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Keywords:

  • interfaces;
  • melting point;
  • polyethylene;
  • thermodynamics

INTRODUCTION

  1. Top of page
  2. INTRODUCTION
  3. MELTING TEMPERATURES
  4. INTERFACE ENERGY
  5. CONCLUDING REMARKS
  6. REFERENCES AND NOTES

The year 2007 marks the 50th anniversary of Keller's seminal proof that thin polyethylene crystals grown from dilute solution are composed of chain-like molecules that are folded.1 Shortly thereafter it was demonstrated that polymer crystals grown from the melt phase also are plate-like with stems oriented in the thin direction, implying that folding must occur in these cases as well.2 The exact nature of this folding remains unknown to this day, although it is agreed that regular folding is favored by slow crystallization at low undercoolings (larger temperatures). Similarly, regular folding is favored by shorter chains and by dilution. One signature presented by the fold surface is the excess free energy (often called “surface energy”) associated with the interface between the large basal surfaces and the surrounding medium. In this article we are concerned with melting and with crystallization from the liquid state, so the surrounding medium is the fully dense amorphous polymer (no solvent).

Crystals composed of chain-like molecules are conspicuously anisotropic, whether folded or not. It is customary to designate the specific energy for basal interfaces as σe (end) and that for lateral interfaces that contain the chain stems as σs (side). The Gibbs free energy of a crystal with transverse dimensions W and thickness lc (see Fig. 1) is written as:

  • equation image(1)
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Figure 1. Sketch of a lamellar crystal of dimensions W × W × lc that indicates the basal surfaces characterized by σe and the lateral surfaces characterized by σs. Crystalline stems are normal to the basal surface in this case. The nature of the basal surface is intentionally not indicated.

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Here ΔGv, the free energy of fusion per unit volume for a defect-free crystal (effectively infinite dimensions), is usually approximated by ΔGv = − ΔHv (Tmath imageT)/Tmath image; ΔHv is the enthalpy of fusion per unit volume (J/m3) and Tmath image is the equilibrium melting temperature for the defect-free crystal. One sees that the equilibrium melting temperature Tmath image and the specific interface energies σe and σs are required for evaluating the free energy ΔGc of a tablet-like crystal of arbitrary dimensions. The free energy of a crystal having the particular volume V = W2lc is minimized by the equilibrium shape having the dimension ratio lc/W = σes.3 For chain-like molecules crystal growth in the lc direction is hindered by kinetic factors, but a nonequilibrium plate-like shape with lc << W in no way invalidates eq 1.

Equation 1 is generally applied in two distinct ways. The one to be considered here involves simplifying and rearranging to express the melting temperature of plate-like crystals in terms of crystal thickness lc. The second application is for crystallization kinetics wherein eq 1 is evaluated for a range of dimensions W and lc at a particular undercooling ΔT = Tmath imageT to establish the smallest free energy barrier ΔG* for nucleation and subsequent growth of small crystals at a particular T. This energy barrier is proportional to σmath imageσenT2 or σsnσenT, hence the three quantities σsn, σen, and Tmath image are each critical to interpreting the absolute values and temperature dependencies of crystal nucleation and growth rates. Here the subscript “n” is for a nucleus; the reader is reminded that the specific interface energies for a mature crystal that is melting (application 1) may differ from those for the formation of a small crystal nucleus (application 2). The symbol σe will be reserved for the specific basal interface energy of a mature crystal, generally at its melting temperature.

Polyethylene is the crystalline polymer that has been studied most for reasons that include commercial significance and a simple, uniform chemical structure. This writer confesses to having subscribed rather uncritically to the values Tmath image = 418.7 K (145.5 °C) and σe = 90 mJ/m2, advocated vigorously by Hoffman and his coworkers for decades.4, 5 This particular Tmath image is furthermore associated with Paul Flory,6 undeniably one of the giants of polymer science. An alternative Tmath image = 414 K7 has not received general acceptance in part because of the prominence of Hoffman and Flory. The occasion to reread a classical study of primary nucleation of polyethylene crystals8 led to the realization that the same value of σe = 90 mJ/m2 is said to apply for both crystal melting and for crystal nucleation, that is σe = σen.4, 5 While coincidences do occur, this unexpected equivalence seems to have been uncritically accepted for decades. Presented here are published melting experiments for various polyethylenes, including oligomeric n-alkanes, that lead to values for both Tmath image and σe that are lower than the established (one might say doctrinaire) ones. First to be considered are melting temperatures and Tmath image, then we discuss the apparent specific basal interface energy σe associated with melting of polyethylene or polyethylene-like crystals.

MELTING TEMPERATURES

  1. Top of page
  2. INTRODUCTION
  3. MELTING TEMPERATURES
  4. INTERFACE ENERGY
  5. CONCLUDING REMARKS
  6. REFERENCES AND NOTES

The melting temperature Tm for a finite-size crystal is that for which the Gibbs energy is the same for the metastable crystal and metastable melt, i.e., ΔGc = 0 in eq 1. The second term on the right side can be safely ignored if σsL << σeW. Rearrangement gives the familiar result:

  • equation image(2)

This expression, often called the Gibbs–Thomson equation, quantifies the destabilization of a thin crystal by the basal interface energy term on the right side of eq 1. It can be used to evaluate simultaneously both Tmath image and σe from the measurement of Tm of crystals of different known thickness lc. The relation between Tm and 1/lc will be linear if the σe is constant or nearly so over the range of interest (ΔHv is assumed to have the constant value for T = Tmath image). At this point it can be mentioned that an entropic contribution to the basal interface energy will cause σe to vary with temperature T.

Presented first are the melting temperatures for extended chain crystals wherein thickness lc is defined by the number of carbon atoms x per chain. Experiments have been done both with n-alkanes that are strictly monodisperse in chain length and with polyethylene fractions having a polydispersity index of about 1.1. Melt crystallization was performed at high enough temperatures to preclude chain folding. Alkane melting data for x ≥ 60 carbon atoms from Takamizawa et al.,9 Stack et al.,10 and Ungar et al.11 are plotted in Figure 2. A chain tilt of 35° has been used to evaluate thickness lc from chain length because it is well established that alkane crystals adopt a “monoclinic” habit with (201) basal planes containing the terminal methyl groups when heated to the melting point.12 The data for n-alkanes synthesized by three different methods are self consistent; a linear fit to eq 2 gives an intercept Tmath image = 411 K. There is no indication that the melting temperatures for x > 100 are described by the curved Flory–Vrij6 dashed line with a limiting Tmath image = 418.5 K.

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Figure 2. Gibbs-Thomson plot for n-alkanes; crystal thickness calculated for chain tilt of 35°. Data are from ref.9 (⋄), ref.10 (□) and ref.11 (×). The solid line is from linear regression of the points, while the dashed line represents the Flory–Vrij analysis.

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Extended chain crystals from polyethylene fractions reported by Mandelkern et al.13 in Figure 3 are similar to the alkanes, but they appear to have smaller Tm in the low x, large 1/lc region. Data for the two sets of extended chain samples are presented separately to avoid obscuring the individual points. There is considerable scatter about the straight line predicted by eq 2; linear regression gives Tmath image = 416 K. It is suspected that molecular weights used to evaluate 1/lc are not so precise as those for the alkanes. Here we employ the weight average Mw ≈ 1.1 Mn to establish chain length, and no chain tilt is assumed for calculating lc. If either Mn or chain tilt is used, the melting temperatures for the PE fractions appear larger than those for alkanes of the same crystal thickness, which is an illogical result. Be reminded that the assumption of constant “chain tilt” or a particular average molecular weight has no effect on the extrapolated Tmath image, although the slope and hence apparent σe will change. The two triangles in Figure 3 are for unfractionated high molecular weight polyethylenes crystallized under high pressure to form extended chain crystals with thicknesses (measured by electron microscopy) of about 1 and 10 μm that correspond roughly to chain lengths based on Mw.7, 14Tm equals 414.6 K (141.4 °C) for the larger crystals; this is the highest value ever reported for polyethylene, and is close to the admittedly imprecise extrapolation of the PE fraction data.

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Figure 3. Gibbs–Thomson plot for extended chain extended chain crystals of polyethylene. Low molecular weight fractions from ref.13 (•). Whole polymer crystallized at high pressure from refs.7 and14 (▴). The solid line is the linear regression of the low M fraction points. The dashed line represents the alkane data in Figure 2.

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Basal surfaces of folded chain crystals have generated the most interest and the most controversy over the years since 1957. Employing the Gibbs–Thomson equation has two distinct problems in such cases; establishment of crystal thickness lc is less certain than for extended chain structures, and thin folded chain polyethylene crystals may thicken during heating to the melting temperature. It is universally agreed that the most regular folded surfaces are obtained by crystallizing high molecular weight polymers from dilute solution. Bair et al.15 used electron irradiation to prevent thickening in sedimented mats of solution grown PE crystals. Thickness lc was equated to the small-angle X-ray scattering (SAXS) period L, ignoring the fact that the samples in question had crystal fraction φc ranging from 0.72 to 0.94 based on experimental heats of fusion. With crystal thickness defined as lc = φcL, the solid diamond points in Figure 4 are clearly displaced from the open diamonds corresponding to lc = L. In fact the (incorrect) open diamonds are those used by Hoffman et al.4 to help establish Tmath image = 418.7 K. With the correct interpretation of SAXS and crystal thickness, these classic experiments give a lower value Tmath image = 416 K. Hocquet et al. recently used atomic force microscopy to measure both lc and Tm for a monolayer PE crystal grown from solution;16 the solid triangle for that datum is gratifyingly near the Bair SAXS/DSC data in Figure 4.

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Figure 4. Gibbs–Thomson plots for folded chain polyethylene crystals. Solution grown crystal data from ref.15 with the incorrect assumption that lc = L (⋄). The same melting points plotted with lc = φcL; see text (♦). Solution crystallized AFM result from ref.16 (▴). Melt crystallized polyethylene from ref.17 (•). Each set of data has a linear regression line.

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What about melt crystallized polyethylene that is folded? There are the possible complications from thickening mentioned above, compounded by even less certainly about the crystal thicknesses. Estimates of lc based on room temperature characterization are questionable for the study of melting (absent cross linking or other procedures to retard thickening), but relatively unambiguous results can be obtained from SAXS patterns recorded during the melting process. Most practitioners of this method unfortunately ignore the portion of the melting range in which the SAXS pattern loses the interference maximum. Cho et al.17 have published the only reliable analysis for melt crystallized polyethylene that we know of. The circles in Figure 4 have an intercept Tmath image = 414 K.

Gibbs–Thomson estimates of Tmath image, together with 95% univariant confidence intervals, are summarized in Table 1. Self-evident is the lack of support for the established value Tmath image = 418.7 K, save for the incorrect analysis of Bair's data for solution grown crystals. Extrapolated polyethylene results are grouped around Tmath image = 415 K ± 1 K, which is consistent with the highest direct experimental Tm = 414.6 K ± 0.5 K.7 The alkane value of 411 K is conspicuously lower than all the PE results and, most interestingly, nearly 8 K below the Flory–Vrij Tmath image = 418.5 K ± 1 K. That extrapolated value was obtained from a complex alternative to the Gibbs–Thomson equation applied to experimental melting points of alkanes with 10 ≤ x ≤ 100.6 While details are not appropriate here, the FV theory has an appreciable entropic contribution to σe, meaning that a Gibbs–Thomson plot is concave upward as 1/lc decreases. It was well established over 25 years ago, however, that FV theory systematically overestimates Tm for alkanes with x > 100.18 This discrepancy has been reinforced as longer alkanes have become available, as indicated by the linearity of the data plotted in Figure 2. It is abundantly clear that the curved FV line is not followed at large x. This writer believes it unwise to cling to a theory that does not conform to experiment; factoring the FV value of 418.5 K into an assessment of Tmath image for polyethylene appears unwarranted. In closing this section it is acknowledged that reason for alkane melting behavior differing from that for polyethylene fractions is not understood. This difference is reflected in the low extrapolated Tmath image that does not “fit” the PE results.

Table 1. Results from Gibbs-Thomson Plots
Chain StateMaterialTmath image (K)σe (mJ/m2)Reference
  1. Ranges represent 95% confidence intervals.

ExtendedAlkanes, soln/melt411 ± 179 ± 4 (35° tilt)9, 10, 11
 PE fractions, melt416 ± 2122 ± 1113
FoldedPE, soln; lc = L420 ± 1.591 ± 715
 PE, soln; lc = φcL416 ± 359 ± 1115
 PE, melt414 ± 144 ± 517

INTERFACE ENERGY

  1. Top of page
  2. INTRODUCTION
  3. MELTING TEMPERATURES
  4. INTERFACE ENERGY
  5. CONCLUDING REMARKS
  6. REFERENCES AND NOTES

Slopes of the plots in Figures 2, 3, and 4 are readily converted to apparent basal interface energies using the conventional enthalpy of fusion of ΔHv = 280 MJ/m3.5 Entries in Table 1 again include the 95% confidence intervals. There is a strong and perhaps unexpected correlation between σe and crystal type; the extended chain crystals have “high” values (80 and 120 mJ/m2) that are approximately twice as large as those for the folded chain crystals (45 and 60 mJ/m2). Here we dismiss 91 mJ/m2 that comes from the incorrect analysis of solution crystallized polyethylene, even though it conforms to the “work of chain folding” model of Hoffman.4, 5

No detailed justification of this division of σe is offered here, but some speculations are appropriate. Without question the best defined surfaces are those in the n-alkanes where end groups may be ordered on well-defined planes. Ungar and coworkers19 have done some very instructive studies of the x = 216 n-alkane in which each 12 carbon end segment is perdeuterated; X-ray diffraction and infra-red spectroscopy were performed at elevated temperatures. The surprising but undeniable finding is that translational chain disorder (poor registration of chain ends) is present in orthorhombic crystals with no chain tilt that were formed at low temperatures as sketched in Figure 5(a). On heating, the alkane chains mutually order so the ends occupy regular inclined surfaces as indicated in Figure 5(b) at the melting temperature. PE fractions, on the other hand, are polydisperse and must have disordered surfaces in extended chain crystals. Longitudinal chain motion in this case can not achieve ordered end planes, and the crystals remain “orthorhombic” through melting. It is not unreasonable that the disordered interface of PE has a higher σe than the ordered end planes of the n-alkane. But part of the difference is geometric; the 35° chain tilt increases the basal interface area by a factor of 1.22 for the alkanes.

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Figure 5. Sketch of chain arrangements in n-alkane crystals based on ref.19. (a) Untilted orthorhombic habit associated with longitudinal disorder. (b) Tilted monoclinic habit is formed when the structure in (a) is heated and longitudinal register is achieved. The transformation from (a) to (b) is not reversible.

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Why is the apparent σe of folded chain polyethylene crystals so much lower than that for extended chain crystals? Two possibilities come to mind. First, the fold surface can become ordered at high temperatures by axial stem diffusion analogous to that in alkane crystals; polydispersity does not lock in an irregular surface as with extended chain PE fractions. Second, folds that are not perfectly regular or “crystallographic” can adopt many conformations, increasing the entropy and hence decreasing the free energy of the surface layer. A reference point is the 90 mJ/m2 ascribed by Hoffman to the “work of chain folding”, which is purely enthalpic.5 Allowing for chain tilt will lower this quantity to about 75 mJ/m2, which, coincidently, is almost identical to σe = 79 mJ/m2 for opposed planes of methyl groups in the n-alkanes (Table 1). Surface or boundary melting has long been advocated to explain in part the reversible increase in SAXS intensity observed when heating folded chain polymers. The entropy per unit of an amorphous chain with two fixed ends increases with the number of units, providing a free energy decrease when the fold is lengthened by converting crystalline units to amorphous ones in the fold. Albrecht and Strobl have outlined nicely various thermodynamic models for this phenomenon and have demonstrated the presence of surface melting in melt crystallized polyethylene.20 This sort of premelting can not occur in extended chain crystals in which an “amorphous” segment is constrained at only one end. Hence entropic stabilization of folded chain crystals is qualitatively consistent with a low apparent basal interface energy σe.

CONCLUDING REMARKS

  1. Top of page
  2. INTRODUCTION
  3. MELTING TEMPERATURES
  4. INTERFACE ENERGY
  5. CONCLUDING REMARKS
  6. REFERENCES AND NOTES

The equilibrium melting temperature for a perfect crystal of [BOND]CH2[BOND] groups is lower than the generally accepted value of 418.7 K that has been promulgated for the past four decades. Melting experiments interpreted with the straightforward Gibbs-Thomson relation give Tmath image = 415 K ± 1 K for polyethyene. This extrapolated value is virtually identical to the experimental Tm = 414.6 K measured for polyethylene with huge crystals more than 10-μm thick. Wunderlich's 1977 arguments7 for Tmath image = 415 K seem to have been right on the mark, and they look even better today. The reason why so many researchers, including this writer, adopted the larger value would make an interesting study in the sociology of science.

The apparent basal interface energy σe can be thought of as a byproduct of the GT analysis for Tmath image. Experiments reviewed here indicate that folded chain crystals have a lower specific interface energy than do extended chain crystals, an effect that is tentatively associated with the entropy of “loose” folds.

We close with some comments about specific interface energies and nucleation barriers. Most kinetic experiments measure the lateral growth rate of lamellar crystals which is controlled by a nucleation barrier ΔG* ∼ σsnσenT. Clearly the inferred value of the product σsnσen will be reduced by about 20% if the undercooling is with respect to the preferred lower Tmath image = 415 K. It is not justified use the mature crystal σe in order to evaluate the lateral specific interface energy from the product σsnσen. There is abundant evidence, particularly from the crystallization of long n-alkanes,12 that folds become more regular on aging during isothermal growth. One way out of this dilemma is to divide σmath imageσen from primary nucleation by the product σsnσen from secondary nucleation (growth rate) to evaluate the two interface energies, as was done by Ross and Folen.21 Here one assumes that the natures of critical three-dimensional and two-dimensional nuclei are the same, perhaps with some correction for temperature dependence. Our more adventurous colleagues are of course entitled to substitute the melting σe for the nucleation σen, provided they state clearly what is being done. The issue then becomes which of the values in Table 1 is to be employed; choices abound!

REFERENCES AND NOTES

  1. Top of page
  2. INTRODUCTION
  3. MELTING TEMPERATURES
  4. INTERFACE ENERGY
  5. CONCLUDING REMARKS
  6. REFERENCES AND NOTES