Thermodynamics of Enthalpy Relaxation and Hole Formation of Polymer Glasses

Crystallization is used at some stage in nearly all process industries as a method of production, purification or recovery of solid materials. In recent years, a number of new applications have also come to rely on crystallization processes such as the crystallization of nano and amorphous materials. The articles for this book have been contributed by the most respected researchers in this area and cover the frontier areas of research and developments in crystallization processes. Divided into five parts this book provides the latest research developments in many aspects of crystallization including: chiral crystallization, crystallization of nanomaterials and the crystallization of amorphous and glassy materials. This book is of interest to both fundamental research and also to practicing scientists and will prove invaluable to all chemical engineers and industrial chemists in the process industries as well as crystallization workers and students in industry and academia.


Introduction
The enthalpy relaxation of the glassy materials has been investigated rheologically for years with a view to approaching the ideal glass 1 -5) . The imaginary liquid at Kauzmann temperature 2,3) , T K , at which the extrapolation line of enthalpy or entropy as a function of temperature for the liquid intersected the enthalpy or entropy line of the crystal, had been considered once to be the ideal glass. However because at T K , the enthalpy or entropy for the liquid was same as that of crystal, the liquid like this was hard to take thermodynamically, bringing the entropy crisis. Fig. 1 depicts the change of the enthalpy difference, ΔH, between the liquid and the crystal upon cooling for a polymer. For stable liquids, ΔH should be almost constant from near T g upon cooling as described below. Therefore, the liquid line can never intersect that of the crystal. The transition from liquid to crystal or vice versa means the emission or absorption of the latent heat accompanying the enthalpy jump. Thus for polymers, T K is merely a temperature parameter.
where f x (= h x -T g s x ), h x and s x are the free energy, enthalpy and entropy per molar structural unit for ordered parts, f flow , h flow and s flow are the free energy, enthalpy and entropy per molar structural unit for flow parts. The molar free energy of holes held photons at the temperature, T, is generally given by f h = −RTln(v f /v 0 ) and then the molar entropy of holes, s h , at a constant pressure, p, is derived: where v f and v 0 are the molar free and core volumes of holes and R is the gas constant.
When v 0 is almost constant, from f h = h h − Ts h and Eq. (3), the molar enthalpy of holes, h ｈ , is derived: When the ordered parts at T g are in equilibrium with the holes; f x (= 0) = f h , from f x = 0 and h x = h g + Δh (see Eq. (13)), s x at T g is derived: s x (= h x /T g ) = Δs g + Δh/T g (5) with Δs g = h g /T g , defining the entropy of unfreezing for the glass parts at T g , where h g is the molar glass transition enthalpy, Δh is the heat per molar structural unit required still to melt ordered parts, relating to the jump of v f (see Eq. (8)). Further from Eq. (2), the h g and the molar glass transition entropy, s g , are derived 13) : h g = RT g 2 (∂lnv f /∂T) p (6) s g (= s g conf + s g int ) = Rln(v f /v 0 ) + RT g (∂lnv f /∂T) p (7) with s g int = (3R/2)ln(2πmkT g /h 2 ) − (1/x)(R/N)lnN! + Rlnq − Rlnv 0 where s g conf and s g int are the conformational and cohesive entropies per molar structural unit for glass parts at T g (see Eq. (16) for s g conf ), m is the mass of a structural unit, k is Boltzmann constant, h is Planck constant, N is the number of chains, x is the degree of polymerization and q (≤ 1) is the packing factor. When v f = v 0 , from Eqs. (6) and (7), h g = 0 and s g = 0 are derived. On the other hand, from f h = 0 (= f x ) at T g , h h is given as: In the case of RT g ln(v f /v 0 ) = Δh, the relation of h x = h h is derived from Eq. (8), because of RT g 2 (∂lnv f /∂T) p = h g . However, when the length distribution by lengthening of ordered parts occurred during the enthalpy relaxation at temperatures below T g , as longer the length of ordered parts, the melting temperature, T x , for ordered parts should be elevated from T g to the higher temperature 23) (see Eq. (29)). Therefore the shortage of Δh (= RT g ln(v f /v 0 )) corresponding to the latent heat of disappearance for the holes at T g is made up by the supply of the heat required to melt all ordered parts: www.intechopen.com

166
where T ℓ is the end temperature of melting for ordered parts, ΔC p is the difference between the observed isobaric heat capacity, C p ℓ , for the equilibrium liquid and C p g for the hypothesized super-heated glass at the glass transition from T g to T ℓ . In the equilibrium liquid, the isobaric heat capacities of ordered parts and flow parts, C p x and C p flow , are equal to C p ℓ , respectively 13) : In the flow parts, the tube-like space exists between a chain and the neighboring chains, behaving as if it is the counterpart of a chain 24) . Therefore when the hole energy at T (> T g ) is given by ε (= 3C v ph T), C p flow is represented as 16) : where C v ph (= 2.701R) 16,18,19) is the constant volume heat capacity for photons, J is the number of holes lost by T and 3 is the degree of freedom for photons. after relaxation shows the subsequent s curve with a jump at T g . Upon cooling in Fig. 2 (lower), the dashed line is the v f curve for the same liquid glass. Upon heating after relaxation, the solid line shows the v f curve with a jump at T g and the dashed line shows a reversible jump of v f between T g and T ℓ .

"Ordered part / hole" pairs
Next whether h x agrees to h h at T g or not is investigated for PET, iPP, PS, PE and N6 glasses.
The agreement provides one of evidences for the generation of "ordered part / hole" pairs during the enthalpy relaxation at T (< T g ). h h at T g is given by 16) : where C v ph = 2.701R. For PET with T g = 342 K, h h = 23.0 kJ/mol was derived. While h x at T g is given by 13,25) : In Eq. (13), h g is given approximately by three expressions 13,25) ; (1) RT g 2 /c 2 (c 2 is the constant of WLF equation 20) ), (2) the molar enthalpy difference between the super-cooled liquid and the crystal at T g , H g a − H g c , and (3) the sum of the conformational and cohesive enthalpies per molar structural unit at T g , h g conf + h g int . Δh is given by either Eq. (14) or (15) 25,26) : with ΔH = H m a -H c a , where H m a is the enthalpy per molar structural unit for the liquid at the equilibrium melting temperature, T m ∞ , H c a is the enthalpy per molar structural unit for the super-cooled liquid at the onset temperature, T c , of a DSC crystallization peak upon cooling and Q is the heat per molar structural unit corresponding to the total area of the DSC endothermic peak upon heating. Or, rewriting Eq. (13), Δh = (h x conf − h g conf ) + Δh int (15) where h x conf is the conformational enthalpy per molar structural unit for ordered parts, Δh int is the molar cohesive enthalpy difference between the ordered parts and the glass parts. Thus when h x conf = h g conf at T g , Δh = Δh int = (RT g lnZ t )/x (see Table 3 for N6) and when h x conf ≠ h g conf = 0 at T g , the another Δh is derived 26,27) .
with s g conf = (RlnZ + RT g dlnZ/dT)/x where Z is the conformational partition function for a chain, Z 0 (= Z/Z t ) and Z t are the component conformational partition function for a chain regardless of temperature and as a function of temperature, respectively. The differential of Eq. (15) by temperature represents the heat capacity jump at the glass transition 15) .  a chain axis 25,30) . For the smectic-c crystals with stretched sequences, h u is 28.5 kJ/mol. Further, DSC revealed 25) that for the crystalline films of smectic-c crystals, the ordered parts in the amorphous regions were like smectic crystals and for the crystalline films of smectic crystals, the ordered parts were like the smectic-c crystals. Fig. 3 shows the sequence models of smectic crystal (A) and smectic-c crystal (B), together with the four conformations (a, b, c and d) that an isolated chain can take preferentially below 10 K. An arrow mark shows the direction of ordering or crystallization for a, b, c and d. From these results, the ordered parts are like the smectic crystal and the hole of a pair should have the free volume coming from the difference of conformation between A and a, b, c or d in Fig. 3. For iPP, h h was 2.5 times as much as h x . This result suggested that the hole of a pair was the inside space of a 3/1 helical ordered part composed of 3 structural units, holding 3 photons, but each photon was concerned in the potential energy of 2.5 structural units in a helical sequence, and that, (h h /2.5)/h x = 1. This was comparable to h h /h x = 1 for PET. The value of  30) . Left: Four conformations taken preferentially below 10 K for an isolated chain; a: TTTTG'T, b: TTTTGT, c: TCTTG'T and d: TCTTGT, by Flory's theory 31) . T, G and G' are the trans, gauche and gauche' isomers, respectively. T and C are the trans and cis isomers of phenylene groups (lower groups). An arrow mark shows the direction of ordering or crystallization.
www.intechopen.com h x was almost equal to h u (= 7.6 kJ/mol for α form). Fig. 4 shows the photon sites in the helix structure and the helical conformation of an isolated sequence with an inversion defect isomer TT, taking preferentially at temperatures below 70 K 26) . For PS, supposing h x = h h , Δh was evaluated. From the value of Δh to be near that of PET, the v f jump at T g should be due to the release between phenyl groups. The T g of PE 15) , producing the entropy of unfreezing for the glass parts; Δs g = h g /T g (see Eq. (5)), was almost dependent on h g int . When a value of h g int was that of the glass with T g = 237 K, h g int /2 gave T g = 135 K. Table 2 shows the values of T g , Δs g , h g , Δh, h x , h h and h h /h x for PE glasses with T g = 135 K (h g int = 2.8/2 kJ/mol) and 237 K (h g int = 2.8 kJ/mol). The above relation in T g and h g int for the glass parts was linked to the ordered parts. For both glasses, h h was about 5 times as much as h g . Thus from Eq. (13), the common relations of h x = h h /4 and Δh = h h /4 − h g were predicted for the ordered parts in both glasses and shown in Table 2. Fig. 5 depicts the sequence models of ordered parts (A and B) and the schematic transition from the glassy state (C: left) to that of the "ordered part / hole" pair (C: right).
For the glass with T g = 135 K, the coarse 4/1 helical ordered parts with GG or G'G' conformation in Fig. 5A and as the hole of a pair, the inside space holding four photons per a helical segmental unit, -(CH 2 ) 4 -, were predicted. Further the length distribution of helical ordered parts in the glass and as the end temperature of melting for the ordered parts, ∼237 K were predicted. In this case, the same value of Δh for both glasses enabled the scheme as depicted in Fig. 6. For the glass with T g = 237 K, the ordered part of fringe-type formed by  bundling TTT parts of four sequences at least, and that, the smallest crystal of PE and the neighboring hole were predicted (see Fig. 5B and C: right), since the value of h x was almost equal to h u = 4.1 kJ/mol. Fig. 6 shows the bar graph of h g at T g = 135 K and 237 K, together with Δh (= 0.5 kJ/mol) at 237 K, for PE. The difference in two complementary lines suggests the supply schedule of heat over the temperature range from 135 K to ∼237 K in order to make up the shortage of Δh required to melt all ordered parts in the glass with T g = 135 K. For the glass with T g = 237 K, the glass transition only at T g is shown. The enthalpy relaxation, accompanied by the generation of stretched segments, at temperatures over 135 K to ∼237 K www.intechopen.com should vitrify or order the melted helical ordered parts and confine the larger helical ordered parts, which were not yet melted even over T g = 135 K, in the glass. The liquid on the way of enthalpy relaxation like this should reach step by step to the glass with T g = 237 K (h g int = 2.8 kJ/mol) at temperatures below T g . Table 3 shows the values of T g , Δs g , h g , Δh, h x , h u and h h for N6. For N6, h g conf = h x conf in Eq. (15), i.e., Δh = (RT g lnZ t )/x was predicted, because of the strong interaction between amido groups. h h was 4.7 times as much as h x in the parenthesis. Accordingly a photon was concerned in the potential energy of a stretched segmental unit, -(CH 2 ) 5 -. Further Δh = 2.5 kJ/mol was 0.5 kJ per molar methylene unit, -CH 2 -, which agreed with Δh = 0.5 kJ/mol for the PE glasses with T g = 135 K and 237 K (see Table 2). In addition, the value of h h was almost equal to h u of the heat of fusion. This agreement suggested that the ordered parts were the stretched segmental units in the smallest crystals of N6. Fig. 7 shows the structural unit of N6 and the photon site in the structural unit.

Hole energy and a photon
A photon has the property as a boson or a wave. Therefore h h is also represented as the vibrational energy of a wave with the quantum number n = 1 (meaning one photon) and the frequency per second (sec), ν: (17) where N A is Avogadro constant. Thus from Eqs. (12) and (17), the zero-point energy, ε 0 (= (1/2)hν), is derived: www.intechopen.com Table 4 shows the values of T g , h h , ν, λ and 1/λ for PE, iPP, PS and PET, where λ is the wavelength and 1/λ is the wavenumber. According to the infrared spectroscopy, for PE, 1/λ = 510 cm − 1 and 893 cm − 1 might be concerned with 720 cm − 1 and 731 cm − 1 bands assigned to the rocking of -CH 2 -32) . For iPP, 1/λ = 1022 cm − 1 almost agreed with 1045 cm − 1 relating to the crystallinity 33) . Also for PS, 1/λ = 1359 cm − 1 almost agreed with one of conformation sensitive bands 34) , i.e., 1365 cm − 1 band. For PET, 1/λ = 1292 cm − 1 was near 1339 cm − 1 and 1371 cm − 1 bands assigned to the wagging of -CH 2 -with trans and gauche conformations, respectively 35) .

Conclusions and introduction to next section
For PET, PS, iPP, PE and N6 glasses, the generation of "ordered part / hole" pairs during the enthalpy relaxation at temperatures below T g and the subsequent disappearance at the glass transition were discussed under the operational definition leading the criterion of T g .
Thus it was concluded that the unfreezing of the glass parts at T g was caused by the first order hole phase transition, accompanied by the jumps of free volume, enthalpy and entropy. h h was concerned with the frequency of the absorption bands in the infrared spectrum. In particularly, 1/λ for iPP and PS coincided closely with the respective sensitive bands with physical meaning. In the next section, the generation of "crystal / anti-crystal hole" pairs from the secondary glass in PE crystal lamella was discussed on the DSC curves. The secondary glass was distinguished from the primary glass discussed in the above sections. The C v ph was also available in the discussion of the cavity radiation from the anticrystal holes filled by photons. The anti-crystal holes were regarded as the lattice crystal made up of photons without the mass.

Thermal analysis
DSC is capable of quantitatively determining by way of standard and dynamical measurements 36) the common thermal phenomena in polymers, e.g., the melting, the crystallization and the glass transition. Such analyses are carried out on the basis of thermodynamics, mathematics and molecular dynamics simulation 36,37) . This section describes an attempt to understand the peculiar DSC curves of PE films containing orthorhombic crystals. DSC demonstrated two indications of the secondary glass in the crystal lamella. One of the underlying reasons was the much larger heat of melting as opposed to the heat of crystallization upon cooling and the other was the fact that the glass www.intechopen.com transition enthalpy was larger than the molar enthalpy of the ordered parts in the amorphous regions; Δh < 0 in Eq. (14). At temperatures above its T g , the generation and disappearance of the "crystal / anti-crystal hole" pairs from the secondary glass were predicted as the simultaneous phenomena in the crystallization and the melting. Hexagonal and monoclinic forms of PE crystals are also well known. However, the hexagonal crystals should not be related to the melting of the orthorhombic crystals since the DSC melting peak of the hexagonal crystals generally cannot be observed for the samples without restraints such as high pressure 38) . Moreover, the DSC melting peak of monoclinic crystals disappears before the melting of the orthorhombic crystals 39,40) . Thus, when the monoclinic crystals are in the bulk state, the heat due to their melting should contribute to the activation heat required to release the secondary glass state in the orthorhombic crystal lamella. for 1 hour. The thin line in T b * and T e * is the curve before division. T c (= 391.5 K) is the onset temperature of crystallization, T b * (≈ T c ) is the intersection between the base line and the extrapolation line from the line segment with the highest slope on the lower temperature side of the melting peak, and that, the onset temperature of the higher temperature side peak and T e * is the end temperature of the lower temperature side peak, and that, the origin of the extrapolation line, respectively. Q m is the heat per molar structural unit corresponding to the endothermic peak area of crystal lamella that starts to melt at T b * and h c (= 0.89 kJ/mol) is the heat of crystallization per molar structural unit corresponding to the area surrounded by the dashed line and the exothermic curve. ΔQ m (= Q m − h c ) corresponds to the area between T b * and T e * of the higher temperature side Fig. 8. The DSC crystallization peak upon cooling and the two peaks divided from a DSC endothermic peak upon heating for the PE film annealed at 416.6 K for 1 hour. dQ/dt is the heat flow rate. The cooling and heating rates are 5 K/min and 10 K/min, respectively.

Secondary glass
peak, which is equal to the area surrounded by the thin line and the lower temperature side peak curve between T b * and T e * . The endothermic peak on the lower temperature side is due to the melting of small crystals around the crystal lamella 41) . The decrease of heat flow rate from T b * to T e * for the peak on the lower temperature side is believed to be due to the crystallization of secondary glass in the inter-grain aggregates belonging to the crystal lamella (see Fig. 9). This precedes the increase of heat flow rate due to the melting of newly crystallized parts from T b * to T e * in the peak on the higher temperature side. The equilibrium melting of the ordered parts in the amorphous regions does not show any peak. Its enthalpy, h x , has been represented as Eq. (13), in which Δh given by Eq. (14) is usually positive; 6.5 kJ/mol and 11.5 kJ/mol for PET with two values of T m ∞ (535 K and 549 K) 25) , respectively. Also for iPP, Δh (= 1.1 kJ/mol) of Eq. (14) was positive, as shown in Table 1. Nevertheless, it was found to be negative for PE (see Table 5). In order to satisfy Δh < 0 (h g > h x ), the glass with a secondary T g , which formed near T c upon cooling and disappeared near T c after melting of the ordered parts in the amorphous regions upon heating, must exist in the crystal lamella. When the secondary T g is approximated to T c , h g * is given by: where h g * is h g at the secondary T g , H c a and H c c are the enthalpy per molar structural unit for the super-cooled liquid and the crystal at T c . h x is given by h g * + Δh (Δh < 0). Here, ΔH (= H m a − H c a ) in Eq. (14) is regarded as the heat emitted when only one single crystal lamella without deformation is formed. According to the ATHAS databank 29) , ΔH is 0.83 kJ/mol, which is close to the value of h c = 0.89 kJ/mol as observed in Fig. 8. The difference in h c and ΔH, 0.06 kJ/mol, might be the additive heat of emission due to the release of lamellar deformation. The spherulites observed in the films are substantially like disks 42) . The twist deformation energy of ribbon-like lamella is believed to originate from the irregular growth of lamella. Fig. 9 shows a schematic structure of the crystal lamella after release of the twist deformation from the ribbon-like lamella. The dark parts between the rectangular parallelepiped blocks correspond to the inter-grain aggregates described above. The crystal lamellae for the samples used here are described in the section 5.5 of "Crystal length distribution function".  Table 5 shows the values of T c (≈ T g * ), T b * , Q, ΔH (=H m a − H c a ), Δh, h g * and h x for the samples annealed at T a = 376.6 K, 416.6 K and 426.6 K for 1 hour, where T g * is the secondary T g and T a is the annealing temperature. h x was found to decrease with an increasing T a . The values of h x for "T a = 416.6 K and 426.6 K"-samples were near h x = 2.3 kJ/mol for the glass with T g = 135 K in Table 2. In the amorphous region of these samples, the ordered parts with the coarse 4/1 helix structure might be formed 5) . The ordered parts in the inter-grain aggregates, being like the crystals of fold-type with h x i instead of h x , could also have the holes as the pair (see Eq. (28)).  Table 5. The values of T c (≈ T g * ), T b * , Q, ΔH, Δh, h g * and h x for PE films annealed at 376.6 K, 416.6 K and 426.6 K for 1 hour.

"Crystal / anti-crystal hole" pairs from secondary glass
The crystal length, ζ, as a function of the melting temperature, T m , is according to Gibbs-Thomson given by: where σ e is the end-surface free energy per unit area for crystals, µ is the conversion coefficient. For PE, µ = (10 6 /14) mol/m 3 , h u = 4.11 kJ/mol 36) and T m ∞ = 415 K 18,36) . σ e is given as 25,43) : with H x = 2h u − Q m , where c * is the cell length of c-axis. The term of the square bracket in Eq. (21) is dimensionless. Table 6 shows the values of σ e for "T a = 376.6 K, 416.6 K and 426.6 K"samples, together with the values of T p , T e * , h x , Q m , ΔQ m (= Q m − h c ) and ΔQ (= Q − Q m ) used in the calculation of σ e , where T p is the melting peak temperature and ΔQ is the heat per molar structural unit corresponding to the area of the lower temperature side peak in Fig. 8, contributing to the activation heat required to release the secondary glass state. ΔQ m is given by: ΔQ m =  Tb* Te* (dQ/dt)(1/α s )dT (22) where α s (= dT/dt) is the heating rate. T e * was derived from Eq. (22) using ΔQ m in Table 6. The fact that T e * was almost equal to that by observation, as shown in Table 6, supported that h c was due only to the formation of the crystal lamella, thus giving rise to the melting peak on the higher temperature side. T e * is also the end temperature of melting for ordered parts of fold-type in the inter-grain aggregates. Upon heating over T e * , the flow parts in the amorphous regions should start to participate directly in the melting of crystals. With an increasing T a , σ e and h x decreased, whereas Q m , ΔQ m and ΔQ increased. ΔQ should influence h x . For all samples, the value of σ e at T p was 1.3 times larger than that at T m ∞ (Q m = 0). The experimental values of σ e from Eq. (20), 30 ∼ 90 mJ/m 2 , differed significantly from those in Table 6, probably due to the length of the lamellae after annealing and the use of cooling as the substitute of ζ at T m 23, 44) . The values in the parentheses of T e * and σ e columns are the apparent Te * by observation and σ e at Tm ∞ = 415 K (Qm = 0), respectively. Tp is corrected by 0.1 K to the lower temperature side. On the other hand, according to Flory's theory 45) on the melting of the fringe-type crystals with a finite ζ, σ e at λ and (df u /dζ) λ = 0 is given by: where λ is the amorphous fraction, f u is the free energy per molar structural unit for the crystals and x is the degree of polymerization. In this context: where P c , given by {(x − ζ + 1)/x} 1/ ζ for fringe-type crystals, is the probability that a sequence occupies the lattice sites of a crystalline sequence. Moreover: Eq. (23) is obtained when lnP c = −1/(x − ζ + 1). From Eq. (26), the relations are derived based on f u and f x at f x ' ≥ 0 and those can be grouped into four equilibrium classes (A ∼ D) and one non-equilibrium class (X) as shown in Table 7. Class A of f x = f u at f x ' = 0 shows the dynamic equilibrium relation between the ordered parts and the crystal parts of equivalent fringetypes, leading to σ e = 0 in Eq. (24) as expected for highly oriented fibers. For class B, f u = −f x ' from Eq. (26) with f x = 0 refers to the anti-crystal holes and f x = 0 is assigned to the ordered parts of ζ = ∞. For class C, f x = f x ' from Eq. (26) with f u = 0 is assigned to the ordered parts of ζ ≠ ∞ (i.e., a kebab structure) and f u = 0 to the crystals of ζ = ∞ (i.e., a shish structure). Class D of f u (= f x ') = f x /2 is related to the equilibrium in crystal and ordered parts. For those with folded chains, the reversible change from crystal or ordered parts to other parts is expected to take place automatically. The relations in class X do not satisfy Eq. (26), suggesting that the holes of class B can not be replaced by the crystals with ζ ≠ ∞.
When f u = −f x ' for class B, the temperature at which the anti-crystal holes disappear (melt), i.e., T m h , is given by: www.intechopen.com where h u , σ e and ζ are imaged for the anti-crystal holes, and that, the photonic crystals made up only of photons without the mass. According to Eq. (27), T m h approaches T m ∞ with an increasing ζ. However, the interface between the anti-crystal holes and the ordered parts, which satisfy f x = 0 (described below), should work as the reflector of photons attached to the anti-crystal holes. In this case, the even interface made of the folded chain segments should be avoided through the random reflection. At such an interface, from Eq. (24), the following relation of energy balance is derived: where s u is the entropy per molar unit for the anti-crystal holes. As well as h u , one mole of units (photons) corresponds to three moles of the oscillators, since three oscillators can be coordinated to each point of the crystal lattice. According to Eq. (28) or (24), the respective interface energies of the hole and the ordered part are compensated each other at the common interface, leading to f x = 0 of class B and further, T m h approaches 0 K with an increasing ζ. From the interface at ζ = ∞, the photons are not reflected, and that, do not exist in the holes. This is exactly the real "dark hole". Therefore when the anti-crystal hole of ζ = ∞ is pairing with the neighboring crystal as shown in Fig. 11C, the crystal is set at 0 K. As opposed to Eq. (27), from f u = f x ' of class D, the T m of the crystals is derived: Eq. (29) is the same as Eq. (20). In Eq. (29), T m is T m ∞ at ζ = ∞ and from Eqs. (27) and (29), T m ∞ = (T m h + T m )/2 is derived. According to the pair relation, the emission of heat from the anticrystal holes after crystallization is necessarily linked to the supply of heat of the same quantity to the newly crystallized parts. However T m h should be depressed down to T m by the emission of heat to the outside. For a model of the inter-grain aggregates shown in Fig. 9, the interaction in the inter-grain aggregates and the a -c face of the crystals must be neglected and the ordered parts in the inter-grain aggregates must satisfy Eqs. (1) and (2) at T g * . It is thus presumed that over T g * , the chains that cross the glass regions give rise to the newly crystallized parts of fringe-type, whereas the folded chain segments around the glass excluded from the ordered parts give rise to the two end-surfaces of the anti-crystal hole with the same ζ as the new crystal. The T m of the crystals from the secondary glass, being equal to T m h , was found to change from T b * to T e * as a function of ζ in Eq. (29). The time spent from T b * to T e * was 0.73 s, in which the probability of observing a spontaneous generation of crystallization or melting should be 1/2, according to the uncertainty principle. Table 7. Relations of equilibrium (A ∼ D) and non-equilibrium (X) in f x and f u at f x ' ≥ 0 for crystalline polymers 46) . Fig. 10 shows the schematic behaviors of sequences and photons on the way of crystallization and melting. The heat of emission, ΔU h , corresponds to ΔQ m of the area between the observed melting curve (thin line) and the lower temperature side peak curve from T b * to T e * and the heat of absorption, ΔU m (= ΔU h ), corresponds to ΔQ m of the under area of the higher temperature side peak curve from T b * to T e * in Fig. 8. Fig. 11 shows the cross sections of the glass (A) and the "crystal / anti-crystal hole" pair (C). The two endsurfaces of the anti-crystal hole in Fig. 11C contact in equilibrium those of the ordered parts. Supposing that this model of aggregates was valid for the "T a = 416.6 K"-sample shown in Fig. 8, a derivation of f x ' (= −f u ) = 0.13 kJ/mol was obtained from Eq. (24) using the values of σ e (see Table 6) and ζ (= 3.1 nm) at T p = 401.8K (see Table 9). Here, f x is rewritten as f   11. The cross sections of the glass (A) and the "crystal / anti-crystal hole" pair (C). The dot in large and small circles is the cross section of a segment and the blank in C is the hole. The arrow mark shows the crystallization from A to C.

Fraction of secondary glass
The anti-crystal holes should be permeated by the photons obeying the frequency distribution function with an upper limit. This is due to the interface between the anticrystal hole and the ordered part be able to act as a filter for the photons. The molar photon energy loss of the anti-crystal holes, ΔU h , due to the cavity radiation from T b * to T e * is given by 18) : On the other hand, the heat change per molar structural unit, ΔU m , due to the melting of the newly crystallized parts from T b * to T e * is given by: where Γ is the fraction of the secondary glass, contributed to the generation of "crystal / anti-crystal hole" pairs, in the inter-grain aggregates at temperatures below T g * . From ΔU h = ΔU m at T e * , Γ is given by 18) :

ζ distribution function, F(ζ)
The occurrence of ζ distribution by crystallization is one of the characteristics of bulk polymers. The conversion of a DSC melting peak into the ζ distribution by Eq.
where δQ m (=ζn ζ Q m /{N c (T e − T b )}) is the heat change per molar structural unit per K, n ζ is the number of crystal sequences with ζ and N c is the number of structural units of crystals melted in the temperature range from T b (= T b * here) to T e . δQ m /Q m is given by: δQ m /Q m = (dQ/dt)/ Tb Te (dQ/dt)dT (34) where dQ/dt is the heat flow rate of the melting curve. Fig. 12 shows F(ζ) of each melting curve from T b * for "T a = 376.6 K, 416.6 K and 426.6 K"-samples. Table 9 lists the values of ζrange, ζ c and ζ p of F(ζ) curve for each sample, where ζ c is ζ at T e * and ζ p is ζ at T p . For "T a = 376.6 K"-sample, ζ p was slightly larger than for other samples. The small value of ζ (< ζ c ) was believed to be caused by the crystallization from the secondary glass in the restricted space of inter-grain aggregates. The large ζ value at the maximum for "T a = 416.6 K and 426.6 K"-samples might be related to the long period change of lamellae at the higher temperature upon heating 36) . Whereas, the very narrow ζ-range for "T a = 376.6 K"-sample might be due to the effective annealing.  The values in the parentheses are ζ at the apparent Te * . Table 9. The values of ζ-range, ζ c and ζ p in F(ζ) for PE films annealed at 376.6 K, 416.6 K and 426.6 K for 1 hour. Fig. 13. The relationship between R (= ±R n ) and L (= ±ζ/2) for PE films (1 g) annealed at 376.6 K (thick line) and 426.6 K (thin line) for 1 hour. The horizontal lines show R of the crystal melting from ζ n or ζ c to ζ = 0.
In the last stage, a single crystal-like image was drawn from F(ζ). Rewriting Eq. (33), n ζ is given by: The number of the crystal sequences from ζ n to ζ, N ζ , is as follows: Accordingly, the number of the crystal sequences from ∼ζ (> ζ) to ζ x , ΔN, is given by: where ζ x and ζ n are the maximum and minimum of ζ, respectively. When the crystal sequences are bundled, like a fringe in a circle, the number of crystal sequences in a radius direction, R n , as a function of ζ is given by: R n = (ΔN/π) 1/2 (38) Fig. 13 shows the relationship between R (= ±R n ) and L (= ±ζ/2) for the samples (1 g) annealed at 376.6 K and 426.6 K for 1 hour. In the ζ-range of 0 ∼ ±ζ n /2, R at ζ n /2 is represented by a solid line, which leads to the supposition of a melting process from the end-surfaces of the crystal with ζ n at T b * . The horizontal line of R at ζ c /2 depicts the same imaginable melting process of the crystal with ζ c at T e * . The distinct difference of R or L between both types of crystals should be available in order to evaluate the annealing effects. The single crystal image from R and L for "T a = 376.6 K"-sample (thick line) was very similar to the electron microscope (EM) image of self-seeded PE crystals 36) .

Conclusions
The generation and disappearance of the "crystal / anti-crystal hole" pairs from the secondary glass in PE crystal lamella were discussed on the DSC curves. Thus the fraction of the secondary glass in the lamella was derived from the molar photon energy loss of the anti-crystal holes upon heating, which agreed with the latent heat of disappearance for the holes at the primary T g of the first order hole phase transition.