± 30˚; segments of torsion angles outside this range such as those in the fold loops were excluded.) This is followed by the extension stage spanning from t ≈ 17 to 55 ns where more segments were reeled-in from the surrounding bulk solution via axial translation and extension of stems up to axial length l* ≈ 3.6 nm.

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ACCEPTED much restricted mobility. All these results appear to support the view that the rotator phase serves as mesomorphic precursor prior to full development of orthorhombic nuclei. Pushing towards improved understanding of the dynamic evolution process in primary nucleation of PE crystals, here we show through MD simulation of a semi-dilute (28 wt%) solution of PE in trichlorobenzene (TCB) up to 100 ns that the nucleation process comprises 3 stages. These include (1) aggregation of short stems, (2) extension of stems via intracluster axial translation of segments and reeling-in of further segments from the surrounding matrix, and (3) solidification of the bundle structure with concomitant step decreases of valence and nonbonding energies, beyond which axial segmental diffusion becomes dramatically slow. During the first two stages, conformer populations remain largely unchanged, meaning that unfavourable gauche conformers are mainly concentrated in the loose fold-loops. Only during the solidification stage are the inner-domain torsions fully adjusted to trans conformation by annihilation of gauche conformers. The evolution process observed lends credibility and further insights to various molecular models previously proposed in the literature. The conservation of conformer populations in the earlier two stages of nucleation and the dominant annihilation (instead of excretion through chain ends) of gauche conformers only in the solidification stage may bear significance in the interpretation of the spinodal-like features in the cold-crystallization of polymers.

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Fig. 2. Evolution profiles of velocity |v(t)| (relative to the center-of-mass of the entire chain) and the corresponding directional correlation χ(t) of a selected methylene group in an embryo developed from the 28 wt% PE/TCB system at 300 K. The v(t) profile demonstrates fast/slow fluctuations in axial motion during aggregation/extension stages (up to t ≈ 55 ns), followed by steadily suppressed axial translation (55 to 65 ns) before static equilibration. The χ(t) profile illustrates improved correlation during axial translation (with χ ≈ 0 at t ≈ 18 ns to χ ≈ 0.8 at t = 22 ns), reaching the fold (χ ≈ −½) at t ≈ 24 ns, followed by repeated axial translationt/fold-traversing steps as revealed by the oscillations between χ ≈ 1 and −½. Note that changes in the directional correlation extend into the solidification stage as long as there are residual movements. The time intervals for calculation of correlation were aperiodically chosen only to demonstrate the characteristics of segmental diffusion within the embryo.

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Correspondingly, there were only minor changes in both the intersegment nonbonding energy and in the polymer-solvent interactions, consistent with the observation that the aggregated domains were only loosely packed. After reaching the critical size l* ≈ 3.6 nm, the loosely packed embryo started to reorganize without further changes in the axial length, resulting in clearly decreased valence energy and also moderate decreases in intersegmental nonbonding energy and polymer-solvent interactions in the period of t = 55 to 65 ns, beyond which metastability is apparently reached. Note that the critical size l* ≈ 3.6 nm here is comparable to the incipient crystal thickness of ca. 6 nm experimentally observed for PE crystallized at 15 °C [21]. During the first two stages, the intrachain valence energy decreased only slightly, indicating insignificant changes in the global population distribution of conformers. Even though transrich stems are reeled into embryonic domains, the global conformer distribution remains largely unchanged; in other words, both the aggregation and the extension stages are mainly associated with the native trans-rich stems (as existed stochastically in the solution state), with gauche-like conformers enriched in the large fold loops. Embryonic domains in these two stages thus comprise near-trans conformers in a liquid-like packing of cylinders, but the stretching is by no means perfect.

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Fig. 1. Evolution of polymer valence energy (squares), polymer nonbonding energy (circles), and polymer-solvent interaction energy (triangles) in the 28 wt% PE/TCB system at 300 K per methylene. The first stage (up to t ≈ 17 ns) corresponds to initial aggregation of trans-rich sequences, as indicated by the stabilization via increased interchain vdW interactions at the cost of decreased polymer-solvent contacts; the stem thickness increases from ca. 2.2 to 2.8 nm mainly by excreting the gauche-like conformers to segments in the large fold loops, as the intrachain valence energy remains little changed. In the subsequent extension stage (t ≈ 17 to 55 ns) where the stem thickness is slightly decreased to ca. 2.5 nm but then more clearly increased to ca. 3.6 nm via axial translation of aggregated chains; concomitantly, there are slight decreases in intrachain valence energy and interchain non-valence energy, along with decreased polymer-solvent contacts. This is followed by the solidification stage (t ≈ 55 to 65 ns) of step-like decrease in intrachain valence, interchain non-valence, and polymer-solvent interaction energies, with correspondingly improved molecular packing in the embryo and fully suppressed axial translation (cf. Fig. 2), reaching apparent metastability thereafter.

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This loosely packed state provides low barriers and/or friction for axial stem translation and domain extension during the reeling-in process, which can be most dramatically demonstrated in terms of the instantaneous segmental velocity |v(t)| (relative to the center-of-mass of the entire chain) and the corresponding orientational correlation. Shown in Fig. 2 are evolution profiles of |v(t)| and the orientational correlation χ(t) ≡½[3cos2θ(t) – 1] of a representative segment, where θ denotes the angle between the velocity directions between t and t+∆t. The v(t) profile demonstrates fast/slow fluctuations in axial motion during the aggregation and extension stages up to t ≈ 55 ns, followed by steadily suppressed axial translation during t ≈ 55 to 65 ns and |v(t)| → 0 for t > 65 ns in the solidification stage. The evolution profile of orientational correlation shows the presence of both oriented axial translation with χ(t) > 0.7 and oscillations of χ(t) values between 1 and −½ when the segment traversed the fold loop during the extension stage (t ≈ 17 to 55 ns). Note that the frequency of traversing the fold loop decreases as the system enters deep into the solidification stage, signifying the slowingdown of axial translation; residual motion nevertheless persists, albeit in a much longer time scale.

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Upon solidification from t = 55 to 65 ns, the steep decrease in intrachain valence energy (Fig. 1) suggests backbone adjustment towards trans conformers. This is confirmed by the intrachain radial distribution function (RDF) (relative to RDF at t = 1 ns) extracted from segments in the embryo as shown in Fig. 3. For t < 55 ns, there are no significant peaks in the intrachain RDF, consistent with the notion that the chain conformation remain fairly irregular in the first two stages. Concomitant with the intrachain ordering, the evolution of corresponding interchain RDF in Fig. 4 indicates transformation from liquid-like packing to solid-like correlations at characteristic intercarbon distances of r = 3.8, 4.4, and 4.8 Å. This solid-like packing, however, deviates significantly from the reference orthorhombic PE crystal where correlation peaks are expected to be around 3.8, 4.0, 4.7, and 5.3 Å. The corresponding vdW density map (cf. inserts in Fig. 4a) bears some resemblance with the rotator phase in n-alkanes or the hexagonal phase of PE in terms of hexagonal-like lateral packing of chains [19,20]. The strong contrast between intra- and interchain RDF reflects clearly the mesomorphic nature of the “solidified” embryo.

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Fig. 3. Evolution of intrachain RDF profile (relative to that at t = 1 ns) of the 28 wt% PE/TCB system at 300 K. There are no correlation peaks during the aggregation/reorganization stages (i.e., up to t ≈ 55 ns), indicating negligible changes in chain conformation. Significantly improved intrachain order (with all-trans conformation apporoached) is observed only in the solidification stage. Given as inset is the expanded fingerprint region of intrachain correlation distances D14 to D16 (from a carbon atom to its 3rd to 5th neighbors along the chain), showing abruptly increased intrachain order (with corresponding orthorhombic crystal values given in parentheses for comparison) around t ≈ 60 ns (which coincides with step-like energy decreases in the interval of t = 55 to 65 ns in Fig. 1) and more delicate improvements thereafter.

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Fig. 4. (a) Evolution of interchain RDF profile (relative to that at t = 1 ns) of 28 wt% PE in TCB at 300 K. In the aggregation stage (up to t ≈ 17 ns), there emerge two broad peaks at carbon- to-carbon distances of ca. 4 and 8 Å, signifying loosely aggregated stems in the embryo. In the subsequent reorganization stage (t ≈ 17 to 55 ns), the interchain packing order is only moderately improved (see also evolution of inter-chain nonbonding energy in Fig. 1) with simultaneously increasing domain size via reelingin of new segments through axial translation of stems. Upon solidification starting at t ≈ 55 ns, the interchain packing is quickly improved with concomitant intrachain reorganization (cf. Fig. 3), leading to 3 sharp peaks at 3.8, 4.4, and 4.8 Å and a group of overlapping peaks at 6.8, 7.8, and 8.4 Å. (b) Comparison of RDF profiles with that of the reference orthorhombic crystal MD trajectory (300 K, 60 ns, cf. section S2 in ESI) suggests significant deviations, particularly in the case of correlation peaks near 4.4 and 4.8 Å. Note that the improvement in interchain packing is somewhat slower than intrachain ordering towards all-trans confomation in Fig. 3, presumably due to surface effects in such a case of small domain size as shown by the vdW density cross-section map where effectively two embryos are in contact through a narrow “neck” of single contact of stems.

This picture is generally in line with the “bundle” model (where the molecular packing in the embryo is strictly hexagonal) proposed by Allegra et al. [9,10], the “molecular nucleation” hypothesis (without specifying mechanistic details) conjectured by Wunderlich et al. [1], the “intramolecular nucleation” model (where bundle formation is limited to segments from the same chain) suggested by Hu et al. [22,23], or the specifically emphasized significance of axial stem diffusion during nucleation

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by Hikosaka et al. [15] The observation ofACCEPTED only mesomorphicMANUSCRIPT embryo demonstrated by the accompanying vdW density crossorder yet with significantly decreased intrachain valence energy section maps. The incomplete agreement between D values of the are consistent with previous SAXS/WAXS/DSC results for cold- 45 “solidified” embryo and the hexagonal phase is not surprising, as crystallization of glassy polymers where a high population of the all-trans sequence length in the hexagonal phase is known [26] to be limited to ca. 5 methylene units. mesomorphic nuclei were identified via SAXS, with nearly half of the crystallization released, yet without observable WAXS signals [3,4,8]. We note further that, in Strobl’s proposal [11] of a mesophase at growth front, the mesomorphic structure is not specified; in Allegra’s bundle theory [9,10], hexagonal packing of strictly trans conformers is assumed. The former conjectures a secondary nucleation process as the growth mechanism whereas the latter a theory for primary nucleation. Both are relevant to the notion that nucleation occurs through an intermediate stage (mesomorphic or hexagonal) before the embryo reaches the final crystalline state. Our MD results demonstrate similar features in a sequential manner to the two proposed structural extremes.

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Fig. 5. Axial diffusion coefficient of a representative segment during embryonic evolution. Corresponding vdW density cross-section maps (at t = 5, 40, and 80 ns, respectively) are given as inserts to demonstrate the state of stem packing at each selected point of time. Solvent molecules are hidden from the view. Note the strong decrease (by nearly 5 orders of magnitude) in the diffusion coefficient during the solidification stage.

Axial Diffusion. Previous results [24] of NMR line shape analysis and spin-lattice relaxation studies at 490 MPa and ca. 510 K have indicated that, while segmental motion is slower and restricted to low-amplitude reorientations in orthorhombic PE crystals, it is faster and axial in character in the mesomorphic hexagonal phase; the measured axial diffusion coefficient D ≈ 10−16 m2 s−1 in orthorhombic crystals and D ≈ 10−13 m2 s−1 in the hexagonal phase. Given in Fig. 5 is the evolution profile of the axial diffusion coefficient (D) calculated from the position-time trajectory of a representative embryonic segment. For both the aggregation and the extension stages, D ≈ 10−10 m2 s−1 which is comparable to values of nematic liquid crystals [25]; upon solidification (t ≈ 55 to 75 ns), there is a step drop of D to 10−14 m2 s−1, closer to the value of the hexagonal phase rather than that of the orthorhombic crystal. These observations are consistent with the loose molecular packing in the aggregation and the extension stages, as well as the hexatic packing in the solidified

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Fig. 6. Evolution of Rg of the 1000-mer chain. The insertted MD snapshots (at 5 and 90 ns, respectively) indicate that the chain (under periodic boundary conditions) passes through several embryonic domains which serve as physical crosslinks. After the initial decrease from ca. 71 to 62 Å during embryo formation via stem bundling in the aggregation stage, Rg is confined to limited fluctuations (61.7 ± 0.7 Å) in the subsequent extension stage and eventually becomes fully locked-in at 60.6 Å after solidification.

Locking-in effects. Upon solidification, embryonic segments become fully immobilized; this renders each embryo a physical crosslink to inhibit all long-range chain motion, with entanglements between embryos effectively locked-in. Fig. 6 shows the evolution profile of the radius of gyration (Rg) of the PE chain as well as MD snapshots of the entire chain at t = 5 and 90 ns. During the aggregation stage, Rg decreased quickly from 71 to 62 Å, followed by moderate fluctuations in the range of 61.7 ± 0.7 Å in the subsequent extension stage and eventually becomes fully locked-in at 60.6 Å for t > 60 ns in the solidification stage. The fluctuating Rg during the extension stage is a reflection of dynamic competition among embryos that are reeling-in new segments, as demonstrated by the inserted MD snapshots where the polymer chain is observed to traverse several developing embryos across the MD periodic boundaries. The solidified structure itself may then be thought of as a tightened network of fully balanced reel-in forces from the connected embryos of mesomorphic order, for which the transformation to true crystalline order is incomplete. This is consistent with previous SAXS/WAXS observations [2−4,7,8] in coldcrystallization of quenched polymer glasses upon heating above the glass transition temperature (Tg), where the emergence of mesomorphic nuclei of fixed size was found to emerge sporadically, reaching a saturated population before the subsequent growth/ripening processes. It is also interesting to note that, the decrease in intrachain valence energy within the

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There are a few points for further elaboration. Firstly, it should be noted that the similarity with the rotator phase of nalkanes or the hexagonal phase of PE is limited to the lateral hexagonal-like packing. Sirota et al. [33,34] reported the absence of rotator phase upon start of folding for longer n-alkanes whereas there are significant population of gauche conformers in the high-pressure hexagonal phase [26], both are at variance with the embryonic features observed here. Regarding the absence of WAXS signal upon emergence of SAXS structure, there could be questions on inadequate WAXS detector sensitivity. This has been addressed by Heeley et al. [35] using a WAXS detector with improved sensitivity by 4 orders of magnitude; their results still indicated substantial time-lag between the appearance of the SAXS peak and the emergence of WAXS reflections. It should also be note that, in the MD simulation of Luo and Sommer [36] using a course-grained PVA model, it was observed that the stem length increases linearly with its lifetime in the earlier “precursory” stage while it increases logarithmically at the following “thickening” stage. These correspond respectively to the “extension” and “solidification” stages here. As a “selfseeding” technique was adopted, they were in effect tackling surface-nucleation for crystal growth rather than primary nucleation, hence our “aggregation” stage was technically skipped. Since their model corresponds to PVA of lower symmetry than PE, no intermediate phase was expected or identified. Nevertheless, the general observation of stage-wise evolution during crystallization is consistent with the present results. Finally, as the “solidified” embryos are laterally small (3 or 4 chains across, cf. Fig. 5) and devoid of strict orthorhombic order, we do expect further reorganization of lateral packing via a much slower process of lateral adjustments of stem positions.

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ACCEPTED 1000-mer system up to t = 100 ns corresponds 6.2 kJ/mol methylene (cf. Fig. 1). This constitutes to ⅔ of the bulk heat of crystallization (9.37 kJ/mol methylene), implying strong stretching in favour of trans conformers not only within the embryos but also in the inter-embryo connecting chains. Extensive studies by Kagi et al. [27] have led to a general scheme describing various paths of structural development in the initial stage of polymer crystallization, in which formation of nematic- and smectic-like mesophases in a broad range of length scales are postulated as intermediates towards the final crystalline state. Similar “global” features of poorly ordered domains have also been suggested in our TEM or SAXS/WAXS studies [5,7]. These cannot be reflected in the present MD computation due to the limited system size here but the observed lock-in effect upon solidification may bear some relevance to features of length scales beyond the embryo size. The role of locked-in entanglements has been extensively demonstrated by Strobl et al. [28−30] in terms of mechanical responses of semicrystalline polymers. In the specific case of cold-crystallization of quenched polymer glass upon reheating above Tg, spinodal-like features have been identified [27] and theoretically handled [31] by introducing a quadratic density-conformation coupling term to the Landau expression of the free energy with the implicit assumption of conserved conformer populations. The present results indicate that the latter assumption appears valid in the early stages but may eventually fail during the solidification stage where annihilation of gauche conformers of opposite rotational sense dominates. Critical length scales. In our previous discussion [6] on MD results for a more diluted case of 6.5 wt% PE (three 300-mer chains) in TCB up to 15 ns at 300 or 400 K, the axial length lo ≈ 3.2 nm for aggregation into a “nematic” cluster, slightly greater than the present value of lo ≈ 2.2 nm; both are comparable with the prediction (lo ≈ 2.2 nm) from the simple Onsager theory [32] where only the geometric effects in the packing of rod-like entities are considered. The duration of the aggregation period was much shorter (2 ns) in the previous case whereas the subsequent extension stage continued till termination of MD runs (up to ca. 15 ns) without apparent signs of solidification, suggesting kinetic effects due to lack of significant entanglements in this case of shorter chains in more diluted solutions. It should be noted that the thickness selection is mainly achieved in the extension stage. The observation that the critical length l*, determined during the extension stage and remained unchanged in the subsequent solidification stage, is indeed characteristic to the nucleation process at a given solution concentration and temperature. Summarized in Table 1 are our results for PE chains of various lengths under different crystallization conditions. From these, it may be summarized that l* decreases with increasing supercooling and concentration while being less sensitive to chain length, consistent with general observations in polymer crystallization. Table 1. Summary of MD simulation results for PE chains under different crystallization conditions. Chain length Concentration Temperature l* 300 6.5 wt% 400 K 9.5 nm 300 6.5 wt% 300 K 6.1 nm 333 28 wt% 300 K 4.2 nm

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Summary By means of MD simulation up to 0.1 µs for a semi-dilute (28 wt%) long-chain PE/TCB solution system, we have demonstrated that the primary nucleation process may be characterized by three stages of (1) stem-aggregation for embryo formation, (2) extension of the embryonic bundle, and (3) the “solidification” of the extended bundle. The incipient formation of embryo requires a minimum stem length lo ≈ 2 nm for aggregation into a molecular cluster of nematic order. During this and the subsequent stage of bundle extension into critical stem length l* via axial gliding of stems, the global distribution of conformer populations remains little changed. The trans conformers become more fully developed mainly in the solidification stage where significant intrachain order increased with the concomitantly increased cluster size, while the lamellar thickness remained unchanged at l* and the interchain packing mesomorphic with hexagonal-like lateral arrangement.

Acknowledgments Financial supports from the National Science Council (grant numbers NSC98 2221 E 007 009 MY3 and NSC99 2113 M 007 018 MY2) are gratefully acknowledged. Thanks are also due to

ACCEPTED the National Centre for High-performance Computing (NCHC) for computer time and facilities.

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Electronic Supplementary Information Primary Nucleation of Polyethylene: Embryogenesis from a Semidilute Solution Yi-Kang Lan and An-Chung Su Department of Chemical Engineering, National Tsing Hua University, Hsinchu 300, Taiwan

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S1. Details of Computational Method The PE/TCB solution system comprised a 1000-mer PE chain and 400 TCB molecules, which corresponded to a semi-dilute (ca. 28 wt %) solution. Commercial package Accelrys® Materials Studio1 was used for initial system generation (Amorphous Cell) and MD simulation engine (Forcite).

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Initial System Generation To generate the initial system, here means the amorphous state of the polymer chain and solvents, there are several methods to approach. For example, one can build up the origin system with the crystal data of polymer and solvent and melt them into the mix-amorphous state with high temperature. Also one can build up the polymer chain and solvents with linear array in a big box, than compress the box and do geometry optimizations per compressing step. Here the commercial package Accelrys® Materials Studio was used for initial system generation (Amorphous Cell). The initial PE chains were stochastically generated from the rotational isomeric state (RIS) model2 by growing the chain molecules inside the three-dimensional box segment by segment, taking into account interactions with all atoms already positioned. Then the solvent molecules were randomly placed into the box with checking the close contacts and ring spearing. The initial system was set to a relative large box, i.e., 0.600 g mL−1 in density, for easier setup of the system. The density was stepwise increased to 1.000 g mL−1 by gradually decreasing box size. The system converged (i.e., <1% in temperature fluctuations) to a stabilized density of ca. 1.252 g mL−1 after geometry optimization and a MD pre-run of 500 ps at 300 K.

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MD Simulation Protocol All the MD simulation was take with the DREIDING force field3, the Verlet integrator, the isobaric-isothermal (NPT) ensemble, a cutoff distance of non-bonding interactions at 1.25 nm and time steps of 2 fs. The partial charge of atoms of TCB was calculated by Gasteiger method4 whereas the partial charge of atoms in PE chain was set to 0. The Nose5 and Berendsen6 methods were adopted for the thermo- and baro-stat purposes. A long-term MD run up to t = 100 ns at 300 K was subsequently performed. Thermo- and Baro-stats There are 4 types of thermostat (Velocity Scale, Nosé, Andersen7 and Berendsen6) and 2 of barostat (Andersen7 and Berendsen) in the Forcite simulation package. The Velocity Scale and Berendsen methods control the system temperature with rescaling the velocity by multiplied a factor; the Andersen method rescaling the velocity with random collisions to all atoms, and the Nosé method add an additional (fictitious) degree of freedom with “mass” to control the velocity of 1

Programs from Accelrys Inc. (San Diego, CA, USA. www.accelrys.com). Flory, P.J. Statistical Mechanics of Chain Molecules, Interscience: New York (1969). 3 Mayo, S. L.; Olafson, B. D.; Goddard III, W. A. J. Phys. Chem. 1990, 94, 8897-98909. 4 Gasteiger, P.-D. D. J.; Saller, D.-C. H. Angew. Chem. Int. Ed. 1985, 24, 687–689. 5 (a) Nosé, S. J. Mol. Phys. 1984, 52, 255-268. (b) Nosé, S. J. Chem. Phys.1984, 81, 511-519 (c) Nosé, S. J. Prog. Theor. Phys., Suppl.1991, 103, 1-46. 6 H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola and J. R. Haak, J. Chem. Phys. 1984, 81, 3684. 7 H. C. Andersen, J. Chem. Phys. 1980, 72 , 2384. 2

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each atom. The adopted Berendsen barostat controls the system pressure by changing the dimensions of the simulation cell size with re-scaling the atom positions during the simulation, where the Andersen method uses an extended fictional volume with a “mass” to control the volume of the simulation cell. Considering that the Velocity Scale and Berendsen methods may trap the system in the localized conformations/correlations and that the Andersen method will turn the continued dynamics into the Markov process, we have chosen the Nosé method for thermostat. A parallel MD run with the Andersen barostat (a modified version of Nosé-Hoover type) was made to check for any barostat effects. Results of different barostats are compared in Figure S1 where the use of Andersen barostat is shown to give the same potential energy evolution as the Berendsen barostat for t < 50 ns, followed by prolonged extension stage to beyond tMD = 130 ns. The change from Berendsen to Anderson barostat bears no impact on the potential energy values but indeed results in a prolonged extension period, leading to delayed solidification.

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Figure S1. Evolution of polymer valence energy (squares), polymer non-bonding energy (circles), and the interaction energy between polymer and solvent (triangles) in the 28 wt%, three 333-mer PE/TCB system at 300 K per methylene. Filled symbols denote results by use of the Andersen barostat whereas open symbols the Berendsen barostat case. The Andersen barostat gives the same potential energy values and evolutions in both the aggregation and the following extension stages, but the latter stage is significantly prolonged such that the solidification step does not appear up to t = 130 ns. Force Field Parameters We employed the DREIDING force field, which has good coverage for organic, biological and main-group inorganic molecules. It is specified for geometries, conformational energies, intermolecular binding energies and crystal packing. The philosophy in DREIDING is to use general force constants and geometry parameters based on simple hybridization considerations rather than individual force constants and geometric parameters that depend on the particular combination of atoms involved in the bond, angle, or torsion terms. The potential form of DREIDING is given as
 
 
 
 
 
 
 


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S2. Reference PE packing without H atoms

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@A 5 @A 5"  10@A 15 2 3  62 3 7  D !C 7 C C The DREIDING force field provides parameters to lump a carbon atom and the attaching hydrogen atoms into a single effective atom (implicit hydrogens). The van der Waals parameters for such cases are estimated by matching calculated methane and benzene crystals with experimentally determined lattice spacing and heat of sublimation extrapolated to 0 K. Here the effective-atom potential (in the form of C32 parameterization) was adopted for each methylene group such that the hydrogen atoms in PE chain were removed to speed up the simulation by effectively reducing the system atom number.

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To check the representativeness of the chain packing in DREIDING force field, we built a crystalline PE system of 2 × 2 × 10 orthorhombic unit cells with all H atoms removed and performed MD simulation with the DREIDING force field for the NPT ensemble. The original b/a ratio of the initial system is 14.776/9.858 = 1.4989, which represents the character of the orthorhombic packing system. After 60 ns MD simulation at 300 K, all 3 crystallographic axes were observed to decrease consistently by 6.2% whereas the orthogonality between crystallographic axes is retained, giving constant ratios between lattice parameters. Although there were significant changes in the azimuthal orientation and slight bending of the backbone plane, the unit cell structure remained orthorhombic under the united-atom DREIDING force field. This is used as the “reference” orthorhombic structure in Figure 4b for comparison with the “solidified” embryonic structure.

Figure S2. The initial snapshot of the crystalline PE system of 2 × 2 × 10 orthorhombic unit cells (upper-left), the snapshot after 60 ns MD simulation at 300 K (upper-right), and the corresponding evolutions of the cell lengths/angles (lower-left/right) of the simulation box.

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For this semi-dilute case of 28 wt% PE in TCB solvent, we have also prepared a system of three 333-mer chains to check the chain-length effects. Evolution processes of the potential energy (Fig. S2) and the radial distribution function (Fig. S3) appear similar to those observed for the single 1000-mer chain case addressed in the main text. The main difference is the moderately shortened aggregation stage, up to t ≈ 13 ns in this case.

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Figure S3. Evolution of polymer valence energy (squares), polymer non-bonding energy (circles), and the interaction energy between polymer and solvent (triangles) in the 28 wt%, three 333mer PE/TCB system at 300 K. The first stage (up to t ≈ 13 ns) corresponds to initial aggregation of trans-rich sequences, as indicated by the stabilization via increased interchain vdW interactions at the cost of decreased polymer-solvent contacts; the stem thickness increases from ca. 1.8 to 3.8 nm mainly by excreting gauche-like conformers to segments in the large fold loops, as the intrachain valence energy remains little changed. In the subsequent extension stage (t ≈ 13 to 50 ns) where the stem thickness is fluctuating, first slightly increased to ca. 4.8 nm but then more clearly decreased to ca. 4.2 nm. Concomitantly, there are decreases in interchain non-valence energy along with decreasing polymer-solvent contacts. This is followed by the solidification stage (t ≈ 50 to 65 ns) of step-like decrease in intrachain valence, interchain non-valence, and polymer-solvent interaction energies, with correspondingly improved molecular packing and substantially suppressed axial translation of stems within the embryo, reaching apparent metastability.

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Figure S4. Evolution of interchain RDF profile (relative to that at t = 1 ns) of 28 wt%, three 333-mer PE chains in TCB at 300 K. Two broad peaks at carbon-to-carbon distances of ca. 4 and 8 Å appears at t = 5 ns, signifying loosely aggregated stems in the embryo. In the second stage (t ≈ 13 to 50 ns), the interchain packing order is only moderately improved with simultaneously increasing domain size via reeling-in of new segments through axial translation of stems. Upon solidification starting at t ≈ 50 ns, the interchain packing is quickly improved with concomitant intrachain reorganization, leading to 3 sharp peaks at 3.7, 4.4, and 5.0 Å and two groups of overlapping peaks around 7.8, and 11.0 Å. Given as inset is the evolution of intrachain RDF profile relative to that at t = 1 ns. There are no correlation peaks during the aggregation/extension stages (i.e., for t < 50 ns), indicating negligible changes in chain conformation. Significantly improved intrachain order towards all-trans conformation is observed only in the solidification stage.

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Figure S5. Evolution of the mean Rg value for the three 333-mer chains, coloured in blue, pink, and black). The inserted MD snapshots (at t = 5 and 80 ns, respectively) demonstrate stem extension concomitant with the steady increase of Rg for t < 55 ns. The chain extension process eventually results in a few long stems (i.e., “tie chains”, ca. 10 nm in length) traversing through the three ordered domains which are ca. 4 nm in axial length. Upon solidification (t > 55 ns), Rg becomes fully locked-in at ca. 41 Å.