We present a new conceptual approach to understanding the glass transition temperature Tg 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 Tg. 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 Tg 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 Tg is called the Twinkling Fractal Theory (TFT).1 The fractal twinkles with a characteristic vibrational density of states g(ω) ∼ ωdf–1, where df = 4/3 is the Orbach fracton dimension.2, 3 Upon further cooling below Tg, the fractal structure existing at Tg 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 αL, 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 Tg 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 Tg and thermal expansion coefficient via the dimensionless number, αL×Tg ≈ 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 Tg where the slope of V versus T changes at Tg. The slopes dV/dT 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 Tg have been advanced by Wolynes et al.,5, 6 who examined cooperatively rearranging regions (CRRs) near the critical temperature Tc 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 Tc.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 Tg (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 ≫ Tg. 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 Tg 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 Tg 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 Tg 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) ∼ ωdf–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 Tg 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 Tg,13–16 the William-Landel-Ferry (WLF) relation17 for the temperature dependence of viscosity near Tg, 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 Tg
We examine the onset of the glass transition temperature Tg 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 Do is the interatomic bond energy, ‘a’ is the anharmonicity factor and x = R − Ro is the bond displacement from its equilibrium value Ro. 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 αL via the mean bond displacement 〈X〉 (Fig. 2).
As Tg 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 Xc at the inflection point of U(x), where the interatomic force f = dU/dx reaches its maximum value and the force constant K = d2U/dX2 = 0. Thus, when X > Xc and T > Tg, the molecules can readily escape with a thermofluctuation to the ergodic liquid state and when X < Xc at T < Tg, 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 (Xc ≈ 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 PS of solid atoms (X < Xc) and a fraction PL of liquid atoms with PS + PL = 1.
The solid fraction Ps at any T is determined from the integral of the Boltzmann energy populations ϕ(x) in the integration limits (0, Xc):
Near Tg, we find that the liquid and solid fractions are well described by eq 1.3 as:
where pc ≈ [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) pc ≈ 0.3, and for vector percolation (force, modulus), pc ≈ 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 Tg, eq 1.5 gives PS ≈ pc. Note that the solid fraction PS is significant at temperatures well above Tg and does not approach zero until T ≈ Tg/(1 – pc), or T ≈ 2 Tg. This is consistent with the dynamic heterogeneity observed at T ≫ Tg.7 Similarly, the liquid fraction PL is significant at T < Tg and only approaches zero as T approaches 0 °K. However, the relations for PL and PS above are approximate near Tg and large extrapolations in temperature would require a more detailed solution of eq 1.3. Note that PS varies smoothly with T in a linear fashion through Tg, such that a first order phase transition does not occur.
As Tg is approached from above, the fraction of solid atoms Ps 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 pc by:
where Df is the fractal dimension of the cluster. As PS approaches pc, a fractal structure of infinite size forms as the correlation length diverges to infinity at pc and percolation occurs, as shown in Figure 3.
The percolation cluster has a fractal dimension Df and is not stationary because the solid and liquid atoms are in dynamic equilibrium. Thus at Tg, 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 Tg, 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 Tg, 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 ETo/(T−To), in which To is the temperature when the relaxation time or viscosity η goes to infinity, e.g., near Pc in the TFT. Others have used the Kohlrausch relation  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) ∼ ωdf–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 Tg.21 The TFT suggests that the dynamics of the glassy state near Tg 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 ωdf–1 is the Orbach vibrational density of states g(ω) ∼ ωdf–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 Uc 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 df = 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 Tg, 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 = 2Doa2 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 = αL2Doa2Ro2. The activation energy ΔE = [U(T) − Uc] 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) ∼ ωdf–1 will have a cluster relaxation function c(t, ω) given by:
Thus, at time t, only vibrations in the range ω = ωo 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 aT and a stretched exponential form of the relaxation function.
At short times or low temperature T < Tg, 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) ∼ ω1/3 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(ωdf)/t solution such that over most of the time range at Tg, we expect:
The transition of the initial time exponent of –1/3 to –4/3 is again a consequence of the fracton dimension df = 4/3 in the fractal density of states.
The spatio-temporal fluctuations of amorphous polyvinyl acetate (PVAc) films near Tg 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 Tg = 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 Tg) that at short times (27–100 s), C(t) ∼ 1.26 t−0.35 with correlation coefficient r2 = 0.97; at intermediate-long times (68–670 s), C(t) ∼ t−1.36 with r2 = 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) ∼ ω2 at low ω or long times. The crossover to the normal Debye density of states g(ω) ∼ ωd–1 (d = 2 or 3) occurs at ωc 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 Tg, 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 Tg), 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 Tg. 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 Tg from the TFT
To obtain Tg from the TFT using eq 1.3, we find that the quasiharmonic approximation for the Morse potential facilitates calculations of thermal expansion coefficients and Tg.19, 22 Using U(x) = Ax2–Bx3 in which the constants A = Doa2 and B = Doa3, the Morse potential can be represented by the quasiharmonic approximation19:
because Pc ≈ [1/2], the glass transition temperature is well approximated by:
For example, if Do = 3.4 kcal/mol, eq 3.4 gives Tg = 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 Do will lead to higher Tg values. For a heteroatom polymer, one would expect that Do 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 ϕi are the mole fractions of the ith atom in the chain and Doi 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 CH2 groups with Do(CH2) ≈ 2 kcal/mol (based on PE) and 2/7 H-bonding sites with Do(HB) ≈ 5 kcal/mol. The average Do 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 Tg (Nylon 6) ≈ 320 °K (47 °C), which is in agreement with experiment. The value of Do(HB) = 5 kcal/mol gives exact agreement with experiment.
The intermolecular bond strength Do can be obtained form Tg in eq 3.4 as:
For polyethylene, Tg = 223 °K which gives Do ≈ 2 kcal/mol, which is similar to the crystallization heat of fusion ΔHf ≈ 2 kcal/mol.25 It can be expected that Do and ΔHf should have comparable magnitude but should not be exactly matched because of the different interatomic configurations between the amorphous and crystalline states. For polypropylene, Tg = 268, Do = 2.4 kcal/mol and ΔHf = 2.13 kcal/mol. With atactic polystyrene, Tg = 373 °K, Do ≈ 3.3 kcal/mol and ΔHf = 2 kcal/mol (isotactic). If we assume that Do ≈ ΔHf, and because the liquid fraction behaves as PL = (1 – pc)T/Tg, we can extrapolate to the melting point T → Tm as PL → 1 which gives the relation between Tm and Tg as:
Because pc ≈ 1/2, this relation suggest that Tm ≈ 2Tg on a Kelvin scale, as is typically observed.4
Thermal Expansion Coefficients near Tg
The TFT predicts a unique relationship between Tg and thermal expansion coefficient αL. In the liquid, the thermal expansion coefficient αL is determined from the average position 〈X〉 of the oscillators with respect to their Boltzmann population distribution ϕ(x):
The latter relation gives typical values of αL for amorphous materials; using k = 1.986 cal/mol °K, a = 2/Å, Ro = 3Å and Do = 3.5 kcal/mol, then αL = 70 ppm/°K and αg ≈ 35 ppm/°K [1 ppm = 10−6].
The existence of the fractal structure at Tg means that at T < Tg, thermal contraction in the glass is constrained by the fractal structure and αg is related to αL by:
Because pc ≈ [1/2], then the glass CTE is typically about [1/2] of the liquid CTE. Physical aging occurs below Tg 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/aT]−1/3, as suggested by eq 2.7. Even well below Tg, 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 Tg 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 ΔV ≈ 0, a critical observation would be that αg = αL and the pseudo second order transition in the heat capacity change ΔCp = 0 at Tg. When the fractal constraints disappear at Tg, 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.
For typical amorphous solids Ro ≈ 3A, and using a ≈ 2/Å for the Morse anharmonicity factor, then we expect that:
Figure 7 shows a plot of αL versus 1/Tg 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 αL → 0, then Tg → ∞ as suggested in Figure 5. Thus, anharmonicity plays a major common role in determining Tg 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 ΔCp at Tg. Cp 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 Df via CpL = dUL/dT and Cpg = dUg/dT. Therefore, when UL = 3 kT and Ug = DfkT, we obtain the simple ratio:
The latter relation highlights the role of the fractal structure on the determination of ΔCp. For bulk samples, d = 3 and Df ≈ 2.5 such that CpL/Cpg = 1.2 and for very thin films, d = 2 and Df = 1.75 and CpL/Cpg = 1.14. This prediction is in excellent agreement with values for ΔCp at Tg for bulk (d = 3) and nano thin film (d = 2) polystyrene samples obtained by Koh et al.23 and summarized in Table 1.
|D||Df||CpL/Cpg TFT||CpL/Cpg Expt||ΔCp TFT||ΔCp Expt|
Table 2 shows a considerable body of Cp 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 Df at Tg via:
We find in Table 2 that most polymers have a fractal dimension of Df ≈ 2.5, as might be expected from a random cluster model of long polymer chains without strong attractions. Organic small molecules have Df ≈ 2 fairly uniformly which is in accord with fractal dimensions of biological colloids with Df = 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 Df ≈ 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 Df ≈ 2–3, which is consistent with experiments on aggregates of inorganic liquids with Df = 1.75, 1.83, and 2.24 for different materials.30 Note that in Table 2 the glasses with Df = 3 should not exhibit an apparent second order transition while going from a glass to a liquid because ΔCp = 0. For these equilibrium glasses, we also expect that αg = αL, which assumes that these materials have reached their equilibrium supercooled state below Tg with ΔV = 0.
|Compound||Figure Label (if Different)||M||Ref.||Tg (K)||Cp(l)/Cf(g,c)||Ref.||Df = 3Cpg/Cpl|
|Organic small molecules|
|Hydrogen bonding liquids|
|ZuCl2 + Py + Cl-||67||31||275||1.415||31||2.12|
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 vf, which according to Simha and Boyer in their Iso-Free Volume model of Tg,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 vf ≈ 0.03, which is quite similar to eq 4.5. However, this free volume is basically equivalent to ΔV extrapolated to 0 °K, which in the TFT is an artifact of the fractal frustration of the glass formation process below Tg and does not accurately describe the state of matter at Tg. In an equilibrium glass with ΔV = 0, we still have a glass transition with the twinkling process even though in eq 6.1 αL = αg and vf = 0. At Tg, the specific volume is associated with the normal thermal expansion process and the true free volume ΔV is created in the glassy state at T < Tg when the contraction process is frustrated by the fractal percolation structure developed at Tg. The magnitude of ΔV at T = 0 °K is equivalent to vf, but this does not exist at Tg 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 Tg to hop into the liquid state and visa versa, in accord with the vibrational density of states, g(ω) ∼ ωdf–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 Tg1 and Tg2 with weight fractions w1 and w2 gives the composite Tg 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 = Tg of the mixture, the Twinkling Fractal percolation cluster forms at the solid fraction PS = pc, or PL = (1 – pc), as
Thus, at T = Tg of the composite, percolation occurs at PL (composite) = (1 – pc) such that:
Dividing across by (1 – pc) and solving for Tg, 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 Tg value, namely Tg (AB), whose magnitude would depend on Do(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 Tg 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 Tg
The Flory-Fox free volume based equation for the effect of polymer molecular weight M on Tg is given by the well know expression14
in which K is a species dependent constant. The TFT provides the liquid fraction near Tg 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 Mend by:
then we obtain from the latter two relations, that when PL = (1 – pc) at percolation:
Comparing the Flory-Fox relation with the TFT relation, we have the constant K as:
For entangled polymers with M > Mc where Mc 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 Mc. The critical entanglement molecular weight Mc is related to the monomer structure via Mc ≈ 60 C∞Mj34 where Mj is the molecular weight per step of the chain along its backbone (e.g., Mj = 14 g/mol for PE). For example, with polystyrene the characteristic ratio C∞ = 10, Mj = 52 g/mol and thus Mc = 31,200 g/mol. If we let Mend = 31,200 g/mol and pc ≈ [1/2], then we obtain from eq 6.10 that K = 1.25 × 105 g/mol. This compares favorably with the Flory-Fox experimental value K = 1.8 × 105 g/mol. Thus, any physical process that increases the liquid fraction will decrease Tg. 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 Tg 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 Tg
The effect of crosslink density v on Tg can be determined from the behavior of the solid fraction PS as a function of temperature T as:
in which PS(v) is the increase in the solid fraction because of crosslinks. The liquid fraction is given by:
where Tg° is the Tg of the corresponding linear polymer with no crosslinks. We let PS(v) be given by the new primary bonded solid content, which replaces the nonbonded interactions as:
where ρ is the density and Mox is the average molecular weight of a repeat unit on the chain and is related to the monomer molecular weight Mo and the number of backbone bonds per monomer j via:
The solid fraction percolates when PS = pc at T = Tg, so we obtain from the above relations:
Solving for Tg, we obtain:
The TFT analysis predicts that Tg will increase linearly with v. This is equivalent to the Fox-Loshaek equation,16 which gives Tg(v) from the free volume theory as a similar linear relation:
in which Kn is a material constant and corresponds to the TFT relation as:
For example, LaScala and Wool31 determined the Tg 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 Tg° = 225 °K and the experimental slope Kn = 0.0055 [°Km3/mol]. As with all the free volume relations, the TFT provides additional levels of information and using eq 6.18, we have Tg° = 225 °K, pc ≈ 1/2, Mox ≈ 14 g/mol for fatty acids, ρ = 1.1 × 106 g/m3, giving the predicted slope Kn = 0.0057 [°Km3/mol], which is in excellent agreement with the experimental Kn 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 aT.17 For example, the viscoelastic stress relaxation modulus G(t) measured at constant strain at a reference temperature To can be related to its value at some other temperature T in the range ΔT = To ± 50 °K via:
This useful relation allows one to obtain a material's relaxation data at just one temperature To 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 C1 and C2 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 Tg and we rewrite in terms of t/aT as:
in which the shift factor aT is
The constant β = αL2Doa2R, as discussed previously in eq 2.3. Letting T = Tg + ΔT and making the approximation that 2Tg + ΔT ≈ 2Tg, we obtain from eq 6.24 the analogous expression for the WLF equation:
in which the constants C1 = 2Tgβ/k and C2 = Tg. 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 Tg because the maximum exchange rate controlling the loss of stored elastic energy between liquid and solid phases in the twinkling fractal process occurs at Tg with ΔE = 0. It can be shown from the TFT that the dependence of Tg on the experimental rate ωx behaves approximately as Tg(ωx) ∼ ωxB, where B ∼ 0.01.32 Depending on the experimental rate ωx, 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 > ωx. Similar rate processes occur in shear thickening fluids and yielding of glassy polymers.1