**Thermodynamics of Enthalpy Relaxation and Hole Formation of Polymer Glasses**

Nobuyuki Tanaka

*Gunma University, Tsutsumicho, Kiryu, Japan* 

## **1. Introduction**

162 Advances in Crystallization Processes

Zhang, S.N., Zhu, T.J. & Zhao, X.B. (2008). Crystallization Kinetics of Si15Te85 and

0921-4526

Si20Te80 Chalcogenide Glasses. *Physica B-Condensed Matter*, 403, pp. 3459-3463,

The enthalpy relaxation of the glassy materials has been investigated rheologically for years with a view to approaching the ideal glass1 - 5). The imaginary liquid at Kauzmann temperature2, 3), TK, at which the extrapolation line of enthalpy or entropy as a function of temperature for the liquid intersected the enthalpy or entropy line of the crystal, had been considered once to be the ideal glass. However because at TK, the enthalpy or entropy for the liquid was same as that of crystal, the liquid like this was hard to take thermodynamically, bringing the entropy crisis. Fig. 1 depicts the change of the enthalpy difference, ΔH, between the liquid and the crystal upon cooling for a polymer. For stable liquids, ΔH should be almost constant from near Tg upon cooling as described below. Therefore, the liquid line can never intersect that of the crystal. The transition from liquid to crystal or vice versa means the emission or absorption of the latent heat accompanying the enthalpy jump. Thus for polymers, TK is merely a temperature parameter.

Fig. 1. The change of ΔH for a polymer liquid or liquid crystal (solid line) upon cooling. The small arrow mark shows the direction of enthalpy relaxation. The base line is that of crystal. The dashed lines are extrapolated to Tg and TK, and the dot line is complementary.

Thermodynamics of Enthalpy Relaxation and Hole Formation of Polymer Glasses 165

 sflow = 0 (∴ hflow = 0) (2) where fx (= hx ‒ Tgsx), hx and sx are the free energy, enthalpy and entropy per molar structural unit for ordered parts, fflow, hflow and sflow are the free energy, enthalpy and entropy per molar structural unit for flow parts. The molar free energy of holes held photons at the temperature, T, is generally given by fh = −RTln(vf/v0) and then the molar

 sh = −(∂fh/∂T)p = Rln(vf/v0) + RT{∂ln(vf/v0)/∂T}p (3) where vf and v0 are the molar free and core volumes of holes and R is the gas constant. When v0 is almost constant, from fh = hh − Tsh and Eq. (3), the molar enthalpy of holes, h<sup>h</sup>

 hh = RT2(∂lnvf/∂T)p (4) When the ordered parts at Tg are in equilibrium with the holes; fx (= 0) = fh, from fx = 0 and

 sx (= hx/Tg) = Δsg + Δh/Tg (5) with Δsg = hg/Tg, defining the entropy of unfreezing for the glass parts at Tg, where hg is the molar glass transition enthalpy, Δh is the heat per molar structural unit required still to melt ordered parts, relating to the jump of vf (see Eq. (8)). Further from Eq. (2), the hg and the

constant, *h* is Planck constant, N is the number of chains, x is the degree of polymerization and q (≤ 1) is the packing factor. When vf = v0, from Eqs. (6) and (7), hg = 0 and sg = 0 are

In the case of RTgln(vf/v0) = Δh, the relation of hx = hh is derived from Eq. (8), because of

2(∂lnvf/∂T)p = hg. However, when the length distribution by lengthening of ordered parts occurred during the enthalpy relaxation at temperatures below Tg, as longer the length of ordered parts, the melting temperature, Tx, for ordered parts should be elevated from Tg to the higher temperature23) (see Eq. (29)). Therefore the shortage of Δh (= RTgln(vf/v0)) corresponding to the latent heat of disappearance for the holes at Tg is made up by the

2(∂lnvf/∂T)p (6)

int) = Rln(vf/v0) + RTg(∂lnvf/∂T)p (7)

conf), m is the mass of a structural unit, k is Boltzmann

2(∂lnvf/∂T)p + RTgln(vf/v0) (8)

Δh = TgTℓΔCpdT (9)

int are the conformational and cohesive entropies per molar structural unit

, is

entropy of holes, sh, at a constant pressure, p, is derived:

hx = hg + Δh (see Eq. (13)), sx at Tg is derived:

molar glass transition entropy, sg, are derived13):

conf + sg

derived. On the other hand, from fh = 0 (= fx) at Tg, hh is given as:

int = (3R/2)ln(2πmkTg/*h*2) − (1/x)(R/N)lnN! + Rlnq − Rlnv0

hg = RTg

sg (= sg

for glass parts at Tg (see Eq. (16) for sg

hh = RTg

supply of the heat required to melt all ordered parts:

conf and sg

derived:

with sg

where sg

RTg

For the glass transition of polymeric or other matter glasses, whether it is the phase transition or not is nowadays yet controversial6 - 11). However for polymers, the criterion of the glass transition temperature, Tg, presented by us is decisive, supporting the first order hole phase transition12 - 14) and the broad heat capacity jump at the glass transition without the enthalpy relaxation has been understood successfully15). The cooled polymers unidentified by the criterion should belong to the liquid on the way of enthalpy relaxation even if they were glassy. When without the enthalpy relaxation at temperatures below Tg, the super-cooled liquid could be stable thermodynamically. Glassy pitch4) might be the stable liquid, which is flowing still at a drop per 9 years from 1927. Recently, for poly(ethylene terephthalate) (PET) 16), polystyrene (PS), isotactic polypropylene (iPP), polyethylene (PE) and nylon-6 (N6), it was predicted that during the enthalpy relaxation at temperatures below Tg, the "ordered part / hole" pairs should generate and consequently, bringing a reduction in the relaxation time to the goal of the ideal glass, e.g., the quasicrystals lacked periodicity but with symmetry17), the alternative and stable glass parts should be formed between their pairs. At Tg, the unfreezing of glass parts is caused by the first order hole phase transition. Then the holes should play the free volume jump as an affair of dynamic equilibrium in the disappearance and generation of "ordered part / hole" pairs. The free volume jump at the glass transition, accompanying the enthalpy and entropy jumps, is the characteristic of the first order hole phase transition. In this chapter, introducing the constant volume heat capacity of photons to the holes16, 18, 19), the generation of "ordered part / hole" pairs during the enthalpy relaxation at temperatures below Tg and the subsequent disappearance at the glass transition, accompanied by the jumps of free volume, enthalpy and entropy, were discussed for PET, PS, iPP, PE and N6 on the basis of thermodynamics. IPP, PE and N6 were investigated as the peculiar cases for the comparison with PET and PS, which are the glassy polymers with the almost same values of the constant, c2, of WLF equation20), i.e., 55.3 K21) and 56.6 K20), respectively. The holes taken in the helix structure of iPP should hold the interaction of partnership with the helical ordered parts. For PE, the glasses with Tg = 135 K and 237 K depending on the structure of the ordered parts15) were discussed. It seems likely that from near 130 K, the growth of open space in the PE glasses occurs22). DSC (Differential Scanning Calorimetry) on PE films18) revealed that ∼38 % of the crystal lamella was constituted by the inter-grain aggregates containing the glass with a secondary Tg. The generation of "crystal / anti-crystal hole" pairs from the secondary glass was discussed. For N6, the ordered parts were the stretched sequences of -(CH2)5- between amido groups. The hole energy of "ordered part / hole" pairs was concerned with the frequency of absorption bands in the infrared spectrum.

#### **2. Enthalpy relaxation and hole formation of PET, PS, iPP, PE and N6 glasses**

#### **2.1 Thermodynamics of glass transition**

The glass transition temperature, Tg, of polymer glasses could be identified as that of the first order hole phase transition by satisfying the criterion consisted of Eqs. (1) and (2), in which fx has been added under the operational definition concluding that the stable glasses could not be formed easily without the generation of "ordered part / hole" pairs during the enthalpy relaxation at temperatures below Tg13):

$$\mathbf{f\_x = f\_{flow}} \ (= \mathbf{h\_{flow}} - \mathbf{T\_{\beta}s\_{flow}}) = \mathbf{0} \tag{1}$$

For the glass transition of polymeric or other matter glasses, whether it is the phase transition or not is nowadays yet controversial6 - 11). However for polymers, the criterion of the glass transition temperature, Tg, presented by us is decisive, supporting the first order hole phase transition12 - 14) and the broad heat capacity jump at the glass transition without the enthalpy relaxation has been understood successfully15). The cooled polymers unidentified by the criterion should belong to the liquid on the way of enthalpy relaxation even if they were glassy. When without the enthalpy relaxation at temperatures below Tg, the super-cooled liquid could be stable thermodynamically. Glassy pitch4) might be the stable liquid, which is flowing still at a drop per 9 years from 1927. Recently, for poly(ethylene terephthalate) (PET) 16), polystyrene (PS), isotactic polypropylene (iPP), polyethylene (PE) and nylon-6 (N6), it was predicted that during the enthalpy relaxation at temperatures below Tg, the "ordered part / hole" pairs should generate and consequently, bringing a reduction in the relaxation time to the goal of the ideal glass, e.g., the quasicrystals lacked periodicity but with symmetry17), the alternative and stable glass parts should be formed between their pairs. At Tg, the unfreezing of glass parts is caused by the first order hole phase transition. Then the holes should play the free volume jump as an affair of dynamic equilibrium in the disappearance and generation of "ordered part / hole" pairs. The free volume jump at the glass transition, accompanying the enthalpy and entropy jumps, is the characteristic of the first order hole phase transition. In this chapter, introducing the constant volume heat capacity of photons to the holes16, 18, 19), the generation of "ordered part / hole" pairs during the enthalpy relaxation at temperatures below Tg and the subsequent disappearance at the glass transition, accompanied by the jumps of free volume, enthalpy and entropy, were discussed for PET, PS, iPP, PE and N6 on the basis of thermodynamics. IPP, PE and N6 were investigated as the peculiar cases for the comparison with PET and PS, which are the glassy polymers with the almost same values of the constant, c2, of WLF equation20), i.e., 55.3 K21) and 56.6 K20), respectively. The holes taken in the helix structure of iPP should hold the interaction of partnership with the helical ordered parts. For PE, the glasses with Tg = 135 K and 237 K depending on the structure of the ordered parts15) were discussed. It seems likely that from near 130 K, the growth of open space in the PE glasses occurs22). DSC (Differential Scanning Calorimetry) on PE films18) revealed that ∼38 % of the crystal lamella was constituted by the inter-grain aggregates containing the glass with a secondary Tg. The generation of "crystal / anti-crystal hole" pairs from the secondary glass was discussed. For N6, the ordered parts were the stretched sequences of -(CH2)5- between amido groups. The hole energy of "ordered part / hole" pairs

was concerned with the frequency of absorption bands in the infrared spectrum.

**2.1 Thermodynamics of glass transition** 

enthalpy relaxation at temperatures below Tg13):

**2. Enthalpy relaxation and hole formation of PET, PS, iPP, PE and N6 glasses** 

The glass transition temperature, Tg, of polymer glasses could be identified as that of the first order hole phase transition by satisfying the criterion consisted of Eqs. (1) and (2), in which fx has been added under the operational definition concluding that the stable glasses could not be formed easily without the generation of "ordered part / hole" pairs during the

fx = fflow (= hflow ‒ Tgsflow) = 0 (1)

$$\mathbf{s}\_{\text{flow}} = \mathbf{0} \left( \vdots \ \mathbf{h}\_{\text{flow}} = \mathbf{0} \right) \tag{2}$$

where fx (= hx ‒ Tgsx), hx and sx are the free energy, enthalpy and entropy per molar structural unit for ordered parts, fflow, hflow and sflow are the free energy, enthalpy and entropy per molar structural unit for flow parts. The molar free energy of holes held photons at the temperature, T, is generally given by fh = −RTln(vf/v0) and then the molar entropy of holes, sh, at a constant pressure, p, is derived:

$$\mathbf{s}^{\text{h}} = - (\partial \mathbf{f}^{\text{h}} / \partial \mathbf{T})\_{\text{F}} = \text{Rln}(\mathbf{v}\_{l} / \mathbf{v}\_{0}) + \text{RT} \{ \partial \ln(\mathbf{v}\_{l} / \mathbf{v}\_{0}) / \partial \mathbf{T} \}\_{\text{F}} \tag{3}$$

where vf and v0 are the molar free and core volumes of holes and R is the gas constant. When v0 is almost constant, from fh = hh − Tsh and Eq. (3), the molar enthalpy of holes, h<sup>h</sup> , is derived:

$$\mathbf{h}^{\text{h}} = \mathbf{RT}^{\text{2}} (\partial \ln \text{nv}/\partial \mathbf{T})\_{\text{F}} \tag{4}$$

When the ordered parts at Tg are in equilibrium with the holes; fx (= 0) = fh, from fx = 0 and hx = hg + Δh (see Eq. (13)), sx at Tg is derived:

$$\mathbf{s}\_{\infty} \left(= \mathbf{h}\_{\infty} / \mathbf{T}\_{\mathcal{G}}\right) = \Delta \mathbf{s}\_{\mathcal{G}} + \Delta \mathbf{h} / \mathbf{T}\_{\mathcal{G}} \tag{5}$$

with Δsg = hg/Tg, defining the entropy of unfreezing for the glass parts at Tg, where hg is the molar glass transition enthalpy, Δh is the heat per molar structural unit required still to melt ordered parts, relating to the jump of vf (see Eq. (8)). Further from Eq. (2), the hg and the molar glass transition entropy, sg, are derived13):

$$\mathbf{h}\_{\mathbb{R}} = \mathbb{RT}\_{\mathbb{R}} \mathbf{2} (\partial \text{lrv} \sqrt{\partial} \mathbf{T})\_{\mathbb{P}} \tag{6}$$

$$\mathbf{s}\_{\mathfrak{F}} \left( \mathbf{=} \mathbf{s}\_{\mathfrak{F}} \mathbf{^{\rm curl}} + \mathbf{s}\_{\mathfrak{F}} \mathbf{^{\rm int}} \right) = \mathbf{R} \ln(\mathbf{v}\_{l}/\mathbf{v}\_{0}) + \mathbf{R} \Gamma\_{\mathfrak{F}} (\mathbf{\hat{\boldsymbol{\sigma}}} \mathbf{l} \mathbf{n} \mathbf{v}\_{l}/\mathbf{\hat{\boldsymbol{\sigma}}} \mathbf{I})\_{\mathfrak{F}} \tag{7}$$

with sg int = (3R/2)ln(2πmkTg/*h*2) − (1/x)(R/N)lnN! + Rlnq − Rlnv0

where sg conf and sg int are the conformational and cohesive entropies per molar structural unit for glass parts at Tg (see Eq. (16) for sg conf), m is the mass of a structural unit, k is Boltzmann constant, *h* is Planck constant, N is the number of chains, x is the degree of polymerization and q (≤ 1) is the packing factor. When vf = v0, from Eqs. (6) and (7), hg = 0 and sg = 0 are derived. On the other hand, from fh = 0 (= fx) at Tg, hh is given as:

$$\mathbf{h}^{\text{h}} = \mathbf{R} \mathbf{T}\_{\mathbb{R}} 2 (\partial \ln \mathbf{v}\_{l} / \partial \mathbf{T})\_{\mathbb{P}} + \mathbf{R} \mathbf{T}\_{\mathbb{R}} \ln (\mathbf{v}\_{l} / \mathbf{v}\_{0}) \tag{8}$$

In the case of RTgln(vf/v0) = Δh, the relation of hx = hh is derived from Eq. (8), because of RTg 2(∂lnvf/∂T)p = hg. However, when the length distribution by lengthening of ordered parts occurred during the enthalpy relaxation at temperatures below Tg, as longer the length of ordered parts, the melting temperature, Tx, for ordered parts should be elevated from Tg to the higher temperature23) (see Eq. (29)). Therefore the shortage of Δh (= RTgln(vf/v0)) corresponding to the latent heat of disappearance for the holes at Tg is made up by the supply of the heat required to melt all ordered parts:

$$
\Delta \mathbf{h} = \left[ \mathbf{r}\_{\mathbb{Z}} \mathbf{T}^{\mathbb{L}} \Delta \mathbf{C}\_{\mathbb{P}} \mathbf{d} \mathbf{T} \right] \tag{9}
$$

Thermodynamics of Enthalpy Relaxation and Hole Formation of Polymer Glasses 167

after relaxation shows the subsequent s curve with a jump at Tg. Upon cooling in Fig. 2 (lower), the dashed line is the vf curve for the same liquid glass. Upon heating after relaxation, the solid line shows the vf curve with a jump at Tg and the dashed line shows a

Next whether hx agrees to hh at Tg or not is investigated for PET, iPP, PS, PE and N6 glasses. The agreement provides one of evidences for the generation of "ordered part / hole" pairs

 hh = 3CvphTg (12) where Cvph = 2.701R. For PET with Tg = 342 K, hh = 23.0 kJ/mol was derived. While hx at Tg

hx = hg + Δh (13)

of WLF equation20)), (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

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

where hxconf is the conformational enthalpy per molar structural unit for ordered parts, Δhint is the molar cohesive enthalpy difference between the ordered parts and the glass parts.

where 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 as a function of temperature, respectively. The differential of Eq. (15) by temperature represents

Table 1 shows the values of Tg, Δsg, hg, Δh, hx, hh and hh/hx for PET, iPP and PS. PET showed the good agreement between hx, i.e., 22.3 ∼ 24.1 kJ/mol, and hh, i.e., 23.0 kJ/mol. The values of them also agreed with the heat of fusion, hu = 23.0 kJ/mol, for the smectic crystals of mesophase with the conformational disorder between the phenylene groups but along

Δh = (hxconf − hg

Δh = Tg{sg

int. Δh is given by either Eq. (14) or (15)25, 26):

a, where Hma is the enthalpy per molar structural unit for the liquid at

conf at Tg, Δh = Δhint = (RTglnZt)/x (see Table 3 for N6) and when hxconf ≠

Δh = ΔH ‒ Q (14)

a is the enthalpy per molar structural unit for

conf) + Δhint (15)

conf ‒ (RlnZ0)/x} (16)

2/c2 (c2 is the constant

reversible jump of vf between Tg and Tℓ.

during the enthalpy relaxation at T (< Tg). hh at Tg is given by16):

In Eq. (13), hg is given approximately by three expressions13, 25); (1) RTg

conf + hg

**2.2 "Ordered part / hole" pairs** 

per molar structural unit at Tg, hg

the equilibrium melting temperature, Tm<sup>∞</sup>, Hc

conf = 0 at Tg, the another Δh is derived26, 27).

conf = (RlnZ + RTgdlnZ/dT)/x

the heat capacity jump at the glass transition15).

DSC endothermic peak upon heating. Or, rewriting Eq. (13),

is given by13, 25):

with ΔH = Hma ‒ Hc

Thus when hxconf = hg

hg

with sg

where Tℓ is the end temperature of melting for ordered parts, ΔCp is the difference between the observed isobaric heat capacity, Cp <sup>ℓ</sup>, for the equilibrium liquid and Cpg for the hypothesized super-heated glass at the glass transition from Tg to Tℓ. In the equilibrium liquid, the isobaric heat capacities of ordered parts and flow parts, Cpx and Cp flow, are equal to Cp <sup>ℓ</sup>, respectively13):

$$\mathbf{C}\_{p}\boldsymbol{\ell} = \mathbf{C}\_{p}\mathbf{x} = \mathbf{C}\_{\mathbf{p}}\mathbf{f}^{\text{flow}} \tag{10}$$

In the flow parts, the tube-like space exists between a chain and the neighboring chains, behaving as if it is the counterpart of a chain24). Therefore when the hole energy at T (> Tg) is given by ε (= 3CvphT), Cp flow is represented as16):

$$\mathbf{C}\_{\text{flow}} = \mathbf{\mathcal{BC}}\_{\text{vhp}} (1 + \mathbf{T} \mathbf{d} \ln \mathbf{I} / \mathbf{q} \mathbf{I}) \tag{11}$$

where Cvph (= 2.701R)16, 18, 19) is the constant volume heat capacity for photons, J is the number of holes lost by T and 3 is the degree of freedom for photons.

Fig. 2 shows the schematic curves of the molar entropy, s, and the vf around Tg upon cooling and heating for polymers. Upon cooling in Fig. 2 (upper), the dashed line is the s curve for the liquid glass frozen partially from the super-cooled liquid and the solid line upon heating

Fig. 2. The schematic curves of the entropy, s, and the free volume, vf, around Tg for polymers. Upper; **1**: the change of s for the liquid glass frozen partially from the supercooled liquid, shown by the dashed line, **2**: the entropy relaxation and **3**: the change of s with a jump at Tg upon heating. Lower; The dashed line upon cooling is the vf curve for the same liquid glass, the solid line upon heating after relaxation shows the vf curve with a jump at Tg and the dashed line shows a reversible jump of vf between Tg and Tℓ.

after relaxation shows the subsequent s curve with a jump at Tg. Upon cooling in Fig. 2 (lower), the dashed line is the vf curve for the same liquid glass. Upon heating after relaxation, the solid line shows the vf curve with a jump at Tg and the dashed line shows a reversible jump of vf between Tg and Tℓ.

#### **2.2 "Ordered part / hole" pairs**

166 Advances in Crystallization Processes

where Tℓ is the end temperature of melting for ordered parts, ΔCp is the difference between

hypothesized super-heated glass at the glass transition from Tg to Tℓ. In the equilibrium

<sup>ℓ</sup> = Cpx = Cp

In the flow parts, the tube-like space exists between a chain and the neighboring chains, behaving as if it is the counterpart of a chain24). Therefore when the hole energy at T (> Tg) is

where Cvph (= 2.701R)16, 18, 19) is the constant volume heat capacity for photons, J is the

Fig. 2 shows the schematic curves of the molar entropy, s, and the vf around Tg upon cooling and heating for polymers. Upon cooling in Fig. 2 (upper), the dashed line is the s curve for the liquid glass frozen partially from the super-cooled liquid and the solid line upon heating

Fig. 2. The schematic curves of the entropy, s, and the free volume, vf, around Tg for polymers. Upper; **1**: the change of s for the liquid glass frozen partially from the supercooled liquid, shown by the dashed line, **2**: the entropy relaxation and **3**: the change of s with a jump at Tg upon heating. Lower; The dashed line upon cooling is the vf curve for the same liquid glass, the solid line upon heating after relaxation shows the vf curve with a jump at Tg and the dashed line shows a reversible jump of vf between Tg and Tℓ.

liquid, the isobaric heat capacities of ordered parts and flow parts, Cpx and Cp

flow is represented as16):

number of holes lost by T and 3 is the degree of freedom for photons.

<sup>ℓ</sup>, for the equilibrium liquid and Cpg for the

flow = 3Cvph(1 + TdlnJ/dT) (11)

flow (10)

flow, are equal

the observed isobaric heat capacity, Cp

Cp

Cp

<sup>ℓ</sup>, respectively13):

given by ε (= 3CvphT), Cp

to Cp

Next whether hx agrees to hh at Tg or not is investigated for PET, iPP, PS, PE and N6 glasses. The agreement provides one of evidences for the generation of "ordered part / hole" pairs during the enthalpy relaxation at T (< Tg). hh at Tg is given by16):

$$\mathbf{h}^{\text{h}} = \mathcal{G} \mathbf{C}^{\text{h}} \mathbf{h} \mathbf{I}^{\text{g}} \tag{12}$$

where Cvph = 2.701R. For PET with Tg = 342 K, hh = 23.0 kJ/mol was derived. While hx at Tg is given by13, 25):

$$\mathbf{h}\_{\mathbf{x}} = \mathbf{h}\_{\mathbf{y}} + \Delta \mathbf{h} \tag{13}$$

In Eq. (13), hg is given approximately by three expressions13, 25); (1) RTg 2/c2 (c2 is the constant of WLF equation20)), (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, hg conf + hg int. Δh is given by either Eq. (14) or (15)25, 26):

$$
\Delta \mathbf{h} = \Delta \mathbf{H} - \mathbf{Q} \tag{14}
$$

with ΔH = Hma ‒ Hc a, where Hma is the enthalpy per molar structural unit for the liquid at the equilibrium melting temperature, Tm<sup>∞</sup>, Hc a 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. Or, rewriting Eq. (13),

$$
\Delta \mathbf{h} = \left( \mathbf{h}\_{\text{x}}{}^{\text{conf}} - \mathbf{h}\_{\text{\text{\textdegree}}}{}^{\text{conf}} \right) + \Delta \mathbf{h}^{\text{int}} \tag{15}
$$

where hxconf is the conformational enthalpy per molar structural unit for ordered parts, Δhint is the molar cohesive enthalpy difference between the ordered parts and the glass parts. Thus when hxconf = hg conf at Tg, Δh = Δhint = (RTglnZt)/x (see Table 3 for N6) and when hxconf ≠ hg conf = 0 at Tg, the another Δh is derived26, 27).

$$
\Delta \mathbf{h} = \mathbf{T}\_{\mathbb{K}} \{ \mathbf{s}\_{\mathbb{K}}^{\text{conf}} - (\mathbf{R} \ln \mathbf{Z}\_0) / \mathbf{x} \} \tag{16}
$$

with sg conf = (RlnZ + RTgdlnZ/dT)/x

where 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 as a function of temperature, respectively. The differential of Eq. (15) by temperature represents the heat capacity jump at the glass transition15).

Table 1 shows the values of Tg, Δsg, hg, Δh, hx, hh and hh/hx for PET, iPP and PS. PET showed the good agreement between hx, i.e., 22.3 ∼ 24.1 kJ/mol, and hh, i.e., 23.0 kJ/mol. The values of them also agreed with the heat of fusion, hu = 23.0 kJ/mol, for the smectic crystals of mesophase with the conformational disorder between the phenylene groups but along

Thermodynamics of Enthalpy Relaxation and Hole Formation of Polymer Glasses 169

hx was almost equal to hu (= 7.6 kJ/mol for α form). Fig. 4 shows the photon sites in the helix structure and the helical conformation of an isolated sequence with an inversion defect

Fig. 4. Upper: The photon sites (the dashed line parts) in the helix structure with TGTG or TG'TG' conformation at temperatures below Tg for iPP. Large circle: C and small circle: H. Lower: The helical conformation of an isolated sequence with an inversion defect isomer *TT*, taking preferentially at temperatures below 70 K. The allow mark shows the shift of *TT* on a

For PS, supposing hx = hh, Δh was evaluated. From the value of Δh to be near that of PET, the vf jump at Tg should be due to the release between phenyl groups. The Tg of PE 15), producing the entropy of unfreezing for the glass parts; Δsg = hg/Tg (see Eq. (5)), was almost

Tg = 135 K. Table 2 shows the values of Tg, Δsg, hg, Δh, hx, hh and hh/hx for PE glasses with Tg

int for the glass parts was linked to the ordered parts. For both glasses, hh was about 5 times as much as hg. Thus from Eq. (13), the common relations of hx = hh/4 and Δh = hh/4 − hg were predicted for the ordered parts in both glasses and shown in Table 2. Fig. 5 depicts the sequence models of ordered parts (A and B) and the schematic transition from the glassy

For the glass with Tg = 135 K, the coarse 4/1 helical ordered parts with GG or G'G' conformation in Fig. 5A and as the hole of a pair, the inside space holding four photons per a helical segmental unit, -(CH2)4-, were predicted. Further the length distribution of helical ordered parts in the glass and as the end temperature of melting for the ordered parts, ∼237 K were predicted. In this case, the same value of Δh for both glasses enabled the scheme as depicted in Fig. 6. For the glass with Tg = 237 K, the ordered part of fringe-type formed by

> Δh kJ/mol

> > 0.5 0.5

hx kJ/mol

> 2.3\*2 4.0\*2

hh

9.1 16.0

kJ/mol hh/hx

4 (1\*3) 4 (1\*3)

hg kJ/mol

> 1.8\*1 3.5\*1

int was that of the glass with Tg = 237 K, hg

int = 2.8 kJ/mol). The above relation in Tg and

int/2 gave

isomer *TT*, taking preferentially at temperatures below 70 K26).

int. When a value of hg

int = 2.8/2 kJ/mol) and 237 K (hg

Δsg J/(K mol)

> 13.8 14.8

Table 2. The values of Tg, Δsg, hg, Δh, hx, hh and hh/hx for PE.

state (C: left) to that of the "ordered part / hole" pair (C: right).

helical sequence.

dependent on hg

Polymer Tg

PE 135

K

237

\*1: hgconf + hgint, \*2: hx = hh/4 and \*3: (hh/4)/hx.

= 135 K (hg

hg


\*1: Hga − Hgc, \*2: RTg2/c2, \*3: hgconf + hgint, \*4: Eq. (14), \*5: Eq. (16), \*6: hx = hh and \*7: (hh/2.5)/hx. The data of iPP used to calculate Δh in Eq. (14) are as follows: Tc = 403.6 K, Tm<sup>∞</sup> = 449 K for the α form crystal28), ΔH (=Hma − Hc a) = 4.89 kJ/mol29) and Q = 3.76 kJ/mol for the sample annealed at 461.0 K for 1 hour.

Table 1. The values of Tg, Δsg, hg, Δh, hx, hh and hh/hx for PET, iPP and PS.

a chain axis25, 30). For the smectic-c crystals with stretched sequences, hu is 28.5 kJ/mol. Further, DSC revealed25) that for the crystalline films of smectic-c crystals, the ordered parts in the amorphous regions were like smectic crystals and for the crystalline films of smectic crystals, the ordered parts were like the smectic-c crystals. Fig. 3 shows the sequence models of smectic crystal (A) and smectic-c crystal (B), together with the four conformations (a, b, c and d) that an isolated chain can take preferentially below 10 K. An arrow mark shows the direction of ordering or crystallization for a, b, c and d. From these results, the ordered parts are like the smectic crystal and the hole of a pair should have the free volume coming from the difference of conformation between A and a, b, c or d in Fig. 3. For iPP, hh was 2.5 times as much as hx. This result suggested that the hole of a pair was the inside space of a 3/1 helical ordered part composed of 3 structural units, holding 3 photons, but each photon was concerned in the potential energy of 2.5 structural units in a helical sequence, and that, (hh/2.5)/hx = 1. This was comparable to hh/hx = 1 for PET. The value of

Fig. 3. Right: The sequence models of the smectic crystal of mesophase with a conformational disorder (A) and the smectic-c crystal with a stretched conformation (B) for PET30). Left: Four conformations taken preferentially below 10 K for an isolated chain; a: T**T**TTG'T, b: T**T**TTGT, c: T**C**TTG'T and d: T**C**TTGT, by Flory's theory31). T, G and G' are the trans, gauche and gauche' isomers, respectively. **T** and **C** are the trans and cis isomers of phenylene groups (lower groups). An arrow mark shows the direction of ordering or crystallization.

Δh kJ/mol

> 6.5\*4 6.2\*5 6.2\*5

> 1.1\*4 1.0\*5

> > 5.3 3.9

hx kJ/mol

> 22.6 23.8 23.7

> > 7.3

24.2\*6

hh

23.0

24.2\*6 24.2 1.0

7.4 18.2 2.5 (1.0\*7)

kJ/mol hh/hx

1.0 1.0 1.0

2.5 (1.0\*7)

1.0

hg kJ/mol

> 16.1\*1 17.6\*2 17.5\*3

6.2\*1 6.4\*3

18.9\*2 20.3\*3

Table 1. The values of Tg, Δsg, hg, Δh, hx, hh and hh/hx for PET, iPP and PS.

(hh/2.5)/hx = 1. This was comparable to hh/hx = 1 for PET. The value of

Fig. 3. Right: The sequence models of the smectic crystal of mesophase with a conformational disorder (A) and the smectic-c crystal with a stretched conformation

(B) for PET30). Left: Four conformations taken preferentially below 10 K for an isolated chain; a: T**T**TTG'T, b: T**T**TTGT, c: T**C**TTG'T and d: T**C**TTGT, by Flory's theory31). T, G and G' are the trans, gauche and gauche' isomers, respectively. **T** and **C** are the trans and cis isomers of phenylene groups (lower groups). An arrow mark shows the direction of ordering or

\*1: Hga − Hgc, \*2: RTg2/c2, \*3: hgconf + hgint, \*4: Eq. (14), \*5: Eq. (16), \*6: hx = hh and \*7: (hh/2.5)/hx. The data of iPP used to calculate Δh in Eq. (14) are as follows: Tc = 403.6 K, Tm<sup>∞</sup> = 449 K for the α form crystal28), ΔH

a) = 4.89 kJ/mol29) and Q = 3.76 kJ/mol for the sample annealed at 461.0 K for 1 hour.

a chain axis25, 30). For the smectic-c crystals with stretched sequences, hu is 28.5 kJ/mol. Further, DSC revealed25) that for the crystalline films of smectic-c crystals, the ordered parts in the amorphous regions were like smectic crystals and for the crystalline films of smectic crystals, the ordered parts were like the smectic-c crystals. Fig. 3 shows the sequence models of smectic crystal (A) and smectic-c crystal (B), together with the four conformations (a, b, c and d) that an isolated chain can take preferentially below 10 K. An arrow mark shows the direction of ordering or crystallization for a, b, c and d. From these results, the ordered parts are like the smectic crystal and the hole of a pair should have the free volume coming from the difference of conformation between A and a, b, c or d in Fig. 3. For iPP, hh was 2.5 times as much as hx. This result suggested that the hole of a pair was the inside space of a 3/1 helical ordered part composed of 3 structural units, holding 3 photons, but each photon was concerned in the potential energy of 2.5 structural units in a helical sequence, and that,

Polymer Tg

(=Hma − Hc

crystallization.

PET 342

K

iPP 270 23.0

PS 359 52.6

Δsg J/(K mol)

> 47.1 51.5 51.2

> 23.7

56.5

hx was almost equal to hu (= 7.6 kJ/mol for α form). Fig. 4 shows the photon sites in the helix structure and the helical conformation of an isolated sequence with an inversion defect isomer *TT*, taking preferentially at temperatures below 70 K26).

Fig. 4. Upper: The photon sites (the dashed line parts) in the helix structure with TGTG or TG'TG' conformation at temperatures below Tg for iPP. Large circle: C and small circle: H. Lower: The helical conformation of an isolated sequence with an inversion defect isomer *TT*, taking preferentially at temperatures below 70 K. The allow mark shows the shift of *TT* on a helical sequence.

For PS, supposing hx = hh, Δh was evaluated. From the value of Δh to be near that of PET, the vf jump at Tg should be due to the release between phenyl groups. The Tg of PE 15), producing the entropy of unfreezing for the glass parts; Δsg = hg/Tg (see Eq. (5)), was almost dependent on hg int. When a value of hg int was that of the glass with Tg = 237 K, hg int/2 gave Tg = 135 K. Table 2 shows the values of Tg, Δsg, hg, Δh, hx, hh and hh/hx for PE glasses with Tg = 135 K (hg int = 2.8/2 kJ/mol) and 237 K (hg int = 2.8 kJ/mol). The above relation in Tg and hg int for the glass parts was linked to the ordered parts. For both glasses, hh was about 5 times as much as hg. Thus from Eq. (13), the common relations of hx = hh/4 and Δh = hh/4 − hg were predicted for the ordered parts in both glasses and shown in Table 2. Fig. 5 depicts the sequence models of ordered parts (A and B) and the schematic transition from the glassy state (C: left) to that of the "ordered part / hole" pair (C: right).

For the glass with Tg = 135 K, the coarse 4/1 helical ordered parts with GG or G'G' conformation in Fig. 5A and as the hole of a pair, the inside space holding four photons per a helical segmental unit, -(CH2)4-, were predicted. Further the length distribution of helical ordered parts in the glass and as the end temperature of melting for the ordered parts, ∼237 K were predicted. In this case, the same value of Δh for both glasses enabled the scheme as depicted in Fig. 6. For the glass with Tg = 237 K, the ordered part of fringe-type formed by


\*1: hgconf + hgint, \*2: hx = hh/4 and \*3: (hh/4)/hx.

Table 2. The values of Tg, Δsg, hg, Δh, hx, hh and hh/hx for PE.

Thermodynamics of Enthalpy Relaxation and Hole Formation of Polymer Glasses 171

should vitrify or order the melted helical ordered parts and confine the larger helical ordered parts, which were not yet melted even over Tg = 135 K, in the glass. The liquid on the way of enthalpy relaxation like this should reach step by step to the glass with Tg =

predicted, because of the strong interaction between amido groups. hh was 4.7 times as much as hx in the parenthesis. Accordingly a photon was concerned in the potential energy of a stretched segmental unit, -(CH2)5-. Further Δh = 2.5 kJ/mol was 0.5 kJ per molar methylene unit, -CH2-, which agreed with Δh = 0.5 kJ/mol for the PE glasses with Tg = 135 K and 237 K (see Table 2). In addition, the value of hh was almost equal to hu of the heat of fusion. This agreement suggested that the ordered parts were the stretched segmental units in the smallest crystals of N6. Fig. 7 shows the structural unit of N6 and

int = 2.8 kJ/mol) at temperatures below Tg. Table 3 shows the values of Tg, Δsg, hg,

Δh\*2 kJ/mol

(2.0) 2.5, 0.5\*3 40.1

conf = hxconf in Eq. (15), i.e., Δh = (RTglnZt)/x was

hx kJ/mol

phTg/NA (18)

hu kJ/mol

(4.5) 21.3 21.1

hh kJ/mol

237 K (hg

Polymer Tg

N6 27)

kJ/mol.

Δh, hx, hu and hh for N6. For N6, hg

the photon site in the structural unit.

<sup>313</sup>120.1

Δsg J/(K mol)

(6.4)

Table 3. The values of Tg, Δsg, hg, Δh, hx, hu and hh for N6.

hg kJ/mol

37.6\*1

\*1: hg = hgconf + hgint, \*2: Δh = (RTglnZt)/x and \*3: the value of Δh per molar methylene unit, -CH2-. The values of Δsg, hg and hx in the parentheses are those without the cohesive energy of amido group, 35.6

Fig. 7. The structure of N6 structural unit and the photon site (the dashed line part) in the unit. The filled circle: N, the shaded circle: O, the large circle: C, and the small circle: H.

A photon has the property as a boson or a wave. Therefore hh is also represented as the vibrational energy of a wave with the quantum number n = 1 (meaning one photon) and the

 hh = NA(3/2)*h*ν (17) where NA is Avogadro constant. Thus from Eqs. (12) and (17), the zero-point energy, ε0 (=

ε0 = Cv

K

**3. Hole energy and a photon** 

frequency per second (sec), ν:

(1/2)*h*ν), is derived:

Fig. 5. The sequence models of ordered parts; A: the 4/1 helix structure and B: the stretched structure in the region surrounded by the dashed line. C (left): the glassy state and C (right): the state of a "ordered part / hole" pair. The plus mark (+) shows the cross section of a stretched segmental unit, -(CH2)4-. An arrow mark shows the direction of ordering.

bundling TTT parts of four sequences at least, and that, the smallest crystal of PE and the neighboring hole were predicted (see Fig. 5B and C: right), since the value of hx was almost equal to hu = 4.1 kJ/mol. Fig. 6 shows the bar graph of hg at Tg = 135 K and 237 K, together with Δh (= 0.5 kJ/mol) at 237 K, for PE. The difference in two complementary lines suggests the supply schedule of heat over the temperature range from 135 K to ∼237 K in order to make up the shortage of Δh required to melt all ordered parts in the glass with Tg = 135 K. For the glass with Tg = 237 K, the glass transition only at Tg is shown. The enthalpy relaxation, accompanied by the generation of stretched segments, at temperatures over 135 K to ∼237 K

Fig. 6. The bar graph of hg at Tg = 135 K and 237 K (the vertical thin lines between the cross marks) and Δh (= 0.5 kJ/mol) at 237 K (the thick line part) for PE. The dot and dot-dashed lines are drawn complementarily.

Fig. 5. The sequence models of ordered parts; A: the 4/1 helix structure and B: the stretched structure in the region surrounded by the dashed line. C (left): the glassy state and C (right): the state of a "ordered part / hole" pair. The plus mark (+) shows the cross section of a stretched segmental unit, -(CH2)4-. An arrow mark shows the direction of ordering.

bundling TTT parts of four sequences at least, and that, the smallest crystal of PE and the neighboring hole were predicted (see Fig. 5B and C: right), since the value of hx was almost equal to hu = 4.1 kJ/mol. Fig. 6 shows the bar graph of hg at Tg = 135 K and 237 K, together with Δh (= 0.5 kJ/mol) at 237 K, for PE. The difference in two complementary lines suggests the supply schedule of heat over the temperature range from 135 K to ∼237 K in order to make up the shortage of Δh required to melt all ordered parts in the glass with Tg = 135 K. For the glass with Tg = 237 K, the glass transition only at Tg is shown. The enthalpy relaxation, accompanied by the generation of stretched segments, at temperatures over 135 K to ∼237 K

Fig. 6. The bar graph of hg at Tg = 135 K and 237 K (the vertical thin lines between the cross marks) and Δh (= 0.5 kJ/mol) at 237 K (the thick line part) for PE. The dot and dot-dashed

lines are drawn complementarily.

should vitrify or order the melted helical ordered parts and confine the larger helical ordered parts, which were not yet melted even over Tg = 135 K, in the glass. The liquid on the way of enthalpy relaxation like this should reach step by step to the glass with Tg = 237 K (hg int = 2.8 kJ/mol) at temperatures below Tg. Table 3 shows the values of Tg, Δsg, hg, Δh, hx, hu and hh for N6. For N6, hg conf = hxconf in Eq. (15), i.e., Δh = (RTglnZt)/x was predicted, because of the strong interaction between amido groups. hh was 4.7 times as much as hx in the parenthesis. Accordingly a photon was concerned in the potential energy of a stretched segmental unit, -(CH2)5-. Further Δh = 2.5 kJ/mol was 0.5 kJ per molar methylene unit, -CH2-, which agreed with Δh = 0.5 kJ/mol for the PE glasses with Tg = 135 K and 237 K (see Table 2). In addition, the value of hh was almost equal to hu of the heat of fusion. This agreement suggested that the ordered parts were the stretched segmental units in the smallest crystals of N6. Fig. 7 shows the structural unit of N6 and the photon site in the structural unit.


\*1: hg = hgconf + hgint, \*2: Δh = (RTglnZt)/x and \*3: the value of Δh per molar methylene unit, -CH2-. The values of Δsg, hg and hx in the parentheses are those without the cohesive energy of amido group, 35.6 kJ/mol.

Table 3. The values of Tg, Δsg, hg, Δh, hx, hu and hh for N6.

Fig. 7. The structure of N6 structural unit and the photon site (the dashed line part) in the unit. The filled circle: N, the shaded circle: O, the large circle: C, and the small circle: H.

#### **3. Hole energy and a photon**

A photon has the property as a boson or a wave. Therefore hh is also represented as the vibrational energy of a wave with the quantum number n = 1 (meaning one photon) and the frequency per second (sec), ν:

$$\mathbf{h}^{\mathrm{h}} = \mathrm{N}\_{\mathrm{A}}(3/2)\mathrm{h}\mathbf{v} \tag{17}$$

where NA is Avogadro constant. Thus from Eqs. (12) and (17), the zero-point energy, ε0 (= (1/2)*h*ν), is derived:

$$\mathfrak{e}\_{\mathfrak{U}} = \mathbb{C}\_{\mathbf{v}} \, ^{\text{fb}}\mathrm{T}\_{\mathfrak{U}} / \mathrm{N}\_{\mathbf{A}} \tag{18}$$

Thermodynamics of Enthalpy Relaxation and Hole Formation of Polymer Glasses 173

transition enthalpy was larger than the molar enthalpy of the ordered parts in the amorphous regions; Δh < 0 in Eq. (14). At temperatures above its Tg, the generation and disappearance of the "crystal / anti-crystal hole" pairs from the secondary glass were predicted as the simultaneous phenomena in the crystallization and the melting. Hexagonal and monoclinic forms of PE crystals are also well known. However, the hexagonal crystals should not be related to the melting of the orthorhombic crystals since the DSC melting peak of the hexagonal crystals generally cannot be observed for the samples without restraints such as high pressure38). Moreover, the DSC melting peak of monoclinic crystals disappears before the melting of the orthorhombic crystals39, 40). Thus, when the monoclinic crystals are in the bulk state, the heat due to their melting should contribute to the activation heat

Fig. 8 depicts the DSC crystallization peak upon cooling and the two peaks divided from a DSC endothermic peak upon heating for the PE film annealed at 416.6 K (near Tm<sup>∞</sup> = 415 K)

onset temperature of crystallization, Tb\* (≈ Tc) is the intersection between the base line and the extrapolation line from the line segment with the highest slope on the lower temperature side of the melting peak, and that, the onset temperature of the higher

and that, the origin of the extrapolation line, respectively. Qm is the heat per molar structural unit corresponding to the endothermic peak area of crystal lamella that starts to melt at Tb\* and hc (= 0.89 kJ/mol) is the heat of crystallization per molar structural unit corresponding to the area surrounded by the dashed line and the exothermic curve. ΔQm (= Qm − hc) corresponds to the area between Tb\* and Te\* of the higher temperature side

Fig. 8. The DSC crystallization peak upon cooling and the two peaks divided from a DSC endothermic peak upon heating for the PE film annealed at 416.6 K for 1 hour. dQ/dt is the heat flow rate. The cooling and heating rates are 5 K/min and 10 K/min, respectively.

\* is the curve before division. Tc (= 391.5 K) is the

\* is the end temperature of the lower temperature side peak,

required to release the secondary glass state in the orthorhombic crystal lamella.

**5.2 Secondary glass** 

for 1 hour. The thin line in Tb\* and Te

temperature side peak and Te

Table 4 shows the values of Tg, hh, ν, λ and 1/λ for PE, iPP, PS and PET, where λ is the wavelength and 1/λ is the wavenumber. According to the infrared spectroscopy, for PE, 1/λ = 510 cm<sup>−</sup>1 and 893 cm<sup>−</sup>1 might be concerned with 720 cm<sup>−</sup>1 and 731 cm<sup>−</sup>1 bands assigned to the rocking of -CH2-32). For iPP, 1/λ = 1022 cm<sup>−</sup>1 almost agreed with 1045 cm<sup>−</sup>1 relating to the crystallinity33). Also for PS, 1/λ = 1359 cm<sup>−</sup><sup>1</sup> almost agreed with one of conformation sensitive bands34), i.e., 1365 cm<sup>−</sup>1 band. For PET, 1/λ = 1292 cm<sup>−</sup>1 was near 1339 cm<sup>−</sup>1 and 1371 cm<sup>−</sup>1 bands assigned to the wagging of –CH2- with trans and gauche conformations, respectively35).


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