Thermodynamics and kinetics of crystallization of flexible molecules


  • B. Wunderlich

    Corresponding author
    1. Department of Chemistry, The University of Tennessee, Knoxville, Tennessee 37996-1600
    2. Chemical Sciences Division, Oak Ridge National Lab., Oak Ridge, Tennessee 37831-6197
    • Department of Chemistry, The University of Tennessee, Knoxville, Tennessee 37996-1600
    Search for more papers by this author

  • This article is a US Government work and, as such, is in the public domain in the United States of America.


Early structure models of crystals go back to the 17th century.1, 2 By studying the external regularity of the crystals and their optical properties, an arrangement of the fundamental particles as shown in Figure 1(a) was suggested without actual knowledge of their size and nature. It took about 200 years until the molecular details could be ascertained, as shown for NaCl in Figure 1(b).3 Note that this was before X-ray diffraction was available for precise crystal structure determination.8 At about the same time, thermodynamics reached the present-day precision.4, 9 It represents the macroscopic tool to describe phases. Thereafter, microscopic experimental crystal structures were amassed by X-ray diffraction. Parallel, large numbers of heat capacities (at constant pressure), Cp, were determined by adiabatic calorimetry, often covering the temperature region from close to absolute zero to beyond the equilibrium melting temperature, Tmath image. Tables of the integral thermodynamic functions enthalpy, H, free enthalpy, G, and entropy, S, were collected for many substances as a function of temperature.10, 11 On the basis of the macroscopic thermodynamics and the microscopic structure, the molecular motion within the crystals was assesses by approximation and in detail.12–14

Figure 1.

(a–d) Development of the knowledge about crystal structure (a2 and b3), phase property and size (c4, 5, 6), and the 10 basic states of matter (d7).

The next step involved the evaluation of the properties of smaller and disordered crystals. As the main change in property, experiments on small crystals showed a lower melting temperature, Tm. This change in Tm could be described by the Gibbs-Thomson equation based on their surface area and the specific surface free energies.5 The same treatment was useful later, when analyzing lamellar polymer crystals.6 Figure 1(c) summarizes the historical definitions of the thermodynamic properties of macrophases and microphases, with the former being, in at least one dimension, larger than 1000 nm (1 μm). The Gibbs-Thomson equation listed in Figure 1(c) applies to the melting point lowering, ΔT, of lamellar, microphase crystals (�� < 1.0 μm). The earlier suggestion that microphases in the form of colloids where a “fourth state of matter” could be discarded after understanding the effects of surface free energy and surface charges. More recently, nanophases are of general interest for objects marginally larger than the limit of a few ångstroms, set by the homogeneity of the phase due to the atomic structure and the fluctuations of the thermodynamic properties because of a too small sampling volume.15 Experiments following the glass transition of unsupported polymer spheres of decreasing size16 suggested that as long as a small phase has unchanged bulk material within its center, there is no reason to apply a different name to a small microphase.17 As one, however, approaches a size so small that the opposing surfaces leave no unchanged bulk material, it was found that there is a size-range for an entirely new phase, a true “nanophase.”15, 17

An updated thermodynamic description of phases is given in Figure 1(d). It contains a list of the possible phase types when considering not only degrees of condensation and ordering, but also the differences in modes of molecular motion. Figure 1(d) expands the basic, classical states of matter to ten.7, 18 The phases known since antiquity, have been linked in the early 19th century to their newly proven atomic nature.19, 20 The expansion with intermediate phases (mesophases) was discovered over the last 150 years.21 On the left side of the figure, it is indicated that the mesophases become increasingly more “solid” when changing in order toward the crystal. Unfortunately, crystals may have a broad range of solidity, so that the term “solid” is not a scientific, operational definition22 for their state. Glasses, in turn, have an easily measured glass transition temperature, Tg, which can be identified as a solid/liquid transition.23 The mesophases often possess a Cp close to that of the melt. On quenching, they undergo a glass transition similar to liquids. This leads to the three mesophase glasses indicated (from top to bottom: liquid crystal glass, plastic crystal glass, and condis crystal glass). Recently, it was observed, that some polymeric crystals, like aliphatic nylons and polyoxides, may even approach liquid-like mobility on heating before melting or undergoing a disordering transition. This change in mobility causes a change in heat capacity as in a glass transition, that is, the crystal displays a glass transition.24 Crystals that remain “truly” solid up to Tm become a mobile liquid on fusion. Turning to the right side of Figure 1(d), one notes that only the gas connects to all condensed phases. The entropy change of a liquid to a gas without change in the molecular structure is expressed empirically by Trouton's rule.25 The disordering of a crystal to a liquid follows the empirical rule of Richards as long as the ordering species are spherical.26 Nonspherical species follow Walden's rule,27 and for conformational disordering, a similar empirical entropy increase was observed as for spheres.28 The possible transitions between the condensed phases are marked in Figure 1(d) and their overall entropy of fusion is expressed in terms of the three types of disorder by the boxed equation.29, 30

Today, this development of the knowledge about crystal structures, phase properties and sizes, and the states of matter are supported by a wide range of experiments. The macroscopic picture is supplied by the equilibrium and nonequilibrium thermodynamics, based on calorimetry, and is supported by direct experimental evidence about the microscopic structure as well as the molecular motion. Furthermore, the enormous increase in computational capability allows to simulate molecular structure and motion. The time scale of importance, the picosecond, however, is far removed from human experience. The present summary is to establish the needed developments to attain a base for the detection of flaws in the still incomplete description of the thermodynamics and kinetics of crystallization of flexible molecules and their phase structures.

In the next three sections, the often neglected problems of nucleation of a new phase of increased order will be analyzed and the nanophase structure of macromolecules will be probed as to its influence across the interfaces. In the conclusions, a view towards the enormous job of supplying details about the resolution of the indicated problems is summarized.


The development of the idea of primary and secondary nucleation as they were ultimately applied to the crystallization of semicrystalline macromolecules are described in this section.31 The classical concept of crystal nucleation was already suggested by Gibbs about 130 years ago.32 The description of small phases as a function of size was discussed in Figure 1(c) and led to the free enthalpy plots describing primary nucleation given in Figure 2(a). The curves are scaled to the free enthalpies of polyethylene crystals. The boxed numbers in the graph on the right of Figure 2(a) represent the free enthalpy in convenient units at an approximately 40 K supercooling. They indicate a saddle point (*), calculated by the given equations. The system must travel across it to become stable. Obviously, at Tmath image the size of the nucleus is infinity, that is, no nucleation is possible. The larger the supercooling, the lower is the barrier for a move into the region of negative ΔG. About 60 years ago, a mathematical expression for the nucleation rate was derived by Turnbull and Fisher33 written in Figure 2(b). This was fitted to experiments on homogeneous nucleation with sufficiently small polyethylene droplets in silicone oil, eliminating the effect of accidental, heterogeneous nuclei. The result is the graph in Figure 2(b), based on the two-dimensional plot of Figure 2(a).36 There is a region of about 30 K in polyethylene where primary, homogeneous nucleation is not observed, followed by increasingly fast nucleation which slows as the melt viscosity, η, increases and reaches zero when the glass transition at 250 K is approached.31

Figure 2.

(a–d) Free enthalpy of primary nucleation of a tetragonal crystal of dimension i = a × a × �� and expressed by the equations (a31); rate of primary nucleation, calculated analogous to part a (b31, 33); and the basic surface effects leading to secondary nucleation (c,34 d35).

To assess the further crystallization after homogeneous or heterogeneous nucleation, the Kossel model of a crystal was used.34 Figure 2(c) illustrates that on a cubic crystal there are five distinguishable locations of different surface free energy for crystallization or melting. Only position 3 has no change in surface free energy, that is, at position 3 crystallization (or melting) should occur at equal rates. Position 1 would require a secondary nucleation on the smooth crystal surface, and position 2, a tertiary nucleation of a new row on a step in the surface. Positions 4 and 5 would be stable and require a positive free enthalpy for removal from the crystal. This model was transferred 30 years later to the crystallization of polymers by Lauritzen and Hoffman35 by simplifying a polymer crystal as shown in Figure 2(d). It was the basis for the description of polymer crystallization for many years with numerous improvements and fine tunings,37 and is often still applied today.

Figure 3(a) shows the calculation for the secondary nucleation of polyethylene, analogous to homogeneous nucleation.31 A special reasoning needed to be given for the observed crystallization path in form of a temperature-dependent, but time-constant length �� with a positive ΔG. Furthermore, it was unlikely that an Io of similar nature as used for crystallization with a single atomic step should fit the experiments on polymers. Early Monte Carlo simulations38 of the polymer conformation, as shown in Figure 3(b), had resulted in a good fit to the experiment of the mean square end-to end distance, 〈r2〉, and distance from the center of gravity, 〈s2〉. Assuming a molecular segment of 50 bonds, one can estimate that a stretched conformation with �� = 6.4 nm which fits onto a crystal is only one of about 10.24 In addition, the conformations vary in energy so that there is not only an entropic, but also an intramolecular energetic contribution which makes Io of Figure 2(b) temperature dependent. In most present-day analyses of secondary nucleation, the thermodynamic functions are furthermore treated as independent of temperature, despite the fact that tables of heat capacities are available for many polymers. For polyethylene, for example, the heat capacity had been measured by 195739 and was linked to its crystalline and glassy vibrational spectrum by 1962.40 Shortly thereafter, extended-chain crystals of close to 100% crystallinity could be produced by crystallization in the condis phase under elevated pressure. After conversion of the condis phase to the orthorhombic phase by removal of the pressure, the equilibrium melting temperature was established by slow dilatometry to avoid superheating.41 Finally, the molar mass extrapolation from well-known paraffins proposed by Flory and Vrij42 could be shown to duplicate the experimental equilibrium melting temperature (Tmath image = 414.6 ± 0.5 K), but only, if the proper temperature-ependence was introduced.43 Assuming temperature-independent thermal functions, usually leads to a 3 K higher Tmath image, a serious error when discussing small degrees of supercooling. Today, the ATHAS Data Bank contains partial and full thermodynamic functions for 200 polymers and low molar mass analogs.44 A final problem with the application of the rate equation of Figure 2(b) to secondary nucleation is the modeling of the secondary polymer nuclei on a smooth crystal surface, as shown in Figure 2(d). Polyethylene, as the most studied polymer, shows prominent growth on its (110) face. A molecular model of such a surface reveals a step-surface with hardly any new bo�� surface. This then leaves the fold or fringe surface as the only hindrance to secondary nucleation. It will be shown in the last section of this article that these surfaces lead to other restrictions to crystal growth. Furthermore, electron-microscopic and thermodynamic evidence of the absence of prominent secondary nucleation will be presented in the next section.

Figure 3.

(a–d) Calculations (a,31 b,38 and d31) and experiments (c31) of possible secondary nucleation of macromolecules.

With all these difficulties, it was surprising that for almost 50 years the premier model for polymer crystallization was the hindrance to secondary nucleation. All attempts at criticism found little attention. The major supporting evidence for secondary nucleation was a large number of data on linear crystal growth rates, as shown in Figure 3(c). All have an exponential appearance and could be fitted to typical nucleation rate expressions, as illustrated in Figure 3(d).31 It was obvious from the experimental results of Figure 3(c), that any improvement in the model would ultimately have to lead to a mathematical description of similar functional appearance.


The first direct experimental proof of the absence of secondary nucleation in polymers is about 40 years old.45 An electron micrograph of a carbon replica of typical extended-chain crystals of polyethylene is shown in Figure 4(a). Many ledges that should be ideal sites for secondary nucleation can be seen on the surface. After the replica was pulled, the sample was heated to melt some polymer, and then quickly cooled to decorate the surface with folded-chain crystals. The identical area as seen in Figure 4(a) is reproduced in Figure 4(b). The remaining sharp ledges on the extended-chain surface should have initiated secondary nucleation, but by following the crystal growth direction, it is clear that these ledges do not serve as crystal nucleation sites. The crystals, however, are in crystallographic register, indicated by their parallel chain directions.45

Figure 4.

(a,b) Electron micrographs suggesting the absence of secondary nucleation. (a) A surface replica of extended chain crystals of polyethylene. (b) Replica after subsequent heating to 393 K for 3 min to crystallize a small amount of folded-chain crystals. Scale bars indicate 1.0 μm. (Reproduced from ref.31, with permission from Academic Press).

Next, the segregation during crystallization of polyethylenes of various molar-mass distributions was investigated by comparing pertinent experiments with the multicomponent, equilibrium phase diagram, obtained by computer iteration.46 First, almost fully crystalline extended-chain crystals were studied by slow dilatometry. A segregation up to a molar mass of about 12,000 Da was found.46 The remaining polymer was melting more sharply than expected from the phase diagram, indicating the presence of a solid solution. This work was extended to folded-chain crystals from solution and from the melt, grown at atmospheric pressure.47 The supernatant after crystallization from solution can simply be separated and studied by size-exclusion chromatography. Literature results were analyzed and the critical mass of 90% exclusion is shown as curve 2 in Figure 5(a). Data on crystallization from the melt are drawn as curve 1 after shifting the abscissa by 36 K due to the increase of the melting temperature on absence of solvent. The melt was crystallized isothermally and then quenched, so that the remaining fractions could only crystallize poorly and were subsequently extracted by dissolution. Differential scanning calorimetry, DSC, was used to determine the dissolution temperature and to check on completion of the dissolution. At the highest crystallization temperature, the two curves show segregation up to 15–20,000 Da, far above the lengths permitted by the phase diagram, drawn as curve 3. How is this possible? Although all four curves approach the same limit at low temperature, at higher temperatures, there must be a reversible process for every molecule to segregate the different chain lengths below the equilibrium melting or dissolution temperature of the given fraction. The assumption that this selection process is the equilibrium at the saddle point of secondary nucleation in Figure 3(a) is not feasible. Only one of many molecules could in this case be rejected by each secondary nucleus. On the basis of these experiments, another solution was suggested: A molecular nucleation, similar to secondary nucleation was needed for every molecule.48 A schematic of the molecular nucleation is drawn as Figure 5(b). It indicates the reversible step of nucleation governed by ΔG* and the decoupled ends which can, if sufficiently mobile, complete crystallization. Longer uncrystallized portions of the molecule could naturally nucleate at a different location. This multiple nucleation of a single molecule could be followed by similar experiments.50 The multiply nucleated molecules, participating in more than one crystal, form tie segments, such as were discovered earlier by morphological observations.51 On partial melting, the molecules crystallized in two different locations form a nonextractable fraction, attached to the higher melting crystal. The melted portions are detectable by DSC after crystallizing by quenching. This picture of the experiment-based thermodynamics and kinetics of crystallization of flexible molecules is by now 35 years old.

Figure 5.

(a–c) Molecular nucleation. (a) Experimental results on the molecular length dependence of the rejection of species below their equilibrium melting temperature (curve 3).47 (b) Schematic of molecular nucleation.48 (c) Results probing the reversibility of the length-dependence of the nucleation and growth of paraffin and polyethylene crystals from the melt.49

Before continuing the description of the experiments, several further developments must be addressed briefly. In 1978, the progress in the physics of crystallization of flexible macromolecules experienced a surprising delay. The possible choice of chain-folds or fringed micelles at the lamellar surfaces of polymer crystals ignited a controversy.52 It turned out, that ultimately, these alternatives were a nonissue. Examples could be found for most hypothetical situations proposed by both sides of the controversy. Because, however, research funding is based on peer review, with success guaranteed usually only by agreement by all reviewers on excellence, it was from then on harder to find support for polymer crystallization, and interest declined. It picked up only with a new generation of researchers and new research tools. The renewed progress is documented in the proceedings of more recent discussion meetings on the subject.53

The new development began with the inclusion of surface roughness into the description of polymer crystals, complicating or negating the secondary nucleation process. The roughness of a growth face was an early topic in the crystallization of small molecules, mainly from the gas phase.54 For flexible macromolecules, roughness considerations were first made for the fold surface.55 Later, a model of rough growth faces was proposed.56 Still, it yielded no way to understand the nonequilibrium molar mass segregation of Figure 5(a). Another suggestion of the crystal-growth mechanism involves a mesophase intermediate.57 Although it is obvious that the path in Figure 5(b) must involve intermediate order before reaching the crystal, and a number of polymers possess stable or metastable mesophases, for polyethylene (at atmospheric pressure) and many other polymers, mesophases are unlikely to be thermodynamically stable at the growth temperature.

Computer simulations capable of following the molecular motion of thousands of coupled atoms on cooling seemed a way to produce a slow-motion view of the crystallization in the 1990s. Molecular dynamics simulations became successful to identify and follow possible defects in polymer crystals and allowed to “see” the motion involved in the sliding diffusion of crystal perfection,58 but the maximum time period of simulation to follow crystallization was still too short, although various conclusions could be drawn.59 To reach the times necessary to yield more tangible results, dynamic Monte Carlo simulation was applied.60 Despite the many simplifications necessary and the unrealistic cubic lattice commonly used [see Fig. 3(b)], an “intramolecular nucleation model” of nucleation and growth could be supported by the simulation results.61 These simulations ultimately also permitted the visualization of molecular segregation.

Temperature-modulated calorimetry was developed in the 1990s as a new tool capable to separate reversible and irreversible processes. It is usually applied as temperature-modulated differential scanning calorimetry, TMDSC.62 Figure 5(c) presents results establishing the limit of the reversibility of crystallization and melting of paraffins and polyethylenes.49 For these experiments standard DSC and quasi-isothermal TMDSC were used. Quasi-isothermal TMDSC is a technique where the modulation is carried out about a fixed temperature with a chosen amplitude (±0.5 K).63 The polyethylenes were three fractions of an average molar mass of the attached number (PE560, PE1150, and PE2150) and two broad molar mass polyethylenes. The segregation is reversible to a length of x ≈ 75 backbone atoms. This limit of chain length agrees also with Figure 5(a), where true reversibility is seen up to about 380 K. It was known, that in the presence of glass or metal surfaces no supercooling occurs in the crystallization of paraffins, normally caused by homogeneous nucleation.64 This found an explanation by the discovery of thin-layer epitaxy of very long alkanes on graphite by atomic force microscopy.65 The epitaxial paraffin melted only 50 K above the bulk Tmath image. Figure 5(c) also shows that the supercooling is not influenced by the chain folding, commonly seen in the crystal morphology above x ≈ 250.

A simple TMDSC with larger amplitudes is produced by linking heating and cooling segments as illustrated with Figure 6(a) for pentacontane.66 As long as Tmath image is within the modulation range, melting (odd numbered peaks) and crystallization (even numbered events) are observed. The peaks 1–2 and 6–7 show incomplete phase transitions due to time limitation and provide a measure of the instrument response. Peaks 3, 4, and 5 are complete, indicated by the latent heats measured by the peaks and the horizontal heat-flow rates which correspond to the respective crystalline and liquid heat capacities. The enlarged construction in Figure 6(b) shows the constancy of the extrapolated onsets of melting (peaks 1, 3, 5) and crystallization (peaks 4, 6) with a difference of less than 0.1 K, the experimental limit of precision. Increasing the heating rate to 30 K min−1 does not change the onset of melting (at Tmath image) beyond the instrument lag which was corrected for in Figure 6(a,b) (about 1 K per 10 K min−1). The onset of crystallization, in contrast, dropped rapidly at cooling rates larger than 10 K min−1 and reached a supercooling of 10 K at 30 K min−1. These experiments give a first experiment-backed observation of the slowing of crystallization by the enormous number of possible conformations of the amorphous molecule [see Fig. 3(b)]. Indium, in contrast, the commonly used metal to calibrate the DSC, shows no slowing of crystal growth with the rate of cooling after primary nucleation. More extensive work on the kinetics of molecular nucleation and its influence on Io in the Turnbull and Fisher equation of Figure 2(b) has not been undertaken as yet. Today, with superfast, thin-film chip calorimetry, cooling rates can be extended to rates as high as 106 K min−1.69 Such range of dynamics may allow a direct study of molecular nucleation kinetics with changing molecular length.

Figure 6.

(a–d) Temperature-modulated calorimetry of melting and crystallization. (a,b) Reversible crystallization of pentacontane (C50H102).66 (c) Evidence for a fraction of locally reversible melting in polyethylene.67 (d) Poly(ethylene-co-octene-1).68

Molecular nucleation as time-determining step of crystallization after homogeneous or heterogeneous nucleation, in addition, points to an intimate connection between crystal and the molecularly connected melt which must influence the crystallization. This effect would affect ΔGη in the Turnbull and Fisher equation of Figure 2(b) and will be illuminated in the last section.


The Overall Phase Structure of Semicrystalline Polymers

On analysis of the amorphous fraction DSC of semicrystalline polymers, it was found when establishing the ATHAS Data Bank,44 that the glass transition was broadened to higher temperature and the increase in Cp at Tg was often smaller than expected from the independently established crystallinity. This deficit in ΔCp was called the “rigid amorphous fraction,”70 and is now commonly designated as RAF. With the development of TMDSC,62 it became possible to separate latent heat effects from the heat capacity in the temperature range between Tg and Tm. Three important results arose from the application of this technique: First, it was found that many folded-chain crystals exhibit a certain amount of reversibility of melting.71 Second, by separating the reversible melting from Cp, a second glass transition could be identified for a number of polymers above the Tg of the bulk-amorphous phase, accounting quantitatively for the RAF.71 By exhibiting a separate transition temperature, the definition of a nanophase, above, suggests that the RAF consists not only of a strained interface, but represents a separate phase located between the crystal and the bulk amorphous phases. Third, separating both, the influence on the heat capacity by reversible melting and the RAF, it became possible to characterize the molecular motion within the crystals in this temperature range. It was found that most polymer crystals show an increasing amount of conformational motion as the melting temperature is approached. In many cases, this molecular motion could also be substantiated and detailed by solid state NMR, X-ray diffraction, neutron scattering, and molecular dynamics calculation.72 In case of polyethylene, which is available as fully crystalline material, this increase in conformational mobility was discovered to start at about room temperature.40 In some cases, this motion even reaches a glass transition of the crystal below Tmath image,24 as mentioned in the discussion of Figure 1(d). In a final group of polymer crystals, a transition to a true, usually pseudohexagonal mesophase occurs below Tmath image, as is long known, for example, for polyethylene at elevated pressure,73 polytetrafluoroethylene, trans-polybutadiene, polydiethylsiloxane, and polyphosphazene.30 To characterize a semicrystalline material and to understand its crystallization, it is thus necessary to identify and characterize three possible phase types: Bulk-amorphous, RAF, and the, possibly more than one, ordered phase. The influence of these phases on the overall thermodynamics and the kinetics of crystallization are analyzed in this final section.

The Specific Reversible Melting and Crystallization

The reversible melting, first observed on poly(ethylene terephthalate),74 is illustrated in Figure 6(c) for polyethylene.67 It occurs at the growth phase and is seen always in the temperature range were irreversible melting is also observed, suggesting a close link between the two.75 A small amount of different surface melting was detected earlier by low-angle X-ray diffraction.76 In this case the reversible process occurs at the fold surface over a wide temperature range, does not reach the crystal core, and develops at a lower temperature than the irreversible, full-strand melting.76, 77 By reviewing a larger number of experiments on reversible melting, it was established that only semicrystalline samples with chain-folds and limited crystallinity show reversible melting.71 The specific, reversible melting is represented by the ratio of reversible to total melting.75 At lower temperature the ratio is larger and it approaches zero at the end of irreversible melting. The less perfect the crystallization, the higher is the reversible melting. It increases with molar mass and with random copolymerization. The copolymerization effect is illustrated in Figure 6(d) on the example of a linear, strictly random poly(ethylene-co-octene-1).68 Crystallization and melting occur down to the glass transition region and are largely reversible at low temperature. The crystallization in Figure 6(d) on cooling at 10 K min−1 shows a sharp peak. On slower cooling with TMDSC, only little of this peak proves reversible because of quick annealing to higher perfection. Analyzing the crystallized sample by subsequent heating, the global irreversibility of the sample is revealed. The final, broad melting peak is located at a higher temperature than the main crystallization peak.

To be linked to the irreversible melting, the reversible melting must be a local process. It can be understood by considering the molecular nucleation schematic of Figure 5(b). The initially melted portions of the molecule are partially decoupled from the remaining, more stable, crystallized portion of the molecule78 so that complete melting can only occur at a somewhat higher temperature. The crystallized portion, however, can serve as a molecular nucleus for the melted portions which recrystallize within the cooling amplitude of modulation.

Crystals of linear macromolecules with little or no reversibility of melting have also been identified. Figure 7(a) shows the results for a 5000 M mass poly (oxyethylene) without any local reversibility. At this molar mass it is easy to grow extended-chain crystals.79 Similarly, extended-chain crystals of high molar mass polyethylene83 and sharply once- or twice-folded poly (oxyethylene)s of high crystallinity show practically no reversible melting.84

Figure 7.

(a–d) Details of the melting and crystallization behavior of a number of different polymers using TMDSC. (a) Evidence for fully irreversible melting of extended-chain crystals of poly(oxyethylene) of 5000 M mass.79 (b) Precisely synthesized poly(ethylene-co-methylmethylene).67 (c) Poly(butylene terephthalate) (PBT).69, 80 (d) Poly(oxy-2,6-dimethyl-1,4-phenylene) (PPO).81, 82

Completely different classes of polymers are precisely-branched or -interrupted copolymers. Again, very little reversible melting was observed for well-crystallized samples of these. Figure 7(b) shows a typical copolymer of similar chemical defect concentration as seen in Figure 6(d).67 A similar reduced reversible melting was also observed for poly(octadecyl acrylate). For poly(ethylene-co-methylmethylene) the crystal morphology was determined by electron microscopy on growth from solution.85 It consisted of a lamellar structure as seen for homopolymers of a thickness sufficient to include layers of aligned defects within the lamellae. The decoupling of the polymer chain78 was in this case not sufficient to allow the homogeneous sequences to act like an oligomer, despite their length of less than the limit of reversibility. The polymeric behavior was maintained, despite the fact that the crystallization and melting temperatures are close to that of a paraffin of similar length. The degree of decoupling of the chemically different segments seems to influence the molecular nucleation and needs to be understood better.

The Rigid-Amorphous Phase

The RAF has a profound influence on the crystallization of flexible macromolecules. Even in case there is no separate Tg of the RAF, as in polyethylene, there is a broadening of the glass transition to higher temperature in all semicrystalline polymers. For polyethylene, the RAF-like intermediate phase was proven by element-specific transmission electron microscopy on RuO4-stained samples.86 This intermediate phase was preferentially stained and appeared on the top and bottom of the crystalline lamellae. This intermediate phase accounted for 20–30% of the sample, and in 13C solid state NMR experiments it shows an intermediate mobility between that of the amorphous and crystalline PE. This difference has also been detected by a discrepancy between X-ray and DSC crystallinity.87 The size of this intermediate phase was estimated to be about 4.0 nm for melt crystallized samples.

More common is a Tg of the RAF below Tm, but separated from the (still broadened) Tg of the bulk-amorphous phase. Figure 7(c) shows such case for poly (butylene terephthalate).69 The Cp for the solid and the various mobile fractions could be calculated from prior calorimetry of the polymer.80 The two separate glass transitions can easily be read from the graph. Above 375 K all noncrystalline polymer is mobile, and the indicated two-phase equation for the crystallinity can be used to represent Cp. Between Tg of the bulk-amorphous and the Tg of the RAF all three phases must be considered when assessing the crystallinity by DSC, while below 314 K no crystallinity can be determined because glass and crystal have close to the same Cp. Naturally, this restriction for the crystallinity determination, and with it the measurement of the crystallization kinetics, does not apply to measurement by X-ray diffraction.

A final possibility is the absence of a mobile-amorphous phase below the melting temperature, as illustrated for poly[oxy-1,4-(2,6-dimethylphenylene)] (PPO) and seen in Figure 7(d). First experiments of this polymer by standard DSC revealed already that there was no glass transition below the melting region, that is, the ≈30% crystalline sample had a 70% RAF content.81 Annealing the sample below the melting peak improved the crystals somewhat, as expected, but it also slowly melted the crystals! The Tg of a bulk-amorphous sample is not far below the melting temperature, but increases sufficiently when in the RAF phase, as indicated by the TMDSC experiments in Figure 7(d).82 The melting rate is governed by the glass transition and the Tmath image is actually below the Tg of the RAF, but melting is inhibited at and below the glass transition region. The melting kinetics could be followed quantitatively and after some softening of the RAF, one repeating unit of PPO could melt for each three repeating units of RAF becoming mobile. Note that there is also no reversibility of melting in PPO. This gives an easy explanation why PPO does not crystallize by cooling from the melt. The glass transition is sufficiently high that any nucleation leads immediately to the formation of the RAF below its glass transition and stops any further crystallization.81 To crystallize PPO, the glass transition needs to be lowered by the addition of a diluent. On orienting fibers, one expects orientation not only of the crystals, but also the mobile-amorphous phase. On sufficient cooling under strain, this orientation is transferred to the RAF. For the case of poly(ethylene terephthalate), it was possible to identify this orientation by full pattern X-ray analysis (Riedvelt Method).88, 89 The scattering pattern of the RAF approached that of a mesophase. A similar result was obtained with gel-spun polyethylene.90


The phase structure of flexible, semicrystalline polymers is a much more complicated system than seen for small molecules. On the basis of the updated phase description in Figure 1 of macroscopic appearance and behavior as well as molecular order and large-amplitude motion, a globally metastable subsystem arrangement of multiple micro and nanophases was derived. With the classical Turnbull-Fishe model of Figure 2, one can understand the homogeneous nucleation of crystals of small as well as large molecules, although the meaning of the Io term may be difficult to assess. Attempts to expand this approach to the growth kinetics of polymer crystals using the secondary nucleation formalism of Figures 2 and 3 failed. Figures 4 and 5 supported the need to introduce a molecular (or intramolecular) nucleation to account for direct observations of crystal growth without secondary nucleation, segregation of fractions of shorter molecules below the equilibrium temperature of their phase diagram, and the independence on chain folding of the limit of molecular nucleation under a given set of experimental parameters. Figures 6 and 7 introduce the key observations on locally reversible melting and their measurement. They indicate first that partial crystallinity of sufficiently long molecules with a proper degree of decoupling along the macromolecules are necessary for locally reversible melting, and illustrate, second, how one can measure with TMDSC the various contributions to heat capacities and latent heats to the thermodynamic functions in the temperature range between Tg and Tm. These analyses lead to the picture that each crystal in semicrystalline polymers is surrounded by an amorphous phase that has at least a very much broadened glass transition from the bulk-amorphous phase, and in many cases leads to a RAF with a separate glass transition representing a separate nanophase subsystem. For oriented semicrystalline samples such as drawn fibers and films, the RAF may remain oriented and take on a mesophase-like structure which relaxes at their Tg with a possible latent heat contribution. Overall, the examples of Figures 6 and 7 show that no one picture applies to all molecules. The details of each sample must be evaluated before a description of the thermodynamics and kinetics of crystallization is possible, an enormous task which has by now only barely begun. The tools are available for this task. For example, calorimetry for evaluation of the thermodynamics, its reversibility, and by superfast measurement perhaps even the estimation of Io. Full-pattern X-ray diffraction can solve the structure analysis of all ordered subsystems, solid-state NMR may directly assess the large-amplitude molecular motion, and computer simulation visualizes the kinetics.

The Turnbull-Fisher expression shown in Figure 2(b) could still describe the experimental linear crystal growth rates seen as a function of temperature in Figure 3(c). One, however, must use a properly modified set of parameters Io, Gη, and ΔG*. Perhaps the rate-determining step of the crystallization process can be modeled by the schematic of Figure 5(b). The temperature-dependence of Io may then be extracted from measurements at very fast rates, assessing the entropy and energy contribution of the conformation change. The format of Gη should take into account the hindrance caused by the RAF because the uncrystallized portions of the molecule in Figure 5(b) are not expected to have the bulk-amorphous viscosity. Finally, ΔG* should be modified, to represent all thermodynamic functions with their proper temperature dependence.


In the past, this work was supported by the Division of Materials Research, National Science Foundation, Polymers Program and the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy at Oak Ridge National Laboratory, managed and operated by UT-Battelle, LLC, for the U.S. Department of Energy, under contract number DOE-AC05-00OR22725. The U.S. Government retains a nonexclusive, royalty-free license to publish, or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes.