The crystallization of polymers has been a topic of continuing interest for close to fifty years.1–11 However, fundamental questions, such as the mechanism of crystallization from the melt, are not still fully understood. Recently, there has been considerable focus on understanding the early stages of crystallization.12–16 Pioneering theoretical work by Olmsted et al.17 suggested that the coupling of orientational (or conformational) and density fluctuations yield a “spinodal” curve which lies “inside” the gas-crystal coexistence curve. When a polymer melt is quenched inside this coexistence curve, Olmsted et al. suggest that phase separation between a low density and a high density disordered phase occurs by a spinodal-type process, followed by crystallization from the dense disordered phase. Imai and coworkers18–21 present experimental evidence for this “spinodal-like” phase separation, which occurs in the induction period of crystallization. In particular, these researchers suggest that the spinodal decomposition is triggered by the parallel ordering of polymer rigid segments. Large scale simulations by Gee et al.22 are consistent with this picture. Another hypothesis, proposed by Strobl,23–26 suggests that the formation and growth of the lamellar crystallites is a multi-step process with several intermediate states, i.e., from the melt via mesomorphic and granular crystalline layers to lamellar crystallites. In contrast to these ideas, Lotz,27 Cheng,28 and in particular Muthukumar and coworkers,29–32 suggest that such a spinodal type phase separation does not occur. Rather, they argue that, while the process may have the apparent signatures of spinodal decomposition at intermediate wavevectors, the dominant mechanism at large length scales is the more conventional nucleation and growth. It is thus immediately clear that the understanding of the early stage of crystallization is currently controversial. To gain another perspective on this problem, here we model the “equilibrium” crystallization of isotactic polypropylene from the melt. Instead of considering the different stages of isothermal crystallization, we emphasize the temperature dependence of this process.
Isotactic polypropylene (iPP) has been investigated for years because of its commercial and scientific importance. By now it is well known that the conformation of chains in a crystal is a threefold (31) helix, which can be either right or left-handed, with a periodicity of 6.5 Å. Depending on different crystallization conditions, such as pressure, thermal treatments, molecular mass, and tacticity, iPP can exhibit three different crystalline morphological forms, namely α-monoclinic, β-hexagonal, and γ-orthorhombic, distinguished by the arrangement of the chains. Moreover, another form of iPP with an intermediate degree of order (between crystalline and amorphous) was found by rapidly quenching iPP from melt into ice water. Natta et al.33 termed this disordered phase as either a “smectic phase,” a “paracrystalline,” or a “mesomorphic phase,” reflecting that it is composed of bundles of parallel helices with more order along the chain axes than in the lateral packing. This disordered phase is stable below 70 °C and transforms into the α-form at higher temperature or under the action of strain.34–37 The smectic phase of iPP is also formed by uniaxial deformation, or under shear even above the melting temperature.38–42
In an effort to understand the crystallization process of iPP, here we take a slightly different perspective and model the “equilibrium” structure of iPP through the use of an on lattice Monte Carlo (MC) simulation. We start with a melt of relatively short iPP chains, and then perform stepwise cooling while ensuring that the system equilibrates at each temperature to the best of our ability. Two characteristic temperatures are observed during cooling procedure: a high temperature “transition” below which the formation of helices is apparently thermodynamically favored. We conjecture that, on further cooling, these helices progressively grow longer, so that at a lower temperature they are long enough to order in a liquid crystalline fashion. Under these conditions the polymer chains appear to crystallize. Our results therefore argue for the importance of conformational ordering (i.e., the formation of helices) which apparently precedes the translational and orientational ordering that is required for the process of crystallization.
In the MC simulations, we closely follow the coarse grained methodology of Mattice and coworkers.43–46 We thus place coarse-grained chains on a high coordination second nearest neighbor diamond lattice (2nnd lattice). The high coordination lattice is derived from a diamond lattice by eliminating every other site. The resulting coarse-grained lattice is a distorted cubic lattice with 10i2 + 2 sites in shell i and with an angle of 60° between any two axes, which is identical to the well-known closest packing of hard spheres. In the case of polypropylene, each coarse-grained bead on the lattice represents a group, the basic repeat unit. Since the carbon–carbon length is 1.53 Å, the step length of the 2nnd lattice is 2.5 Å.
The box length of the simulated 2nnd lattice is 50 Å along each axis, with periodic boundary conditions in all directions. Eighteen coarse-grained chains, each with 53 beads, are placed on this lattice. Since each bead represents a propylene unit, the chain can be reverse mapped to C159H320 in a fully atomistic system. The fraction of 2nnd lattice sites occupied is 0.11925, which corresponds to a density of 0.7528 g/cm3, comparable to the density of an iPP melt.47 All simulations are performed at this density, thus ignoring the densification that is produced by cooling. This neglect of densification will tend to delay the onset of crystallization in the simulation. The systems were gradually cooled from 585 K (well above the melting temperature) to 365 K which is just below melting temperature of this model of iPP.
The simulations are examined with a Metropolis MC algorithm, using a Hamiltonian that contains two parts. One part accounts for the intramolecular torsional potential between 1 and 4 neighbors, which is calculated through an extended rotational isomeric state (RIS) model derived from the classic three-state RIS model. (The bond length and the bond angle characterizing 1-2-3 bonds are fixed by the lattice definition.) The second part accounts for intermolecular interactions, as well as for interactions along monomers on the same chain which are separated by more than 3 bonds. These interactions are modeled by a discrete version of a continuous Lennard-Jones potential energy function with ε/kB = 237.1 K and σ = 5.118 Å, which describes the interaction between pairs of propane molecules. The temperature dependent effective interaction energies for the first three shells, which are the only shells used in the calculations, are listed in Table 1. The first two shells are repulsive, while the potential in the third shell alone is attractive. The elementary MC moves on the 2nnd lattice can either be single bead local moves, or a “pivot” which moves the whole chain beyond a randomly selected point. A Monte Carlo Step (MCS) is defined as the length of the simulation where one attempt is made, on average, for each bead.
Table 1. Effective Interaction Energies (kJ/mol) for the First Three Shells
RESULTS AND DISCUSSION
Since this study does not consider the kinetics of crystallization, and only focuses on equilibrium aspects, it is critical that we carefully establish that equilibration has been achieved at the different temperatures examined. The equilibration of chain conformations is monitored using the end-to-end distance autocorrelation function 〈R(t) · R(0)〉/〈R2〉. The temporal decay of this function is depicted in Figure 1 for a range of selected temperatures. The system is well equilibrated for all temperatures up to 375 K. While substantial relaxation is obtained at the lowest temperature, that is, 365 K, it is clear that this function does not decay completely to zero during our (relatively long) simulations. Thus, while the data at the lowest temperature needs to be viewed with some caution, we shall show below that this lack of equilibrium probably reflects the fact that this system is completely crystallized under these conditions.
Equilibration is also analyzed using a more local metric, that is, the dynamic structure factor with a selected wavevector value. The static structure factor is defined as
ρ is the monomer density, q is the wave vector, and g(r) is the radial distribution function. We define q* as the wavevector corresponding to the first peak in S(q). The dynamic structure factor, , then provides a measure of (local) structural relaxation in the system. Figure 2 shows this structural relaxation for a range of temperatures, and on this basis, we conclude that the system has equilibrated over the times simulated for all temperatures.
In summary, the system is definitely equilibrated at all but the lowest temperature. At T = 365 K, we suggest that the lack of equilibration is a direct manifestation of the fact that the system has crystallized.
Helix Survival Probability Function
It has been reported that the chain conformation of iPP in crystals is a threefold (31) helix, which can be either right or left-handed, with a periodicity of 6.5 Å. We find similar behavior in our simulations, as shown in a representative snapshot in Figure 3. The torsional states responsible for the formation of helices are trans-gauche pairs.
To investigate the thermodynamic (and hence the temporal) stability of these helices over the range of temperatures considered, we apply a concept proposed by Odagaki in the context of glasses.47 Odagaki define a waiting time distribution (WTD), that is, he looks for the first time when a particle moves from its original lattice position. Various moments of this WTD are then calculated. The second moment diverges at the onset of glassy behavior; the first moment diverges at the glass temperature, while even the zeroth moment diverges at the Vogel temperature. By analogy, here we propose to calculate the helix survival probability (HSP) function, ψ(t), which is the probability that a repeat unit of a iPP chain involved in a helix at time t = 0 remains helical at some later time t. A segment is considered a part of a helix when we have at least one full helix turn composed of a minimum of four repeat units. Based on the glass work, we suggest that various moments of the HST should indicate “critical” temperatures involved in helix formation/melting, which we deem as being crucial to the crystallization process for this polymer.
Figure 4 is a plot of the HSP function, ψ(t), as a function of simulation time. Three different regimes can be seen during stepwise cooling of the system. At high temperatures, T ≥ 455 K, the HSP of iPP helical structures follow an approximately exponential time dependence, whereas it becomes stretched at lower temperatures. Below 395 K, the HSP does not decay over the time scale of our simulations. Thus, helices do not appear to “melt” at these lower temperatures. Mathematically, it is clear that all moments of this distribution diverge at these lowest temperatures. We identify T = 395 K as a “critical” temperature on this basis, and we conjecture (as we shall show below) that this corresponds to the equilibrium melting point for this model.
Next, following Odagaki, we calculate the second moment of the HSP . We plot this quantity in Figure 5 as a function of temperature for T > 390 K. In addition to the divergence at ∼390 K, there is another signature in the vicinity of T ∼ 450 K. We thus have two characteristic temperatures in our system. We shall show below that this higher temperature reflects a weaker “transition”, below which helices are stable in a thermodynamic sense. However, the helices are not long enough to undergo any ordering (liquid crystalline or crystalline). We therefore propose that the crystallization of iPP is “nucleated” by the growth of helices: as the temperature is lowered they become long enough that they order into a smectic phase and then into a crystal.
Growth of Helices
The temperature dependence of the average helix length and the fraction of repeat units in helical structures are summarized in Figure 6(a,b). While both the average helix length and fraction of monomers in helices increase with decreasing temperature, no dramatic “transition” behavior is found at the critical temperatures reported above.
Imai et al. applied the Doi-Edward theory to understand the nematic ordering of such polypropylene helices. The critical helix length, L, beyond which ordering is predicted to occur is:
b is the diameter of the polymer segment, ρ is the density, NA is Avogadro's number, and M0 and l0 are the molecular weight and length of the repeat unit, respectively. For the iPP melt, M0, b, l0, and ρ can be taken to be 42 g/mol, 0.665 nm, 0.25 nm, and 0.7528 g/cm3, respectively. With these parameters, the critical length L was found to be 2.33 nm, corresponding to 11 repeat units of iPP. This result suggests that when the helical sequence length is equal to or larger than 11 repeat units, the level of parallel ordering of helices starts to grow, and then crystallization occurs. Figure 6(c) thus plots the average number of helices (that are longer than L) per chain as a function of temperature. The results are consistent with the hypothesis we proposed above. The number of such “long” helices increases significantly below T ∼ 450 K, consistent with the fact that nematic ordering of these helices probably precedes their crystallization.
Ordering of Helices
We have suggested above that the parallel ordering of helical structures is a basic but crucial requirement for the formation of iPP crystallites. Hence, we study the translation order and orientation order of helices at different temperatures. For simplicity, we consider a single helix turn and examine the packing and the order of intermolecular helices alone: thus, we ignore any intramolecular ordering of the helices. The translation order is described by the radial distribution function of the center of mass of the helical unit. The orientation order is characterized by the second Legendre polynomial, , where θ is the angle between the helical units. The physically accessible range for this parameter is from −0.5 to 1. S = 0 corresponds to random ordering, S = −0.5 indicates perpendicular ordering, and S = 1 indicates parallel ordering. As Figure 7(a,b) show, there is effectively no long ranged translational or orientational ordering at high temperatures. This implies that, at most, some small domains (15 Å or less in size) of relatively ordered helices might be formed in the melt. Long range translational and orientational order only develop below ∼375 K, which we then identify clearly as being the crystallization temperature of this system. Clearly, while there are helices forming at higher temperatures, these helices do not form a smectic (or a liquid crystal) phase under these conditions. This is consistent with the threshold L values reported as being necessary for the formation of such liquid crystal phases.
To conclusively establish that the lower temperature is indeed the equilibrium melting temperature, we perform simulations where the samples annealed at 365 K are reheated. We find that the ordering of helices does not show obvious qualitative changes. Thus, for all temperatures up to 385 K, we find that both orientational and translational order persist, with annealing perfecting this ordering. However, they both simultaneously disappear above this temperature, giving credence to the notion that 385 K is close to the true melting point of this model.
There are few features of this data that need further elaboration. First, the melting point of this model iPP is rather low, that is, 385 K. In contrast, the experimental equilibrium melting point for iPP is closer to 433 K. We speculate that this difference might stem from the relatively short chains simulated. Further, we have fixed the density at some value (=0.7528 g/cm3) and do not account for the densification that occurs on cooling. This densification would reduce the entropy of the melt, and presumably result in a higher value of the melting temperature.
The 2nnd simulations are performed on a lattice, therefore, changing the density requires a careful interplay between number of chains and simulation box size. Performing constant pressure simulations for on-lattice simulations is not very easy. It has been done for cubic lattices but it would be harder to perform this type of a simulation for the 2nnd lattice that we are using. Addition or removal of chains changes the density at 1/18 intervals (there are 18 chains in the current simulations). Therefore, although it is easy to change the density by adding or deleting chains, obtaining the correct density at each temperature of the simulation is not possible. To check the effect of density, we performed two additional simulations at 585 K using 17 chains and at 405 K using 19 chains. Using 17 chains at 585 K and 19 chains at 405 K yields the correct density at these temperatures.47 We chose these temperatures mainly because at these temperatures, we were able to get closest to the experimental density values. Table 2 shows the comparison of these new simulations and the fixed density simulations. It is seen that the values obtained from the fixed and correct density simulations are very close. In fact, the standard deviation values in the data (1.2% for helix size and 2% for number of helices and fraction of beads in helices) are greater than the errors between these simulations.
Table 2. Comparison of Fixed Density and Experimentally Correct Density Simulations at Two Temperatures
Second, we consider the recent experiments of de Jeu and coworkers, who have shown that the smectic phase of iPP is apparently stabilized even above the melting temperature under the action of shear. We have not simulated these situations, but we note that helices are stabilized in our quiescent calculations at temperatures well above the equilibrium melting point of this model. However, the helices are too short on average to form a smectic phase. We suggest that shear would have two consequences under these conditions. First, the helices will tend, on average, to line up along the shear direction. Second, since fluctuations are expected to be reduced along the shear direction, there might be a possibility that helices would show a smaller propensity to “melt” in this direction. These two facts, in combination, could help to explain the experiments. While these arguments are not conclusive, and we are in the process of conducting simulations to verify these conjectures, nevertheless they offer a plausible mechanism to understand the role of flow fields in this context. While well outside the purview of experiments and simulations, we also suggest that such anisotropies might also play an important role when one considers the crystallization/ordering of iPP in the vicinity of flat and curved surfaces, which might be of relevance to the field of nanocomposites.
Lastly, we would like to state that our simulations do not tell if the intermediate crystalline order is due to chain packing or if there are other factors that affect it. This aspect of the problem is not addressed in the current study.
The analysis of on lattice MC simulations suggest rather persuasively that iPP chains undergo a “transition” in the vicinity of T ∼ 450 K, below which helices are thermodynamically (and hence temporally) stable. These helices grow longer with decreasing temperature, culminating in this sample crystallizing in the vicinity of T ∼ 375–385 K. Since the crystallization temperature corresponds to the point where the mean helix length becomes long enough to undergo liquid crystalline ordering, we additionally conjecture that the temporal evolution of crystallization might include a “smectic” intermediate.22 This ansatz is consistent with recent experiments which support the fact that smectic phases are formed under the action of shear even above the equilibrium melting temperature.
This work is supported in part by the NSF Nanoscale Science and Engineering Center initiative (DMR-0117792), CMS-0310596, and a GOALI grant. We thank Proctor and Gamble, whose interest in this problem motivated this research.