The values of T_{g}, ∆C_{p}^{exp}, ∆C_{p} (= 2C_{v}^{ph}), ∆C_{p}/∆C_{p}^{exp}, and h^{h}/h_{x} for iPS, PET, and iPP.

## 1. Introduction

The glass transition for polymers has been investigated for a long time as the mysterious physical phenomena of solid or liquid phases from the initial studies on the equation of state in pressure (P), volume (V), and temperature (T) to the recent thermal analyses with the temperature modulated differential scanning calorimeter (TMDSC) [1 – 9]. Polystyrene (PS) is one of polymers taking a leading part in the studies on the glass transition of polymers, so far showing the heat capacity jump of 28 ∼ 31 J/(K mol) at the glass transition. The temperature modulation of TMDSC emerged the latent heat capacity jump at the glass transition temperature (T_{g}), confirming the heat capacity jump data on the basis of PVT relations for PS. Also for poly(ethylene terephthalate) (PET), the abrupt heat capacity jump at T_{g} was observed on TMDSC curves, being not found with the standard DSC [8]. Recently, in the advances of the studies on the photonic contribution to the glass transition of polymers, the mysterious glass transition has been reasonably understood as the quantum phenomena [10 – 16]. For frozen polymer glasses, the heat capacity jump at T_{g} should start from the first order hole phase transition and then the glass parts should be unfrozen accompanying with the enthalpy and entropy jumps [10]. The holes are generally neighboring with the ordered parts, which are formed as pairs during the enthalpy relaxation at temperatures below T_{g}. First in this chapter, for isotactic PS (iPS) and PET, the heat capacity jump at the glass transition was discussed as the discontinuous change of energy in quantum state of the photon holes, followed by unfreezing of the glass parts. IPS and PET have the benzene rings being able to cause the resonance by neighboring in the side groups or the skeletal chains, respectively. Further, the details on the heat capacity jump found for iPS were also investigated for isotactic polypropylene (iPP) with methyl groups of the same 3/1 helix structure [16]. The resonance suggests the presence of remarkable photons in holes. The dimension of them is characterized by the geometric molecular structure, e.g., the distance between reflectors such as benzene rings, affecting to the amplitude as a wave. While for the photon holes, the constant volume heat capacity could be defined as the differential coefficient of the internal energy of holes [10 – 16]. So for iPS, PET, and iPP, in order to confirm the identity in two heat capacities of ordered parts and holes in pairs, the heat capacity jump data per molar structural unit at the glass transition were compared with that per molar photon for the holes in ordered part / hole pairs. Here it should be noted that in the ordered part / hole pairs, the molar photon used for photon holes is equivalent to the molar structural unit for ordered parts numerically.

For iPS, PET, and iPP, surely the heat capacity jump at the glass transition was due to the discontinuous change of energy in quantum state of the resonant photon holes between neighboring benzene rings, but methyl groups for iPP, followed by unfreezing of the glass parts [14 – 16]. For iPS, the substance of the helix–coil transition with the enthalpy of 16.1 kJ/mol, but being smaller than the glass transition enthalpy of 18.9 kJ/mol, was shown as the ordered part / hole pairs. For PET, the ordered part / hole pairs were like the mesophase crystals with the glassy conformational disorder of ethylene glycol parts. For iPP, the helical sequences with the enthalpy of 7.4 kJ/mol or the nodules of mesophase with the enthalpy of 12.1 kJ/mol, interchanging between ordered parts and crystals automatically, were shown as the ordered part / hole pairs, depending on the presence of the crystallization upon cooling from the melt. According to above results, it could be understood that the glass transition of polymers investigated for a long time was only the collateral unfreezing phenomena of the glass parts starting by the disappearance of ordered part / hole pairs formed during the enthalpy relaxation at temperatures below T_{g}.

On the other hand, for iPS and iPP, from the quantum demand of hole energy at regular temperature intervals of 120 K for iPS and 90 K for iPP, the homogeneous glasses free from ordered part / hole pairs with T_{g}= 240 K and 180 K have been predicted, respectively [15, 16]. T_{g} of them could be understood as the first order glass phase transition temperature of the homogeneous glass [17, 18]. But, as one of other quantum possibilities for these polymers, the liquids with T_{g}= 0 K have also been predicted. In this connection, the equilibrium melting temperature, T_{m}^{∞}= 450 K, for α form crystals of iPP was corresponding to 5 times the interval of 90 K. The sift of melting from α to γ form crystals between two peaks of a DSC double melting peak curve observed upon heating was discussed, relating to the formation and then disappearance of crystal / hole pairs.

## 2. Theoretical treatments and discussion

When the hole energy in the ordered part / hole pairs excited at the glass transition, being in equilibrium with the flow parts, is given by h^{h} (= 3C_{v}^{ph}T), the heat capacity per molar photon for holes, C_{p}^{h*} (= C_{p}^{flow}), is given by [10 – 16] (see section **4**):

where C_{p}^{flow} is the heat capacity per molar structural unit for the flow parts, being equal to C_{p}^{x} of the heat capacity per molar structural unit for the excited ordered parts [19], C_{v}^{ph} (= 2.701R) is the constant volume molar heat capacity for photons [20], R is the gas constant, J_{h} is the number of holes lost by T, and 3 is the degree of freedom for photons. When dJ_{h}/dT = 0 at T_{g} and the end temperature, T_{l}, of the glass transition, C_{p}^{h*} at those temperatures is given by 3C_{v}^{ph}. Thus, the heat capacity jump per molar photon, ∆C_{p}, at the glass transition is given by:

In Eq. (2), ∆C_{p} should be due to the discontinuous energy change from a quantum ground level for photons in the holes, that is, (1/2)*h*ν (= h _{0}^{h}/N_{A}) to (3/2)*h*ν (= 3h_{0}^{h}/N_{A}), where *h* is Planck constant, ν (= c/λ) is the frequency per second, c is the velocity of light, λ is the wavelength, N_{A} is Avogadro constant, and h_{0}^{h} is the zero– point energy per molar photon, which also is used as the energy unit bellow. The holes in the excited ordered part / hole pairs should be in dynamical equilibrium with the spatial tubes between a chain and neighboring chains in the flow parts [21]. At the glass transition, the sigmoidal mean heat capacity curve of C_{p} as shown in Fig. 1 is observed generally. Even in this case, the C_{p} for ordered parts should be equal to the C_{p}^{h} of the mean heat capacity for the holes in ordered part / hole pairs at the glass transition:

where α and 1 – α are the fractions of ordered part / hole pairs with the respective holes of C_{v}^{ph} and C_{p}^{ph} (= 3C_{v}^{ph}), and C_{p}^{ph} is the adiabatic molar heat capacity for photons. On the other hand, the C_{p} for ordered parts in pairs could be divided into two components [22]:

under

where ∆C_{p}^{x} (= C_{p}^{h*} – 3C_{v}^{ph}) is the relative component heat capacity per molar structural unit for the excited ordered parts and C_{p}^{r} is the heat capacity change per molar structural unit due to the crystallization followed by the melting. At the glass transition, ∆C_{p}^{x} shows a peak against T, reflecting the size distribution of ordered parts. Fig. 1 shows the representative C_{p} curve composed of ∆C_{p}^{x} and C_{p}^{r} at the glass transition for polymers.

Thus, for iPS, PET, and iPP, ∆C_{p} (= 2C_{v}^{ph}) per molar photon (mol*) was compared with the reference value of heat capacity jump, ∆C_{p}^{exp}, per molar structural unit (mol) [8]. The results deviated from ∆C_{p}/∆C_{p}^{exp}= 1. Table 1 shows the comparison of ∆C_{p} (= 2C_{v}^{ph}) with ∆C_{p}^{exp} for these polymers, together with h^{h}/h_{x} at T_{g}, where h_{x} is the enthalpy per molar structural unit for ordered parts [14 – 16].

Polymer | TgK | ∆Cp exp J/(K mol) | ∆CpJ/(K mol*) | ∆Cp/∆Cp exp | hh/hx |

iPS | 359^{*1} | 30.8 | 44.9 | 1.5 | 1.5 or 1.0 |

PET | 342 | 77.8 (80.4, 46.5 ^{*2}) | 44.9 | 0.6 (0.6) | 1.0 |

iPP | 270 | 19.2 | 44.9 | 2.3 | 1.5 or 2.5 |

However, the values of ∆C_{p}/∆C_{p}^{exp} were correlated to h^{h}/h_{x} of the number of structural units holding one photon potentially (described below). h_{x} at T_{g} is given by [10, 23]:

where h_{g} {= RT_{g}^{2}(∂lnv_{f}/∂T)_{p}} is the glass transition enthalpy per molar structural unit due to the discontinuous free volume change of v^{*} from v_{f}= v_{0} to v_{0}+ v^{*} at T_{g}, v_{f} and v_{0} are the free volume and the core free volume per molar structural unit. h_{g} is given approximately by three expressions; (1) RT_{g}^{2}/c_{2} or φ_{g}E_{a} (in WLF equation [24], φ_{g} {= 1/(2.303c_{1})} is the fraction of the core free volume in glasses, c_{1} and c_{2} are constant, and E_{a} is the activation energy), (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}. For PET and iPP, the additional heat per molar structural unit, ∆h, needed to melt all ordered parts by T_{l} in Fig. 1 is given by [10, 23]:

where ∆H= H_{m}^{a} ‒ H_{c}^{a}, H_{m}^{a} is the enthalpy per molar structural unit for the liquid at 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. While, in the case of h_{x}^{conf} ≠ h_{g}^{conf}= 0 at T_{g}, ∆h is derived as [10]:

with

where h_{x}^{conf} is the conformational enthalpy per molar structural unit for ordered parts, s_{g}^{conf} is the conformational entropy per molar structural unit at T_{g}, 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 depending on the temperature, respectively, and x is the degree of polymerization. For PET and iPP, the values of ∆h from Eq. (7) were a little smaller than those from Eq. (6), respectively. In the case of h_{x}^{conf}= h_{g}^{conf}, ∆h= (RT_{g}lnZ_{t})/x was derived, applying to nylon 6 [10].

### 2.1. Isotactic polystyrene

From h^{h}= (3/2)N_{A}*h*ν and h^{h} (= 3C_{v}T_{g})= 24.2 kJ/mol* at T_{g}= 359 K, the wavenumber of 1/λ= 1350 cm^{– 1} was derived for a photon in holes [10]. This agreed nearly with the conformation sensitive band of 1365 cm^{– 1} assigned to benzene rings [25, 26]. Further from the assigned relation of one photon to one structural unit numerically, the unity of h^{h}/h_{x}= ∆C_{p}/∆C_{p}^{exp}= 1 at the glass transition was expected [10], applying to the ordered sequences of– *TTTT*– (see Fig. 2), where *T* is the trans isomer. However, ∆C_{p}/∆C_{p}^{exp} was 1.5 (see Table 1), where T_{l} in ∆C_{p}^{exp} is ∼381 K [8]. Accordingly, the number of structural units holding one photon potentially in holes, n (= h^{h}/h_{x}), is defined here necessarily. Fig. 2 shows the sequence models of– *TTTT*– (n= 1) and unstable– *TCTC*– (n = 2) for iPS, where *C* is the cis isomer.

According to h^{h}/h_{x}= 1.5, h_{x} (= 2h_{0}^{h}) = 16.1 kJ/mol was derived. This corresponded to ∆C_{p}T_{g}= 16.1 kJ/mol* of the C_{p} jump energy for holes at T_{g}. While h_{x} can be also derived from theδ solubility parameter, δ {= (h_{0}/V)^{1/2}}, where h_{0} is the latent cohesive energy per molar structural unit, corresponding to the heat of vaporization or sublimation and V is the molar volume of structural units. The relations among h_{0}, h_{u}, h_{x}, and h_{g} at temperatures before and after T_{g} are given by [19, 27]:

where h_{u} is the heat of fusion per molar structural unit. For crystalline polymers, h_{u} is contained in Eqs. (8) and (9). For iPS, h_{0}= 35.0 kJ/mol was derived from δ = 9.16 (cal/cm^{3})^{1/2} of the mean of 12 experimental values (≥ 9.0 (cal/cm^{3})^{1/2}) [28], and the value of h_{x} from Eq. (8), 16.1 kJ/mol, agreed with that from h^{h}/h_{x}= 1.5 perfectly. However it was smaller than h_{g} (= RT_{g}^{2}/c_{2}) = 18.9 kJ/mol. The difference in h_{g} and h_{x}, 2.8 kJ/mol, agreed with the cohesive energy of methylene residues of h_{m}^{int}= 2.8 kJ/mol [29], suggesting that the ordered part / hole pairs might fill softly the parts in glassy bulks.

TgK | h0kJ/mol | hgkJ/mol | ∆hkJ/mol | hxkJ/mol | hhkJ/mol* | hh/hx |

359^{*1} (360) | 43.1^{*2} | 18.9 | 5.3 | 24.2^{*4} | 24.2 | 1 |

359^{*1} (360) | 35.0^{*2,*3} | 18.9 | –2.8 | 16.1 | 24.2 | 1.5 |

240 | 1.7^{*2} | 1.7^{*5} | –1.7 | 0 | 0 (16.1) | --- |

Table 2 shows the values of T_{g}, h_{0}, h_{g}, ∆h, h_{x}, and h^{h} at h^{h}/h_{x}= 1 and 1.5 for iPS. In the 4th line, h^{h}= h_{x}= 0 and h_{g}= 1.7 kJ/mol at 240 K are shown (discussed below). The relation of h^{h}= h_{x}= 0 is brought by the energy radiation of 2h_{0}^{h} (= 16.1 kJ/mol*) at T_{g} and the energy loss of h_{0}^{h} (= 8.1 kJ/mol*) upon cooling from T_{g} obeying:

In Eq. (10), the specific temperature of 240 K at h^{h}= 0 agreed with the hole temperature at h^{h} (= 3C_{v}^{ph}T)= 16.1 kJ/mol*. In the glasses upon heating from 0 K, the generation of ordered part / hole pairs at 240 K and succeedingly, the instant radiation of the hole energy of 16.1 kJ/mol* should bring the same state as that of h^{h}= 0 at 240 K upon cooling, suggesting T_{g}= 240 K for the homogeneous glass free from the ordered part / hole pairs. Altering 3C_{v}^{ph} (= C_{p}^{ph}) in Eq. (10) to C_{v}^{ph}, the temperature at h^{h}= 0 was T_{g}= 0 K. While for the glasses including the ordered part / hole pairs, T_{g}= 360 K (see Table 2) was expected from the quantum demand of hole energy at regular temperature intervals of 120 K.

For iPS, the rotational isomeric 2–state (RIS) model of *T* (trans) and *G* (gauche) or *G’* (gauche’) is known well [31]. Fig. 3 depicts the helix structure of – *TGTGTG* – (n = 3) for iPS. From h^{h}/h_{x} (= n) = 1.5, the intermediate sequences (n = 1.5) between the right or left handed helical sequence (n = 3) and the aperiodic sequence (n = 0), displaying the helix–coil transition, were predicted as the sequences of ordered parts [14, 15]. The frequency of occurrence, Γ, of the helix–coil transition should be given approximately by h^{h}/h_{0}^{h} in Eq. (10) with h^{h}/h_{0}^{h}= 0 at T = 240 K. The ∆C_{p} (= 3C_{v}^{ph}) at T_{g}= 240 K in a glassy state of Γ = 0 was 67.4 J/(K mol*) [15].

Fig. 4 shows the relation between f (= f^{conf} or f^{conf}+ 0.81 kJ/mol) and T calculated for the RIS model chains (x = 100) with the normalized statistical weight of η = 1 applied to *TG* isomer of iPS, where f^{conf} is the conformational free energy per molar structural unit, being minus and decreasing with an increase in temperature. Adding the value of – f^{conf} (= 0.81 kJ/mol) at 240 K to all values of the original thin line, the thick line of f^{conf}+ 0.81 kJ/mol is depicted. In the case of – f^{conf}= h^{int}, i.e., (h^{conf}+ h^{int}) – Ts^{conf}= 0 at 240 K, the sum of – f^{conf} (= 0.81 kJ/mol) and h^{conf} (= 0.89 kJ/mol), 1.70 kJ/mol, should be the ultimate h_{g} at the first order glass phase transition for the homogeneous glass free from the ordered part / hole pairs (see Table 2), where h^{int} is the cohesive enthalpy per molar structural unit, and h^{conf} and s^{conf} are the conformational enthalpy and entropy per molar structural unit.

Fig. 5 shows the schematic chart of the instantaneous state changes at T_{g} (= 360 K) upon cooling and heating as a working hyposesis. The ordered part / hole pairs formed instantaneously at T_{g} upon cooling have h^{h}= h_{0}^{h} and h_{x}= 2h_{0}^{h}. At T_{g} upon heating, the ordered part / hole pairs are excited by absorbing the photon energy of 2h_{0}^{h} for the holes and adding the energy of h_{0}^{h} for the ordered parts, followed by the absorption of h_{g} for the glass parts. The equilibrium relation at the melting transition among the ordered parts, the holes, and the flow parts is shown by the dashed lines in Fig. 5. In order to melt the excited ordered part / hole pairs perfectly, further the latent heat of h_{0}^{h} is needed at T_{g}.

### 2.2. Poly(ethylene terephthalate)

For PET, h_{x} (= h_{g}+ ∆h) = 24.1 kJ/mol was obtained from Eq. (5), being almost equal to h^{h} (= 3C_{v}^{ph}T_{g}) = 23.0 kJ/mol* at T_{g} (= 342 K) [10, 23]. Thus, h^{h}/h_{x} (= n) = 0.95 was shown experimentally. However as shown in Table 1, ∆C_{p}/∆C_{p}^{exp} was 0.6. Table 3 shows the values of T_{g}, h_{0}, h_{u}, h_{g}, ∆h, h_{x}, h^{h}, and h^{h}/h_{x} for PET. The two values of h_{u}, 23.0 and 28.5 kJ/mol, are assigned to the crystals with the conformational disorder of ethylene glycol parts and the smectic–c crystals with the stretched sequences, respectively [23, 32]. ∆C_{p}/∆C_{p}^{exp}= 0.6 at the glass transition meant that one photon was situated in the neighboring phenylene residues comprising ∼60 % of the structural unit length and 40 % of ∆C_{p}^{exp} was brought by unfreezing of the ethylene glycol parts in a glass state [14]. This was predicted also from the data by TMDSC [8]. From h^{h}= (3/2) N_{A}*h*ν and h^{h} (= 3C_{v}^{ph}T_{g}) = 23.0 kJ/mol* at T_{g}= 342 K, 1/λ = 1290 cm^{–1} was derived, agreeing with 1288 cm^{–1} observed for the un–oriented samples [33].

TgK | h0kJ/mol | hukJ/mol | hgkJ/mol | ∆hkJ/mol | hxkJ/mol | hhkJ/mol* | hh/hx | |

342 | 64.7^{*1} | 23.0 (535 K) | 17.6^{*3} | 6.5 | 24.1 | 23.0 | 0.95 | |

342 | 70.2^{*1}, 68.2^{*2} | 28.5 (549 K) | 17.6^{*3} | 6.5 | 24.1 | 23.0 | 0.95 |

Fig. 6 shows the C_{p} curve converted from DSC curve data for the non–annealed PET film cooled to 323 K (50 °C) at 5 K/min from 573 K (300 °C). T_{g} agreed almost with 342 K of [8]. T_{e} of the end temperature of melting is 535 K (262 °C). The parts of a, b, and a – b of C_{p} jump to the liquid line at T_{g} and T_{e} were correlated to the structural unit length, the lengths of phenylene and glassy ethylene glycol residues, respectively. Fig. 7 shows the parts in the structural unit related to a, b, and a – b.

### 2.3. Isotactic polypropylene

According to the scheme of the formation of ordered part / hole pairs at T_{g} upon cooling (see Fig. 5), for iPP with T_{g}= 270 K, h_{x} (= 2h_{0}^{h}) = 12.1 kJ/mol was derived, being much larger than h_{g} ≈ H_{g}^{a} – H_{g}^{c}= 6.2 kJ/mol [34] and h_{x} (= h_{g}+ ∆h) = 7.4 kJ/mol, where ∆h = ∆H – Q, ∆H = H_{m}^{a} – H_{c}^{a} (see Eqs. (5) and (6)). The used data are as follows: T_{c}= 403.6 K, T_{m}^{∞}= 450 K for α form crystals, ∆H = 4.89 kJ/mol [34], and Q = 3.76 kJ/mol for the sample annealed at 461.0 K for 1 hour [10, 35]. However, h_{0} (= h_{g}+ h_{x}) = 18.3 kJ/mol from Eq. (8) was almost equal to h^{h} (= 3h_{0}^{h}) = 18.2 kJ/mol*, meaning the appearance of frozen glasses with h_{g}= h_{0}^{h}+ 0.1 kJ/mol. For holes with C_{p}^{ph} even upon cooling from T_{g}, Eq. (10) showed the specific temperature of 180 K, at which all ordered part / hole pairs should be disappeared because of h^{h}= 0, corresponding to 240 K for iPS [15]. At temperatures below 180 K, all should be in a state of the homogeneous glass with T_{g}= 180 K. The ∆C_{p} (= 3C_{v}^{ph}) = 67.4 J/(K mol*) at T_{g}= 180 K was the same as that of iPS with T_{g}= 240 K. Fig. 8 shows the relation between f (= f^{conf} or f^{conf}+ 0.1 kJ/mol) and T calculated for RIS model chains (x = 100) with the normalized statistical weight of σ = 1 applied to *TT* isomer of iPP [36, 37]. The f^{conf} at temperatures below and above 180 K is minus and decreases with an increase in temperature. The absolute value of f^{conf}= – 0.102 kJ/mol at 180 K equaled to h_{g} – h_{0}^{h}= 0.1 kJ/mol at 270 K. Adding the cohesive enthalpy of h^{int}= 0.102 kJ/mol to f^{conf} (= – 0.102 kJ/mol), from f = f^{conf}+ h^{int}= 0 and h^{conf}= 0.18 kJ/mol at T_{g}= 180 K, h_{g} (= h^{conf}+ h^{int}) = 0.28 kJ/mol is derived as the first order glass phase transition enthalpy for the homogeneous glass composed of isolated chains, but with the cohesive energy of h^{int} and free from ordered part / hole pairs (see Table 5).

On the other hand, altering 3C_{v}^{ph} (= C_{p}^{ph}) in Eq. (10) to C_{v}^{ph}, the temperature at h^{h}= 0 was T_{g}= 0 K as well as iPS. From s^{conf}= 0.38 J/(K mol) of constant at temperatures below 70 K, the sequence model of – *TGTGTGTTG’TG’TG’T*– in a liquid state was predicted, where *T* is trans, *G* is gauche, *G’* is gauche’ isomer, and *TT* is the trans–trans isomer shifting always to the left or right direction on a sequence [38]. From h_{x}= 12.1 kJ/mol, the nodules of mesophase interchanging between crystals and ordered parts automatically were predicted in the glasses. According to the equilibrium relation in crystals and ordered parts of this class (D in Table 4) given by f_{x}= 2f_{u} [10, 39], 2h_{u} – h_{x}= 2.9 kJ/mol was derived using h_{u}= 7.5 kJ/mol for α form crystals, corresponding to 2T_{m}(s_{u} – s_{x}/2), which was almost equal to h_{m}^{int}= 2.8 kJ/mol of the cohesive energy of methylene residues in the sequences, where f_{x} and s_{x} are the free energy and entropy per molar structural unit for ordered parts, and f_{u} and s_{u} are those for crystals.

According to Flory’s theory [40] on the melting of the fringe–type crystals with a finite crystal length of ζ, the end surface free energy of crystals per unit area, σ_{e}, at (df_{u}/dζ)_{ϕ}= 0 is given by:

where ϕ is the amorphous fraction and μ is the conversion coefficient of mol/m^{3}. 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. (11) is obtained when lnP_{c}= – 1/(x – ζ+ 1). From Eq. (14), 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 4. 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 same fringe–type, leading to σ_{e}= 0, and that, ζ = 0/0 in Eq. (16) (described below). For class B, f_{u}= – f_{x}’ from Eq. (14) with f_{x}= 0 refers to the anti–crystal holes and f_{x}= 0 is assigned to the ordered parts of ζ = ∞. The interface between the anti–crystal holes and the ordered parts should work as the reflector of photons. In this case, the even interface made of the folded chain segments should be avoided through the random reflection. According to Eq. (12) with h_{x} – h_{u}= σ_{e}/(μζ), the respective interface energies of the hole and the ordered part are compensated each other at the common interface, thus leading to f_{x}= 0 [10]. For class C, f_{x}= f_{x}’ from Eq. (14) 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. (14), suggesting that the holes of class B cannot be replaced by the crystals with ζ ≠ ∞. Fig. 9 shows the schematic structure models of bulk polymers conformable to A ∼ X classes in Table 4.

fx’ | fx | fu | Class |

f_{x}’ = 0 | f_{x} = f_{u} | f_{u} = f_{x} | A |

f_{x}’ > 0 | f_{x} = 0 | f_{u} = –f_{x}’ | B |

f_{x} = f_{x}’ | f_{u} = 0 | C | |

f_{x} = 2f_{u} | f_{u} = f_{x}/2 = f_{x}’ | D | |

f_{x}’> 0 | f_{x} = 0 | f_{u} = f_{x}’ | X |

At the rapid glass transition absorbing the photon energy of 2h_{0}^{h} at T_{g} upon heating, the ordered part / hole pairs should be excited immediately and then melted, followed by unfreezing of the glass parts. At the slow glass transition, the disappearance and then crystallization of ordered part / hole pairs should occur upon heating, bringing the new crystal parts [16]. In the closed system that the both heats of crystallization and melting should be cancelled out according to Eq. (4), those should be melted by T_{l} in Fig. 1. While in the open system that the heat irradiated by crystallization was escaped out of the system, T_{l} corresponded to T_{m}^{∞} (450 K for α form crystals) and h_{0} (= h_{g}+ h_{x}+ h_{u})= 21.0 kJ/mol in Eq. (8) agreed perfectly with the value of h^{h} (= 3h_{0}^{h})+ h_{m}^{int}, where h_{x} (= h_{g}+ ∆h) is 7.4 kJ/mol, being larger than h_{0}^{h} (= h^{h}/3)= 6.1 kJ/mol*. In this context, h_{g} – h_{0}^{h}= 0.1 kJ/mol, h_{x} – h_{0}^{h}= 1.3 kJ/mol, and h_{u} – h_{0}^{h}= 1.4 kJ/mol. The sum of them was equal to h_{m}^{int}= 2.8 kJ/mol. Thus, n (= h^{h}/h_{x}) = 2.5 was shown, almost corresponding to ∆C_{p}/∆C_{p}^{exp}= 2.3. Table 5 shows the values of T_{g}, h_{0}, h_{u}, h_{g}, h_{x}, h^{h}, and h^{h}/h_{x} for iPP. From h^{h} (= 3h_{0}^{h}) = 18.2 kJ/mol* at T_{g} (= 270 K), the wavenumber of 1/λ = 1022 cm^{– 1} was derived for a photon in holes [10]. This agreed nearly with 1045 cm^{– 1} relating to the crystallinity [41]. Accordingly, one photon should be situated between the neighboring methyl groups in the helical sequence.

TgK | h0kJ/mol | hukJ/mol | hgkJ/mol | hxkJ/mol | hhkJ/mol* | hh/hx |

270 | 18.3^{*1} | --- | 6.2^{*3} | 12.1 | 18.2 | 1.5 |

270 | 21.1^{*1} | 7.5^{*2} | 6.2^{*3} | 7.4 | 18.2 | 2.5 |

180 | 0.28^{*1} | --- | 0.28^{*4} | 0 | 0 | --- |

#### 2.3.1. Equilibrium melting temperature, T_{m}^{∞}

For PET discussed in the previous section **2.2**, T_{l} in Fig. 1 corresponded to T_{m}^{∞}= 535 K for the crystals with conformational disorder of ethylene glycol parts, but with a chain axis parallel to c-axis of a cell. This finding in PET was also discussed in iPP with T_{m}^{∞}= 435 K, 450 K, and 462 K for β, α, and γ form crystals, respectively, which were found by DSC measurements [35]. Above all, T_{m}^{∞}= 450 K for α form crystals could be T_{l} (in Fig.1) of the temperature expected from the quantum demand of hole energy at regular temperature intervals of 90 K. Experimentally, T_{m}^{∞} is determined as the intersection temperature between an extrapolation line of T_{m}= T_{e} in the melting peaks without superheating and a T_{m}= T_{a} line, where T_{m} is the melting temperature from T_{a} to T_{e}, T_{a} is the annealing (crystallization) temperature, and T_{e} is the end temperature of melting peak. While at T_{m}^{∞}, the processes of melting and crystallization should occur reversibly, so that T_{m}^{∞} equals to both temperatures of T_{e} and T_{b} of the onset temperature of melting, and that, there are two points of T_{m}^{∞} at T_{a} (= T_{e}) and T_{b} (= T_{e}) on a T_{m}= T_{e} line. The line through two points of T_{m}^{∞} should be parallel to the abscissa of T_{a}, because T_{m}^{∞} is only one [23]. For the bulk contained α form crystals with h_{u}= 7.46 kJ/mol, the sum of h_{g} – h_{0}^{h}= 0.1 kJ/mol, h_{x} – h_{0}^{h}= 1.3 kJ/mol, and h_{u} – h_{0}^{h}= 1.4 kJ/mol was equal to h_{m}^{int}= 2.8 kJ/mol (see section **2.3**). While for the bulk contained only γ form crystals with h_{u}= 8.70 kJ/mol, the difference in h_{m}^{int} and (h_{u} – h_{0}^{h}) was 0.2 kJ/mol, suggesting h_{g}= h_{x}= h_{0}^{h}+ 0.1 kJ/mol. Figs. 10 and 11 show the DSC single and double melting peak curves for the iPP films annealed at T_{a}= 461.0 K and 441.5 K for 1 hour, respectively. Here the single melting peak curve in Fig. 10 is divided into α and β peaks, and the double melting peak curve in Fig. 11 is divided into γ, α, and β peaks. In both Figures, T_{b} is the temperature that the extrapolation line from the line segment with a highest slope in the lower temperature side of the melting peak intersects the base line. The onset temperature of the extrapolation line, T*, is also the end temperature of the residual peak appeared by subtracting the area of single or double peak with T_{b} from the total endothermic peak area. The β peak is considered to be due to the melting of small crystals attached around the crystal lamellae of α form. The mean of end temperatures in β peaks found for some annealing samples agreed closely with T_{e}= 435 K of β form crystals [42].

Table 6 shows the values of T_{b}, T*, Q_{m}, ∆Q_{m}, ∆h^{h}, and ∆h^{h}/∆Q_{m} in α and γ peak curves for the iPP films annealed at 461.0 K and 441.5 K for 1 hour. Where Q_{m} is the heat per molar structural unit corresponding to the area of α or γ peak from T_{b} and ∆Q_{m} is the heat per molar structural unit corresponding to the area from T_{b} to T* of α or γ peak and relating to the melting of crystals recrystallized newly from β to α form or α to γ form. For the holes of crystal / hole pairs formed newly by recrystallization from T_{b} to T* of β or α peak, the hole energy per molar photon, ∆h^{h}, is given by [10]:

As shown in Table 6, the small difference in ∆Q_{m} and ∆h^{h} could be regarded as significant for the formation and then disappearance of crystal / hole pairs from T_{b} to T*. For the shift from β to α peak in Fig. 11, ∆h^{h}/∆Q_{m} was 1.21 contrary to our expectation, but at T*= 435 K of the mean of T* (T_{m}^{∞} for β form crystals), ∆h^{h}/∆Q_{m}= 0.98 was derived. For the shift from α to γ peak, it was 0.61, meaning the melting of original γ form crystals with 39 % of ∆Q_{m}; 0.13 kJ/mol. The relay of melting from α to γ form crystals between two peaks of a DSC double melting peak curve should be done through the mediation of the formation and then disappearance of the crystal / hole pairs with 61 % of ∆Q_{m}; 0.20 kJ/mol (= ∆h^{h}), which agreed with the difference in h_{m}^{int} and (h_{u} – h_{0}^{h}) perfectly, corresponding to (h_{g}+ h_{x}) – 2h_{0}^{h}= 0.2 kJ/mol suggested above.

Ta K | Form | TbK | T*K | Qm kJ/mol | ∆QmkJ/mol | ∆hh kJ/mol | ∆hh/∆Qm |

461.0 | α | 429.6 | 434.2 | 2.05 | 0.38 | 0.24 | 0.63 |

441.5 | α | 423.7 | 437.7 (435) | 2.16 | 0.78 | 0.94 (0.76) | 1.21 (0.98) |

γ | 446.9 | 449.9 | 1.64 | 0.33 | 0.20 | 0.61 |

#### 2.3.2. ζ distribution function, F (ζ)

Next, the α peak curve in Fig. 10 and the two divided peak curves of α and γ in Fig. 11 starting to melt at T_{b} were converted into the crystal length (ζ) distribution function, F(ζ). The ζ is according to Gibbs–Thomson given by:

where T_{m} is the corrected melting temperature. In the calculation of ζ, the values of T_{m}^{∞}, h_{u}, h_{x}, σ_{e}, and the corrected T_{m} are needed previously. For h_{u}, the reference value was used (see Table 8). The value of σ_{e} was evaluated by [43, 44]:

with

where c* is the cell length of c–axis. The term of square blanket in Eq. (17) is dimensionless. h_{u} refers to the heat of fusion of crystals with a crystal form liking to evaluate σ_{e}. h_{x} could be calculated from Eqs.(5) and (6), but in Fig. 11, using H_{m}^{a} at T_{m}^{∞} of the other α or γ form crystals (sub–crystals). Table 7 shows the values of h_{x} for the iPP films annealed at 461.0 K and 441.5 K for 1 hour, together with the values of T_{c}, T_{b}, T_{e}, Q, ΔH, Δh, and h_{g} used in the calculation of h_{x}. Q and ∆H are defined in Eq. (6). The value of h_{x} for the iPP sample of T_{a}= 441.5 K was smaller than h_{u} of the sub–crystals (see Table 8). Further, the value of h_{x} in the row of γ form, 6.85 kJ/mol, was ∼0.7 kJ/mol larger than h_{g} (= 6.22 kJ/mol). Also the value of h_{x} in the line of α form, 8.07 kJ/mol, was ∼0.7 kJ/mol larger than h_{x} (= 7.35 kJ/mol) for the sample of T_{a}= 461.0 K. Therefore, the value of h_{0} (= h_{g}+ h_{x}+ h_{u}) in Eq. (8) for the iPP sample annealed at 441.5 K was 0.7 kJ/mol larger than that for the iPP sample of T_{a}= 461.0 K. The cause could be attributed to ∆h affected by F(ζ) (see Eq. 18) of α form crystals, leading the characteristic R–L image (see Fig. 15).

Form*1 (Ta/K) | Tc K | TbK | TeK | QkJ/mol | ΔHkJ/mol | ΔhkJ/mol | hgkJ/mol | hxkJ/mol |

α (461.0) | 403.6 | 429.6 | 449.2 | 3.76 | 4.89 | 1.13 | 6.22 | 7.35 |

α (441.5) | 403.6 | 423.7 | 449.5 | 4.41 | 6.26 (γ)^{ *2} | 1.85 | 6.22 | 8.07 |

γ (441.5) | 403.6 | 446.9 | 461.9 | 4.41 | 5.04 (α)^{ *2} | 0.63 | 6.22 | 6.85 |

TaK | Form | TpK | hukJ/mol | hxkJ/mol | h0kJ/mol | Qm kJ/mol | σeJ/m2 |

461.0 | α | 435.9 | 7.46 | 7.35 | 14.8 | 2.05 | 36.0×10^{−3}(26.7×10 ^{−3}) |

441.5 | α | 442.8 | 7.46 | 8.07 | 15.5 | 2.16 | 45.2×10^{−3}(30.8×10 ^{−3}) |

γ | 450.9 | 8.70 | 6.85 | 15.6 | 1.64 | 26.5×10^{−3}(22.2×10 ^{−3}) |

Table 8 shows the values of σ_{e} at T_{p} for α and γ form crystals contained in the iPP films annealed at 461.0 K and 441.5 K for 1 hour, together with the values of T_{p}, h_{u}, h_{x}, h_{0}, and Q_{m} used in the calculation of σ_{e}, where T_{p} is the melting peak temperature. The σ_{e} of α form was larger than that of γ form, because according to Eq. (17), the σ_{e} was mainly dependent on h_{x}. For α and γ forms in the sample of T_{a}= 441.5 K, h_{0} (= h_{u}+ h_{x}) at T (> T_{g}) of Eq. (9) was ∼15.5 kJ/mol, nevertheless the values of h_{u} were different. T_{p} is corrected by 0.6 K (436.5 K → 435.9 K) for the sample of T_{a}= 461.0 K and 0.2 K (443.0 K → 442.8 K) for α form and 0.8 K (451.7 K → 450.9 K) for γ form in the sample of T_{a}= 441.5 K to the lower temperature side, according to our concept [45].

F(ζ) is defined as [23]:

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, N_{c} is the number of structural units of crystals melted in the temperature range from T_{b} to T_{e}. δQ_{m}/Q_{m} is given by:

where dQ/dt is the heat flow rate of DSC melting curve and t is time (see Figs. 10 and 11). Figs. 12 and 13 show F(ζ) of α and γ peak curves converted from the DSC single and double melting peak curves for the iPP films annealed at 461.0 K and 441.5 K for 1 hour.

Table 9 shows the ζ range and ζ_{p} in F(ζ) of α and γ peak curves obtained for the samples of T_{a}= 461.0 K and 441.5 K, where ζ_{p} is ζ at the maximum of F(ζ). For α peak, F(ζ) in Fig. 12 showed a sharp peak with the ζ range of 10 nm ∼ 3870 nm and ζ_{p}= 14.6 nm, and in Fig 13, F(ζ) showed the roundish curve with the ζ range of 10 nm ∼ 250 nm and ζ_{p}= 19.5 nm, whereas for γ peak, F(ζ) showed the sharp peak with the ζ range of 8 nm ∼ 840 nm and ζ_{p}= 11.2 nm. The maximum of ζ was calculated using T_{e} (∼T_{m}^{∞}) observed actually for each sample. At T_{e}= T_{m}^{∞}, the maximum of ζ should be infinite at σ_{e} ≠ 0, because the melt at T_{e} could be interchanged in equilibrium to the imaginary crystals of ζ = ∞. In the σ_{e}= 0 of the class A in Table 4, ζ = 0/0 of indetermination at T_{m}^{∞} is derived from Eq. (16). The refraction point at ζ = 55 nm on the thick line of α peak in Fig. 13 is that of dQ/dt at T_{b} (= 446.9 K) in Fig. 11. For the sample of T_{a}= 441.5 K, the ζ range of α peak was narrower than that of γ peak, because upon cooling, α form crystals should be formed around γ form crystals. As the result, the value of h_{x} at the interfaces between α and γ form crystals increased only ∼0.7 kJ/mol (derived above). The ζ range of α peak calculated for the sample of T_{a}= 461.0 K was much larger than those of α and γ form crystals for the sample of T_{a}= 441.5 K.

TaK | Form | ζ rangenm | ζpnm |

461.0 | α | 10 - 3870 | 14.6 |

441.5 | α | 10 - 250 | 19.5 |

γ | 8 - 840 | 11.2 |

In the last stage, the ζ distribution of a single–crystal like image was drawn from F(ζ). The number of crystal sequences in a radius direction, R_{n}, is given by [10]:

with

where ζ_{x} and ζ_{n} are the maximum and the minimum of ζ, respectively. Figs. 14 and 15 show the representation of R (= ±R_{n}) and L (= ±ζ/2) for α and γ form crystals in the iPP films (per 1g) annealed at 461.0 K and 441.5 K for 1 hour.

From the comparison of both figures, the change of image in α form crystal lamellae by annealing, and further in Fig. 15, the difference in packing states of α and γ form crystals in same ζ range can be seen at the view of 2D disk image.

## 3. Conclusion

For iPS, PET, and iPP, the heat capacity jump at the glass transition was due to the discontinuous change of energy in quantum state of the photon holes between neighboring benzene rings, but methyl groups for iPP, followed by unfreezing of glass parts. For iPS and iPP, the homogeneous glasses free from ordered part / hole pairs with T_{g}= 240 K and 180 K were predicted, respectively. For iPP, the cohesive energy of methylene residues was subdivided into the transition enthalpies of glasses, ordered parts, and crystals, whereas for iPS, it agreed with the difference between the transition enthalpies of glasses and ordered parts, but the transition enthalpy of glasses was larger than that of ordered parts. The photonic contribution of 60 % to the heat capacity jump at the glass transition found for PET meant that one photon was situated in the neighboring phenylene residues comprising ∼60 % of the structural unit length and the residual jump of 40 % was brought by unfreezing of the ethylene glycol parts in a glass state. T_{m}^{∞}= 450 K for α form crystals of iPP could be the temperature of the quantum demand of hole energy at regular temperature intervals of 90 K. The shift of melting from α to γ form crystals by DSC measurements was done through the mediation of the formation and then disappearance of crystal / hole pairs. The interface parts formed in α and γ form crystals by annealing brought the excess energy of ∼0.7 kJ/mol to the enthalpy of the ordered parts. This result was reflected clearly to the single crystal like image depicted on the basis of the crystal length distribution function.

## 4. A list of abbreviations (italic in Eqs.)

α : fraction of ordered part / hole pairs with C_{v}^{ph}

1 – α : fraction of ordered part / hole pairs with C_{p}^{ph}

C_{p} : mean heat capacity per molar structural unit for ordered parts in ordered part / hole pairs

C_{p}^{h} : mean heat capacity per molar photon for holes in ordered part / hole pairs

C_{p}^{h*}: heat capacity per molar photon for holes in excited ordered part /hole pairs

C_{p}^{ph} : adiabatic molar heat capacity for photons

C_{v}^{ph} : constant volume molar heat capacity for photons

C_{p}^{flow} : heat capacity per molar structural unit for flow parts

C_{p}^{r} : heat capacity change per molar structural unit due to crystallization followed by melting

C_{p}^{x}: heat capacity per molar structural unit for ordered parts in excited ordered part /hole pairs

c : velocity of light

c* : cell length of c–axis

c_{1} and c_{2} : constants in WLF equation

dQ/dT : heat flow rate of DSC melting curves

δQ_{m} : heat change per molar structural unit per K

ΔC_{p} : heat capacity jump per molar photon at the glass transition

ΔC_{p}^{exp} : experimental heat capacity jump per molar structural unit at the glass transition

ΔC_{p}^{x} : relative component heat capacity per molar structural unit for excited ordered parts

Δh : additional heat per molar structural unit needed to melt all ordered parts

Δh^{h} : hole energy of crystal / hole pairs formed newly by recrystallization

δ : solubility parameter

E_{a} : activation energy

F(ζ) : crystal length (ζ) distribution function

f : free energy per molar structural unit

f^{conf} : conformational free energy per molar structural unit

f_{g}^{conf} : conformational free energy per molar structural unit at T_{g}

f_{x} : free energy per molar structural unit for ordered parts

f_{u} : free energy per molar structural unit for crystals

ϕ : amorphous fraction

Γ : frequency of occurrence of the helix–coil transition

H_{m}^{a} : enthalpy per molar structural unit for the liquid at T_{m}^{∞}

H_{c}^{a} : enthalpy per molar structural unit for the super–cooled liquid at T_{c}

H_{g}^{a} : enthalpy per molar structural unit for the super–cooled liquid at T_{g}

H_{g}^{c} : enthalpy per molar structural unit for the crystal at T_{g}

h^{h} : hole energy per molar photon for holes in ordered part / hole pairs

h_{0} : latent cohesive energy per molar structural unit

h_{0}^{h} : zero–point energy per molar photon, or energy unit per molar photon

h_{u} : heat of fusion per molar structural unit

h_{x} : enthalpy per molar structural unit for ordered parts

h_{g} : glass transition enthalpy per molar structural unit

h^{conf} : conformational enthalpy per molar structural unit

h_{g}^{conf} : conformational enthalpy per molar structural unit at T_{g}

h_{x}^{conf} : conformational enthalpy per molar structural unit for ordered parts

h^{int} : cohesive enthalpy per molar structural unit

h_{g}^{int} : cohesive enthalpy per molar structural unit at T_{g}

h_{m}^{int} : cohesive energy per molar structural unit for methylene residues

*h* : Plank constant

η : statistical weight

J_{h} : number of holes lost by T at the glass transition

φ_{g} : fraction of core free volume in glasses

λ and 1/λ : wavelength and wave number

mol : molar structural unit

mol* : molar photon

μ : conversion coefficient of mol/m^{3}

N_{A} : Avogadro constant

N_{c} : number of structural units of crystals melted in the temperature range from T_{b} to T_{e}

n : number of structural units holding one photon potentially

n_{ζ} : number of crystal sequences with ζ

ν : frequency per second

P : pressure

P_{c} : probability that a sequence occupies the lattice sites of a crystalline sequence

Q : heat per molar structural unit corresponding to the total area of DSC endothermic curve

Q_{m} : heat per molar structural unit corresponding to the area of a DSC melting curve from T_{b}

R : gas constant

R_{n} : number of crystal sequences at the radius direction of an imaginary single crystal lamella depicted on the basis of F(ζ)

s_{u} : entropy of fusion per molar structural unit

s_{x} : entropy per molar structural unit for ordered parts

s^{conf} : conformational entropy per molar structural unit

s_{g}^{conf} : conformational entropy per molar structural unit at T_{g}

σ_{e} : end surface free energy of a crystal per unit area

σ : statistical weight

T : temperature

T_{g} : glass transition temperature

T_{m} : melting temperature

T_{m}^{∞} : equilibrium melting temperature

T_{e} and T* : end temperature of DSC melting peak curve

T_{l} : end temperature of the glass transition

T_{c} : onset temperature of DSC crystallization peak curve upon cooling

T_{b} : onset temperature of DSC melting peak curve upon heating

T_{a} : annealing temperature

T_{p} : DSC melting peak temperature

V : volume per molar structural unit

v_{f} : free volume per molar structural unit

v_{0} : core free volume per molar structural unit

x : degree of polymerization

Z : conformational partition function for a chain

Z_{0} : component conformational partition function for a chain regardless of temperature

Z_{t} : component conformational partition function for a chain depending on temperature

ζ : crystal length

ζ_{p} : crystal length at the maximum of F(ζ)

ζ_{n} : crystal length at the minimum of ζ

ζ_{x} : crystal length at the maximum of ζ

## Acknowledgments

The author would like to thank the late Professor em. B. Wunderlich of the University of Tennessee and Rensseler Polytechnic Institute for the long time encouragement.