**5. Crystallization of secondary glass in PE lamella**

## **5.1 Thermal analysis**

DSC is capable of quantitatively determining by way of standard and dynamical measurements36) 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 simulation36, 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 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 Tg, 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 pressure38). Moreover, the DSC melting peak of monoclinic crystals disappears before the melting of the orthorhombic crystals39, 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.

#### **5.2 Secondary glass**

172 Advances in Crystallization Processes

Table 4 shows the values of Tg, hh, 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<sup>−</sup>1 and 893 cm<sup>−</sup>1 might be concerned with 720 cm<sup>−</sup>1 and 731 cm<sup>−</sup>1 bands assigned to the rocking of -CH2-32). For iPP, 1/λ = 1022 cm<sup>−</sup>1 almost agreed with 1045 cm<sup>−</sup>1 relating to the

sensitive bands34), i.e., 1365 cm<sup>−</sup>1 band. For PET, 1/λ = 1292 cm<sup>−</sup>1 was near 1339 cm<sup>−</sup>1 and 1371 cm<sup>−</sup>1 bands assigned to the wagging of –CH2- with trans and gauche conformations,

PE 135 9.1 15.3 × 1012 19.6 510

iPP 270 18.2 30.6 × 1012 9.78 1022 PS 359 24.2 40.6 × 1012 7.36 1359 PET 342 23.0 38.6 × 1012 7.74 1292

For PET, PS, iPP, PE and N6 glasses, the generation of "ordered part / hole" pairs during the enthalpy relaxation at temperatures below Tg and the subsequent disappearance at the glass transition were discussed under the operational definition leading the criterion of Tg. Thus it was concluded that the unfreezing of the glass parts at Tg was caused by the first order hole phase transition, accompanied by the jumps of free volume, enthalpy and entropy. hh 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

crystal holes filled by photons. The anti-crystal holes were regarded as the lattice crystal

DSC is capable of quantitatively determining by way of standard and dynamical measurements36) 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 simulation36, 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

ν sec <sup>−</sup><sup>1</sup>

237 16.0 26.9 × 1012 11.2 893

ph was also available in the discussion of the cavity radiation from the anti-

hh kJ/mol

Table 4. The values of Tg, hh, ν, λ and 1/λ for PE, iPP, PS and PET.

**4. Conclusions and introduction to next section** 

**5. Crystallization of secondary glass in PE lamella** 

almost agreed with one of conformation

λ μm

1/λ cm<sup>−</sup><sup>1</sup>

crystallinity33). Also for PS, 1/λ = 1359 cm<sup>−</sup><sup>1</sup>

K

respectively35).

sections. The Cv

**5.1 Thermal analysis** 

made up of photons without the mass.

Polymer Tg

Fig. 8 depicts 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 (near Tm<sup>∞</sup> = 415 K) for 1 hour. The thin line in Tb\* and Te \* is the curve before division. Tc (= 391.5 K) is the onset temperature of crystallization, Tb\* (≈ Tc) 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 Te\* is the end temperature of the lower temperature side peak, and that, the origin of the extrapolation line, respectively. Qm is the heat per molar structural unit corresponding to the endothermic peak area of crystal lamella that starts to melt at Tb\* and hc (= 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. ΔQm (= Qm − hc) corresponds to the area between Tb\* and Te\* 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.

Thermodynamics of Enthalpy Relaxation and Hole Formation of Polymer Glasses 175

135 K in Table 2. In the amorphous region of these samples, the ordered parts with the coarse 4/1 helix structure might be formed5). The ordered parts in the inter-grain

> ΔH kJ/mol

376.6 391.5 392.4 1.79 0.83 −0.96 3.99 3.03 416.6 391.5 392.5 2.38 0.83 −1.55 3.99 2.44 426.6 391.5 393.3 2.85 0.83 −2.02 3.99 1.97 Table 5. The values of Tc (≈ Tg\*), Tb\*, Q, ΔH, Δh, h g\* and hx for PE films annealed at 376.6 K,

The crystal length, ζ, as a function of the melting temperature, Tm, is according to Gibbs-

where σe is the end-surface free energy per unit area for crystals, μ is the conversion coefficient. For PE, μ = (106/14) mol/m3, hu = 4.11 kJ/mol 36) and Tm<sup>∞</sup> = 415 K18, 36). σe is given

with Hx = 2hu − Qm, 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 σe for "Ta = 376.6 K, 416.6 K and 426.6 K"-

in the calculation of σe, where Tp 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. ΔQm is given

that hc was due only to the formation of the crystal lamella, thus giving rise to the melting

amorphous regions should start to participate directly in the melting of crystals. With an increasing Ta, σe and hx decreased, whereas Qm, ΔQm and ΔQ increased. ΔQ should influence hx. For all samples, the value of σe at Tp was 1.3 times larger than that at Tm<sup>∞</sup> (Qm = 0). The experimental values of σe from Eq. (20), 30 ∼ 90 mJ/m2, differed significantly from those in Table 6, probably due to the length of the lamellae after annealing and the use of cooling as

parts of fold-type in the inter-grain aggregates. Upon heating over Te

Δh kJ/mol

ζ = {Tm<sup>∞</sup>/(Tm<sup>∞</sup> − Tm)}{2σe/(μhu)} (20)

\*, hx, Qm, ΔQm (= Qm − hc) and ΔQ (= Q − Qm) used

\* was derived from Eq. (22) using ΔQm in Table 6.

\* is also the end temperature of melting for ordered

\*, the flow parts in the

ΔQm = Tb\*Te\*(dQ/dt)(1/αs)dT (22)

σe = μhuc\*[{RTm2 + (Hx − hx)(Tm<sup>∞</sup> − Tm)}/{2(Hx − hx)Tm<sup>∞</sup>}] (21)

\* was almost equal to that by observation, as shown in Table 6, supported

instead of hx, could also have the

hx kJ/mol

hg\* kJ/mol

aggregates, being like the crystals of fold-type with hxi

Q kJ/mol

**5.3 "Crystal / anti-crystal hole" pairs from secondary glass** 

Tb\* K

holes as the pair (see Eq. (28)).

Tc K

416.6 K and 426.6 K for 1 hour.

samples, together with the values of Tp, Te

where αs (= dT/dt) is the heating rate. Te

peak on the higher temperature side. Te

the substitute of ζ at Tm23, 44).

Thomson given by:

as25, 43):

by:

The fact that Te

Sample Ta/K

peak, which is equal to the area surrounded by the thin line and the lower temperature side peak curve between Tb\* and Te\*. The endothermic peak on the lower temperature side is due to the melting of small crystals around the crystal lamella41). The decrease of heat flow rate from Tb\* to Te \* 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 Tb\* to Te \* 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, hx, 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 Tm<sup>∞</sup> (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 (hg > hx), the glass with a secondary Tg, which formed near Tc upon cooling and disappeared near Tc after melting of the ordered parts in the amorphous regions upon heating, must exist in the crystal lamella. When the secondary Tg is approximated to Tc, hg\* is given by:

$$\mathbf{h}\_{\emptyset}^{\*} = \mathbf{H}\_{\emptyset}^{\*} - \mathbf{H}\_{\emptyset}^{\*} \tag{19}$$

where hg\* is hg at the secondary Tg, Hc a and Hc c are the enthalpy per molar structural unit for the super-cooled liquid and the crystal at Tc. hx is given by hg\* + Δh (Δh < 0). Here, ΔH (= Hma − Hc a ) in Eq. (14) is regarded as the heat emitted when only one single crystal lamella without deformation is formed. According to the ATHAS databank29), ΔH is 0.83 kJ/mol, which is close to the value of hc = 0.89 kJ/mol as observed in Fig. 8. The difference in hc 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 disks42). 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*".

Fig. 9. A schematic structure of the crystal lamella after release of the twist deformation from the ribbon-like lamella. a, b and c (chain axis) correspond to the three cell axes of the orthorhombic crystal and the dark parts represent the inter-grain aggregates. The cell lengths of a, b and c axes for PE are 0.74 nm, 0.49 nm and 0.25 nm, respectively.

Table 5 shows the values of Tc (≈ Tg\*), Tb\*, Q, ΔH (=Hma − Hc a), Δh, hg\* and hx for the samples annealed at Ta = 376.6 K, 416.6 K and 426.6 K for 1 hour, where Tg\* is the secondary Tg and Ta is the annealing temperature. hx was found to decrease with an increasing Ta. The values of hx for "Ta = 416.6 K and 426.6 K"-samples were near hx = 2.3 kJ/mol for the glass with Tg =

peak, which is equal to the area surrounded by the thin line and the lower temperature side peak curve between Tb\* and Te\*. The endothermic peak on the lower temperature side is due to the melting of small crystals around the crystal lamella41). The decrease of heat flow

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

melting of the ordered parts in the amorphous regions does not show any peak. Its enthalpy, hx, 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 Tm<sup>∞</sup> (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 (hg > hx), the glass with a secondary Tg, which formed near Tc upon cooling and disappeared near Tc after melting of the ordered parts in the amorphous regions upon heating, must exist in

> <sup>a</sup> − Hc c

a ) in Eq. (14) is regarded as the heat emitted when only one single crystal lamella

the super-cooled liquid and the crystal at Tc. hx is given by hg\* + Δh (Δh < 0). Here, ΔH (=

without deformation is formed. According to the ATHAS databank29), ΔH is 0.83 kJ/mol, which is close to the value of hc = 0.89 kJ/mol as observed in Fig. 8. The difference in hc 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 disks42). 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* 

Fig. 9. A schematic structure of the crystal lamella after release of the twist deformation from

annealed at Ta = 376.6 K, 416.6 K and 426.6 K for 1 hour, where Tg\* is the secondary Tg and Ta is the annealing temperature. hx was found to decrease with an increasing Ta. The values of hx for "Ta = 416.6 K and 426.6 K"-samples were near hx = 2.3 kJ/mol for the glass with Tg =

the ribbon-like lamella. a, b and c (chain axis) correspond to the three cell axes of the orthorhombic crystal and the dark parts represent the inter-grain aggregates. The cell lengths of a, b and c axes for PE are 0.74 nm, 0.49 nm and 0.25 nm, respectively.

Table 5 shows the values of Tc (≈ Tg\*), Tb\*, Q, ΔH (=Hma − Hc

the crystal lamella. When the secondary Tg is approximated to Tc, hg\* is given by:

a and Hc c

\* for the peak on the lower temperature side is believed to be due to the

\* in the peak on the higher temperature side. The equilibrium

(19)

a), Δh, hg\* and hx for the samples

are the enthalpy per molar structural unit for

rate from Tb\* to Te

Hma − Hc

*distribution function*".

crystallized parts from Tb\* to Te

hg\* ≈ Hc

where hg\* is hg at the secondary Tg, Hc

135 K in Table 2. In the amorphous region of these samples, the ordered parts with the coarse 4/1 helix structure might be formed5). The ordered parts in the inter-grain aggregates, being like the crystals of fold-type with hxi instead of hx, could also have the holes as the pair (see Eq. (28)).


Table 5. The values of Tc (≈ Tg\*), Tb\*, Q, ΔH, Δh, h g\* and hx for PE films annealed at 376.6 K, 416.6 K and 426.6 K for 1 hour.

#### **5.3 "Crystal / anti-crystal hole" pairs from secondary glass**

The crystal length, ζ, as a function of the melting temperature, Tm, is according to Gibbs-Thomson given by:

$$\mathcal{L} = \{ \mathbf{T\_m}^{\circ} \,/\langle \mathbf{T\_m}^{\circ} - \mathbf{T\_m} \rangle \} \{ \mathbf{2} \sigma\_e / \langle \mu \mathbf{h}\_u \rangle \} \tag{20}$$

where σe is the end-surface free energy per unit area for crystals, μ is the conversion coefficient. For PE, μ = (106/14) mol/m3, hu = 4.11 kJ/mol 36) and Tm<sup>∞</sup> = 415 K18, 36). σe is given as25, 43):

$$\mathbf{w}\_c = \mu \mathbf{h}\_w \mathbf{c}^\* \left[ \left\{ \mathbf{R} \mathbf{T}\_\mathbf{m} \mathbf{^2} + (\mathbf{H}\_\mathbf{x} - \mathbf{h}\_\mathbf{x}) (\mathbf{T}\_\mathbf{m} \, \prescript{\circ}{} - \mathbf{T}\_\mathbf{m}) \right\} / \left\{ \mathbf{2} (\mathbf{H}\_\mathbf{x} - \mathbf{h}\_\mathbf{x}) \mathbf{T}\_\mathbf{m} \, \prescript{\circ}{} \right\} \right] \tag{21}$$

with Hx = 2hu − Qm, 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 σe for "Ta = 376.6 K, 416.6 K and 426.6 K" samples, together with the values of Tp, Te \*, hx, Qm, ΔQm (= Qm − hc) and ΔQ (= Q − Qm) used in the calculation of σe, where Tp 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. ΔQm is given by:

$$\Delta \mathbf{Q}\_{\rm m} = [\_{\rm Tb^\*} \text{Tr}^\*(\mathbf{d} \mathbf{Q}/\mathbf{d} \mathbf{t}) (1/\alpha\_\*) \text{d} \mathbf{T} \tag{22}$$

where αs (= dT/dt) is the heating rate. Te \* was derived from Eq. (22) using ΔQm in Table 6. The fact that Te \* was almost equal to that by observation, as shown in Table 6, supported that hc was due only to the formation of the crystal lamella, thus giving rise to the melting peak on the higher temperature side. Te \* is also the end temperature of melting for ordered parts of fold-type in the inter-grain aggregates. Upon heating over Te \*, the flow parts in the amorphous regions should start to participate directly in the melting of crystals. With an increasing Ta, σe and hx decreased, whereas Qm, ΔQm and ΔQ increased. ΔQ should influence hx. For all samples, the value of σe at Tp was 1.3 times larger than that at Tm<sup>∞</sup> (Qm = 0). The experimental values of σe from Eq. (20), 30 ∼ 90 mJ/m2, 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 Tm23, 44).

Thermodynamics of Enthalpy Relaxation and Hole Formation of Polymer Glasses 177

where hu, σe 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), Tmh approaches Tm<sup>∞</sup> with an increasing ζ. However, the interface between the anti-crystal holes and the ordered parts, which satisfy fx = 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

 hx − hu = Tmh(su − sx) = σe/(μζ) (28) where su is the entropy per molar unit for the anti-crystal holes. As well as hu, 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 fx = 0 of class B and further, Tmh 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

 Tm = Tm<sup>∞</sup>{1 − 2σe/(μζhu)} (29) Eq. (29) is the same as Eq. (20). In Eq. (29), Tm is Tm<sup>∞</sup> at ζ = ∞ and from Eqs. (27) and (29), Tm<sup>∞</sup> = (Tmh + Tm)/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 Tmh should be depressed down to Tm 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 Tg\*. It is thus presumed that over Tg\*, 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 Tm of the crystals from the secondary glass, being

generation of crystallization or melting should be 1/2, according to the uncertainty

fx' fx fu Class fx' = 0 fx = fu fu = fx A fx' > 0 fx = 0 fu = −fx' B fx = fx' fu = 0 C fx = 2fu fu = fx/2 = fx' D fx' > 0 fx = 0 fu = fx' X Table 7. Relations of equilibrium (A ∼ D) and non-equilibrium (X) in fx and fu at fx' ≥ 0 for

\* as a function of ζ in Eq. (29). The time

\* was 0.73 s, in which the probability of observing a spontaneous

opposed to Eq. (27), from fu = fx' of class D, the Tm of the crystals is derived:

following relation of energy balance is derived:

equal to Tmh, was found to change from Tb\* to Te

spent from Tb\* to Te

crystalline polymers46).

principle.


The values in the parentheses of T e\* and σ e columns are the apparent Te \* by observation and σ e at Tm<sup>∞</sup> = 415 K (Qm = 0), respectively. Tp is corrected by 0.1 K to the lower temperature side.

Table 6. The values of Tp, Te \*, hx, Qm, ΔQm, ΔQ and σe for PE films annealed at 376.6 K, 416.6 K and 426.6 K for 1 hour.

On the other hand, according to Flory's theory45) on the melting of the fringe-type crystals with a finite ζ, σe at λ and (dfu/dζ)λ= 0 is given by:

$$\sigma\_{\rm e} = \mu (\text{RT}\zeta / 2) [1/(\chi - \zeta + 1) + (1/\zeta)\ln[(\chi - \zeta + 1)/\chi]] \tag{23}$$

where λ is the amorphous fraction, fu is the free energy per molar structural unit for the crystals and x is the degree of polymerization. In this context:

$$2\sigma\_{\rm c}/\zeta = \mu(\mathbf{f}\_{\rm x} - \mathbf{f}\_{\rm u})\tag{24}$$

$$\mathbf{f\_x}' = \text{RT}[(1/\zeta)\text{ln}\{(\mathbf{x}-\zeta+1)/\mathbf{x}\} - \text{lnP\_c}] \tag{25}$$

where Pc, given by {(x − ζ + 1)/x}1/<sup>ζ</sup> for fringe-type crystals, is the probability that a sequence occupies the lattice sites of a crystalline sequence. Moreover:

$$\mathbf{f}\_{\mathbf{u}} - (\mathbf{f}\_{\mathbf{x}} - \mathbf{f}\_{\mathbf{x}}') = \mathbf{0} \tag{26}$$

Eq. (23) is obtained when lnPc = −1/(x − ζ + 1). From Eq. (26), the relations are derived based on fu and fx at fx' ≥ 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 fx = fu at fx' = 0 shows the dynamic equilibrium relation between the ordered parts and the crystal parts of equivalent fringetypes, leading to σe = 0 in Eq. (24) as expected for highly oriented fibers. For class B, fu = −fx' from Eq. (26) with fx = 0 refers to the anti-crystal holes and fx = 0 is assigned to the ordered parts of ζ = ∞. For class C, fx = fx' from Eq. (26) with fu = 0 is assigned to the ordered parts of ζ ≠ ∞ (i.e., a kebab structure) and fu = 0 to the crystals of ζ = ∞ (i.e., a shish structure). Class D of fu (= fx') = fx/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 fu = −fx' for class B, the temperature at which the anti-crystal holes disappear (melt), i.e., Tmh, is given by:

$$\mathbf{T\_m}^{\mu} = \mathbf{T\_m}^{\prime\prime} \{ \mathbf{1} + \mathbf{2} \mathbf{c}\_{\mathbf{c}} / (\mu \zeta \mathbf{h}\_{\mathbf{u}}) \} \tag{27}$$

and σ e columns are the apparent Te

On the other hand, according to Flory's theory45) on the melting of the fringe-type crystals

is the amorphous fraction, fu is the free energy per molar structural unit for the

2σe/ζ = μ(fx − fu) (24)

 fx' = RT[(1/ζ)ln{(x − ζ + 1)/x} − lnPc] (25) where Pc, given by {(x − ζ + 1)/x}1/<sup>ζ</sup> for fringe-type crystals, is the probability that a

 fu − (fx − fx') = 0 (26) Eq. (23) is obtained when lnPc = −1/(x − ζ + 1). From Eq. (26), the relations are derived based on fu and fx at fx' ≥ 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 fx = fu at fx' = 0 shows the dynamic equilibrium relation between the ordered parts and the crystal parts of equivalent fringetypes, leading to σe = 0 in Eq. (24) as expected for highly oriented fibers. For class B, fu = −fx' from Eq. (26) with fx = 0 refers to the anti-crystal holes and fx = 0 is assigned to the ordered parts of ζ = ∞. For class C, fx = fx' from Eq. (26) with fu = 0 is assigned to the ordered parts of ζ ≠ ∞ (i.e., a kebab structure) and fu = 0 to the crystals of ζ = ∞ (i.e., a shish structure). Class D of fu (= fx') = fx/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

When fu = −fx' for class B, the temperature at which the anti-crystal holes disappear (melt),

Tmh = Tm<sup>∞</sup>{1 + 2σe/(μζhu)} (27)

= 0 is given by:

Qm kJ/mol

(397.7) 3.03 1.37 0.48 0.42 16.3

(399.0) 2.44 1.42 0.53 0.96 14.3

(389.7) 1.97 1.52 0.63 1.33 12.9

\*, hx, Qm, ΔQm, ΔQ and σe for PE films annealed at 376.6 K, 416.6

σe = μ(RTζ/2)[1/(x − ζ + 1) + (1/ζ)ln{(x − ζ + 1)/x}] (23)

ΔQm kJ/mol

ΔQ kJ/mol

\* by observation and σ e at

σe at Tp mJ/m2

(12.4)

(11.1)

(10.1)

hx kJ/mol

Tm<sup>∞</sup> = 415 K (Qm = 0), respectively. Tp is corrected by 0.1 K to the lower temperature side.

Sample Ta/K

Tp K

376.6 401.2 399.2

416.6 401.8 400.1

426.6 401.9 400.7

The values in the parentheses of T e\*

λ

and (dfu/dζ)

crystals and x is the degree of polymerization. In this context:

λ

sequence occupies the lattice sites of a crystalline sequence. Moreover:

the holes of class B can not be replaced by the crystals with ζ ≠ ∞.

Table 6. The values of Tp, Te

K and 426.6 K for 1 hour.

with a finite ζ, σe at

i.e., Tmh, is given by:

where λ Te \* K

where hu, σe 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), Tmh approaches Tm<sup>∞</sup> with an increasing ζ. However, the interface between the anti-crystal holes and the ordered parts, which satisfy fx = 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:

$$\mathbf{h\_{x} - h\_{u} = T\_{m}} h (\mathbf{s\_{u} - s\_{x}}) = \mathbf{c\_{e}} / (\mu \mathbf{f\_{z}}) \tag{28}$$

where su is the entropy per molar unit for the anti-crystal holes. As well as hu, 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 fx = 0 of class B and further, Tmh 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 fu = fx' of class D, the Tm of the crystals is derived:

$$\mathbf{T\_m} = \mathbf{T\_m}^\* \{ \mathbf{1} - \mathfrak{D}\mathbf{c} / (\mathfrak{h}\mathcal{L}\mathfrak{h}\_u) \}\tag{29}$$

Eq. (29) is the same as Eq. (20). In Eq. (29), Tm is Tm<sup>∞</sup> at ζ = ∞ and from Eqs. (27) and (29), Tm<sup>∞</sup> = (Tmh + Tm)/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 Tmh should be depressed down to Tm 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 Tg\*. It is thus presumed that over Tg\*, 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 Tm of the crystals from the secondary glass, being equal to Tmh, was found to change from Tb\* to Te \* as a function of ζ in Eq. (29). The time spent from Tb\* to Te \* 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 fx and fu at fx' ≥ 0 for crystalline polymers46).

Thermodynamics of Enthalpy Relaxation and Hole Formation of Polymer Glasses 179

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

ph(Te

On the other hand, the heat change per molar structural unit, ΔUm, due to the melting of the

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 Tg\*. From ΔUh =

\* is given by:

ph(Te

for PE films annealed at 376.6 K, 416.6 K and 426.6 K for 1 hour. From ΔQm/Qm and Γ, 35 ∼ 41 % of the lamella was constituted of the inter-grain aggregates and approximately 79 ∼ 96 % of this was the glass at T ≤ Tg\*. The difference in ΔQm and ΔUm was believed to represent the irreversible heat change due to the melting of the ordered parts of fold-type, since the folded segments have the excess defect energy. The values of ΔUh were almost same as Δh (= 0.5 kJ/mol) for the glasses with Tg = 135 K and 237 K in Table 2, giving the latent heat

> ΔQm kJ/mol

376.6 6.8 1.37 0.48 0.46 0.35 0.95 416.6 7.6 1.42 0.53 0.51 0.37 0.96 426.6 7.4 1.52 0.63 0.50 0.41 0.79

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. (20) needs the values of σe and Tm<sup>∞</sup>. The σe can be evaluated by Eq. (21) using only the DSC data. On Tm<sup>∞</sup>, the inherent temperature of the crystal form should be selected. The F(ζ) is defined

F(ζ) = (δQm/Qm)/ ζ = nζ/{Nc(Te − Tb)} (33)

\* is given

\* − Tb\*) (30)

ΔUm = ΓΔQm (31)

\* − Tb\*)/ΔQm (32)

kJ/mol ΔQm/Qm <sup>Γ</sup>

\* − Tb\*, Qm, ΔQm, ΔUh (= ΔUm), ΔQm/Qm and Γ (= ΔUh/ΔQm)

ΔUh

\* − Tb\*, Qm, ΔQm, ΔUh (= ΔUm), ΔQm/Qm and Γ (=

energy loss of the anti-crystal holes, ΔUh, due to the cavity radiation from Tb\* to Te

ΔUh = 3Cv

Γ = 3Cv

Qm kJ/mol

ΔUh/ΔQm) for PE films annealed at 376.6 K, 416.6 K and 426.6 K for 1 hour.

**5.4 Fraction of secondary glass** 

newly crystallized parts from Tb\* to Te

\*, Γ is given by18):

Table 8 shows the values of Te

required to disappear the holes.

Te \* − Tb\* K

Table 8. The values of Te\* − Tb\* → Te

**5.5** ζ **distribution function, F(ζ)** 

Sample Ta/K

as25):

by18):

ΔUm at Te

Fig. 10 shows the schematic behaviors of sequences and photons on the way of crystallization and melting. The heat of emission, ΔUh, corresponds to ΔQm of the area between the observed melting curve (thin line) and the lower temperature side peak curve from Tb\* to Te \* and the heat of absorption, ΔUm (= ΔUh), corresponds to ΔQm of the under area of the higher temperature side peak curve from Tb\* to Te \* 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 "Ta = 416.6 K"-sample shown in Fig. 8, a derivation of fx' (= −fu) = 0.13 kJ/mol was obtained from Eq. (24) using the values of σe (see Table 6) and ζ (= 3.1 nm) at Tp = 401.8K (see Table 9). Here, fx is rewritten as fxi for the ordered parts in the inter-grain aggregates. Furthermore, under the assumption that the strain energy in the glass should be spent to build the interface between the anti-crystal hole and the ordered part, by substituting hc − ΔH (= 0.06 kJ/mol) for hxi − hu, it was possible to derive su − sxi = 0.15 J/(K mol) at Tc (= 391.5 K), where hxi and sxi are the enthalpy and entropy per molar structural unit for the ordered parts in the inter-grain aggregates. The relation of hxi − hu ≈ hc − ΔH (= 0.06 kJ/mol) could be supported by determining σe/(μζ ) = 0.06 ∼ 0.07 kJ /mol in Eq. (28), which was obtained for all samples using ζ and σe at Tp. From fxi = 0, hxi = 4.17 kJ/mol and sxi = 10.7 J/(K mol) at Tc were obtained. Moreover, using hu = 4.11 kJ/mol, su was 10.9 J/(K mol) at Tc.

Fig. 10. The schematic process from the glass (A) to the generation of "crystal / anti-crystal hole" pairs (B → C) by emission of ΔUh and then the disappearance of them (M) by absorption of ΔUm (= ΔUh). The filled circle (•) is the segmental unit, the arrow mark (↔) is the oscillator and the large circle is the photon.

Fig. 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.

#### **5.4 Fraction of secondary glass**

178 Advances in Crystallization Processes

Fig. 10 shows the schematic behaviors of sequences and photons on the way of crystallization and melting. The heat of emission, ΔUh, corresponds to ΔQm of the area between the observed melting curve (thin line) and the lower temperature side peak curve

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 "Ta = 416.6 K"-sample shown in Fig. 8, a derivation of fx' (= −fu) = 0.13 kJ/mol was obtained from Eq. (24) using the values of

ordered parts in the inter-grain aggregates. Furthermore, under the assumption that the strain energy in the glass should be spent to build the interface between the anti-crystal hole and the ordered part, by substituting hc − ΔH (= 0.06 kJ/mol) for hxi − hu, it was possible to

entropy per molar structural unit for the ordered parts in the inter-grain aggregates. The relation of hxi − hu ≈ hc − ΔH (= 0.06 kJ/mol) could be supported by determining σe/(μζ ) = 0.06 ∼ 0.07 kJ /mol in Eq. (28), which was obtained for all samples using ζ and σe at Tp. From

Fig. 10. The schematic process from the glass (A) to the generation of "crystal / anti-crystal

absorption of ΔUm (= ΔUh). The filled circle (•) is the segmental unit, the arrow mark (↔) is

Fig. 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.

hole" pairs (B → C) by emission of ΔUh and then the disappearance of them (M) by

σe (see Table 6) and ζ (= 3.1 nm) at Tp = 401.8K (see Table 9). Here, fx is rewritten as fxi

= 0.15 J/(K mol) at Tc (= 391.5 K), where hxi

area of the higher temperature side peak curve from Tb\* to Te

\* and the heat of absorption, ΔUm (= ΔUh), corresponds to ΔQm of the under

\* in Fig. 8. Fig. 11 shows the

are the enthalpy and

and sxi

= 10.7 J/(K mol) at Tc were obtained. Moreover, using hu =

for the

from Tb\* to Te

derive su − sxi

= 0, hxi

= 4.17 kJ/mol and sxi

the oscillator and the large circle is the photon.

The arrow mark shows the crystallization from A to C.

4.11 kJ/mol, su was 10.9 J/(K mol) at Tc.

fxi

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, ΔUh, due to the cavity radiation from Tb\* to Te \* is given by18):

$$
\Delta \mathbf{U}\_{\rm h} = \mathfrak{JC}\_{\rm v} \mathbb{P}^{\rm h} (\mathbf{T}\_{\rm c}^\* - \mathbf{T}\_{\rm b}^\*) \tag{30}
$$

On the other hand, the heat change per molar structural unit, ΔUm, due to the melting of the newly crystallized parts from Tb\* to Te \* is given by:

$$
\Delta \mathbf{U}\_{\rm m} = \Gamma \Delta \mathbf{Q}\_{\rm m} \tag{31}
$$

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 Tg\*. From ΔUh = ΔUm at Te \*, Γ is given by18):

$$
\Gamma = \Im C\_{\rm v} \mathbb{P}^{\rm h} (\Gamma\_{\rm c}^{\rm \*} - \Gamma\_{\rm b}^{\rm \*}) / \Delta \mathbb{Q}\_{\rm m} \tag{32}
$$

Table 8 shows the values of Te \* − Tb\*, Qm, ΔQm, ΔUh (= ΔUm), ΔQm/Qm and Γ (= ΔUh/ΔQm) for PE films annealed at 376.6 K, 416.6 K and 426.6 K for 1 hour. From ΔQm/Qm and Γ, 35 ∼ 41 % of the lamella was constituted of the inter-grain aggregates and approximately 79 ∼ 96 % of this was the glass at T ≤ Tg\*. The difference in ΔQm and ΔUm was believed to represent the irreversible heat change due to the melting of the ordered parts of fold-type, since the folded segments have the excess defect energy. The values of ΔUh were almost same as Δh (= 0.5 kJ/mol) for the glasses with Tg = 135 K and 237 K in Table 2, giving the latent heat required to disappear the holes.


Table 8. The values of Te\* − Tb\* → Te \* − Tb\*, Qm, ΔQm, ΔUh (= ΔUm), ΔQm/Qm and Γ (= ΔUh/ΔQm) for PE films annealed at 376.6 K, 416.6 K and 426.6 K for 1 hour.

#### **5.5** 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. (20) needs the values of σe and Tm<sup>∞</sup>. The σe can be evaluated by Eq. (21) using only the DSC data. On Tm<sup>∞</sup>, the inherent temperature of the crystal form should be selected. The F(ζ) is defined as25):

$$\mathbf{F}(\mathbf{\zeta}) = (\mathbf{\hat{\mathbf{Q}}} \mathbf{\hat{m}}/\mathbf{Q}\_{\mathbf{m}}) / \ \mathbf{\zeta} = \mathbf{r}\_{\mathbf{\zeta}} / \left\{ \mathbf{N}\_{\mathbf{c}} (\mathbf{T}\_{\mathbf{c}} - \mathbf{T}\_{\mathbf{b}}) \right\} \tag{33}$$

Thermodynamics of Enthalpy Relaxation and Hole Formation of Polymer Glasses 181

376.6 2.1 - 14 2.9 (2.7) 3.3 416.6 1.8 – 810 ∼ 2.7 (2.6) 3.1 426.6 1.7 – 730 ∼ 2.5 (2.2) 2.8

\* . Table 9. The values of ζ-range, ζc and ζp in F(ζ) for PE films annealed at 376.6 K, 416.6 K and

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

In the last stage, a single crystal-like image was drawn from F(ζ). Rewriting Eq. (33), nζ is

nζ = F(ζ)Nc(Te − Tb) (35)

Accordingly, the number of the crystal sequences from ∼ζ (> ζ) to ζx, ΔN, is given by:

ζ

<sup>ζ</sup>xF(ζ)dζ − ζ<sup>n</sup>

ζ

F(ζ)dζ (36)

F(ζ)dζ) (37)

The number of the crystal sequences from ζn to ζ, Nζ, is as follows:

ΔN = Nc(Te − Tb)(ζ<sup>n</sup>

Nζ = Nc(Te − Tb) ζ<sup>n</sup>

ζc nm

ζp nm

ζ-range nm

The values in the parentheses are ζ at the apparent Te

crystal melting from ζn or ζc to ζ = 0.

given by:

Sample Ta/K

426.6 K for 1 hour.

where δQm (=ζnζQm/{Nc(Te − Tb)}) is the heat change per molar structural unit per K, n<sup>ζ</sup> is the number of crystal sequences with ζ and Nc is the number of structural units of crystals melted in the temperature range from Tb (= Tb\* here) to Te. δQm/Qm is given by:

$$\delta \mathbf{Q}\_{\rm m}/\mathbf{Q}\_{\rm m} = (\mathbf{d} \mathbf{Q}/\mathbf{d}t)/[\mathbf{r}^{\rm I} \mathbf{r}^{\rm I} (\mathbf{d} \mathbf{Q}/\mathbf{d}t) \mathbf{d}^{\rm I} \tag{34}$$

where dQ/dt is the heat flow rate of the melting curve. Fig. 12 shows F(ζ) of each melting curve from Tb\* for "Ta = 376.6 K, 416.6 K and 426.6 K"-samples. Table 9 lists the values of ζrange, ζc and ζp of F(ζ) curve for each sample, where ζc is ζ at Te \* and ζp is ζ at Tp. For "Ta = 376.6 K"-sample, ζp was slightly larger than for other samples. The small value of ζ (< ζc) 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 "Ta = 416.6 K and 426.6 K"-samples might be related to the long period change of lamellae at the higher temperature upon heating36). Whereas, the very narrow ζ-range for "Ta = 376.6 K"-sample might be due to the effective annealing.

Fig. 12. F(ζ) for PE films annealed at 376.6 K (right), 416.6 K (middle) and 426.6 K (left) for 1 hour. ×; F(ζc).


The values in the parentheses are ζ at the apparent Te \* .

180 Advances in Crystallization Processes

where δQm (=ζnζQm/{Nc(Te − Tb)}) is the heat change per molar structural unit per K, n<sup>ζ</sup> is the number of crystal sequences with ζ and Nc is the number of structural units of crystals

where dQ/dt is the heat flow rate of the melting curve. Fig. 12 shows F(ζ) of each melting curve from Tb\* for "Ta = 376.6 K, 416.6 K and 426.6 K"-samples. Table 9 lists the values of ζ-

376.6 K"-sample, ζp was slightly larger than for other samples. The small value of ζ (< ζc) 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 "Ta = 416.6 K and 426.6 K"-samples might be related to the long period change of lamellae at the higher temperature upon heating36). Whereas, the very narrow ζ-range for "Ta = 376.6 K"-sample

Fig. 12. F(ζ) for PE films annealed at 376.6 K (right), 416.6 K (middle) and 426.6 K (left) for

δQm/Qm = (dQ/dt)/TbTe(dQ/dt)dT (34)

\* and ζp is ζ at Tp. For "Ta =

melted in the temperature range from Tb (= Tb\* here) to Te. δQm/Qm is given by:

range, ζc and ζp of F(ζ) curve for each sample, where ζc is ζ at Te

might be due to the effective annealing.

1 hour. ×; F(ζc).

Table 9. The values of ζ-range, ζc and ζp 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 ζn or ζc to ζ = 0.

In the last stage, a single crystal-like image was drawn from F(ζ). Rewriting Eq. (33), nζ is given by:

$$\mathbf{n}\_{\zeta} = \mathbf{F}(\zeta)\mathbf{N}\_{\xi}(\mathbf{T}\_{\mathbf{e}} - \mathbf{T}\_{\mathbf{b}}) \tag{35}$$

The number of the crystal sequences from ζn to ζ, Nζ, is as follows:

$$\mathbf{N}\_{\zeta} = \mathbf{N}\_{\mathbf{f}} (\mathbf{T}\_{\mathbf{c}} - \mathbf{T}\_{\mathbf{b}}) \int\_{\tilde{\zeta}^{\mathbf{n}}} \mathbf{F}(\zeta) d\zeta \tag{36}$$

Accordingly, the number of the crystal sequences from ∼ζ (> ζ) to ζx, ΔN, is given by:

$$
\Delta\Delta\mathbf{N} = \mathbf{N}\_{\mathbf{c}}(\mathbf{T}\_{\mathbf{c}} - \mathbf{T}\_{\mathbf{b}}) \left( \mathbb{J}^{n}{}\_{\mathbf{c}\mathbf{F}} \mathbf{F}(\mathbf{\zeta})\mathbf{d}\mathbf{\zeta} - \mathbb{J}^{\zeta\_{\mathbf{c}}n}{}\_{\mathbf{c}} \mathbf{F}(\mathbf{\zeta})\mathbf{d}\mathbf{\zeta} \right) \tag{37}
$$

Thermodynamics of Enthalpy Relaxation and Hole Formation of Polymer Glasses 183

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where ζx and ζn 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:

$$R\_n = (\Delta \mathbf{N} / \pi)^{1/2} \tag{38}$$

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 ∼ ±ζn/2, *R* at ζn/2 is represented by a solid line, which leads to the supposition of a melting process from the end-surfaces of the crystal with ζn at Tb\*. The horizontal line of *R* at ζc/2 depicts the same imaginable melting process of the crystal with ζc at Te \*. 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 "Ta = 376.6 K"-sample (thick line) was very similar to the electron microscope (EM) image of self-seeded PE crystals36).

#### **6. 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 Tg of the first order hole phase transition.

### **7. Acknowledgements**

The author wishes to thank Professor em. B. Wunderlich of the University of Tennessee and Rensseler Polytechnic Institute for the encouragement during this work.

#### **8. References**


where ζx and ζn 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

 *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 ∼ ±ζn/2, *R* at ζn/2 is represented by a solid line, which leads to the supposition of a melting process from the end-surfaces of the crystal with ζn at Tb\*. The horizontal line of *R* at ζc/2 depicts the same

between both types of crystals should be available in order to evaluate the annealing effects. The single crystal image from *R* and L for "Ta = 376.6 K"-sample (thick line) was very

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

The author wishes to thank Professor em. B. Wunderlich of the University of Tennessee and

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similar to the electron microscope (EM) image of self-seeded PE crystals36).

\*. The distinct difference of *R* or L

direction, *R*n, as a function of ζ is given by:

**6. Conclusions** 

**7. Acknowledgements** 

[1] C. A. Angell, Science, 267, 1924(1995). [2] F. H. Stillinger, Science, 267, 1935(1995).

**8. References** 

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Rensseler Polytechnic Institute for the encouragement during this work.


**Crystallization Behavior and** 

Lai-Chang Zhang

*Australia* 

*The University of Western Australia* 

**Control of Amorphous Alloys** 

Since the discovery in 1960 by Duwez (Klement et al., 1960), considerable effort has been devoted to form amorphous (or glassy) alloys either by rapid solidification techniques or by solid-state amorphization techniques (Inoue, 2000; Johnson, 1999; Suryanarayana & Inoue, 2011; Wang et al., 2004). However, the geometry of the amorphous samples has long time been limited in the form of ribbons or wires. The first "bulk" amorphous alloys, arbitrarily defined as the amorphous alloys with a dimension no less than 1 mm in all directions, was discovered by Chen and Turnbull (Chen & Turnbull, 1969) in ternary Pd-Cu-Si alloys. These ternary bulk glass-forming alloys have a critical cooling rate of about 102 K s-1 and can be obtained in amorphous state with a thickness up to 1 mm and more. Since then, especially after the presence of new bulk metallic glasses (BMGs) in La55Al25Ni20 (Inoue et al., 1989) and Zr41.2Ti13.8Cu12.5Ni10.0Be22.5 (Peker & Johnson, 1993), multicomponent BMGs, which could be prepared by direct casting from molten liquid at low cooling rates, have been drawing increasing attention in the scientific community. A great deal of effort has been devoted to developing and characterizing BMGs with a section thickness or diameter of a few millimetres to a few centimetres (Suryanarayana & Inoue, 2011). A large variety of multicomponent BMGs in a number of alloy systems, such as Pd-, Zr-, Mg-, Ln-, Ti-, Fe-, and Ni-based BMGs, have been developed via direct casting method with low cooling rates of the order of 1 – 102 K s-1 (Inoue, 2000; Johnson, 1999; Suryanarayana & Inoue, 2011; Wang, et al., 2004). In this method, the alloy compositions were carefully designed to have large glassforming ability (GFA) so that "bulk" amorphous alloys can be formed at a low cooling rate to frustrate crystallization from molted liquid state. A number of parameters/indicators have been proposed to evaluate the GFA of multicomponent alloy systems to search for BMGs with larger dimensions (Suryanarayana & Inoue, 2011). So far, the "record" size of the BMGs is 72 mm diameter for a Pd40Cu30Ni10P20 bulk metallic glass (Inoue et al., 1997). The discovery of amorphous alloys has attracted widespread research interests because of their technological promise for practical applications and scientific importance in understanding

Arising from their disordered atomic structure and unique glass-to-supercooled liquid transition, amorphous alloys represent a new class of structural and functional materials with excellent properties (Eckert et al., 2007; Inoue, 2000; Johnson, 1999; Suryanarayana & Inoue, 2011; Wang, 2009; Xu et al., 2010), e.g. high strength about 2–3 times of their

**1. Introduction** 

glass formation and glass phenomena.

