## INTRODUCTION

We present a new conceptual approach to understanding the glass transition temperature *T*_{g} of glass forming liquids, which is applicable to amorphous polymers, ceramics and metals. As a liquid cools towards the glass, atoms interacting via their interatomic anharmonic potential *U*(*x*) form dynamic metastable clusters which eventually become stable and percolate near *T*_{g}. The percolating cluster is a stable, persistent fractal and its solid clusters are in dynamic equilibrium with the surrounding liquid; when viewed by the observer, this fractal at *T*_{g} would appear to “twinkle” with a frequency spectrum *F*(ω) as solid clusters of vibrational frequency ω interchange with the liquid and visa versa. Hence, the name of this new approach to *T*_{g} is called the Twinkling Fractal Theory (TFT).1 The fractal twinkles with a characteristic vibrational density of states *g*(ω) ∼ ω^{df–1}, where *d*_{f} = 4/3 is the Orbach fracton dimension.2, 3 Upon further cooling below *T*_{g}, the fractal structure existing at *T*_{g} frustrates the formation of the contracting equilibrium supercooled liquid and “free volume” is generated for the first time. The apparent second order transition in the volume-temperature cooling curve and changes in the thermal expansion coefficients of the liquid to glass are a consequence of the fractal structure formation. The TFT approach is nested in the understanding of the coefficient of thermal expansion (CTE) of the liquid α_{L}, and how the average interatomic bond distance 〈*X*〉 is controlled by the Boltzmann population of excited energy levels in the anharmonic potential energy function *U*(*x*). The glass transition occurs in a rate dependent manner when rigidity can propagate through the percolating dynamic fractal structure. The twinkling frequencies *F*(ω) determine the kinetics of *T*_{g} and the dynamics of the glassy state. In this paper, we will show that the TFT provides considerable new insight into the physics of the glass formation process and predicts a unique relationship between *T*_{g} and thermal expansion coefficient via the dimensionless number, α_{L}×*T*_{g} ≈ 0.03.

The understanding of the complexity of glass formation during cooling of noncrystallizing liquids remains as one of the major intellectual challenges of solid state physics. This field has recently been reviewed by Angell et al.4 and it was found that about 50 important unresolved questions could be framed in the context of the traditional volume-temperature (*V*-*T*) curve. Figure 1 shows the classical onset of the glass transition temperature *T*_{g} where the slope of *V* versus *T* changes at *T*_{g}. The slopes d*V*/d*T* are the bulk coefficients of thermal expansion α′_{g} and α′_{L} for the glass and liquid respectively. The bulk α′-values are related to the linear thermal expansion coefficients via α′(bulk) ≈ 3α(linear). In this paper, we will use the linear α-values in all calculations and comparisons with experimental data. The glass transition, structure and relaxation of the glassy state remain as unresolved fundamental issues. New insights into *T*_{g} have been advanced by Wolynes et al.,5, 6 who examined cooperatively rearranging regions (CRRs) near the critical temperature *T*_{c} via the random first order transition theory (RFOT) of glasses. The CRRs change from being compact at low temperatures to more fractal-like strings near *T*_{c}.6 Bakai and Fischer7 have also reported the formation of fractal-like dynamic structures in glass forming liquids such as *ortho*-terphenyl (OTP) at temperatures up to 100 °K above *T*_{g} (244 °K). The “Fischer Cluster” could be observed by polarized light scattering and Wide angle X-Ray Scattering (WAXS) and is reminiscent of liquid crystal-like ordering at *T* ≫ *T*_{g}. Such studies reviewed in reference7 have lead to concepts of dynamic heterogeneity, length scale of cooperativity and rate exchange effects between the liquid and solid-like fractal structures in glass forming fluids. The concept of percolating dynamic fractal structures because of density fluctuations in Van der Waals fluids has been used by Long et al.8 to examine the effects of film thickness on *T*_{g} elevation (with strong adhesion) or depression (no adhesion) in polymer thin films. Such models agree with experiments by Keddie and Jones9 and were supported by computer simulations of bead-spring models of polymer dynamics by Baljon et al.10 who observed slow moving percolating clusters near *T*_{g} of the thin films. In this paper, we will give precise physical meaning to the several related concepts of dynamic heterogeneity, cooperativity and fractal structure in a manner to provide useful new interpretations of many physical properties of glass forming liquids.

The development of percolating dynamical fractal structures near *T*_{g} is the key element of the Twinkling Fractal Theory (TFT) presented herein and for a single specie fluid, the fractal twinkles with a frequency spectrum *F*(ω) ∼ ω^{df–1}. Interestingly, the twinkling fractal is mostly “invisible” to typical scattering experiments (neutrons, X-ray, light) because its structure is in dynamic equilibrium with the surrounding liquid and little scattering contrast exists. However, the twinkling fractal can be seen by such methods as AFM dielectric spectroscopy where changes in the local dielectric force can be detected as spatio-temporal fluctuations11, 12 and basically show the fractal and its twinkling frequencies. We shall see that the fractal structure at *T*_{g} has a profound influence on many physical properties such as thermal expansion coefficients, mechanical damping, heat capacity, viscosity and yield stress. The TFT will be shown to reproduce the well know empirical free-volume relations for *T*_{g},13–16 the William-Landel-Ferry (WLF) relation17 for the temperature dependence of viscosity near *T*_{g}, and the Stretched Exponential relaxation functions while providing additional physics and fundamental quantitative information on the glass formation process.

### Twinkling Fractal Theory

#### Fractal Structure at *T*_{g}

We examine the onset of the glass transition temperature *T*_{g} in Figure 1 by considering the anharmonic interaction potential *U*(*x*) of diatomic oscillators (Fig. 2) in the liquid state as a function of temperature. As an example of *U*(*x*), consider the Morse function18:

in which *D*_{o} is the interatomic bond energy, ‘a’ is the anharmonicity factor and *x* = *R* − *R*_{o} is the bond displacement from its equilibrium value *R*_{o}. We use the Morse potential or its quasiharmonic equivalent to generate quantitative data later, but any anharmonic interatomic potential function, such as the Lennard-Jones potential, will suffice. The anharmonicity controls the vibrational frequencies ω, with temperature *T*, hydrostatic pressure *P* and stress σ,19 and the thermal expansion coefficient α_{L} via the mean bond displacement 〈*X*〉 (Fig. 2).

As *T*_{g} is approached from above, the average bond distance 〈*X*〉 between the oscillators contracts, the volume *V* decreases and the bond expansion factor 〈*X*〉 approaches its critical value *X*_{c} at the inflection point of *U*(*x*), where the interatomic force *f* = d*U*/d*x* reaches its maximum value and the force constant *K* = *d*^{2}*U*/*dX*^{2} = 0. Thus, when *X* > *X*_{c} and *T* > *T*_{g}, the molecules can readily escape with a thermofluctuation to the ergodic liquid state and when *X* < *X*_{c} at *T* < *T*_{g}, the molecules on average exist in the nonergodic solid state. The latter idea parallels the Lindemann (1910) theory of melting, which states that when the vibrational amplitudes exceed a critical value of the bond length (*X*_{c} ≈ 0.11 for a Lennard-Jones potential), then melting occurs.20 In our case we have a Boltzmann distribution of oscillators ϕ(*x*) at various quantized energy levels in the anharmonic potential energy function *U*(*x*) such that the number of oscillators at energy level *U*(*x*) is:

where *k* is the Boltzmann constant. At any temperature *T*, there exists a fraction *P*_{S} of solid atoms (*X* < *X*_{c}) and a fraction *P*_{L} of liquid atoms with *P*_{S} + *P*_{L} = 1.

The solid fraction *P*_{s} at any T is determined from the integral of the Boltzmann energy populations ϕ(*x*) in the integration limits (0, *X*_{c}):

Near *T _{g}*, we find that the liquid and solid fractions are well described by eq 1.3 as:

and

where *p*_{c} ≈ [1/2]. The latter value for the percolation threshold depends on the method of observation and aspect ratio *A* of the percolating particles; for scalar percolation of particles with A ≈ 1 (e.g., electrical or heat conductivity) *p*_{c} ≈ 0.3, and for vector percolation (force, modulus), *p*_{c} ≈ 0.4–0.5. The percolation threshold will not have a universal value for all glasses but will be of order [1/2] which we use throughout this paper to facilitate quantitative comparisons of the TFT with experimental data. As *T* approaches *T*_{g}, eq 1.5 gives P_{S} ≈ *p*_{c}. Note that the solid fraction *P*_{S} is significant at temperatures well above *T*_{g} and does not approach zero until *T* ≈ *T*_{g}/(1 – *p*_{c}), or *T* ≈ 2 *T*_{g}. This is consistent with the dynamic heterogeneity observed at *T* ≫ *T*_{g}.7 Similarly, the liquid fraction *P*_{L} is significant at *T* < *T*_{g} and only approaches zero as *T* approaches 0 °K. However, the relations for *P*_{L} and *P*_{S} above are approximate near *T*_{g} and large extrapolations in temperature would require a more detailed solution of eq 1.3. Note that *P*_{S} varies smoothly with *T* in a linear fashion through *T*_{g}, such that a first order phase transition does not occur.

As *T*_{g} is approached from above, the fraction of solid atoms *P*_{s} grows with a dynamic cluster size distribution function determined by percolation theory with a correlation length:

in which ν ≈ 0.8 is the critical exponent. The average cluster mass *M* is determined near *p*_{c} by:

where *D*_{f} is the fractal dimension of the cluster. As *P*_{S} approaches *p*_{c}, a fractal structure of infinite size forms as the correlation length diverges to infinity at *p*_{c} and percolation occurs, as shown in Figure 3.

The percolation cluster has a fractal dimension *D*_{f} and is not stationary because the solid and liquid atoms are in dynamic equilibrium. Thus at *T*_{g}, to the observer, there appears to exist a fractal structure which “twinkles” with a frequency spectrum *F*(ω) as liquid and solid atoms exchange. This “twinkling” fractal structure is expected to be “invisible” to the usual scattering experiments because the liquid and solid are essentially indistinguishable. Above *T*_{g}, the dynamics of the twinkling nonpercolating fractal clusters will affect the fluid viscosity and the apparent activation energies in terms of *fragile* (*T*-dependent) or strong (*T*-independent) activation energies. When *T* is above but near *T*_{g}, the concept of twinkling fractal clusters should have impact on the nature of *fragile* and *strong* fluids. Here, the concept of a *fragile* fluid is determined by the extent of deviation of the relaxation times beyond that predicted from a simple Arrhenius activated state (*strong*). Some strong liquids will obey simple relaxation process τ ∼ exp *E*/*kT* while others such as polymers follow the empirical Vogel-Fulcher relation, τ ∼ exp *ET*_{o}/(*T*−*T*_{o}), in which *T*_{o} is the temperature when the relaxation time or viscosity η goes to infinity, e.g., near *P*_{c} in the TFT. Others have used the Kohlrausch relation [1847] which is often referred as a *stretched exponential* as τ ∼ exp −(*t*/τ)^{β}, where β ≤ 1. Mathematically this is the infinite sum of a continuous relaxation spectrum suggested by the TFT vibrational density of states *g*(ω) ∼ ω^{df–1} in which τ ∼ 1/ω. The twinkling fractal is the “dynamics engine” of the amorphous state and suggests a broad range of frequencies and relaxation times with heterogeneous structures varying from nanoscale to microns and plays an important role in complex processes such as physical aging, self-healing1 and polymer welding below *T*_{g}.21 The TFT suggests that the dynamics of the glassy state near *T*_{g} can be obtained from the spatio-temporal thermal fluctuation auto correlation function *C*(*t*), as shown in the next section.

### Twinkling Fractal Frequencies and Dynamics

Significant support for the TFT is provided by spatio-temporal thermal fluctuation experiments of Israeloff et al.11, 12 To compare his experiments to the TFT, we need to calculate the temperature dependent autocorrelation function *C*(*t,T*) for the TFT spatio-temporal fluctuations. The twinkling fractal frequency spectrum *F*(ω) representing solid-to-liquid cluster transitions of frequency ω are given by:

Here the first term ω^{df–1} is the Orbach vibrational density of states *g*(ω) ∼ ω^{df–1} for a particular fractal cluster2 and the second exponential term is the probability that the vibration will cause a “twinkle” or change from solid to liquid, or visa versa. *U*(*T*) is the mean energy of the atoms in the cluster at temperature *T* and *U*_{c} is the energy at the critical inflection point in the anharmonic potential *U*(*x*). Note that this energy difference is positive. Happily, the *fracton* dimension has the same value *d*_{f} = 4/3 in dimension *d* = 2 and *d* = 3, which as we will show later, conveniently facilitates the analysis of 3D data with an experiment which is 2D in nature. Near *T*_{g}, we can approximate the temperature dependence of *U*(*x*) for a Morse oscillator in the harmonic approximation, as *U*(*x*) = [1/2] *K*〈*X*〉^{2}, where the force constant *K* = 2*D*_{o}*a*^{2} and the average bond expansion 〈*X*〉 ∼ *T* such that the energy of the oscillator becomes:

Hence, the temperature dependence of *F*(ω) is obtained with a temperature dependent activation energy:

in which the constant β = α_{L}^{2}*D*_{o}*a*^{2}*R*_{o}^{2}. The activation energy Δ*E* = [*U*(*T*) − *U*_{c}] is temperature dependent (in the harmonic oscillator approximation) as:

The TFT predicts that a cluster with frequency ω and relaxation time τ ∼ 1/ω with density of states *g*(ω) ∼ ω^{df–1} will have a cluster relaxation function *c*(*t*, ω) given by:

Combining eq 2.3 and eq 2.5, the temperature dependent global autocorrelation function *C*(*t, T*) for the relaxation dynamics is given for all twinkling fractal frequencies by:

Thus, at time *t*, only vibrations in the range ω = ω_{o} to ω = 1/*t* can contribute to spatio-temporal fluctuations. Note that the temperature dependence of ω in the exponent creates a double exponential dependence on temperature for *C*(*t*). We will see later that this leads to a WLF type dependence on the temperature shift factor a_{T} and a stretched exponential form of the relaxation function.

At short times or low temperature *T* < *T*_{g}, *C*(*t*) is dominated by the higher frequencies with faster relaxation times and solutions to eq 2.6 give:

The exponent of –1/3 comes directly from the Orbach density of states for a fractal *g*(ω) ∼ ω^{1/3} and its experimental observation would be a useful test of both the TFT and the Orbach theory.

At intermediate and long times, eq 2.6 approaches the (gamma) Γ(ω^{df})/*t* solution such that over most of the time range at *T*_{g}, we expect:

The transition of the initial time exponent of –1/3 to –4/3 is again a consequence of the fracton dimension *d*_{f} = 4/3 in the fractal density of states.

The spatio-temporal fluctuations of amorphous polyvinyl acetate (PVAc) films near *T*_{g} were imaged at the nanoscale using dielectric force microscopy by Israeloff et al.11, 12 By depositing the PVAc films on a metal substrate, a conducting AFM tip was able to detect thermal fluctuations at the nanoscale by measuring changes in the local dielectric force response, as shown in Figure 4 for temperatures 301.5 and 305.5 °K near *T*_{g} = 308 °K. The space-time images of the dielectric force fluctuations were generated by repeated scanning across the same location. This is similar to taking a single horizontal line intensity sampling with time of the fractal structure in Figure 3. The autocorrelation functions *C*(*t*) for the spatio-temporal fluctuations at both temperatures are shown in Figure 4(c) and cover a spatial range up to 700 nm and times of order 2000 s.

Analyzing the autocorrelation *C*(*t*) data in Figure 4(c), as shown in Figure 5, we find that for the 305.5 °K data (2.5 °K below *T*_{g}) that at short times (27–100 s), *C*(*t*) ∼ 1.26 *t*^{−0.35} with correlation coefficient *r*^{2} = 0.97; at intermediate-long times (68–670 s), *C*(*t*) ∼ *t*^{−1.36} with *r*^{2} = 0.94. These results are in compelling agreement with the TFT predictions *C*(*t*) ∼ *t*^{−1/3} and *C*(*t*) ∼ *t*^{−4/3}. At still longer times (300–1000 s), we find that *C*(*t*) ∼ *t*^{−2}, which is representative of the fracton-phonon crossover effect on the density of states for percolating systems discussed by Yakubo et al.,3 where *g*(ω) ∼ ω^{2} at low ω or long times. The crossover to the normal Debye density of states *g*(ω) ∼ ω^{d}^{–1} (*d* = 2 or 3) occurs at ω* _{c}* when the phonons no longer “recognize” the fractal structure at a phonon wavelength λ equal to the cluster correlation length ξ.

When the temperature is lowered below *T*_{g}, the TFT predicts via eq 2.6 that the initial relaxation behavior with *C*(*t*) ∼ *t*^{−1/3} is expanded in time when the lower vibrational frequencies are suppressed. For example, at 301.5 °K (6.5 °K below *T*_{g}), the data in Figure 4(c) in the entire time range (68–1164 s) gives *C*(*t*) ∼ *t*^{−0.39} as shown in Figure 6 and at short times (68–300 s) the exponent of –1/3 is more exact. The *C*(*t*) autocorrelation function can also be fit with an empirical Kohlrausch stretched exponential of the type *C*(*t*) ∼ exp − (*t*/τ)^{β}, where β ≈ 0.5 and τ is a characteristic time12 However, though useful for analysis purposes, the stretched exponential function is strictly empirical and does not carry the same physical interpretation as the full TFT autocorrelation function in eq 2.6.

The above spatio-temporal experiments are in excellent accord with the TFT and clearly point to the dynamic heterogeneity and the fractal nature of the liquid near *T*_{g}. Cooperative relaxation is occurring with larger clusters on the scale of hundreds of nanometers involving very long timescales of order 30 mins. Simultaneously, smaller clusters of nanometer length scale are exchanging solid-to-liquid and visa versa in fractions of a second. Rate effects will obviously affect the manner in which *C*(*t*) manifests itself to the observer. At high shear rates for example, the material will appear more “solid-like” because little relaxation can occur and at slow rates, the material appears more liquid-like.1

### Determination of *T*_{g} from the TFT

To obtain *T*_{g} from the TFT using eq 1.3, we find that the quasiharmonic approximation for the Morse potential facilitates calculations of thermal expansion coefficients and *T*_{g}.19, 22 Using *U*(*x*) = *Ax*^{2}–*Bx*^{3} in which the constants *A* = *D*_{o}a^{2} and *B* = *D*_{o}a^{3}, the Morse potential can be represented by the quasiharmonic approximation19:

The solid fraction as a function of temperature is derived using eq 3.1 in eq 1.3 in the harmonic approximation as:

When rigidity percolation occurs at *P*_{S} = *P*_{c}, as shown in Figure 3 then we obtain *T*_{g} from eq 3.2 as:

because *P*_{c} ≈ [1/2], the glass transition temperature is well approximated by:

For example, if *D*_{o} = 3.4 kcal/mol, eq 3.4 gives *T*_{g} = 380 °K (107 °C), which is a typical value for polymers such as polystyrene and PMMA.23, 24 Thus, any mechanism such as hydrogen bonding, polar groups, or *para*-bonded aromatic rings which raise *D*_{o} will lead to higher *T*_{g} values. For a heteroatom polymer, one would expect that *D*_{o} would be the sum of all the fractional interactions. The latter calculation is nontrivial and requires specific knowledge of the interatomic pairings. An approximate form would give:

where ϕ_{i} are the mole fractions of the *i*th atom in the chain and D_{oi} are the individual interatomic bond energies. For example consider Nylon 6 with the molecular structure:

Of the seven atoms in the repeat unit, we have 5/7 CH_{2} groups with *D*_{o}(CH_{2}) ≈ 2 kcal/mol (based on PE) and 2/7 H-bonding sites with *D*_{o}(HB) ≈ 5 kcal/mol. The average *D*_{o} value is thus obtained from eq 3.5 as (2 × 5 + 5 × 2)/7 = 2.86 kcal/mol. Using eq 3.4, this value predicts that *T*_{g} (Nylon 6) ≈ 320 °K (47 °C), which is in agreement with experiment. The value of *D*_{o}(HB) = 5 kcal/mol gives exact agreement with experiment.

The intermolecular bond strength *D*_{o} can be obtained form *T*_{g} in eq 3.4 as:

For polyethylene, *T*_{g} = 223 °K which gives *D*_{o} ≈ 2 kcal/mol, which is similar to the crystallization heat of fusion Δ*H*_{f} ≈ 2 kcal/mol.25 It can be expected that *D*_{o} and Δ*H*_{f} should have comparable magnitude but should not be exactly matched because of the different interatomic configurations between the amorphous and crystalline states. For polypropylene, *T*_{g} = 268, *D*_{o} = 2.4 kcal/mol and Δ*H*_{f} = 2.13 kcal/mol. With atactic polystyrene, *T*_{g} = 373 °K, *D*_{o} ≈ 3.3 kcal/mol and Δ*H*_{f} = 2 kcal/mol (isotactic). If we assume that *D*_{o} ≈ Δ*H*_{f}, and because the liquid fraction behaves as *P*_{L} = (1 – *p*_{c})*T*/*T*_{g}, we can extrapolate to the melting point *T* → *T*_{m} as *P*_{L} → 1 which gives the relation between *T*_{m} and *T*_{g} as:

Because *p*_{c} ≈ 1/2, this relation suggest that *T*_{m} ≈ 2*T*_{g} on a Kelvin scale, as is typically observed.4

### Thermal Expansion Coefficients near *T*_{g}

The TFT predicts a unique relationship between *T*_{g} and thermal expansion coefficient α_{L}. In the liquid, the thermal expansion coefficient α_{L} is determined from the average position 〈*X*〉 of the oscillators with respect to their Boltzmann population distribution ϕ(*x*):

Substituting eq 1.2 with eq 3.1 in eq 4.1, we obtain the linear coefficient of thermal expansion approximately as:

The latter relation gives typical values of α_{L} for amorphous materials; using *k* = 1.986 cal/mol °K, *a* = 2/Å, *R*_{o} = 3Å and *D*_{o} = 3.5 kcal/mol, then α_{L} = 70 ppm/°K and α_{g} ≈ 35 ppm/°K [1 ppm = 10^{−6}].

The existence of the fractal structure at *T*_{g} means that at *T* < *T*_{g}, thermal contraction in the glass is constrained by the fractal structure and α_{g} is related to α_{L} by:

Because *p*_{c} ≈ [1/2], then the glass CTE is typically about [1/2] of the liquid CTE. Physical aging occurs below *T*_{g} through the relaxation of ΔV via the twinkling frequencies and is complex. We expect the low temperature glass dynamics will be influenced by the twinkling fractal frequencies with *C*(*t*) ∼ [*t*/*a*_{T}]^{−1/3}, as suggested by eq 2.7. Even well below *T*_{g}, there is a nonzero fraction of liquid atoms remaining, which will cause the twinkling process to continue at an ever-slowing pace but allow the nonequilibrium structure to eventually approach a new equilibrium value near *V*_{∞}. A near-equilibrium glass can be made by removing the fractal constraints in 3D at *T*_{g} by forming the material using 2D vapor deposition, as recently reported by Ediger et al.26 They used vapor deposition of indometracin to make films, which were reported to have maximum density and possess exceptional kinetic and thermodynamic stability, consistent with the liquid structure extrapolated into the glass (along the *V*_{∞} line in Fig. 1). We have suggested that in this case with Δ*V* ≈ 0, a critical observation would be that α_{g} = α_{L} and the pseudo second order transition in the heat capacity change Δ*C*_{p} = 0 at *T*_{g}. When the fractal constraints disappear at *T*_{g}, the glass transition also disappears on a *V*-*T* curve, while the tan δ mechanical transition remains, which is a most unusual prediction of the TFT.

Another interesting prediction of the TFT is that from eq 3.4 and 4.2 for α_{L} and *T*_{g} respectively, we have the simple result:

For typical amorphous solids *R*_{o} ≈ 3A, and using *a* ≈ 2/Å for the Morse anharmonicity factor, then we expect that:

Figure 7 shows a plot of α_{L} versus 1/*T*_{g} for a variety of materials including amorphous metals,27 ceramics,28 Pyrex glass,29 polymers25 and glycerol24 and this prediction is found to be well supported. The overall slope of this line is about 0.04 with a correlation coefficient of 0.98. The polymers taken alone have a slope of 0.31. In the harmonic limit, we expect that as α_{L} → 0, then *T*_{g} → ∞ as suggested in Figure 5. Thus, anharmonicity plays a major common role in determining *T*_{g} as well as its normal influence on thermal expansion. Equation 4.5 could be useful in making approximations for thermal expansion coefficients when they are not readily available or measurable.

### Heat Capacity dependence on Fractal Structure

The fractal structure of the glass is expected to dominate the change in heat capacity Δ*C*_{p} at *T*_{g}. *C*_{p} is determined from the internal energy U via the temperature derivative of the sum of phonon energy contributions weighted with their vibrational density of states. This TFT calculation is extremely complex. However, in the high temperature classical Debye limit, the ratio of the heat capacities for the liquid and glass will be controlled by the dimensionality d and *D*_{f} via *C*_{pL} = *dU*_{L}/*dT* and *C*_{pg} = *dU*_{g}/*dT*. Therefore, when *U*_{L} = 3 kT and Ug = *D*_{f}*kT*, we obtain the simple ratio:

The latter relation highlights the role of the fractal structure on the determination of Δ*C*_{p}. For bulk samples, *d* = 3 and *D*_{f} ≈ 2.5 such that *C*_{pL}/*C*_{pg} = 1.2 and for very thin films, *d* = 2 and *D*_{f} = 1.75 and *C*_{pL}/*C*_{pg} = 1.14. This prediction is in excellent agreement with values for Δ*C*_{p} at *T*_{g} for bulk (*d* = 3) and nano thin film (*d* = 2) polystyrene samples obtained by Koh et al.23 and summarized in Table 1.

D | D_{f} | C_{pL}/C_{pg} TFT | C_{pL}/C_{pg} Expt | ΔC_{p} TFT | ΔC_{p} Expt |
---|---|---|---|---|---|

3 (bulk) | 2.5 | 1.2 | 1.2 | 0.32 | 0.32 |

2 (thin) | 1.75 | 1.14 | 1.13 | 0.21 | 0.22 |

Table 2 shows a considerable body of C* _{p}* data on glass forming materials summarized by Huang and McKenna,24 where we have added the far right column, which investigates the utility of eq 5.1 to obtain the fractal dimension

*D*

_{f}at

*T*

_{g}via:

We find in Table 2 that most polymers have a fractal dimension of *D*_{f} ≈ 2.5, as might be expected from a random cluster model of long polymer chains without strong attractions. Organic small molecules have *D*_{f} ≈ 2 fairly uniformly which is in accord with fractal dimensions of biological colloids with *D*_{f} = 1.99 and 2.14.30 The “cloud point” of biodiesel at low temperatures should also have a similar fractal dimension as the more saturated fatty acids form a solid fractal aggregate embedded in the more unsaturated liquid fatty acids. This could be a useful experimental system to study the TFT in more detail. In Table 2, the hydrogen bonding liquids have a lower *D*_{f} ≈ 1.75 suggestive of a more open fractal structure and lower percolation threshold expected for aggregating polar materials. The inorganic liquids have a broader range of *D*_{f} ≈ 2–3, which is consistent with experiments on aggregates of inorganic liquids with *D*_{f} = 1.75, 1.83, and 2.24 for different materials.30 Note that in Table 2 the glasses with *D*_{f} = 3 should not exhibit an apparent second order transition while going from a glass to a liquid because Δ*C*_{p} = 0. For these equilibrium glasses, we also expect that α_{g} = α_{L}, which assumes that these materials have reached their equilibrium supercooled state below *T*_{g} with Δ*V* = 0.

Compound | Figure Label (if Different) | M | Ref. | T_{g} (K) | C_{p}(l)/C_{f}(g,c) | Ref. | D_{f} = 3C_{pg}/C_{pl} |
---|---|---|---|---|---|---|---|

Polymer | |||||||

Polystyrene | PS | 116 | 46 | 373 | 1.188 | 32 | 2.52 |

Polypropylene | PP | 137 | 5 | 260 | 1.268 | 32 | 2.37 |

1,4 polybutadiene | PBD | 107 | 24 | 216 | 1.293 | 32 | 2.32 |

Poly(methyl acrylate) | PMA | 102 | 5 | 387 | 1.398 | 32 | 2.15 |

Polydimethyl siloxane | PDMS | 79 | 48 | 149.5 | 1.387 | 32 | 2.16 |

Poly(vinyl acetate) | PVAc | 95 | 5 | 311 | 1.520 | 32 | 1.97 |

Poly(oxide propylene) | POP | 74 | 61 | 198 | 1.510 | 32 | 1.99 |

Polyisobutylene | PIB | 46 | 5 | 200 | 1.352 | 32 | 2.22 |

Polycarbonate | PC | 132 | 5 | 423.5 | 1.113 | 32 | 2.70 |

Polysulfone | PSF | 141 | 52 | 459 | 1.142 | 32 | 2.63 |

Polyethylene(amorphous) | PE | 46 | 9 | 237 | 1.538 | 32 | 1.95 |

Polyethylene terephthalate | PET | 156 | 111 | 342 | 1.304 | 32 | 2.30 |

Polyvinylchloride | PVC | 191 | 5 | 354 | 1.278 | 32 | 2.35 |

Poly(ethyl acrylate) | PEA | 83 | 53 | 249 | 1.370 | 32 | 2.19 |

Polyethermide | PEI | 214 | 29 | 478.5 | 1.100 | 29 | 2.73 |

Poly(oxyethylene) | POE | 23 | 54 | 232 | 1.928 | 32 | 1.55 |

Poly(ethylene Sebacate) | PESb | 35.5 | 55 | 245 | 1.605 | 32 | 1.87 |

Poly(methyl methacrylate) | PMMA | 103 | 56 | 367 | 1.231 | 56 | 2.44 |

Poly(ethyl Methacrylate | PEMA | 81 | 56 | 344 | 1.189 | 56 | 2.52 |

Poly(N-propyl methacrylate) | PprMA | 63 | 56 | 323.5 | 1.169 | 56 | 2.57″ |

Poly(N-butyl methacrylate) | PBMA | 56 | 56 | 305 | 1.187 | 56 | 2.53 |

Poly(N-pentyl methacrylate) | PpenMA | 49 | 56 | 291 | 1.155 | 56 | 2.60 |

Poly(N-hexyl methacrylate) | PHMA | 45 | 56 | 280.5 | 1.147 | 56 | 2.62 |

Organic small molecules | |||||||

2-methylpentane | 2-MP | 58 | 57 | 79.5 | 1.790 | 91 | 1.68 |

Dibutylphthalate | DBP | 69 | 58 | 179 | 1.527 | 92 | 1.96 |

Triphenylchloromethane | TPCM | 93 | 59 | 243 | 1.402 | 59 | 2.14 |

Tri-2-ethylhexylphthalate | T-2-NB | 66 | 60 | 447 | 1.249 | 60 | 2.40 |

o-terphenyl | OTP | 76 | 31 | 241 | 1.472 | 93 | 2.04 |

Di-2-ethylhexlphthalate | D-2-EHP | 67 | 62 | 187 | 1.505 | 92 | 1.99 |

m-xylene | MX | 56 | 31 | 120 | 1.323 | 94,95 | 2.28 |

o-xylene | OX | 55 | 31 | 123 | 1.192 | 94,95 | 2.52 |

Toluene | T | 59 | 31 | 110 | 1.512 | 96 | 1.98 |

3-bromopentane | 3-BP | 53 | 63 | 125.5 | 1.798 | 97 | 1.67 |

a-phenyl-o-cresol | APOC | 83 | 58 | 220 | 1.457 | 15 | 2.06 |

m-fluorotulene | MFT | 45 | 64 | 117 | 1.935 | 64 | 1.55 |

5-phenyl-4-ether | 5P4E | 85 | 65 | 243 | 1.421 | 65 | 2.11 |

Hydrogen bonding liquids | |||||||

Ethyl alcohol | Ethanol | 55 | 31 | 97 | 1.718 | 98 | 1.75 |

Glycerol | 53 | 66,68 | 190 | 1.847 | 99 | 1.62 | |

n-propanol | 35 | 67 | 109 | 1.898 | 100 | 1.58 | |

Propylene glycol | 52 | 66,68 | 167 | 1.848 | 101 | 1.62 | |

Sorbitol | 93 | 69 | 274 | 1.886 | 102 | 1.59 | |

Salol | 63 | 70 | 218 | 1.714 | 15 | 1.75 | |

m-cresol | 57 | 64 | 198.5 | 1.837 | 64 | 1.63 | |

m-fluoroaniline | 70 | 64 | 173 | 1.896 | 64 | 1.58 | |

m-toluidine | 79 | 64 | 187 | 1.836 | 64 | 1.63 | |

Inorganic liquids | |||||||

2BiCl_{3} KCl- | 85 | 31 | 306 | 1.647 | 31 | 1.82 | |

ZuCl2 + Py + Cl- | 67 | 31 | 275 | 1.415 | 31 | 2.12 | |

Na_{2}O.2Sio_{2} | 38 | 71 | 713 | 1.237 | 103 | 2.42 | |

As_{2}Se_{3} | 36 | 72 | 450 | 1.560 | 104 | 1.92 | |

Se | 87 | 23 | 307 | 1.498 | 105 | 2.00 | |

B_{2}O_{3} | 32 | 74 | 521 | 1.449 | 88,106 | 2.07 | |

ZBLA^{4} | 134 | 31 | 552 | 1.659 | 107 | 1.81 | |

SiO_{2} | 20 | 75 | 1446 | 1.005 | 108 | 2.99 | |

GeO_{2} | 24 | 76 | 818 | 1.073 | 109 | 2.80 | |

BeF_{2} | 24 | 77 | 590 | 1.000 | 31,77 | 3.00 | |

ZnCl_{2} | 30 | 2 | 370.5 | 1.279 | 110 | 2.35 | |

0.244Na_{2}O.0.756SiO_{2} | 30 | 56 | 764 | 1.216 | 73 | 2.47 | |

K_{3}Ca_{2}(NO_{3})_{7} | 93 | 79 | 332 | 1.568 | 37,88 | 1.19 |

### Relation of the TFT to Free Volume Theories

The TFT can predict most of the well known empirical relations that are based on *Free Volume* theories discussed by Fox and Flory13, 14 and Simha and Boyer15 in the last 60 years. The free volume theory postulated that a number of vacancies or holes existed into which atoms could hop and promote relaxation and diffusion. The specific volume of these holes was the free volume *v*_{f}, which according to Simha and Boyer in their Iso-Free Volume model of *T*_{g},15 was described by:

where the α′ values were the volumetric bulk expansion coefficients [α′(bulk ≈ 3α(linear)]. Thus, in terms of the linear α-values, we obtain *v*_{f} ≈ 0.03, which is quite similar to eq 4.5. However, this free volume is basically equivalent to Δ*V* extrapolated to 0 °K, which in the TFT is an artifact of the fractal frustration of the glass formation process below *T*_{g} and does not accurately describe the state of matter at *T*_{g}. In an equilibrium glass with Δ*V* = 0, we still have a glass transition with the twinkling process even though in eq 6.1 α_{L} = α_{g} and *v*_{f} = 0. At *T*_{g}, the specific volume is associated with the normal thermal expansion process and the true free volume Δ*V* is created in the glassy state at *T* < *T*_{g} when the contraction process is frustrated by the fractal percolation structure developed at *T*_{g}. The magnitude of Δ*V* at *T* = 0 °K is equivalent to *v*_{f}, but this does not exist at *T*_{g} as the free volume theory proposes. In the TFT approach, the free volume holes are replaced by the Twinkling Fractal process where the twinkling frequencies *F*(ω) allow large solid clusters in the glassy state at *T*_{g} to hop into the liquid state and visa versa, in accord with the vibrational density of states, *g*(ω) ∼ ω^{df–1}. In the TFT, the spatio-temporal fluctuation hopping process is mediated by the Boltzmann distribution of excited energy levels operating near the critically connected bonding state of atoms with an anharmonic potential energy function. The TFT provides new insight on the dynamics and structure of the glassy state which we demonstrate in the following brief examples.

#### The Fox-Flory Equation for Polymer Mixtures

The important Fox13 Rule of mixtures for two miscible polymers of individual glass transition temperatures *T*_{g}_{1} and *T*_{g}_{2} with weight fractions *w*_{1} and *w*_{2} gives the composite *T*_{g} of the single phase mixture as the well known formula:

Using the TFT, from eq 1.4 we have the sum of the liquid fractions at temperature *T* as:

At *T* = *T*_{g} of the mixture, the Twinkling Fractal percolation cluster forms at the solid fraction *P*_{S} = *p*_{c}, or *P*_{L} = (1 – *p*_{c}), as

Thus, at *T* = *T*_{g} of the composite, percolation occurs at *P*_{L} (composite) = (1 – *p*_{c}) such that:

Dividing across by (1 – *p*_{c}) and solving for *T*_{g}, we obtain the Fox eq 6.2 above. In a more detailed treatment,32 one should ask the specific system, e.g., copolymer or physical mixture of atoms A and B as to the state of aggregation between A and B. Thus, for random mixing of A,B atoms, the fractions of AA, AB and BB contacts would need to be considered and this approach would correctly introduce another *T*_{g} value, namely *T*_{g} (AB), whose magnitude would depend on *D*_{o}(*AB*) via eq 3.4. At the nanoscale, one could have multiple states of aggregation while maintaining a bulk single-phase system. In two-phase systems, we have two separate *T*_{g} values, which will manifest separately, assuming phase separation. However, with nanoscale phase separation, the twinkling processes of each phase will interact and one may see some interesting departures from the anticipated outcome predicted by the Fox equation.

#### Molecular Weight Effects on T_{g}

The Flory-Fox free volume based equation for the effect of polymer molecular weight *M* on *T*_{g} is given by the well know expression14

in which *K* is a species dependent constant. The TFT provides the liquid fraction near *T*_{g} as:

If we assume that the liquid fraction contribution from the two chain ends is given in terms of the molecular weights of the ends M_{end} by:

then we obtain from the latter two relations, that when *P*_{L} = (1 – *p*_{c}) at percolation:

Comparing the Flory-Fox relation with the TFT relation, we have the constant *K* as:

For entangled polymers with *M* > *M*_{c} where *M*_{c} is the critical entanglement molecular weight, we expect the liquid contribution of the chain end in a reptating chain dynamics environment to be associated with the local Rouse like motion of the entanglement length *M*_{c}. The critical entanglement molecular weight *M*_{c} is related to the monomer structure via *M*_{c} ≈ 60 *C*_{∞}*M*_{j}34 where *M*_{j} is the molecular weight per step of the chain along its backbone (e.g., *M*_{j} = 14 g/mol for PE). For example, with polystyrene the characteristic ratio *C*_{∞} = 10, *M*_{j} = 52 g/mol and thus *M*_{c} = 31,200 g/mol. If we let *M*_{end} = 31,200 g/mol and *p*_{c} ≈ [1/2], then we obtain from eq 6.10 that *K* = 1.25 × 10^{5} g/mol. This compares favorably with the Flory-Fox experimental value *K* = 1.8 × 10^{5} g/mol. Thus, any physical process that increases the liquid fraction will decrease *T*_{g}. These include solvents, low molecular weight blends, lengthy side groups, plasticizers, nano-thin films with free surfaces, low hydrostatic pressure etc. Conversely, any process that decreases the liquid fraction will increase *T*_{g} accordingly and these include antiplasticizers, short rigid side groups, high hydrostatic pressure, hydrogen bonding, nanoconfined polymers with strong adhesion to a rigid substrate, and crosslinks.

#### Effect of crosslinks on T_{g}

The effect of crosslink density *v* on *T*_{g} can be determined from the behavior of the solid fraction *P*_{S} as a function of temperature *T* as:

in which *P*_{S}(v) is the increase in the solid fraction because of crosslinks. The liquid fraction is given by:

where *T*_{g}° is the *T*_{g} of the corresponding linear polymer with no crosslinks. We let *P*_{S}(*v*) be given by the new primary bonded solid content, which replaces the nonbonded interactions as:

where ρ is the density and *M*_{ox} is the average molecular weight of a repeat unit on the chain and is related to the monomer molecular weight *M*_{o} and the number of backbone bonds per monomer *j* via:

The solid fraction percolates when *P*_{S} = *p*_{c} at *T* = *T*_{g}, so we obtain from the above relations:

Solving for *T*_{g}, we obtain:

The TFT analysis predicts that *T*_{g} will increase linearly with *v*. This is equivalent to the Fox-Loshaek equation,16 which gives *T*_{g}(*v*) from the free volume theory as a similar linear relation:

in which *K*_{n} is a material constant and corresponds to the TFT relation as:

For example, LaScala and Wool31 determined the *T*_{g} versus crosslink density for a model series of acrylated triglycerides and found the experimental linear relation in good accord with the Fox-Loshaek theory:

in which *T*_{g}° = 225 °K and the experimental slope *K*_{n} = 0.0055 [°Km^{3}/mol]. As with all the free volume relations, the TFT provides additional levels of information and using eq 6.18, we have *T*_{g}° = 225 °K, *p*_{c} ≈ 1/2, *M*_{ox} ≈ 14 g/mol for fatty acids, ρ = 1.1 × 10^{6} g/m^{3}, giving the predicted slope *K*_{n} = 0.0057 [°Km^{3}/mol], which is in excellent agreement with the experimental *K*_{n} value.

#### Relation of TFT to WLF

The William-Landel-Ferry (WLF) equation has been used in polymer rheology to relate viscosities and relaxation times to temperature using a time-temperature shift factor *a*_{T}.17 For example, the viscoelastic stress relaxation modulus *G*(*t*) measured at constant strain at a reference temperature *T*_{o} can be related to its value at some other temperature *T* in the range Δ*T* = *T*_{o} ± 50 °K via:

This useful relation allows one to obtain a material's relaxation data at just one temperature *T*_{o} and then predict how this material will behave at either very short times at high temperature, or for very long times at low temperature. This is done by the generation of a master curve for the viscoelastic property where one simply shifts the data set along the log time axis using eq 6.20. The shift factor as a function of temperature is described empirically by the WLF equation as:

in which *C*_{1} and *C*_{2} are constants.

The temperature dependence of the TFT derives from the spatio-temporal autocorrelation function *C*(*t*) in eq 2.6, which describes the twinkling frequencies *F*(ω) near *T*_{g} and we rewrite in terms of *t*/*a*_{T} as:

in which the shift factor *a*_{T} is

and

The constant β = α_{L}^{2}*D*_{o}*a*^{2}*R*, as discussed previously in eq 2.3. Letting *T* = *T*_{g} + Δ*T* and making the approximation that 2*T*_{g} + Δ*T* ≈ 2*T*_{g}, we obtain from eq 6.24 the analogous expression for the WLF equation:

in which the constants *C*_{1} = 2*T*_{g}β/*k* and *C*_{2} = *T*_{g}. The correspondence of the TFT with the WLF equation is gratifying and worth further study.32 Physical aging in the glassy state should be influenced by the higher twinkling frequencies described by eq 2.7 such that their relaxation function is:

The TFT also provides useful information on rate effects on the mechanical damping tan δ peaks and yield stress.1 The tan δ peak should occur near *T*_{g} because the maximum exchange rate controlling the loss of stored elastic energy between liquid and solid phases in the twinkling fractal process occurs at *T*_{g} with Δ*E* = 0. It can be shown from the TFT that the dependence of *T*_{g} on the experimental rate ω_{x} behaves approximately as *T*_{g}(ω_{x}) ∼ ω_{x}^{B}, where *B* ∼ 0.01.32 Depending on the experimental rate ω_{x}, only certain parts of the twinkling frequency spectrum *F*(ω) will be able to relax in time, e.g., those twinkles in *C*(*t*) with ω > ω_{x}. Similar rate processes occur in shear thickening fluids and yielding of glassy polymers.1