Effects of Anisotropy and Disorder on Crystal–Melt Tensions in

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Effects of Anisotropy and Disorder on Crystal−Melt Tensions in Polyolefins Qin Chen and Scott T. Milner*

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Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States ABSTRACT: Interfacial tensions between a polymer crystal and its melt play a central role in nucleation theory. While the tension on the end face (lamellar surface) can be obtained from melting-point suppression, tensions on the sides of a crystal nucleus have not been measured. We use our recently developed “plunger” simulation method to obtain melt− crystal interfacial tensions for polyethylene (PE) and isotactic polypropylene (iPP) ordered phases against their melts. We find PE orthorhombic crystals have strong anisotropy in their side-face interfacial tensions, which leads to a rather oblong crystal cross section under the Wulff construction, far from an idealized hexagonal or cylindrical crystal nucleus. We also investigate rotationally disordered “rotator” or “condis” phases in PE and iPP, which have smaller side-face tensions than their crystalline counterparts. This finding is consistent with Strobl’s hypothesis that rotator phases have lower nucleation barriers and nucleate first in many common polymers including PE and iPP.

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INTRODUCTION Crystallization of a polymer from the melt is a thermodynamic transition, but the process is dominated by kinetics. In the molten state, the sluggish movement of long entangled polymer chains prevents the formation of crystals with fully extended chains and hence to achieve the minimum free energy state. As a result, crystallizable long polymers are universally observed to be semicrystalline. Their solid-state structure typically consists of lamellar crystallites of nanoscale thickness, separated by amorphous regions of similar dimensions. Because the polymer structure is populated by crystal−melt interfaces, surface effects play an important role in the crystallization mechanisms of polymers. According to nucleation theory, tiny crystalline nuclei arise randomly as a result of thermal fluctuations from a melt supercooled below the equilibrium melting temperature. These nuclei grow or shrink depending on the competition between two factors: the lower bulk free energy of the nucleus compared to the surrounding melt and the free energy of the interface between crystal and melt. For a sufficiently large nucleus, the bulk free energy decrease dominates the interfacial tension, and further growth decreases the overall free energy. The thermodynamic nucleation barrier ΔG is the free energy required to build a nucleus of the critical size from the melt. Within the approximation of a cylindrically shaped nucleus, ΔG is given by1 ΔG* =

In the most straightforward nucleation scenario, only one ordered phase is stable with respect to the melt in the supercooled state, so the phase that crystallizes is the phase that nucleates. More complicated scenarios are possible, in which a second phase is stable with respect to the melt but only metastable with respect to the crystal. If this phase is more kinetically favorable to nucleate, perhaps because its interfacial tensions with the melt are significantly lower than for the crystal, it can happen that the metastable phase nucleates and afterward transforms to the stable crystalline phase. This scenario corresponds to Ostwald’s “rule of stages”.2 Accumulated evidence suggests this scenario applies to nucleation of many common polymers, including polyethylene (PE). For example, observations of crystallization in intermediate-length n-alkanes (oligomeric PE) with X-ray scattering find that a mesomorphic phase first forms immediately upon nucleation and then quickly transforms to the more stable crystal structure.3,4 This mesomorphic phase has been identified experimentally as the RII “rotator phase”, consisting of a hexagonal packing of parallel, nearly all-trans PE chain segments or “stems”, randomly rotated about their own axes. This rotator phase has also been observed and characterized in atomistic MD simulations.5 Strobl and collaborators have amassed data on crystallization and melting temperatures versus lamellar thickness in many systems, which support his hypothesis that for many common polymers some less-ordered mesophase nucleates first and transforms to the final observed crystalline phase.6−8

8πγs 2γe [ΔS(ΔT )]2

(1)

Here γe and γs denote the “end” and “side” interfacial tensions, while ΔS and ΔT represent the transition entropy and undercooling. © XXXX American Chemical Society

Received: June 11, 2018 Revised: August 23, 2018

A

DOI: 10.1021/acs.macromol.8b01248 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Because the rotator phase free energy is higher than that of the stable crystalline phase, the only way it can have a lower nucleation barrier at a given undercooling is for its interfacial tensions with the melt to be lower than for the crystal. One might expect this to be the case because the rotator phase structure is less dense and more disordered than the crystal, so that the rotator−melt interface involves a less severe adjustment of phase properties on both sides of the interface than for the crystal−melt interface (see Figure 1). A fluffier,

Figure 2. Top view of PE orthorhombic crystal, displaying (110), (100), and (001) facets.

with a still higher value for the highly corrugated (010) interface. From the interfacial tension values for different facets, the equilibrium crystal shape is given by the Wulff construction,11,12 which states that the distance from the crystal center normal to a given facet is proportional to the tension on that facet. Low-tension facets lie close to the crystal center and thus have large interfacial areas. The more anisotropic is the interfacial tension, the farther from round is the equilibrium crystal shape. The equilibrium shape is relevant for nucleation because it has the lowest interfacial free energy at a given volume and thus gives the lowest nucleation barrier. In previous work, we developed a “plunger” method to obtain the interfacial free energy of polymer crystal−melt interfaces from molecular dynamics simulation.13 This method exploits the mechanical definition of surface tension, which quantifies the tendency of a liquid to wet a solid surface. In brief, we simulate a polymer melt confined in a planar channel bounded on the sides by crystalline slabs, with a reservoir of melt above, and confined by a graphene-like “plunger” below. We measure the force on the plunger required to restrain the melt from advancing in the channel and wetting more of the crystalline surface. In this way, we measure the difference Δγ between the crystal−vacuum and crystal−melt interfacial free energies γcv and γcm. Separately, we determine the crystal−vacuum interfacial free energy γcv by measuring the work to pull a crystal apart. We measure the force required to hold two crystalline slabs at a given distance and integrate the force to obtain the work. Combining these results, we have γcm = γcv − Δγ. The values we obtain are evidently independent of nucleation theory. Because we measure the static force on the plunger to get Δγ, and obtain γcv from the integral of the static force to hold two crystalline slabs apart, our values are equilibrium interfacial free energies. They can therefore be used as inputs to compute nucleation barriers for comparison to experimentally observed nucleation rates. Our results can also be used for comparing predicted barriers for crystalline and mesophase nuclei. However, our present work is incomplete in that we cannot obtain end-face tensions between ordered lamellae and adjacent melt. End-face interfaces have a complex structure of chain folds and loops, with some chain segments spanning continuously from the ordered to the amorphous phase. Assembling such interfaces

Figure 1. Schematic of PE orthorhombic crystal nucleus (left) and rotator phase nucleus (right), viewed along the chain stems. Thick lines depict in-plane projection of all-trans stems; solid and dashed curves denote interface between melt and crystal or rotator phase.

more disordered rotator phase would suffer less reduction of favorable packing energies at the interface, and the melt would suffer less reduction of favorable configurational entropy immediately adjacent to a less well-packed rotator phase. From the above discussion, it is apparent that the interfacial tensions between melt and ordered phases are key parameters in nucleation theory and in predicting the phase that nucleates. The end-face melt−crystal tension can be obtained from melting-point suppression by plotting the melting temperature versus inverse lamellar thickness and applying the Gibbs− Thomson relation.1,3 But the various side-face tensions have not been measured, and it is unclear how they could be measured directly. Side-face tension values have been inferred from painstaking homogeneous nucleation experiments9,10 by fitting the observations to nucleation theory. The nucleation rate depends exponentially on the nucleation barrier as I = I0 exp( −ΔG*/kT )

(2)

Here I denotes the homogeneous nucleation rate, I0 is a kinetic prefactor, and T is the crystallization temperature. In this approach, the shape of the nucleus is taken to be a cylinder with some assumed cross section (e.g., circular or hexagonal), so that an effective side-face tension σs can be inferred from the barrier obtained from nucleation rate data by applying eq 2. Evidently this approach cannot be used to validate nucleation theory nor to compare crystal and rotator phases to predict which phase would nucleate more readily. In a more realistic view of a crystalline nucleus, there are different crystalline facets, each with its own distinct interfacial free energy with the melt (see Figure 2). We may expect that facets with smoother surfaces will have lower interfacial free energies with the melt because the energy penalty for missing neighbors of crystalline stems at the interface will be less severe. Hence, we expect that the (110) interface may have a lower melt tension than the slightly corrugated (100) interface, B

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Figure 3. Schematic of a representative cylindrical iPP nucleus in the α phase and in a hypothetical rotator phase.

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SIMULATION DETAILS At first sight, the equilibrium interfacial tension between an ordered phase and its melt is only properly defined right at the melting temperature Tm. However, as long as the ordered phase and melt are both metastable, the ordered phase can be restrained from melting, the melt prevented from ordering, and the interfacial tension sensibly defined. Hence, the plunger method is straightforward conceptually but requires some care in two respects: (1) keeping the melt melted and the ordered phase ordered as the simulation progresses and (2) constructing the initial configurations. In this section, we describe how this is accomplished. Readers less concerned with these details may proceed to the next section. Constructing Ordered Phases. In our plunger and pulling simulations for PE, we use ordered phases of n-alkanes as a proxy for PE lamellae. For interfacial tensions with crystalline PE relevant to the construction of crystalline nuclei, in the present work we investigate low-index facets (010) and (110) to extend our first results on the (100) facet13 (see Figure 2.) Here we also examine the (001) facet, which forms the “end face” of extended-chain alkane crystals. (The (001) facet is in the plane of the page in Figure 2.) The orthorhombic crystal unit cell contains two alkane chains in all-trans conformations, rotated ±90° about the chain axes from their reference orientation aligned with the xyz axes. The initial unit cell dimensions are taken as a = 7.13 Å and b = 4.64 Å, slightly smaller than those found in previous simulation studies of PE rotator phases. 5 However, these initial configurations then undergo equilibration at constant-stress conditions, which allows the structure to reach its equilibrium dimensions. The RII rotator phase of PE is a hexagonal packing of alkane chains in all-trans conformations, rotated randomly about their chain axes. The unit cell can again be taken as containing two chains, with cell dimensions in the xy plane taken as 8.21 Å along x and 4.77 Å along y, obtained from MD simulation using the flexible Williams potential at 335 K.5 For iPP crystalline nuclei, we assume the α phase as the structure, which has been well-characterized.15 The unit cells contains two left-handed and two right-handed iPP helices, arranged to form alternating bilayers of “up” or “down” helices (defined by whether the methyl groups on the helix surface point up or down). The cell dimensions are a = 6.65 Å, b = 20.96 Å, c = 6.04 Å, and β = 99.33°. Figure 3 depicts a small iPP nucleus that displays two lowangle facets: (010) and (110). In contrast to PE (see Figure 2), both low-angle facets are reasonably smooth, suggesting that the corresponding interfacial tensions will be similar. For

from bulk phases is a more complex process than simply cleaving a crystal to expose a crystalline facet and allowing a melt to come into contact. In the present work, we shall investigate the effects of anisotropy and disorder on the interfacial tensions between ordered phases and adjacent melt for polyethylene (PE) and isotactic polypropylene (iPP), the two most common commercial polymers. For PE, we shall first obtain crystal− melt interfacial free energies of low-index “side-face” crystal facets (100), (110), and (010), which to our knowledge have not been measured experimentally. From these values, we can assess how anisotropic the side-face tensions are and apply the Wulff construction to determine the shape and interfacial energy of the lowest free-energy nucleus. For completeness, we also examine the end-face surface of an extended-chain alkane crystal against its melt. Because this face is covered in CH3 groups, we may expect the corresponding binding energies between crystal facets and between crystal and melt both to be lower than for other facets. Next, we apply our methods to the rotationally disordered RII “rotator” phase of PE as input to calculations of barriers for nucleation from crystal and rotator phases. Although we cannot yet complete these calculations with values for the endface tension of rotator phase lamellae, if side-face tensions are lower for rotator phases than for crystalline PE, this will lend additional support for the Strobl hypothesis. Correspondingly, we shall obtain side-face tensions of iPP ordered phase nuclei against melt for both the crystalline α phase and a hypothetical rotator phase structure. In iPP, the phase analogous to a rotator phase in PE is the conformationally disordered “condis” phase.14 Unlike PE, for which rotator phases can be studied in equilibrium for intermediatelength n-alkanes, there is no known way to prepare iPP condis phases other than by deep quenching of entangled melts. Hence, structural and thermodynamic information about iPP condis phases is less certain. In the absence of a detailed structure for the condis phase from experiment, we make an educated guess as to its structure. We assert that the condis phase consists of iPP helices of random helicity, packed in a hexagonal array, with random rotations of each helix about its axis. To determine the appropriate cross-sectional area per helix, we equilibrate the above-described random packing in simulations with constantstress boundary conditions, in which the transverse dimensions can vary. On the basis of this guess for the condis structure, we can compare side-face tensions for iPP crystalline and rotator nuclei against melt and again ask whether the results lend support for the Strobl hypothesis for iPP. C

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connected by stiff harmonic bonds of about 1.42 Å in length, with bonds as well across the xy periodic boundaries of the simulation box. To keep the grid flat, we apply improper dihedral potentials to every atom and its three bonded neighbors. Additionally, we choose the bond length so that the sheet is under moderate tension as it spans the system dimensions, which are kept fixed during the simulation. The spring constant of the harmonic potential is taken to be K = 100 kJ/(mol nm2). Together with the mass of the grid (of 1776 carbon-like atoms), we can compute the oscillation frequency of the grid in its potential as ω = K /M , which gives a period of about 100 ps. We must average for times much longer than this to obtain a good value for the average force. Evidently a stiffer spring will give not only a shorter period but also larger thermal force fluctuations σF, which can be estimated using equipartition as σF2 = ⟨ΔF2⟩ = kBTK. For the values above, this leads to σF = 15.8 kJ/(mol nm). Given the total edge length 2L of about 10 nm, this corresponds to a typical swing in the inferred tension Δγ of about 2.6 mN/m. Thus, the spring constant K = 100 kJ/(mol nm2) is a convenient choice, in that it gives an oscillation period roughly 1000 times smaller than the simulation time (of order 100 ns) and force fluctuations roughly 10 times smaller than the signal we are measuring (Δγ around 20 mN/m). In plunger simulations, the system dimensions normal to the gap (along y) and along the flow direction (along z) must be held constant to avoid closing the gap or pressing on the top of the melt reservoir. In practice, we run plunger simulations under NVT conditions, with periodic boundary conditions in x and y. The dimensions of the crystalline walls are equilibrated before assembly, with an NσT simulation in which the transverse dimensions of the system can vary. The plunger simulation requires that the melt not crystallize and the walls not melt. In alkane systems, with no special precautions, chains from the melt tend to crystallize onto the walls at temperatures somewhat below Tm, whereas slightly above Tm, chains on the wall surface tend to melt away. In iPP systems, the melt does not readily crystallize even at low temperatures; however, wall melting does occurs readily above T m. We remedy this situation in two ways. First, all plunger systems are simulated at temperatures above Tm to prevent melt chains from crystallizing, especially onto the crystalline wall. Some alkane crystal facets have such a strong tendency to align and crystallize the adjacent melt that conservatively we simulate at temperatures as much as 100 K above the experimental alkane melting point. Of course, simulating at temperatures above Tm will melt the crystalline walls. To prevent this, we place the carbon atoms of crystalline chains in harmonic restraining potentials, with spring constants of 0.1kBT mol−1 Å−2, which is weak enough to let the crystal vibrate freely but prevents melting. It is important to let the crystal vibrate rather than simply freezing it in place because surface vibrations contribute importantly to the surface entropy and thereby reduce the surface free energy significantly.13 Likewise, alkanes show some tendency to align with and order at the grid, if the interactions are attractive LennardJones as usual between carbon atoms. To prevent this, we design the interactions between the grid and the melt to be purely repulsive, which keeps the melt at a greater distance and minimizes orientational alignment. This precaution is not

simplicity, we examine the (010) facet only and neglect any small effects of anisotropy on nucleus shape for iPP. For the iPP rotator phase, we assume a structure consisting of chains in helical configurations on a hexagonal lattice. Each helix has random chirality (left-handed or right-handed) and is randomly assigned to be either an “up” or “down” helix. Finally, the iPP helices are randomly rotated about their axes. For consistency with the construction of the α phase, we construct the iPP rotator phase from unit cells composed of four chains, with unit cell dimensions a = 6.32 Å, b = 25.15 Å, c = 6.04 Å, and β = 90°. These dimensions correspond to a hexagonal packing, with area per chain about 15% larger than in the α phase and length per helical repeat the same as for α. (As for other ordered phases we construct, an interval of simulation at constant-stress conditions suffices to equilibrate the dimensions of the structure.) Because the rotator phase has hexagonal symmetry, a single low-angle facet (110) suffices to construct a nucleus. Plunger Simulation Details. We measure the difference Δγ = γcv − γcm between the crystal−vacuum and crystal−melt interfacial free energies using a nanoscale plunger, shown in Figure 4. Polymer melt (blue) flows from a reservoir into the

Figure 4. Plunger configuration for alkane crystal (001) facet: C50H102 melt (blue), amorphous C2 H6 “capping” layer (red), C 40 H82 orthorhombic crystal (yellow), and graphene-like grid (green).

gap between crystalline polymer slabs (yellow), the surface of which is the facet under investigation. Sometimes a capping layer of frozen amorphous polymer (red) is used to prevent the melt from crystallizing atop the slabs. The channel walls are separated by about 5 nm. The melt phase flows into the gap by capillary action; cohesive forces prevent chains from escaping into the vacuum above. Flow into the gap is halted by an impermeable grid (green). The grid atoms interact repulsively with chains in the melt but do not interact with chains in the crystal. The grid is restrained in the z direction by a harmonic “umbrella” potential U(z); the simulation measures the average force FU(z) = −U′(z). The driving force for the melt to push down on the grid is F = 2ΔγL, where L is the linear dimension of the gap in the x direction. At equilibrium, the driving force F is balanced by the restraining force FU. The plunger grid is a graphene-like sheet, composed of carbons atoms in a honeycomb lattice. Adjacent atoms are D

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Figure 5. Schematic of the crystal pulling simulation setup to obtain γcv.

constraint. The integral of the average force with respect to distance gives 2Aγcv, where A is the area of one interface (two interfaces are created, on the surface of each half). When investigating side-face crystal−vacuum tensions, we use molecules bonded to themselves across the periodic boundary in the y direction (see Figure 5) to simulate crystal surfaces of fully extended infinite chains with no free ends. In pulling simulations for some high-energy crystal−vacuum alkane interfaces, the surface structure sometimes undergoes restructuring, particularly at room temperature. This can be prevented by performing the pulling simulations at lower temperatures. In such cases, we have performed simulations at different temperatures to assess the temperature dependence of the crystal−vacuum tension. Force Fields. All our simulations are performed using Gromacs (version 4.6.1).16 Alkanes are simulated using the “flexible Williams” force field, which reasonably represents the structure of crystalline and rotator phases as well as the transitions between ordered phases.5 All iPP simulations are performed using the OPLS all-atom force field, which accurately describes the structure of the iPP alpha phase. In previous work, we used this force field in conjunction with our “slab melting” technique to obtain melting temperatures for iPP oligomers that compare well with experimental values.17 Although these prior studies do not directly validate our present results, they do depend similarly on accurate representations of the packing, energy, and entropy of the same ordered phases. This gives us reason to be hopeful that our present simulations for interfacial tensions may reasonably describe real polymers.

required for iPP, which exhibits much less tendency to order at a flat interface. (001) Plunger Setup. The above approach is generally sufficient for plunger simulations to determine crystal side-face tensions. However, plunger simulations for the (001) “end” face of extended-chain crystals is more challenging. For this facet, the chains segments are aligned with the normal to the face, which is covered in methyl groups. The plunger walls are composed of a single 10 × 20 array of C40H82 alkane molecules in the orthorhombic phase, connected through the walls of the periodic box in the y direction, rather than two separate crystal slabs. The top surface of the slab presents the “side” of the chain stems to the melt reservoir immediately above, which tends to induce freezing of chains in the reservoir. To prevent this, we insert a thin amorphous layer of C2H6 molecules, frozen in place between the top of the wall and the reservoir (red atoms in Figure 4). (We create this layer by melting a 14 × 11 × 4 array of C2H6 molecules at 450 K for 1 ns under NVT conditions. The simulation box has transverse dimensions matching the plunger walls and an average density matched to a C50 alkane melt. We take a 5 Å thick slice of the equilibrated C2H6 system and place it atop the plunger walls.) Because the chains are oriented differently in the walls for the (001) facet, the mechanism of melting is different. For this facet, chains tend to diffuse through the crystal along their own axes, poking their ends into the adjacent melt. To prevent wall melting for the (001) facet, we first stabilize the top and bottom of the crystal walls using restraints. The bottom two layers of molecules on each wall are completely frozen, while a softer restraint is applied to the top two layers. We use a harmonic potential to restrict the movement of carbon atoms in the top layers with spring constants of ky = kz = 40 kJ nm−2. Then, we restrain the position of the center carbon atom of the remaining wall molecules, restricting its movement in the y direction (ky = 40 kJ nm−2). The result is that chain ends exposed at the gap surface can still vibrate freely, but the molecule can no longer make large excursions into the gap. Crystal Pulling Details. To evaluate the crystal−vacuum surface free energy γcv, we measure the work required to creating crystal−vacuum interface by pulling apart two halves of a crystal slab (see Figure 5). We split a single equilibrated crystal slab along the facet of interest and prepare a series of initial configurations by translating the two halves normal to the cleaving plane by increasing distances. We then simulate each configuration with a rigid constraint on the distance between the centers of mass of the two halves and measure the average force required to enforce the

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RESULTS FOR ALKANES Measuring Δγ. We first examine the results of plunger simulations for alkane crystal facets (010), (110), and (001) and the low-angle (110) facet of the RII rotator phase. The (110) and (010) systems are composed of C50 alkane molecules. The crystalline walls contain 120 and 112 chains respectively for (110) and (010), and the melt phase contains 153 chains for both. For the RII system, the ordered walls are likewise composed of 100 C50 alkanes (two slabs of 5 rows each, with 10 chains per row). For the (001) system, the crystal phase contains 200 C40 chains, and the melt contains 153 C50 molecules as before. For the (110), (010), and RII systems, walls built with C50 alkanes give a sufficiently long gap in the z direction for the melt to flow into, with sufficient extra room beyond the grid. E

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Figure 6. Δγ = γcv − γcm at 475 K of C50 plunger systems for the (a) (110), (b) (010), and (c) (001) crystal facets and (d) the RII rotator phase. Dashed lines indicates equilibrium Δγ values: 19.8 ± 0.56 mN/m (110), 20.1 ± 0.44 mN/m (010), 4.4 ± 0.78 mN/m (001), and 18.4 ± 0.5 mN/ m (RII).

Figure 7. Crystal−melt interfacial density profiles of the (a) (110), (b) (010), and (c) (001) plunger systems. Crystal density is shown in blue, and melt density is shown in red. The average interfacial separation for each system is (110) 0.36 nm, (010) 0.38 nm, and (001) 0.45 nm.

In all cases, the z coordinate of the grid center of mass was placed in a harmonic potential with a spring constant K = 100

All the alkane systems have constant simulation box dimensions, with cross-sectional area of 5.11 nm × 9.10 nm. F

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Macromolecules kJ mol−1 nm−2, and the restraint force was recorded every picosecond. Figure 6 displays the restraint force versus time interpreted as Δγ = γcv − γcm, for the (110) and (010) crystal side faces and the (001) crystal end face. The constraint force equilibrated in about 40 ns, which reflects the time needed for the melt to equalize pressure and stress throughout the gap. Averaging the final 40 ns of the simulation, we find Δγ values of 19.8 ± 0.56 mN/m (110), 20.1 ± 0.44 mN/m (010), 4.4 ± 0.78 mN/m (001), and 18.4 ± 0.5 mN/m (RII). The (010) and (110) alkane crystal side facets, the (100) facet from previous work (Δγ(100) = 18.3 ± 0.66 mN/m),13 and the RII facet all have similar Δγ values. This means the polymer melt tends to wet all three vacuum surfaces with similar strength. In contrast, the much lower Δγ value for the (001) end face means the driving force for the melt to wet this facet is much weaker. A complementary approach to investigating the strength of attractive interactions between a given alkane facet and the melt is to examining the density profile of the crystal−melt interface. Using a short 10 ns trajectory from an equilibrated plunger simulation, we compute the density profile normal to the gap for both crystal and melt phases, averaged over time and transversely (in the yz plane). Figure 7a−c displays the number-average density profile for the (110), (010), and (001) systems. In each case, the crystal density exhibits regularly spaced peaks on both sides of the box, corresponding to the highly ordered crystal walls. At the edge of the walls, the crystal density quickly decays to zero. The melt confined within the gap results in a higher average melt density in the region between the walls. (The reservoir above the gap also contributes to the average, resulting in a melt nonzero density outside the gap.) The melt phase approaches the wall more closely for the side faces (110) and (010) than for the end face (001). The separation between the melt and crystal phase density peaks nearest to the interface is smaller for the (110) and (010) facets (0.36 and 0.38 nm, respectively) than for the (001) facet (0.45 nm). The side facets also exhibit strong templating effects on the adjacent melt, evident in Figure 7a,b as the sharp peaks in the melt density profile adjacent to the crystal walls. Altogether, the increased melt density at the wall for these side facets results in greater molecular attractions across the interface and hence a greater driving force for the melt to wet the crystal. In contrast, the (001) surface is decorated with methyl groups, which interact more weakly with the melt, leading to weaker ordering of the adjacent melt and a smaller driving force to wet the surface. Measuring γcv. We perform pulling simulations to measure the crystal-vacuum interfacial free energy γcv for the (110), (010), and (001) alkane crystal surfaces as well as the (110) RII surface. System configurations for the (110) and (010) facets were composed of 64 periodically bonded C40 molecules in the orthorhombic phase. The RII system was built from 50 periodically bonded C40 chains in hexagonal arrays (two slabs of 5 rows each, 5 chains per row). The system was first equilibrated for 1 ns at 250 K and 1 bar under NσT conditions. The slab was then cleaved into two halves along facet, and the two halves were translated normal to the facet in steps of 0.05 nm. The constraint force to maintain the two halves at the given center-of-mass distance is then measured for each configuration.

The (001) end face system construction is slightly different, consisting of two crystalline slabs containing C40H82 molecules, each in a 8 × 8 array. One slab is placed atop the other with AB stacking, and the resulting system was equilibrated for 5 ns at 300 K and 1 bar under NσT conditions. Then a series of initial configurations at increasing separations are prepared as above, and the average constraint force was measured. All pulling simulations were performed under NVT conditions for 5 ns each, with constraint forces measured every picosecond. After the system was equilibrated under NσT conditions to allow the crystalline dimensions to adjust (before the sample was cleaved), all dimensions are held constant thereafter. The transverse dimensions of the (110), (010), and (001) crystalline systems were 3.54 nm × 5.06 nm, 2.94 nm × 5.06 nm, and 3.01 nm × 3.91 nm; the RII system had transverse dimensions of 2.38 nm × 5.06 nm. Figure 8 summarizes the average constraint force with respect to distance h for all alkane crystal planes, including

Figure 8. Crystal splitting results for alkane crystal facets: (100) at 300 K [squares], (110) at 150 K [triangles], (010) at 250 K [crosses], and (001) at 300 K [circles]. Error bars are smaller than the data point markers.

results for (100) from previous work.13 The crystal−vacuum interfacial energy equals the area under the curve. We carry out the integration by fitting each set of discrete data points with a continuous function, consisting of a sum of inverse power laws (broadly based on the form of the force to pull apart two semiinfinite Lennard-Jones slabs). The integrals under each curve are 67.3 ± 1.4, 69.5 ± 1.05, and 27.4 ± 7.2 mN/m, respectively. The (110) and (010) planes have the highest γcv. In simulation, these high-energy vacuum surfaces tend to be unstable at room temperature to restructuring. Particularly for small separations between slabs, there is apparently a very large driving force to eliminate the exposed surfaces, which occurs by migration of chains at the surface to close portions of the gap. When this happens, it is no longer possible to measure the interfacial free energy by pulling; instead, we lower the simulation temperature to decrease chain mobility, increase chain order, and thereby increase intermolecular attractions. We find that the (110) surface is sufficiently stable below 150 K, and the (010) surface below 250 K, such that we can perform pulling simulations. Because we were obliged to perform pulling simulations on high-energy facets at lower temperature, we assess the temperature dependence of γcv by performing pulling simulations for (010) at 150 and 250 K. Figure 9 compares G

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anisotropic shape and an effective interfacial tension in the cylindrical model rather different from the interfacial tension on any particular facet. Figure 10 shows the shape of the minimum-energy PE crystal assumed to lie in the xy plane, relevant to the cross section of a nucleus consisting of parallel all-trans stems.

Figure 9. Crystal−vacuum (010) force−distance curves at 250 K (red) and 150 K (blue). Intergrals are 69.5 ± 1.05 mN/m at 250 K and 75.3 ± 0.86 mN/m at 150 K.

the resulting constraint force at 150 and 250 K. A greater pull force is required at 150 K compared to 250 K, consistent with the argument that thermal excitation gives rise to entropy at the crystal surface, thereby lowering the surface free energy. The integrals under the curves are 75.3 ± 0.86 mN/m (150 K) and 69.5 ± 1.05 mN/m (250 K). Despite the large temperature variation, its effect on the surface free energy is relatively weak. Our simulations require analytical corrections to account for the artifacts introduced by simulating a finite system and by using a finite cutoff for nonbonded interactions.13 Both effects makes it easier to create crystal−vacuum interfaces by cleaving a crystal because of missing molecular attractions. We approximate the missing material beyond each type of cutoff as a continuum and correct our results by adding back the attractive Lennard-Jones potential contributed by this continuum to the simulated energy. Details of these corrections are presented in the Appendix; Table 1 presents our corrected values for crystal−vacuum tension γcv and crystal−melt tension γcm. Table 1 shows there is considerable variation in PE crystal− melt surface tensions. The (001) end surface has the lowest γcv and is thus the weakest interface to cleave an alkane extended chain crystal. This is expected because of the weak attractions between methyl groups across the (001) interface compared to the attractions between CH2 groups across the (110) and (010) surfaces. In fact, on the basis of our present results, we estimate it takes almost the same amount of work to pull a melt off of the (001) surface, as to cleave an alkane crystal along (001). There is considerable anisotropy even among different planes parallel to the chain axis [i.e., (110), (010), and (100)]. This means the Wulff construction11,12 will give a rather

Figure 10. Shape of minimum free energy PE crystal constrained to lie in the xy plane.

We can define an “effective tension” γeff for the side faces in the cylindrical model, such that a cylinder and the Wulff minimum shape with unit cross-sectional area have the same surface free energy. To obtain γeff, it is useful to define the ratio r = P / A of the perimeter P and the square root of the crosssectional area A for a given shape. This ratio is independent of the size of the nucleus because both P and A scale linearly with increases in linear dimensions. The larger r is the more out-of-round the shape. For a circular cross section, this ratio is rc = 2 π = 3.545; for a hexagon, the ratio is rh = 3.722. For the Wulff minimum shape here, rW = 3.866. In terms of rc, we define γeff by

∫ γ (s ) d s A

= γeff rc

(3)

For the results above, γeff = 49.6 mN/m. The value of Δγ, which measures the wetting force pulling melt across a vacuum surface, is nearly the same for RII as for the in-plane crystalline interfaces (100), (110), and (010). This reflects the fact that the RII facet and the in-plane crystalline interfaces present similar substrates to the melt, consisting in all cases of “side views” of all-trans alkanes.

Table 1. Crystal−Vacuum γcv and Crystal−Melt γcm Interfacial Free Energies (in mN/m) for Various Alkane Crystal and Rotator Facets from Simulationa (110) γcv γcorr cv Δγ γcm γcorr cm

67.3 74.3 19.8 46.9 48.0

± ± ± ± ±

1.4 (150 K) 1.4 0.56 (475 K) 1.5 1.5

(010) 69.5 76.6 20.1 49.0 50.1

± ± ± ± ±

1.0 (250 K) 1.4 0.44 (475 K) 1.1 1.1

(001) 27.4 ± 7.2 (300 K) 34.4 ± 7.2 4.4 ± 0.78 (450 K) 22.9 ± 7.2 24.0 ± 7.2

(100) 46.0 53.1 18.3 27.7 28.8

± ± ± ± ±

1.4 (300 K) 1.4 0.66 (450 K) 1.5 1.5

RII 54.0 59.5 18.4 35.6 36.3

± ± ± ± ±

1.2 (300 K) 1.2 0.5 (475 K) 1.3 1.3

Δγ = γcv − γcm; corr includes analytical corrections for finite size and finite cutoff effects. Results for (100) are from previous work.13

a

H

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Figure 11. Left: iPP plunger system. Right: wall structure for α phase (010) facet and hypothetical rotator phase, viewed along the chain axis.

Figure 12. Δγ of iPP (30-mer) plunger system for the (a) crystal (010) and (b) rotator surfaces. Dashed line indicates equilibrium Δγ values, 9.15 ± 0.10 mN/m [crystal (010) 418 K] and 7.8 ± 0.2 mN/m [rotator 373 K].

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RESULTS FOR iPP In contrast to PE, for iPP the low-index (110) and (010) facets of the α phase in the xy plane are both relatively smooth (see Figure 3). Thus for iPP, we focus on the effect of disorder on the side-face interfacial tensions, comparing the (010) α phase facet to the low-index facet in a hypothetical iPP rotator phase. Measuring Δγ. We use the plunger method to obtain Δγ = γcv − γcm for the α phase (010) facet and the rotator phase facet. For both systems, the crystal and melt phases are composed of C90H182 iPP oligomers (30-mers). The wall structures and plunger configuration are shown in Figure 11. The transverse dimensions of the crystal and rotator phase systems are 9.4 nm × 5.7 nm and 5.1 nm × 10.1 nm, respectively. The crystal and rotator phase plungers are simulated at 418 and 373 K, respectively. These temperatures are 25 K above the melting points of iPP 30-mers in each solid phase, obtained from simulation using our previously described slab-melting method.17 As for the PE plunger simulations, the plunger grid was held in a harmonic “umbrella” potential with a spring constant of kz = 100 kJ mol−1 nm−2. Figure 12 displays the iPP plunger simulation time series. The force in the rotator system equilibrates in about 50 ns, while only 20 ns is required for the crystal system. The longer

The RII tension against vacuum is 59.5 mN/m; this is slightly higher than the smoothest in-plane crystalline interface (100) but lower than the more corrugated crystalline interfaces (110) and (010). Correspondingly, the RII value of γcm is slightly higher than for (100) but substantially lower than for the (110) and (010) interfaces. Assuming a hexagonal cross section for a RII nucleus, we can translate the γcm value of 36.3 mN/m into an effective tension for a hypothetical circular cross section by using

γcmrh = γeff rc

(4)

With the above values, this gives γeff = 38.1 mN/m for RII treated as an effective cylindrical nucleus. This effective tension value for RII is 23% lower than the corresponding value for a crystalline nucleus. A complete comparison of the nucleation barrier for PE crystal versus RII nuclei would require in addition the end-face interfacial tension for RII and the bulk free energy of RII versus PE as a function of undercooling, but the substantially lower side-face tension result does lend support for the Strobl hypothesis of nucleation via mesophase for PE. I

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Figure 13. Interfacial density profiles of the (a) crystal−melt and (b) rotator−melt interface in plunger simulations. Solid phase density is shown in blue, and melt density is shown in red. The average interfacial separation for the crystal and rotator systems is 0.45 and 0.54 nm, respectively.

equilibration time is the result of slower chain dynamics in the melt at the lower simulation temperature. At equilibrium, we obtain average Δγ values of 9.2 ± 0.1 and 7.8 ± 0.2 mN/m for the crystal and rotator phases. The rotator phase Δγ is slightly lower than for the crystal phase, meaning that the driving force for the melt to wet rotator phase is slightly weaker. Perhaps since the rotator phase surface is more disordered, the melt cannot approach the plunger walls as closely as if they were crystalline. This would lead to weaker attractions across the interface, and so a lower propensity for the melt to wet the walls. We can check this story by comparing the density profiles of the rotator and crystal plunger systems at the melt−solid interface. We average the cross-sectional density of the melt and the solid walls over 10 ns of equilibrated simulation trajectory. The results are presented in Figure 13a,b. Figure 13 shows the melt indeed comes closer to the (010) facet than to the rotator phase surface. The separation between the nearest melt and crystal peaks in each case is 0.45 and 0.54 nm, respectively. Both values are large compared to the corresponding separations at alkane interfaces. Alkane chains in the melt tend to align at the solid surface, thereby achieving closer packing at the interface. This sort of chain alignment is not observed in iPP. Chains in an iPP melt near a flat surface in our simulations do not adopt helical conformations, which would permit them to lie close to the surface. Otherwise, the methyl side groups on “denatured” (nonhelical) iPP chains presumably disrupt close alignment to the surface. Measuring γcv. To obtain the interfacial free energy of iPP surfaces against vacuum, we perform pulling simulations analogous to those for alkane systems. We start with an α phase crystal slab composed of iPP 30-mers in an 8 × 8 array. To effectively simulate an “infinite” polymer chain. the molecules are bonded at their ends across the periodic y boundary. The crystal is oriented so that the (010) cleaving plane is normal to the z-axis (see Figure 14). We equilibrate the initial crystal structure for 1 ns under NσT conditions at 300 K and 1 bar, before cleaving the structure in half along the (010) plane. We generate a sequence of initial configurations by separating the two halves along the z direction in increments of 0.05 nm. The periodic boundary condition in z is then removed, resulting in two crystal sheets of finite thickness in z and infinite extent in the transverse periodic dimensions. In a similar manner, we construct a set of initial structures for the rotator phase. Each configuration serves as an initial configuration for a pulling simulation, in which the center-of-mass distance between the

Figure 14. Initial and later configuration of iPP crystal pulling simulation at average separation between slabs of 0.3 nm (center-ofmass distance between blue molecules on each slab is rigidly constrained).

two halves is constrained. All snapshots are simulated at 200 K for 5 ns each, with the constraint force measured every picosecond. In our simulations, the thin α phase crystal slabs at small separations apparently have such large cohesive forces between them relative to the elastic constants of the material, that the slabs sometimes undergo a bending deformation to bring a portion of the surfaces into contact (see Figure 14). However, these bending deformations at small separations are not fatal for computing the work of adhesion, as long as the crystal packing is not disrupted, and the system at larger separations recovers its undeformed shape. Then, the average force at different separations still represents the progression of pulling apart two slabs of constant structure. Pulling simulations for the iPP rotator phase are performed in the same way as for the crystal. Figure 15 shows a typical configuration, for which the separation between slabs is 0.2 nm, as measured by the distance between the center of mass of the first two layers on each slab (blue and red molecules). Compared to the crystal phase, rotator phase slabs subjected to large cohesive forces at small separations are susceptible to drastic molecular rearrangements, which permanently disrupt the chain packing. This susceptibility may be the result of lessdense packing in the rotator phase as compared to the dense, interdigitating packing of the α phase. In any case, such rearrangements are fatal for computing the work of adhesion, as the succession of simulations at increasing distance no longer correspond to pulling apart two slabs of fixed packing structure. To prevent such rearrangements, we restrain alternating molecules in the first layer of each slab (red molecules in Figure 15) by applying harmonic potentials to J

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Table 2. Crystal−Vacuum γcv and Crystal−Melt γcm Surface Free Energies (in mN/m) for αiPP (010) Facet and iPP Rotator Phase Side Surfacea αiPP (010) γcv γcorr cv Δγ γcm γcorr cm

54.2 61.3 9.15 45.1 45.4

± ± ± ± ±

0.91 (200 K) 0.91 0.10 (418 K) 0.92 0.92

rotator iPP 34.8 ± 0.97 (200 K) 41.4 ± 0.97 7.8 ± 0.15 (373 K) 27.0 ± 0.98 27.0 ± 0.98

a Δγ = γcv − γcm; corr indicates analytical corrections for finite size and/or cutoff.

lower than for the α phase. We do not yet have values for the end-face tension between rotator phase and adjacent melt, which would require a different simulation technique, because of the complex structure of the end face with its carpet of chain folds and loops. Still, our present results lend quantitative support for the Strobl hypothesis, insofar as the rotator phase side-face tension is substantially smaller than its crystalline counterpart, which would favor the rotator phase as the preferred nucleation pathway in isotactic polypropylene.

Figure 15. Typical configuration of a pulling simulation for iPP rotator phase, with center-of-mass distance between colored chains on each slab constrained at 0.2 nm. To prevent molecular rearrangements, y coordinates of red chain centers of mass are restrained by harmonic potentials.

their center of mass y coordinates, with a spring constant of 5000 kJ/nm2. Figure 16 displays the average force (normalized by interfacial area) versus slab separation, for both iPP crystal

■

CONCLUSIONS In this work, we use MD simulations to explicitly measure free energies of interfaces between melt and ordered crystalline or rotator phases for polyethylene (PE) and isotactic propylene (iPP). We employ our recently developed “plunger” method, which measures the force required to restrain a melt from wetting the surface of an ordered phase, in combination with a “pulling” method, which measures the work required to pull a slab of ordered phase apart along a given plane. Combining these two methods allows us to measure the free energies of interfaces between melt and crystal or rotator phases, in which the ordered phase is oriented with chains parallel to the interface. Interfaces of this type form the sides of ordered-phase nuclei in an undercooled melt undergoing crystallization via nucleation and growth. The traditional view of nucleation in polymer melts has been that crystalline nuclei form by thermal fluctuations and grow spontaneously if their size exceeds a critical size. More recently, Strobl and others have amassed considerable evidence that for many polymers nucleation proceeds via a less-ordered mesophase intermediate, which later transforms to the final stable crystal. For PE, evidence is strong that this intermediate is the RII rotator phase, which is stable in intermediate-length alkanes and well studied both experimentally and in MD simulation. For iPP, the conformationally disordered or “condis” phase, metastable with respect to the crystal and accessible only by deep quench from the melt, is a candidate for a nucleating mesophase. In the absence of detailed structural characterization of the condis phase, we hypothesize that it consists of a hexagonal packing of parallel iPP helices, with random helicity, and randomly rotated about the helix axes. The side-face interfacial tensions are not presently accessible to experiment but have only been inferred by comparing nucleation theory to the temperature dependence of homogeneous nucleation rates. This approach evidently cannot serve as a test of nucleation theory, nor can compare the barriers to nucleation via rotator or crystal phase. We find that the side-face tensions of PE are rather anisotropic, varying by 50% between different plausible low-

Figure 16. Force per interfacial area versus slab separation h from pulling simulations for iPP α-phase (010) facet (blue) and rotator phase (red).

and rotator phases. The curves are fits to sums of inverse power laws; the interfacial tension with vacuum γcv is the area under the curve, 54.2 ± 0.91 mN/m for α (010) facet and 34.8 ± 0.97 mN/m for iPP rotator phase. As for alkanes/PE above, we correct our plunger and pulling results to account for finite slab thickness and finite LennardJones cutoffs. (In these corrections, we approximate the density of iPP as close to 1 g/cm3 and use the fact that the simulated iPP melt density is 95% of the crystal density. We also assume that the rotator and melt phase densities are roughly equal.) Table 2 summarizes our results for iPP systems. The value of γcv for the (010) facet of α-phase iPP is similar to values for crystal−vacuum side-face tensions in alkanes (see Table 1). As expected, γcv for crystalline iPP is larger than for the rotator phase. Denser packing of helices in the α crystal compared to the rotator phase results in greater cohesive forces between molecules and thus higher energy surfaces in a vacuum. The contrast between crystal and rotator side-face tensions with melt is striking; the value for the iPP rotator phase is 40% K

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term in S is proportional the product of species concentrations, we multiply the nominal value of S = 2.29 (mN/m) nm2 by (ρRII/ρc)2, or 0.9392 = 0.88. For our RII pulling simulations, the finite slab thickness d is about 2 nm. The cutoff length rc is 1 nm in all our simulations. With these values, we correct the value of γRII cv derived from pulling by adding a finite-slab correction of 0.23 and a finitecutoff correction of 5.28, for a total of 5.5 mN/m. Now we discuss corrections to the plunger results for Δγ = γcv − γcm. In the plunger simulations, attractions between the ordered-phase walls and the adjacent melt are weakened by missing interactions beyond the cutoff. However, because of the periodic geometry, the melt that enters the channel has attractive interactions from distant material in the transverse direction, without interruptions. Hence, there are no finite-slab corrections to Δγ. To compute the finite-cutoff correction for Δγ, we observe that the forces between crystal and melt that draw the melt into the channel are slightly weaker because of the finite cutoff. We correct for the density of the wall and melt relative to our nominal value of S by multiplying by (ρw/ρc)(ρm/ρc), where ρw and ρm are the wall and melt density, respectively. Hence, we have a correction to Δγ for RII of 4.785 mN/m. When we use this value of Δγ to compute γcm from γcv − Δγ, we observe that the corrections to γcv and Δγ nearly cancel. This is because the finite-cutoff corrections are dominant, and the densities of the melt and ordered phase are so close. The net correction to γcm for RII is 5.5 − 4.785 = 0.71 mN/m.

angle facets. Applying the Wulff construction, we determine the lowest free energy shape of a crystalline nucleus of parallel chain segments. The RII nucleus has an average side-face tension 23% lower than for a crystalline nucleus. Likewise, for iPP we find the free energy of an interface between rotator phase and melt is about 40% lower than for a melt−crystal interface. A full comparison of barriers for rotator and crystalline nuclei requires a value for the interfacial tension against melt of the rotator “end face”, i.e., the top and bottom of a growing rotator lamella. Still, a substantially lower value of the side-face tension for rotator−melt versus crystal−melt interfaces lends support for nucleation via rotator phase in PE and iPP.

■

APPENDIX We make two analytical corrections to our interfacial tension results to account for (1) the use of finite thickness slabs in our pulling simulations and (2) the use of a finite cutoff for attractive Lennard-Jones interactions. The theory behind these corrections was given in our first paper on the plunger method;13 this brief Appendix presents a simplified version of those corrections and focuses on how to use the expressions to obtain final values. The finite-slab correction accounts for missing interactions with material beyond the far end of the slab, in the attractive energy per area of two slabs of thickness d in contact: Δγ slab =

7πS 48d 2

■

(5)

Similarly, the finite-cutoff correction accounts for missing interactions with material beyond the range of the LennardJones cutoff, in the attractive energy per area of two semiinfinite regions in contact: Δγ cutoff =

5πS 6rc 2

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (S.T.M.). ORCID

Scott T. Milner: 0000-0002-9774-3307 (6)

Notes

The authors declare no competing financial interest.

In both expressions, S is a sum over atomic species i and j on either side of an interface, across which two regions of material are interacting by van der Waals attractions. S is given by

■

(7)

ACKNOWLEDGMENTS We acknowledge support from National Science Foundation DMR-1507980.

where ρi is the concentration of species i, and ϵij and σij are the Lennard-Jones parameters for atomic species i interacting with j. For saturated hydrocarbons at a nominal crystalline density of 1 g/cm2, S equals 2.29 (mN/m) nm2. From this value, we can determine corrections to our pulling and plunger results, taking account of the difference in densities of the crystal, rotator, and melt phases. In brief, we scale the nominal value of S by the mass densities on either side of the relevant interface, relative to the crystal density. As an example, we consider results for PE. The PE melt density is about 0.85 times that of the crystal, so ρm/ρc = 0.85. For the RII rotator phase of PE, the area per chain is 0.196 nm2 versus 0.184 nm2 in the crystal while chain length per repeat is nearly the same in both phases,5 so density ratio ρRII/ρc = 0.939. Both finite-slab and finite-cutoff corrections apply to the pulling simulation results for interfacial tensions between vacuum and ordered phase, either crystal or rotator. To account for the lower density of the RII phase, because each

(1) Lodge, T. P.; Hiemenz, P. C. Polymer Chemistry, 2nd ed.; CRC Press: Boca Raton, FL, 2007. (2) Ostwald, W. Studien über die Bildung und Umwandlung fester Körper. Z. Phys. Chem. 1897, 22, 289. (3) Kraack, H.; Deutsch, M.; Sirota, E. B. n-Alkane Homogeneous Nucleation: Crossover to Polymer Behavior. Macromolecules 2000, 33, 6174−6184. (4) Sirota, E. B.; Herhold, A. B. Transient phase-induced nucleation. Science 1999, 283, 529−532. (5) Wentzel, N.; Milner, S. T. Crystal and rotator phases of nalkanes: A molecular dynamics study. J. Chem. Phys. 2010, 132, 044901. (6) Iijima, M.; Strobl, G. Isothermal Crystallization and Melting of Isotactic Polypropylene Analyzed by Time- and TemperatureDependent Small-Angle X-ray Scattering Experiments. Macromolecules 2000, 33, 5204−5214. (7) Strobl, G. From the melt via mesomorphic and granular crystalline layers to lamellar crystallites: A major route followed in polymer crystallization? Eur. Phys. J. E: Soft Matter Biol. Phys. 2000, 3, 165−183.

S=

∑ ϵijρi ρj σij i,j

6

■

L

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