Kinetic interpretation of the glass transition: Glass temperatures of n-alkane liquids and polyethylene

On the basis of an isoviscosity criterion for the glass transition (η_g ? 10¹³ poise) in liquids of low molecular weight, theoretical T_g values were calculated for the n-alkane series by the equation log η = log A + B/(T ? T₀), with the use of values reported by Lewis for the parameters. The T_g/T₀ ratio reaches a limiting value of 1.25 and ?_g = (T_g ? T₀)/2.3B = 0.027, a constant. Extrapolation to (CH₂)_∞ gives T_g = 200°K., T₀ = 160°K., and B = 640°K. This T_g is consistent with other estimates for poly-ethylene, and T₀ coincides with the temperature at which the “excess” liquid entropy for (CH₂)_∞ becomes zero from thermodynamic data. For polymer liquids it is proposed that E₀ = 2.3RB is determined by the internal barriers to rotation for the “isolated” polymer chains. Thus, E₀ = 2.9 kcal./mole for polyethylene, 3.0 kcal./mole for polystyrene, 5.7 kcal./mole for polyisobutylene, and 1.9 kcal./mole for polydimethylsiloxane.     

Free volumes in simple and polymeric liquids

Values of ε₀ν₀ the vaporization energy and volume in the hypothetical liquid state at 0°K., are derived for some simple polar and nonpolar molecules used as models for vinyl polymers. The following empirical relationship between the free volume fraction, f = (v ? v₀)/v, and the liquid compressibility coefficient β is demonstrated: ?f² ∝? This is applied to several vinyl polymer liquids near their glass transition temperatures, T_g, giving. f_g ? 0.17, if the “hard-core” volume v^* is considered to be independent of pressure and temperature, (i.e., v^* = v₀); or, f_g ?0.12, if the P,T dependence of v^* is considered to be the same as that of the glass. These agree with f_g values derived by Simha and Boyer from thermal expansion coefficients for the two analogous cases. An empirical viscosity-free volume equation of the Doolittle form: η = ATⁿe^b/f is applied to the glass transition, on assuming that this is an isoviscosity state and with the use of reported values for the expansion and compressibility coefficients and dT_g/dP for three polymers: polystyrene, poly(methyl methacrylate), and poly(vinyl acetate). Reasonable values of b/n are thus obtained. This viscosity equation is critically examined in the light of molecular theories of liquid viscosity.     

The dependence of dielectric response of an elastomer on hydrostatic pressure

Dielectric methods have been employed to study the high-pressure behavior of a polyurethane elastomer (Solithane 113) in the vicinity of its α transition. The α-loss peak is shifted to higher temperatures and broadened somewhat with the application of hydrostatic pressure up to 6.4 kbars. The slope of T_α vs. P, or dT_α/dP, obtained at low frequencies was found to be equal to dT_g/dP obtained by a volumetric method. Moreover, it attained a nonzero limiting value at high pressures for each frequency tested (3—30,000 Hz) and the limiting value itself increased with increasing frequency from 10.5°C/kbar at 3 Hz to 18°C/kbar at 30,000 Hz. The activation enthalpy ΔH* was found to be nearly constant over the pressure range tested, but the activation volume ΔV* decreased with increasing pressure. The relation dT_α/dP = T (ΔV*/ΔH*) was shown to hold for the elastomer.     

Pressure dependence of the viscosity of dilute polystyrene solutions in toluene

The effect of pressure on the viscosity of dilute solutions of anionically polymerized polystyrene (M?_w = 209,000; M_w/M_n = 1.12) in toluene has been studied at different temperatures and concentrations using a falling-body viscometer. Measurements were performed in the concentration range from 0.0025 to 0.02 g/mL and at temperatures from 25 to 45°C under pressure up to 1057 bars. The viscosity coefficient η increases exponentially with pressure at a given temperature and concentration, while the apparent volume of activation V^? decreases with increasing temperature. The hypothesis that the pressure dependence of η is given by the pressure dependence of the activation energy holds true under the prevailing thermodynamic conditions. Log η increases linearly with increasing concentration at a given pressure. Intrinsic viscosity increases with increasing pressure, whereas the Huggins constant decreases.     

Pressure dependence of dielectric properties of chlorinated polyethylene vulcanizate. I. α relaxation and shift factors

The complex dielectric constant was measured under elevated pressure for the α relaxation of vulcanized chlorinated polyethylene. Both temperature and pressure effects on the static dielectric constants, the activation enthalpy, and volume, and the pressure dependence of the glass-transition temperature were obtained. The dependence of shift factors on temperature was expressed by the Vogel–Fulcher–Tamman–Hesse (VFTH) equation: ?log a_T = A ? B/(T ? T₀). The parameters A, B, and T₀ for each pressure applied were calculated by minimizing the standard deviation between log a_T and experiments. The values of the parameters in the Williams–Landel–Ferry (WLF) equation: ?log a_T = C₁(T ? T_g)/[C₂ + (T ? T_g)], were also estimated from the resulting values of the VFTH parameters. All these parameters depended on pressure. The activation volume plotted against T ? T_g decreased with increasing pressure.     

Dependence of viscosity of concentrated polymer solutions upon molecular weight and concentration

Molecular weight M and concentration c dependencies of the zero-shear viscosity (η) were measured over wide ranges of M and c for concentrated solutions of linear and branched poly(vinyl acetate) as well as of polystyrene under θ conditions. The log η versus log M and log η versus log c curves for a given system can be superposed by the horizontal shift along the abscissa, giving smooth master curves. From the shift factors the ratio of two exponents β and α, which appear in the following equation, can be evaluated: η = K′(cρ)^αM^β, where ρ is the density of the solution and K′ is a constant at constant temperature. The evaluated values of β/α for the systems under θ conditions are equal to or very close to 0.50 as was anticipated from the previous work. The above superposition method was also applied to available viscosity data, and it was found that β/α had a good correlation with a in [η] = KM^a. This indicates that the individual molecules in concentrated solutions maintain the same individuality as in dilute solutions, and might be a positive support to the packed sphere model proposed previously by the authors. The effect of solvent on the molecular weight and the concentration dependencies of viscosity was also discussed.     

Moderately concentrated solutions of polystyrene. I. Viscosity as a function of concentration,temperature, and molecular weight

Data on the viscosity η of moderately concentrated solutions of polystyrene are reported. Several solvents were investigated, including cyclopentane solutions over a temperature span between θ_U = 19.5°C and θ_L = 154.5°C. The data were analyzed in terms of a relation giving η as a function of αφM, where α_φ is the expansion factor for the chain dimensions in a solution with volume fraction φ of polymer with molecular weight M. It is shown that values of α_φ so determined decrease as ? lnα_φ/? lnφ = (1 ? 2μ)/6μ for φ greater than φ^* = 0.2M/s³ for moderately concentrated solutions, where s is the root-mean-square radius of gyration and μ = ? ln[η]/? lnM with [η] the intrinsic viscosity.     

Comments on “some evidence against the existence of the liquid-liquid transition. IV. The temperature dependence of shear viscosity”

A formal definition of T_LL as a function of M?ⁿ for polystyrene was prepared with literature T_LL values from torsional braid analysis (TBA), differential scanning calorimetry (DSC), and zero-shear melt viscosity η₀. Data from six authors using anionically prepared PS and blends thereof were involved. The resultant linear least-squares regression line, T_LL(°C) = 148.5 ? 11.487 × 10⁴M? [standard error in T_LL (calculated) 4.056 K, correlation coefficient R² = 0.9534] is considered valid from M?_n = 2000 to the entanglement molecular weight M_c = 35,000. The “best” T_LL values reported by Orbon and Plazek from double Arrhenius plots are well below this line for M?_v = 47,000, 16,400, 3400, and above it for M?_v = 1100. These best T_LL values are artifacts arising from no or insufficient data points above or below T_LL and/or too many data points near T_g. The associated high enthalpies of activation which they report confirm this diagnosis. The fact that these artificial T_LL values tend to disappear when checked by the three-parameter Vogel equation, logη = logA + B exp[(T ? T_∞)^?1], has no relevance to the controversy concerning the existence and meaning of T_LL. The claim by Orbon and Plazek that T_LL values obtained by TBA, DSC, and melt viscosity are all artifacts of the individual methods by which they were obtained is inconsistent with the excellent master plot which they generate. Alternative plotting devices which reveal T_LL > T_g from η₀ vs. T^?1 data, as developed by van Krevelen and Hoftyzer and by Utracki and Simha (not previously considered by either party), are reviewed. A statistical examination of the nature of the Vogel–Fulcher–Tammann–Hesse equation, based on synthetic data, is presented. Evidence for T_LL in atactic polypropylene is offered based on published data by Plazek and Plazek. T_LL is considered to possess both relaxational and quasiequilibrium attributes, just as T_g does.     

Photon correlation spectroscopy of polystyrene as a function of temperature and pressure

Photon correlation spectroscopy is employed to study the slowly relaxing density and anisotropy fluctuations in bulk atactic polystyrene as a function of temperature from 100 to 160°C and pressure from 1 to 1330 bar. The light-scattering relaxation function is well described by the empirical function ?(t) = exp[?(t/τ)^β], where for polystyrene β = 0.34. The average relaxation time is determined at each temperature and pressure according to 〈τ〉 = (τ/β)Γ(1/β) where Γ(x) is the gamma function. The data can be described by the empirical relation 〈τ〉 = 〈τ〉₀ exp[(A + BP)/R(T ? T₀)] where R is the gas constant and T₀ is the ideal glass transition temperature. The empirical constant A/R is in good agreement with that determined from the viscosity or the dielectric relaxation data (1934 K). The empirical constant B can be interpreted as the activation volume for the fundamental unit involved in the relaxation and is found to be comparable to one styrene subunit (100 mL/mol). The quantity B appears to be a weak function of temperature. The use of pressure as a tool in the study of light scattering near the glass transition now has been established.     

Poly(2,6-diphenyl-1,4-phenyl oxide): Characterization by gel-permeation chromatography,light scattering,and solution viscosity

Ten unfractionated poly(2,6-diphenyl-1,4-phenylene oxide) samples were examined by gel permeation chromatography (GPC) and intrinsic viscosity [η] at 50°C in benzene, by intrinsic viscosity at 25°C in chloroform, and by light scattering at 30°C in chloroform. The GPC column was calibrated with ten narrow-distribution polystyrenes and styrene monomer to yield a “universal” relation of log ([η]M) versus elution volume. GPC-average molecular weights, defined as M?gpc = \documentclass{article}\pagestyle{empty}\begin{document}$\Sigma w_i [\eta ]_i M_i /\Sigma w_i [\eta ]_i$\end{document}, w_i denoting the weight fraction of polymer of molecular weight M_i, were computed from the GPC and [η] data on the polyethers. The M?GPC were then compared with the weight-average M?_w from light scattering. The intrinsic viscosity (dl/g) versus molecular weight relations for the unfractionated poly(2,6-diphenyl-1,4-phenylene oxides) determined over the molecular weight range 14,000 ≤ M?_w ≤ 1,145,000 are log [η] = ?3.494 + 0.609 log M?_w (chloroform, 25°C) and log [η] = ?3.705 + 0.638 log M?_w (benzene, 50°C). The M?_w(GPC)/M?_n(GPC) ratios for the polymers in the molecular weight range 14,000 ≤ M?_w ≤ 123,000 approximate 1.5 according to computer integrations of the GPC curves with the use of the “universal” calibration and the measured log [η] versus log M?_w relation. The higher molecular weight polymers (326,000 ≤ M?_w ≤ 1,145,000) show slightly broadened distributions.     

Pressure dependence of dielectric properties of chlorinated polyethylene vulcanizate. II. Free volume and configurational entropy theories

The P–V–T relation and the heat capacity of a chlorinated polyethylene vulcanizate were measured. Several tests on the validity of thermodynamic treatments of the α relaxation process and the glass-transition temperature were performed by using dielectric properties. From a study using excess variables, it was shown that the entropy theories represented by the equations of Goldstein and Adam–Gibbs' were slightly better than the free volume theory represented by Doolittle's equation. However the study provided no distinction between the two entropy theories. Some tests were also performed on the pressure dependence of the glass-transition temperature, dT_g/dP, and on H^*_V/H^*_V where H^*_V is the isochoric activation enthalpy and H^*_P is the isobaric activation enthalpy. Here, too, the entropy theories were better than the free volume theory. Goldstein's expression gave values of both dT_g/dP and H^*_V/H^*_P closest to those from the dielectric experiments. The Adam–Gibbs' equation gave a temperature dependence for dT_g/dP and H^*_V/H^*_P most similar to those from the experiments.     

Twinkling fractal theory of the glass transition: Rate dependence and time–temperature superposition

The twinkling fractal theory (TFT) of the glass transition temperature T_g provides a new method of analyzing rate effects and time–temperature superposition in amorphous materials. The rate dependence of T_g was examined in the light of new experimental and theoretical evidence for the nature of the dynamic heterogeneity near T_g. As T_g is approached from above, dynamic solid fractal clusters begin to form and eventually percolate rigidity at T_g. The percolation cluster is a solid fractal and to the observer, appears to “twinkle” as solid and liquid clusters interchange in dynamic equilibrium with a vibrational density of states g(ω) ∼ ω. The solid-to-liquid twinkling frequencies ω_TF are controlled by the Boltzmann population of intermolecular oscillators in excited energy levels of their anharmonic potential energy functions U(x) such that ω_TF = ω exp −B(T*² − T²)/kT in which T* ≈ 1.2T_g. An oscillator changes from a solid to a liquid when a thermal fluctuation causes it to expand beyond its inflection point in the anharmonic potential. This leads to a continuous solid fraction P_s near T_g given by P_S ≈ 1−[(1 − p_c) T/T_g] where p_c ≈ 1/2 is the rigidity percolation threshold. Since g(ω) is continuous from very low to very high frequencies, the complex twinkling dynamics existing near T_g produces a continuous relaxation spectrum with many different length scales and times associated with the fractal clusters. The twinkling frequencies control the kinetics of T_g such that for a given observation time t when the rate γ > 1/t, only those parts of the twinkling spectrum with ω > γ can contribute to relaxation or percolation upto time t. The most important results in this article are as follows: The TFT describes the rate dependence of T_g, both for DSC thermal heating/cooling rates and DMA frequencies as the classic T_g − lnγ law as T_g(γ) = T_go + (k/2B) ln γ/γ_o in which the constant B = 0.3 cal/mol K². The constant B appears quite universal for the 17 thermoset polymers investigated in this study and 18 linear polymers investigated by others. Many other amorphous metal and ceramic glass materials exhibited the same rate law but required a new B value approximately half that for polymers. The same B = 0.3 value was also used to successfully describe the TTS shift factors using the twinkling fractal frequencies ω_TF = ωexp −B(T*² − T²)/kT, as ln a_T(TFT) = exp B(T_R² − T²)/kT, which gave comparable results with the classical WLF equation, log a_T = [−C₁(T − T_R)]/[C₂ + (T − T_R)]. The advantage of the TFT over the WLF is that C₁ and C₂ are not universal constants and must be determined for every material, whereas the TFT uses one known constant B which appears to be the same for all polymers. The TFT has also been found to describe the strong and fragile nature of the viscosity behavior of liquids and the rate and temperature dependence of the yield stress in polymers. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2578–2590, 2009     

Kinetics of the iodine atom catalyzed gas phase geometrical isomerization of diiodoethylene. Rotation about a single bond

The spectrophotometric determination of the rate of iodine atom catalyzed geometrical isomerization of diiodoethylene in the gas phase from 502.8 to 609.1°K leads to a rate constant for the bimolecular reaction between I and trans-diiodoethylene of log k_t?c(M^?1 sec^?1) = 8.85 ± 0.12 ? (11.01 ± 0.30)/θ. Estimates of the entropy and enthalpy change for the addition of I atoms to trans-diiodoethylene (process a.b) lead to log K_a.b(M^?1) = ?2.99 ? 4.0/θ, and thus to log k_c (sec^?1) = log k_t?c – log K_ab = 11.8 ?7.0/θ for the rate constant for rotation about the single bond in the adduct radical. The theory for calculation of the rotation rate constant is presented and it is shown that while the exact value depends on the barrier height, a value of 6.8 kcal/mole for this quantity leads to log k (sec^?1) = 11.8 ?6.7/θ. The activation energy points to a better value of the group contribution to heat of formation of the group C -(I)₂(H)(C) than one based on bond additivity.     

Viscosity of polydimethylsiloxane–pentamer systems

Viscosities of polydimethylsiloxane–pentamer systems were measured over the whole range of concentration. Twelve samples having molecular weights from about 1000 to 5 × 10⁵ were studied. The empirical reduction scheme, plots of log η versus log _cM^0.68, suggested by Ferry and co-workers is applicable to samples of M?_v ≥ 22,000 over the entire concentration. Such satisfying superposition of data may be attributed to the systems being the homologous mixtures in which glass temperatures of polymers are very low. On the basis of the treatment of Fox and Allen, the effects of the number and weight-average molecular weight on viscosity were examined, and the friction coefficient ζ per chain atom at constant M?_n was calculated over a wide range of M?_n. The value ζ is almost constant (ζ = 7.4 × 10^?9 dyne-sec./cm.) in the region of M?_n ≥ M_c, and where otherwise it decreases rapidly with decreasing M?_n. The length of the chainend segment was tentatively calculated.     

Styrene and isoprene friction factors in styrene–isoprene matrices

Monomeric friction factors, Ξ, for polystyrene (PS), polyisoprene (PI), and a polystyrene–polyisoprene (SI) diblock copolymer have been determined as a function of temperature in four poly(styrene-b-isoprene-b-styrene-b-isoprene) tetrablock copolymer matrices. The Rouse model has been used to calculate the friction factors from tracer diffusion coefficients measured by forced Rayleigh scattering. Within the experimental temperature range the tetrablock copolymers are disordered, allowing for measurement of the diffusion coefficient in matrices with average compositions determined by the tetrablock copolymers (23, 42, 60, and 80% styrene by volume). Remarkably, for a given matrix composition the styrene and isoprene friction factors are essentially equivalent. Furthermore, at a constant interval from the system glass transition temperature, T_g, all of the friction factors (obtained from homopolymer, diblock copolymer, and tetrablock copolymer dynamics) agree to within an order of magnitude. This is in marked contrast to results for miscible polymer blends, where the individual components generally have distinct composition dependences and magnitudes at constant T − T_g. The homopolymer friction factors in the tetrablock matrices were systematically slightly higher than those of the diblock, which in turn were slightly higher than those of the homopolymers in their respective melts, when all compared at constant T − T_g. This is attributed to the local spatial distribution of styrene and isoprene segments in the tetrablocks, which presents a nonuniform free energy surface to the tracer molecules. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 3079–3086, 1998     

Extension of the chain‐end,free‐volume theory for predicting glass temperature as a function of conversion in hyperbranched polymers obtained through one‐pot approaches

Compared with linear polymers, more factors may affect the glass‐transition temperature (T_g) of a hyperbranched structure, for instance, the contents of end groups, the chemical properties of end groups, branching junctions, and the compactness of a hyperbranched structure. T_g's decrease with increasing content of end‐group free volumes, whereas they increase with increasing polarity of end groups, junction density, or compactness of a hyperbranched structure. However, end‐group free volumes are often a prevailing factor according to the literature. In this work, chain‐end, free‐volume theory was extended for predicting the relations of T_g to conversion (X) and molecular weight (M) in hyperbranched polymers obtained through one‐pot approaches of either polycondensation or self‐condensing vinyl polymerization. The theoretical relations of polymerization degrees to monomer conversions in developing processes of hyperbranched structures reported in the literature were applied in the extended model, and some interesting results were obtained. T_g's of hyperbranched polymers showed a nonlinear relation to reciprocal molecular weight, which differed from the linear relation observed in linear polymers. T_g values decreased with increasing molecular weight in the low‐molecular‐weight range; however, they increased with increasing molecular weight in the high‐molecular‐weight range. T_g values decreased with increasing log M and then turned to a constant value in the high‐molecular‐weight range. The plot of T_g versus 1/M or log M for hyperbranched polymers may exhibit intersecting straight‐line behaviors. The intersection or transition does not result from entanglements that account for such intersections in linear polymers but from a nonlinear feature in hyperbranched polymers according to chain‐end, free‐volume theory. However, the conclusions obtained in this work cannot be extended to dendrimers because after the third generation, the end‐group extents of a dendrimer decrease with molecular weight. Thus, it is very possible for a dendrimer that T_g increases with 1/M before the third generation; however, it decreases with 1/M after the third generation. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1235–1242, 2004     

Pressure dependence of dielectric properties of chlorinated polyethylene vulcanizate. III. β relaxation

The effects of temperature and pressure on the shift factor and the dielectric increment of the β relaxation process were measured for vulcanized chlorinated polyethylene. The isobaric and isochoric activation enthalpies, H^*_P and H^*_V, the activation volume V^*, the pressure dependence of the glass–glass transition temperature, T_gβ/dP, and the apparent extinction temperature T_0β were obtained. The pressure dependences of both V^* and the dielectric increment would reach very small values near the liquid–glass transition temperature T_g, and the β process seems to be affected by the transition near T_g. The value of H^*v/H^*p for the β process is larger than that for the α process, and it is suggested that the molecular motions pertaining to the β process are more strongly restricted than those pertaining to the α process. The ratio T_0β/T₀, where T₀ is the characteristic temperature in the Vogel–Fulcher–Tammann–Hesse equation for the α process, follows the empirical relation of Matsuoka and Ishida, T_gβ/T_g ～0.75. The value of dT_gβ/dP estimated from T_g and T_0β/T₀ is consistent with the experimental value.     

Pressure and volume dependence of thermal conductivity and isothermal bulk modulus up to 1 GPa for poly(isobutylene)

The thermal conductivity λ and heat capacity per unit volume ρc_p of poly(isobutylene)s, one 2.8 in weight average molecular weight and one 85 kg mol⁻¹ in viscosity average molecular weight (PIB-2800 and PIB-85000), have been measured in the temperature range 170–450 K at pressures up to 2 GPa using the transient hot-wire method. At 297 K and atmospheric pressure, λ = 0.115 W m⁻¹ K⁻¹ for PIB-2800 and λ = 0.120 W m⁻¹ K⁻¹ for PIB-85000. The bulk modulus B_T has been measured in the temperature range 170–297 K up to 1 GPa. At atmospheric pressure, the room temperature bulk moduli B_T are 2.0 GPa for PIB-2800 and 2.5 GPa for PIB-85000 with dB_T/dp = 10 for both. These data were used to calculate the volume dependence of λ, At room temperature and atmospheric pressure (liquid phase) we find g = 3.4 for PIB-2800 and g = 3.9 for PIB-85000, but g depends strongly on temperature for both molecular weights. The difference in g between the glassy state and liquid phase is small and just outside the inaccuracy of g of about 8%. The best predictions for g are given by the theoretical model of Horrocks and McLaughlin. We have found that PIB exhibits two relaxations, where one is associated with the glass transition. The value for dT_g/dp at atmospheric pressure (for the main glass transition) is about 0.21 K MPa⁻¹ for both molecular weights. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 1781–1792, 1998