Open access peer-reviewed chapter

Photonic Contribution to the Glass Transition of Polymers

By Nobuyuki Tanaka

Submitted: May 17th 2014Reviewed: October 24th 2014Published: May 6th 2015

DOI: 10.5772/59717

Downloaded: 743

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) [19]. 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 (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 [1016]. 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 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 [1016]. 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 [1416]. 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 Tg.

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 Tg= 240 K and 180 K have been predicted, respectively [15, 16]. Tg 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 Tg= 0 K have also been predicted. In this connection, the equilibrium melting temperature, Tm= 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 hh (= 3CvphT), the heat capacity per molar photon for holes, Cph* (= Cpflow), is given by [1016] (see section 4):

Cph*= Cpflow=3Cpph(1 +TdlnJh/dT) E1

where Cpflow is the heat capacity per molar structural unit for the flow parts, being equal to Cpx of the heat capacity per molar structural unit for the excited ordered parts [19], Cvph (= 2.701R) is the constant volume molar heat capacity for photons [20], R is the gas constant, Jh is the number of holes lost by T, and 3 is the degree of freedom for photons. When dJh/dT = 0 at Tg and the end temperature, Tl, of the glass transition, Cph* at those temperatures is given by 3Cvph. Thus, the heat capacity jump per molar photon, ∆Cp, at the glass transition is given by:

Cp = Cph* - Cvph=2Cvph (=44.9 J/(K mol*)) E2

In Eq. (2), ∆Cp should be due to the discontinuous energy change from a quantum ground level for photons in the holes, that is, (1/2)hν (= h 0h/NA) to (3/2)hν (= 3h0h/NA), where h is Planck constant, ν (= c/λ) is the frequency per second, c is the velocity of light, λ is the wavelength, NA is Avogadro constant, and h0h 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 Cp as shown in Fig. 1 is observed generally. Even in this case, the Cp for ordered parts should be equal to the Cph of the mean heat capacity for the holes in ordered part / hole pairs at the glass transition:

Cp= Cph = αCvph +1 - αCpph E3

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

Cp= Cpx + Cpr E4

under TgTlCpdT = TgTlCpxdTand TgTlCprdT =0,

where ∆Cpx (= Cph* – 3Cvph) is the relative component heat capacity per molar structural unit for the excited ordered parts and Cpr is the heat capacity change per molar structural unit due to the crystallization followed by the melting. At the glass transition, ∆Cpx shows a peak against T, reflecting the size distribution of ordered parts. Fig. 1 shows the representative Cp curve composed of ∆Cpx and Cpr at the glass transition for polymers.

Figure 1.

The components of the Cp curve (the thick line) at the glass transition for polymers. The dash – dotted line is the relative component ∆Cpx curve for the excited ordered parts with a size distribution and the thin line is the component Cpr curve due to the crystallization and then melting.

Thus, for iPS, PET, and iPP, ∆Cp (= 2Cvph) per molar photon (mol*) was compared with the reference value of heat capacity jump, ∆Cpexp, per molar structural unit (mol) [8]. The results deviated from ∆Cp/∆Cpexp= 1. Table 1 shows the comparison of ∆Cp (= 2Cvph) with ∆Cpexp for these polymers, together with hh/hx at Tg, where hx is the enthalpy per molar structural unit for ordered parts [1416].

PolymerTg
K
∆Cp exp J/(K mol)∆Cp
J/(K mol*)
∆Cp/∆Cp exphh/hx
iPS359*130.844.91.51.5 or 1.0
PET34277.8
(80.4, 46.5*2)
44.90.6
(0.6)
1.0
iPP27019.244.92.31.5 or 2.5

Table 1.

The values of Tg, ∆Cpexp, ∆Cp (= 2Cvph), ∆Cp/∆Cpexp, and hh/hx for iPS, PET, and iPP.

The values in ( ): our data of Fig. 6. *1: see Table 2. *2: ∆Cpexp at Tm= 535 K.


However, the values of ∆Cp/∆Cpexp were correlated to hh/hx of the number of structural units holding one photon potentially (described below). hx at Tg is given by [10, 23]:

hx= hg + h E5

where hg {= RTg2(∂lnvf/∂T)p} is the glass transition enthalpy per molar structural unit due to the discontinuous free volume change of v* from vf= v0 to v0+ v* at Tg, vf and v0 are the free volume and the core free volume per molar structural unit. hg is given approximately by three expressions; (1) RTg2/c2 or φgEa (in WLF equation [24], φg {= 1/(2.303c1)} is the fraction of the core free volume in glasses, c1 and c2 are constant, and Ea is the activation energy), (2) the molar enthalpy difference between the super – cooled liquid and the crystal at Tg; Hga – Hgc, and (3) the sum of the conformational and cohesive enthalpies per molar structural unit at Tg; hgconf + hgint. For PET and iPP, the additional heat per molar structural unit, ∆h, needed to melt all ordered parts by Tl in Fig. 1 is given by [10, 23]:

h= H -Q E6

where ∆H= Hma ‒ Hca, Hma is the enthalpy per molar structural unit for the liquid at Tm, Hca is the enthalpy per molar structural unit for the super– cooled liquid at the onset temperature, Tc, 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 hxconf ≠ hgconf= 0 at Tg, ∆h is derived as [10]:

h= Tg{sgconf -(RlnZ0)/x) E7

with sgconf=(RlnZ +RTgdlnZ/dT)

where hxconf is the conformational enthalpy per molar structural unit for ordered parts, sgconf is the conformational entropy per molar structural unit at Tg, Z is the conformational partition function for a chain, Z0 (= Z/Zt) and Zt 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 hxconf= hgconf, ∆h= (RTglnZt)/x was derived, applying to nylon 6 [10].

2.1. Isotactic polystyrene

From hh= (3/2)NAhν and hh (= 3CvTg)= 24.2 kJ/mol* at Tg= 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 hh/hx= ∆Cp/∆Cpexp= 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, ∆Cp/∆Cpexp was 1.5 (see Table 1), where Tl in ∆Cpexp is ∼381 K [8]. Accordingly, the number of structural units holding one photon potentially in holes, n (= hh/hx), 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.

Figure 2.

The sequence models of– TTTT– (n = 1) (Left) and unstable– TCTC– (n = 2) (Right) for iPS. The dashed line shows one of the photon sites between benzene rings.

According to hh/hx= 1.5, hx (= 2h0h) = 16.1 kJ/mol was derived. This corresponded to ∆CpTg= 16.1 kJ/mol* of the Cp jump energy for holes at Tg. While hx can be also derived from theδ solubility parameter, δ {= (h0/V)1/2}, where h0 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 h0, hu, hx, and hg at temperatures before and after Tg are given by [19, 27]:

h0= hg + hx or hg + hx + hu  at T  TgE8
h0= hx or hx + hu  at T > TgE9

where hu is the heat of fusion per molar structural unit. For crystalline polymers, hu is contained in Eqs. (8) and (9). For iPS, h0= 35.0 kJ/mol was derived from δ = 9.16 (cal/cm3)1/2 of the mean of 12 experimental values (≥ 9.0 (cal/cm3)1/2) [28], and the value of hx from Eq. (8), 16.1 kJ/mol, agreed with that from hh/hx= 1.5 perfectly. However it was smaller than hg (= RTg2/c2) = 18.9 kJ/mol. The difference in hg and hx, 2.8 kJ/mol, agreed with the cohesive energy of methylene residues of hmint= 2.8 kJ/mol [29], suggesting that the ordered part / hole pairs might fill softly the parts in glassy bulks.

TgKh0
kJ/mol
hg
kJ/mol
∆hkJ/molhx
kJ/mol
hh
kJ/mol*
hh/hx
359*1 (360)43.1*218.95.324.2*424.21
359*1 (360)35.0*2,*318.9–2.816.124.21.5
2401.7*21.7*5–1.700 (16.1)---

Table 2.

The values of Tg, h0, hg, ∆h, hx, hh, and hh/hx for iPS.

*1: experimental value [30]. *2: from Eq. (8). *3: from δ = 9.16 (cal/cm3)1/2. *4: hx for the excited ordered parts or – TTTT– sequences. *5: hg= hgint+ hgconf from hgint= – fgconf at Tg= 240 K (see Fig. 4, described below). The value in ( ) of hh column is hh (= 3CvphT) at 240 K.


Table 2 shows the values of Tg, h0, hg, ∆h, hx, and hh at hh/hx= 1 and 1.5 for iPS. In the 4th line, hh= hx= 0 and hg= 1.7 kJ/mol at 240 K are shown (discussed below). The relation of hh= hx= 0 is brought by the energy radiation of 2h0h (= 16.1 kJ/mol*) at Tg and the energy loss of h0h (= 8.1 kJ/mol*) upon cooling from Tg obeying:

hh= h0h -3CvphTg -TE10

In Eq. (10), the specific temperature of 240 K at hh= 0 agreed with the hole temperature at hh (= 3CvphT)= 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 hh= 0 at 240 K upon cooling, suggesting Tg= 240 K for the homogeneous glass free from the ordered part / hole pairs. Altering 3Cvph (= Cpph) in Eq. (10) to Cvph, the temperature at hh= 0 was Tg= 0 K. While for the glasses including the ordered part / hole pairs, Tg= 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 hh/hx (= 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 hh/h0h in Eq. (10) with hh/h0h= 0 at T = 240 K. The ∆Cp (= 3Cvph) at Tg= 240 K in a glassy state of Γ = 0 was 67.4 J/(K mol*) [15].

Figure 3.

The 3/1 helix structure of – TGTGTG– (n = 3) and the photon sites (dashed line parts) between benzene rings for iPS.

Figure 4.

The relation between f and T calculated for the RIS model chains (x = 100) of iPS. The thin line is f = fconf and the thick line is f = fconf+ 0.81 kJ/mol.

Fig. 4 shows the relation between f (= fconf or fconf+ 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 fconf is the conformational free energy per molar structural unit, being minus and decreasing with an increase in temperature. Adding the value of – fconf (= 0.81 kJ/mol) at 240 K to all values of the original thin line, the thick line of fconf+ 0.81 kJ/mol is depicted. In the case of – fconf= hint, i.e., (hconf+ hint) – Tsconf= 0 at 240 K, the sum of – fconf (= 0.81 kJ/mol) and hconf (= 0.89 kJ/mol), 1.70 kJ/mol, should be the ultimate hg at the first order glass phase transition for the homogeneous glass free from the ordered part / hole pairs (see Table 2), where hint is the cohesive enthalpy per molar structural unit, and hconf and sconf are the conformational enthalpy and entropy per molar structural unit.

Fig. 5 shows the schematic chart of the instantaneous state changes at Tg (= 360 K) upon cooling and heating as a working hyposesis. The ordered part / hole pairs formed instantaneously at Tg upon cooling have hh= h0h and hx= 2h0h. At Tg upon heating, the ordered part / hole pairs are excited by absorbing the photon energy of 2h0h for the holes and adding the energy of h0h for the ordered parts, followed by the absorption of hg 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 h0h is needed at Tg.

Figure 5.

The schematic chart of the instantaneous state changes at Tg for iPS. The arrow marks of ↓ and ↑ show the cooling and heating directions, respectively. Tg+ and Tg– are the glass transition temperatures upon cooling and heating, respectively. The hx= 2h0h 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.

2.2. Poly(ethylene terephthalate)

For PET, hx (= hg+ ∆h) = 24.1 kJ/mol was obtained from Eq. (5), being almost equal to hh (= 3CvphTg) = 23.0 kJ/mol* at Tg (= 342 K) [10, 23]. Thus, hh/hx (= n) = 0.95 was shown experimentally. However as shown in Table 1, ∆Cp/∆Cpexp was 0.6. Table 3 shows the values of Tg, h0, hu, hg, ∆h, hx, hh, and hh/hx for PET. The two values of hu, 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]. ∆Cp/∆Cpexp= 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 ∆Cpexp 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 hh= (3/2) NAhν and hh (= 3CvphTg) = 23.0 kJ/mol* at Tg= 342 K, 1/λ = 1290 cm–1 was derived, agreeing with 1288 cm–1 observed for the un–oriented samples [33].

Tg
K
h0
kJ/mol
hu
kJ/mol
hg
kJ/mol
∆hkJ/molhx
kJ/mol
hh
kJ/mol*
hh/hx
34264.7*123.0 (535 K)17.6*36.524.123.00.95
34270.2*1, 68.2*228.5 (549 K)17.6*36.524.123.00.95

Table 3.

The values of Tg, h0, hu, hg, ∆h, hx, hh, and hh/hx for PET.

*1: from Eq. (8). *2: from δ = 10.7 (cal/cm3)1/2 [19]. *3: hg= RTg2/c22. The values in () are Tm of the respective crystals [23].


Figure 6.

The C p curve for the non–annealed PET film. The parts of a and b show the C p jump to the liquid line at Tg and Te.

Fig. 6 shows the Cp 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). Tg agreed almost with 342 K of [8]. Te of the end temperature of melting is 535 K (262 °C). The parts of a, b, and a – b of Cp jump to the liquid line at Tg and Te 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 Cp 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).

2.3. Isotactic polypropylene

According to the scheme of the formation of ordered part / hole pairs at Tg upon cooling (see Fig. 5), for iPP with Tg= 270 K, hx (= 2h0h) = 12.1 kJ/mol was derived, being much larger than hg ≈ Hga – Hgc= 6.2 kJ/mol [34] and hx (= hg+ ∆h) = 7.4 kJ/mol, where ∆h = ∆H – Q, ∆H = Hma – Hca (see Eqs. (5) and (6)). The used data are as follows: Tc= 403.6 K, Tm= 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, h0 (= hg+ hx) = 18.3 kJ/mol from Eq. (8) was almost equal to hh (= 3h0h) = 18.2 kJ/mol*, meaning the appearance of frozen glasses with hg= h0h+ 0.1 kJ/mol. For holes with Cpph even upon cooling from Tg, Eq. (10) showed the specific temperature of 180 K, at which all ordered part / hole pairs should be disappeared because of hh= 0, corresponding to 240 K for iPS [15]. At temperatures below 180 K, all should be in a state of the homogeneous glass with Tg= 180 K. The ∆Cp (= 3Cvph) = 67.4 J/(K mol*) at Tg= 180 K was the same as that of iPS with Tg= 240 K. Fig. 8 shows the relation between f (= fconf or fconf+ 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 fconf at temperatures below and above 180 K is minus and decreases with an increase in temperature. The absolute value of fconf= – 0.102 kJ/mol at 180 K equaled to hg – h0h= 0.1 kJ/mol at 270 K. Adding the cohesive enthalpy of hint= 0.102 kJ/mol to fconf (= – 0.102 kJ/mol), from f = fconf+ hint= 0 and hconf= 0.18 kJ/mol at Tg= 180 K, hg (= hconf+ hint) = 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 hint and free from ordered part / hole pairs (see Table 5).

Figure 8.

The relation between f and T calculated for RIS model chains (x = 100) of iPP. The thin line is f = fconf and the thick line is f = fconf+ 0.1 kJ/mol.

On the other hand, altering 3Cvph (= Cpph) in Eq. (10) to Cvph, the temperature at hh= 0 was Tg= 0 K as well as iPS. From sconf= 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 hx= 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 fx= 2fu [10, 39], 2hu – hx= 2.9 kJ/mol was derived using hu= 7.5 kJ/mol for α form crystals, corresponding to 2Tm(su – sx/2), which was almost equal to hmint= 2.8 kJ/mol of the cohesive energy of methylene residues in the sequences, where fx and sx are the free energy and entropy per molar structural unit for ordered parts, and fu and su 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, σe, at (dfu/dζ)ϕ= 0 is given by:

σe= μ(RTζ/2)[1/(x - ζ +1) +(1/ζ)ln{(x  ξ +1)/x}] E11

where ϕ is the amorphous fraction and μ is the conversion coefficient of mol/m3. In this context:

2σe/ζ =μfx - fuE12
fx'=RT[(1/ζ)lnx - ζ +1/x-lnPc]E13

where Pc, 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:

fu -fx - fx'=0E14

Eq. (11) is obtained when lnPc= – 1/(x – ζ+ 1). From Eq. (14), 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 4. Class A of fx= fu at fx’= 0 shows the dynamic equilibrium relation between the ordered parts and the crystal parts of same fringe–type, leading to σe= 0, and that, ζ = 0/0 in Eq. (16) (described below). For class B, fu= – fx’ from Eq. (14) with fx= 0 refers to the anti–crystal holes and fx= 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 hx – hu= σe/(μζ), the respective interface energies of the hole and the ordered part are compensated each other at the common interface, thus leading to fx= 0 [10]. For class C, fx= fx’ from Eq. (14) 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. (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’fxfuClass
fx’ = 0fx = fufu = fxA
fx’ > 0fx = 0fu = –fxB
fx = fxfu = 0C
fx = 2fufu = fx/2 = fxD
fx’> 0fx = 0fu = fxX

Table 4.

Relations of equilibrium (A ∼ D) and non–equilibrium (X) in fx and fu at fx’ ≥ 0 for crystalline polymers [10, 39].

Figure 9.

Schematic structure models of bulk polymers. A ∼ X correspond to the classes in Table 4, respectively. •••: ordered parts, ¾¾ : crystals, ⊃ and ⊂: folded segments, the space between ⊃ and ⊂ of B: anti–crystal hole. B is equivalent to B’.

At the rapid glass transition absorbing the photon energy of 2h0h at Tg 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 Tl in Fig. 1. While in the open system that the heat irradiated by crystallization was escaped out of the system, Tl corresponded to Tm (450 K for α form crystals) and h0 (= hg+ hx+ hu)= 21.0 kJ/mol in Eq. (8) agreed perfectly with the value of hh (= 3h0h)+ hmint, where hx (= hg+ ∆h) is 7.4 kJ/mol, being larger than h0h (= hh/3)= 6.1 kJ/mol*. In this context, hg – h0h= 0.1 kJ/mol, hx – h0h= 1.3 kJ/mol, and hu – h0h= 1.4 kJ/mol. The sum of them was equal to hmint= 2.8 kJ/mol. Thus, n (= hh/hx) = 2.5 was shown, almost corresponding to ∆Cp/∆Cpexp= 2.3. Table 5 shows the values of Tg, h0, hu, hg, hx, hh, and hh/hx for iPP. From hh (= 3h0h) = 18.2 kJ/mol* at Tg (= 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.

TgKh0
kJ/mol
hu
kJ/mol
hg
kJ/mol
hx
kJ/mol
hh
kJ/mol*
hh/hx
27018.3*1---6.2*312.118.21.5
27021.1*17.5*26.2*37.418.22.5
1800.28*1---0.28*400---

Table 5.

The values of Tg, h0, hu, hg, hx, hh, and hh/hx for iPP.

*1: from Eq. (8). *2: hu for α form crystals. *3: hg= Hga – Hgc, *4: hg= hgint+ hgconf from hgint= – fgconf at Tg= 180 K.


2.3.1. Equilibrium melting temperature, Tm

For PET discussed in the previous section 2.2, Tl in Fig. 1 corresponded to Tm= 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 Tm= 435 K, 450 K, and 462 K for β, α, and γ form crystals, respectively, which were found by DSC measurements [35]. Above all, Tm= 450 K for α form crystals could be Tl (in Fig.1) of the temperature expected from the quantum demand of hole energy at regular temperature intervals of 90 K. Experimentally, Tm is determined as the intersection temperature between an extrapolation line of Tm= Te in the melting peaks without superheating and a Tm= Ta line, where Tm is the melting temperature from Ta to Te, Ta is the annealing (crystallization) temperature, and Te is the end temperature of melting peak. While at Tm, the processes of melting and crystallization should occur reversibly, so that Tm equals to both temperatures of Te and Tb of the onset temperature of melting, and that, there are two points of Tm at Ta (= Te) and Tb (= Te) on a Tm= Te line. The line through two points of Tm should be parallel to the abscissa of Ta, because Tm is only one [23]. For the bulk contained α form crystals with hu= 7.46 kJ/mol, the sum of hg – h0h= 0.1 kJ/mol, hx – h0h= 1.3 kJ/mol, and hu – h0h= 1.4 kJ/mol was equal to hmint= 2.8 kJ/mol (see section 2.3). While for the bulk contained only γ form crystals with hu= 8.70 kJ/mol, the difference in hmint and (hu – h0h) was 0.2 kJ/mol, suggesting hg= hx= h0h+ 0.1 kJ/mol. Figs. 10 and 11 show the DSC single and double melting peak curves for the iPP films annealed at Ta= 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, Tb 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 Tb 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 Te= 435 K of β form crystals [42].

Figure 10.

DSC single melting peak curve composed of α and β peaks for the iPP film annealed at 461.0 K for 1 hour. The thin line is the part of an original DSC curve. Tb 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. Tb is the onset temperature of γ or α peak and T* is the end temperature of α or β peak.

Table 6 shows the values of Tb, T*, Qm, ∆Qm, ∆hh, and ∆hh/∆Qm in α and γ peak curves for the iPP films annealed at 461.0 K and 441.5 K for 1 hour. Where Qm is the heat per molar structural unit corresponding to the area of α or γ peak from Tb and ∆Qm is the heat per molar structural unit corresponding to the area from Tb 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 Tb to T* of β or α peak, the hole energy per molar photon, ∆hh, is given by [10]:

hh=3Cvph(T* - Tb)E15

As shown in Table 6, the small difference in ∆Qm and ∆hh could be regarded as significant for the formation and then disappearance of crystal / hole pairs from Tb to T*. For the shift from β to α peak in Fig. 11, ∆hh/∆Qm was 1.21 contrary to our expectation, but at T*= 435 K of the mean of T* (Tm for β form crystals), ∆hh/∆Qm= 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 ∆Qm; 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 ∆Qm; 0.20 kJ/mol (= ∆hh), which agreed with the difference in hmint and (hu – h0h) perfectly, corresponding to (hg+ hx) – 2h0h= 0.2 kJ/mol suggested above.

Ta KFormTb
K
T*KQm kJ/mol∆Qm
kJ/mol
∆hh kJ/mol∆hh/∆Qm
461.0α429.6434.22.050.380.240.63
441.5α423.7437.7
(435)
2.160.780.94
(0.76)
1.21
(0.98)
γ446.9449.91.640.330.200.61

Table 6.

The values of Tb, T*, Qm, ∆Qm, ∆hh, ∆hh/∆Qm of α and γ peak curves for the iPP films annealed at 461.0 K and 441.5 K for 1 hour.

The values in ( ) show Tm, ∆hh, and ∆hh/∆Qm at Tm for β form crystals.


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 Tb were converted into the crystal length (ζ) distribution function, F(ζ). The ζ is according to Gibbs–Thomson given by:

ζ={Tm/(Tm -Tm)}{2σe/(μhu)}E16

where Tm is the corrected melting temperature. In the calculation of ζ, the values of Tm, hu, hx, σe, and the corrected Tm are needed previously. For hu, the reference value was used (see Table 8). The value of σe was evaluated by [43, 44]:

σe= μhuc*[{RTm2 +Hx - hxTm - Tm}/{2Hx - hxTm}]E17

with Hx=2hu - Qm

where c* is the cell length of c–axis. The term of square blanket in Eq. (17) is dimensionless. hu refers to the heat of fusion of crystals with a crystal form liking to evaluate σe. hx could be calculated from Eqs.(5) and (6), but in Fig. 11, using Hma at Tm of the other α or γ form crystals (sub–crystals). Table 7 shows the values of hx for the iPP films annealed at 461.0 K and 441.5 K for 1 hour, together with the values of Tc, Tb, Te, Q, ΔH, Δh, and hg used in the calculation of hx. Q and ∆H are defined in Eq. (6). The value of hx for the iPP sample of Ta= 441.5 K was smaller than hu of the sub–crystals (see Table 8). Further, the value of hx in the row of γ form, 6.85 kJ/mol, was ∼0.7 kJ/mol larger than hg (= 6.22 kJ/mol). Also the value of hx in the line of α form, 8.07 kJ/mol, was ∼0.7 kJ/mol larger than hx (= 7.35 kJ/mol) for the sample of Ta= 461.0 K. Therefore, the value of h0 (= hg+ hx+ hu) 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 Ta= 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 KTb
K
Te
K
QkJ/molΔHkJ/molΔhkJ/molhg
kJ/mol
hx
kJ/mol
α (461.0)403.6429.6449.23.764.891.136.227.35
α (441.5)403.6423.7449.54.416.26 (γ) *21.856.228.07
γ (441.5)403.6446.9461.94.415.04 (α) *20.636.226.85

Table 7.

The values of hx, Tc, Tb, Te, Q, ΔH, Δh, and hg for the iPP films annealed at 461.0 K and 441.5 K for 1 hour.

*1: The form of main crystals liking to evaluate σe. *2: ΔH calculated using Hma at Tm of the sub–crystals with the form of α or γ shown in ( ).


Ta
K
FormTp
K
hu
kJ/mol
hx
kJ/mol
h0
kJ/mol
Qm kJ/molσe
J/m2
461.0α435.97.467.3514.82.0536.0×10−3
(26.7×10−3)
441.5α442.87.468.0715.52.1645.2×10−3
(30.8×10−3)
γ450.98.706.8515.61.6426.5×10−3
(22.2×10−3)

Table 8.

The values of σe, Tp, hu, hx, h0 (= hx+ hu), and Qm for the iPP films annealed at 461.0 K and 441.5 K for 1 hour.

The values in ( ) are σe at Tm (Qm= 0).


Table 8 shows the values of σe at Tp 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 Tp, hu, hx, h0, and Qm used in the calculation of σe, where Tp is the melting peak temperature. The σe of α form was larger than that of γ form, because according to Eq. (17), the σe was mainly dependent on hx. For α and γ forms in the sample of Ta= 441.5 K, h0 (= hu+ hx) at T (> Tg) of Eq. (9) was ∼15.5 kJ/mol, nevertheless the values of hu were different. Tp is corrected by 0.6 K (436.5 K → 435.9 K) for the sample of Ta= 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 Ta= 441.5 K to the lower temperature side, according to our concept [45].

F(ζ) is defined as [23]:

Fζ=(δQm/Qm)/ζ= nζ/{NcTe - Tb}E18

where δQm (= ζnζQm/{Nc(Te – Tb)}) is the heat change per molar structural unit per K, nζ is the number of crystal sequences with ζ, Nc is the number of structural units of crystals melted in the temperature range from Tb to Te. δQm/Qm is given by:

δQm/Qm = (dQ/dt)/TbTe(dQ/dt)dT E19

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.

Figure 12.

F(ζ) of α peak curve for the iPP film annealed at 461.0 K for 1 hour.

Figure 13.

f(ζ) of α (thick line) and γ (thin line) peak curves for the iPP film annealed at 441.5 K for 1 hour.

Table 9 shows the ζ range and ζp in F(ζ) of α and γ peak curves obtained for the samples of Ta= 461.0 K and 441.5 K, where ζp 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 ζp= 14.6 nm, and in Fig 13, F(ζ) showed the roundish curve with the ζ range of 10 nm ∼ 250 nm and ζp= 19.5 nm, whereas for γ peak, F(ζ) showed the sharp peak with the ζ range of 8 nm ∼ 840 nm and ζp= 11.2 nm. The maximum of ζ was calculated using Te (∼Tm) observed actually for each sample. At Te= Tm, the maximum of ζ should be infinite at σe ≠ 0, because the melt at Te could be interchanged in equilibrium to the imaginary crystals of ζ = ∞. In the σe= 0 of the class A in Table 4, ζ = 0/0 of indetermination at Tm 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 Tb (= 446.9 K) in Fig. 11. For the sample of Ta= 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 hx 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 Ta= 461.0 K was much larger than those of α and γ form crystals for the sample of Ta= 441.5 K.

Ta
K
Formζ rangenmζp
nm
461.0α10 - 387014.6
441.5α10 - 25019.5
γ8 - 84011.2

Table 9.

The ζ range and ζp 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, Rn, is given by [10]:

Rn=(N/π)1/2E20

with N= Nc(Te  - Tb)(ζnζxFζdζ - ζnζFζdζ)

where ζx and ζn are the maximum and the minimum of ζ, respectively. Figs. 14 and 15 show the representation of R (= ±Rn) 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.

Figure 14.

Representation of R (= ±Rn) and L (= ±ζ/2) for α form crystals in the iPP film (per 1g) annealed at 461.0 K for 1 hour. The horizontal lines show R of the imaginary crystals melting from ζn to ζ= 0.

Figure 15.

Representation of R (= ±Rn) and L (= ±ζ/2) for α (thick line) and γ (thin line) form crystals in the iPP film (per 1g) annealed at 441.5 K for 1 hour. The horizontal lines show R of the imaginary crystals melting from ζ n to ζ = 0.

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 Tg= 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. Tm= 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 Cvph

1 – α : fraction of ordered part / hole pairs with Cpph

Cp : mean heat capacity per molar structural unit for ordered parts in ordered part / hole pairs

Cph : mean heat capacity per molar photon for holes in ordered part / hole pairs

Cph*: heat capacity per molar photon for holes in excited ordered part /hole pairs

Cpph : adiabatic molar heat capacity for photons

Cvph : constant volume molar heat capacity for photons

Cpflow : heat capacity per molar structural unit for flow parts

Cpr : heat capacity change per molar structural unit due to crystallization followed by melting

Cpx: 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

c1 and c2 : constants in WLF equation

dQ/dT : heat flow rate of DSC melting curves

δQm : heat change per molar structural unit per K

ΔCp : heat capacity jump per molar photon at the glass transition

ΔCpexp : experimental heat capacity jump per molar structural unit at the glass transition

ΔCpx : 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

Δhh : hole energy of crystal / hole pairs formed newly by recrystallization

δ : solubility parameter

Ea : activation energy

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

f : free energy per molar structural unit

fconf : conformational free energy per molar structural unit

fgconf : conformational free energy per molar structural unit at Tg

fx : free energy per molar structural unit for ordered parts

fu : free energy per molar structural unit for crystals

ϕ : amorphous fraction

Γ : frequency of occurrence of the helix–coil transition

Hma : enthalpy per molar structural unit for the liquid at Tm

Hca : enthalpy per molar structural unit for the super–cooled liquid at Tc

Hga : enthalpy per molar structural unit for the super–cooled liquid at Tg

Hgc : enthalpy per molar structural unit for the crystal at Tg

hh : hole energy per molar photon for holes in ordered part / hole pairs

h0 : latent cohesive energy per molar structural unit

h0h : zero–point energy per molar photon, or energy unit per molar photon

hu : heat of fusion per molar structural unit

hx : enthalpy per molar structural unit for ordered parts

hg : glass transition enthalpy per molar structural unit

hconf : conformational enthalpy per molar structural unit

hgconf : conformational enthalpy per molar structural unit at Tg

hxconf : conformational enthalpy per molar structural unit for ordered parts

hint : cohesive enthalpy per molar structural unit

hgint : cohesive enthalpy per molar structural unit at Tg

hmint : cohesive energy per molar structural unit for methylene residues

h : Plank constant

η : statistical weight

Jh : number of holes lost by T at the glass transition

φg : 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/m3

NA : Avogadro constant

Nc : number of structural units of crystals melted in the temperature range from Tb to Te

n : number of structural units holding one photon potentially

nζ : number of crystal sequences with ζ

ν : frequency per second

P : pressure

Pc : 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

Qm : heat per molar structural unit corresponding to the area of a DSC melting curve from Tb

R : gas constant

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

su : entropy of fusion per molar structural unit

sx : entropy per molar structural unit for ordered parts

sconf : conformational entropy per molar structural unit

sgconf : conformational entropy per molar structural unit at Tg

σe : end surface free energy of a crystal per unit area

σ : statistical weight

T : temperature

Tg : glass transition temperature

Tm : melting temperature

Tm : equilibrium melting temperature

Te and T* : end temperature of DSC melting peak curve

Tl : end temperature of the glass transition

Tc : onset temperature of DSC crystallization peak curve upon cooling

Tb : onset temperature of DSC melting peak curve upon heating

Ta : annealing temperature

Tp : DSC melting peak temperature

V : volume per molar structural unit

vf : free volume per molar structural unit

v0 : core free volume per molar structural unit

x : degree of polymerization

Z : conformational partition function for a chain

Z0 : component conformational partition function for a chain regardless of temperature

Zt : component conformational partition function for a chain depending on temperature

ζ : crystal length

ζp : crystal length at the maximum of F(ζ)

ζn : crystal length at the minimum of ζ

ζx : 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.

© 2015 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Nobuyuki Tanaka (May 6th 2015). Photonic Contribution to the Glass Transition of Polymers, Advanced Topics in Crystallization, Yitzhak Mastai, IntechOpen, DOI: 10.5772/59717. Available from:

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