Photonic Contribution to the Glass Transition of Polymers

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 temper‐ ature (Tg), 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 Tg 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 Tg 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 Tg. 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 polypro‐ pylene (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


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. 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, 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]: 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]. The values in ( ): our data of Fig. 6. *1: see Table 2. *2: ∆C p exp at T m ∞ = 535 K. 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 s g 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].

Isotactic polystyrene
From h h = (3/2)N A hν 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.   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].   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 . Figure 5. The schematic chart of the instantaneous state changes at T g for iPS. The arrow marks of ↓ and ↑ show the cooling and heating directions, respectively. T g + and T g -are the glass transition temperatures upon cooling and heating, respectively. The h x = 2h 0 h shows the substance of helix-coil transition between the helical sequence (n = 3) and the aperiodic sequence (n = 0). The arrow marks of ⇔ show the interaction between the ordered parts and the holes in the pairs. The Γ is the frequency of occurrence of the helix-coil transition. The dashed lines show the equilibrium relation of melting among ordered parts, flow parts, and holes in the excited state.

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 hν 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].     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. Figure 7. The parts in the structural unit related to the C p jumps of a, b, and a -b shown in Fig. 6 for PET. The part attached to the spindle mark shows the phenylene residue holding one photon together with the neighboring same residue (omitted here).

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 nonequilibrium 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. Table 4. Relations of equilibrium (A ∼ D) and non-equilibrium (X) in f x and f u at f x ' ≥ 0 for crystalline polymers [10,39].
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 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.     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].  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]: (15) 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. The thin line is the part of an original DSC curve. T b is the onset temperature of α peak and T* is the end temperature of β peak. Figure 11. DSC double melting peak curve composed of α, β, and γ peaks for the iPP film annealed at 441.5 K for 1 hour. The thin lines are the parts of an original DSC curve. T b is the onset temperature of γ or α peak and T* is the end temperature of α or β peak. The values in ( ) show T m ∞ , ∆h h , and ∆h h /∆Q m at T m ∞ for β form crystals. Table 6. The values of T b , T*, Q m , ∆Q m , ∆h h , ∆h h /∆Q m of α and γ peak curves for the iPP films annealed at 461.0 K and 441.5 K for 1 hour.

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]: (17) with H x = 2h u -Q m 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).  The values in ( ) are σ e at T m ∞ (Q m = 0).  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: (19) 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.  2   Table 9. The ζ range and ζ p in F(ζ) of α and γ peaks for the iPP films annealed at 461.0 K and 441.5 K for 1 hour.
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]: 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.

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.

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