Diluent effects on molecular motions and glass transition in polymers. I. Polystyrene–toluene

The effects of diluent on molecular motions and glass transition in the polystyrene–toluene system was studied by means of dielectric, thermal, and NMR measurements. Three dielectric relaxations were observed between 80 and 400°K. On the basis of NMR measurements on solutions in toluene and in deuterated toluene, relaxation processes were assigned to segmental motions of polystyrene, rotations of toluene, and the local motions of polystyrene and toluene in order of appearance from the high-temperature side. The concentration dependence of the relaxation strength and of the activation energy for the primary relaxation (that at the highest temperature) show a step increment at about 50% by weight. The activation plots for the primary process were expressed by the Vogel–Tamman equation. With this equation, the temperatures at which the mean dielectric relaxation time becomes 100 sec is determined. This agrees well with the glass-transition temperature T_g and hence T_g in concentrated solution is expressed by in terms of the parameters A, B, and T₀ of the Vogel–Tamman equation. The values of A and B are, respectively, about 12 and 0.65 and independent of the concentration. The physical meaning of these parameters is discussed.     

Analysis of the melt viscosity and glass transition of polystyrene

The melt viscosity, the glass transition, and the effect of pressure on these are analyzed for polystyrene on the basis of the Tammann-Hesse viscosity equation: log η = log A + B/(T ? T₀). Evidence that the glass transition is an isoviscosity state (log η_g ? 13) for lower molecular weight fractions (M < M_c) is reviewed. For a polystyrene fraction of intermediate molecular weight (M ? 19,000; t_g = 89°C.), it is shown that B is independent of the p–v–T state of the polymer liquid and that dT₀/dP = dT_g/dP. This is consistent with the postulate that B is determined by the internal barriers to rotation in the isolated polymer chain. Relationships are derived for flow “activation energies” at constant pressure and at constant volume, and for the “activation volume.” Values for polystyrene along the zero-pressure isobar and along the constant viscosity, glasstransition line are reported. For the latter, ΔV_g* is constant and corresponds to about 10 styrene units. The “free volume” viscosity equation: log η = log A + b/2.3?, is reexamined. For polystyrene and polyisobutylene, ?_g/b = 0.03, but ?_g and b themselves differ appreciably in these polymers. The parameter b is the product of an equilibrium term Δα and the kinetic term B, and none of these is a “universal” constant for different polymers. The physical significance of the free volume parameter ?, particularly with regard to the “excess” liquid volume, remains undefined. Two new relationships for dT_g/dP, one an exact derivation and the other an empirical correlation, are presented.     

Effects of diluent on molecular motion and glass transitions in polymers. III. The system poly(vinyl acetate)–toluene

Dielectric measurements, differential thermal analyses (DTA), and broad-line proton magnetic resonance (NMR) measurements are reported on the system poly(vinyl acetate)–toluene. Four dielectric relaxations were observed between 80 and 400°K. From proton NMR measurements on solutions in toluene and in deuterated toluene, the relaxation processes can be assigned, respectively, to segmental motion of poly(vinyl acetate), α; motion of side group, β′ rotation of toluene, β; local motions of poly(vinyl acetate) and toluene, γ, in order of appearance with decreasing temperature. Two stepwise changes in DTA traces have been observed and can be assigned as glass transition points T_gI and T_gII. Comparison of these glass transition points with temperatures at which dielectric relaxation times for the α and β processes are 100 sec, indicate that segmental motion of poly(vinyl acetate) and rotation of toluene are frozen-in at T_gI and T_gII, respectively. Activation plots for the α process conform to the Vogel–Tamman equation. In terms of the parameters A, B, and T₀ of the equation, T_gI can be represented by an expression of the form T_gI ≈ T₀ + B/(A + 3). In the range of concentration above 50% by weight, A and B are almost independent of concentration but T₀ varies strongly. The nature of the secondary dispersions is also discussed.     

Effects of diluent on molecular motion and glass transitions in polymers. II. The system polyvinylchloride–tetrahydrofuran

Dielectric measurements, differential thermal analyses, and density measurements are reported on concentrated solutions of polyvinylchloride in tetrahydrofuran. The relaxation processes observed between 80 and 400°K have been classified into four types. From the analysis of experimental data, the primary process at the highest temperature and the process at the lowest temperature are assigned, respectively, to segmental motion of the polymer and motion of the solvent. Activation plots for the primary process conform to the Vogel–Tamman equation. The dielectric glass-transition temperature T'_g (defined as the temperature at which the dielectric relaxation time is 100 sec) determined with this equation agrees well with the glass-transition temperature T_g from thermal analysis. Therefore, T_g can be represented by an expression of the form The parameters of the Vogel–Tamman equation A and B are nearly independent of concentration, whereas T_o depends strongly on concentration. The dipole moment per monomeric unit calculated from the experimental data changes with concentration and exhibits steep increments around 30% and 90% by weight. The width of the distribution of the relaxation time also increases with the concentration. The results were compared with those for the system polystyrene–toluene studies in the same way.     

Some evidence against the existence of the liquid-liquid transition. IV. The temperature dependence of shear viscosity

The viscosity data available for four anionically polymerized polystyrenes ranging in molecular weight from 1100 to 47,000 for the temperature range T_g to T_g + 100°C have been fitted by computer programs to both the Vogel, Fulcher, Tamman, and Hesse (VFTH) equation and to two optimum intersecting Arrhenius equations. The intersection point has been interpreted as a manifestation of a liquid-liquid transition. The fits to the VFTH equation were in every case found to be far superior. Systematic deviations of the residuals were observed for the best Arrhenius fits which indicate the lack of any validity for such a representation of the data.     

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.     

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.     

The temperature dependence of the viscoelastic softening and terminal dispersions of linear amorphous polymers

Thermorheological simplicity is shown to hold for poly(vinyl acetate) in the temperature range extending from T_g + 25°C to T_g + 80°C. Between T_g and T_g + 25°C the softening (glass to rubberlike) viscoelastic dispersion exhibits time-scale shift factors a_T different from those of the terminal (rubberlike to steady-state) dispersion. The a_T values calculated from zero-shear viscosities coincide with those from the terminal dispersion in the temperature range 60–154°C (T_g ? 35°C). The a_T shifts obtained from the response in the terminal dispersion can be fitted to the Williams, Landel, and Ferry equation over the entire temperature range 42–154°C. The a_T obtained from the softening dispersion is shown to exhibit a different functionality. An empirical modification of the Doolittle equation yields a very flexible relation which can be fitted to some a_Ts which cannot be represented by the usual Doolittle free-volume expression.     

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     

Simultaneous kinetic and microdielectric studies of some epoxy–amine systems

Three reactive epoxy–amine systems based on diglycidyl ether of bisphenol A (DGEBA) with 4,4′-diaminodiphenylsulfone (DDS), 4,4′-methylenebis [3-chloro 2,6-diethylaniline] (MCDEA), and 4,4′-methylenebis [2,6-diethylaniline] (MDEA), were studied during isothermal curings at 140 and 160°C. The simultaneous kinetic and dielectric studies allow to express conductivity, σ, in terms of conversion, x, and of glass transition temperature, T_g. The conductivity, σ₀, of the initial monomer mixture and, σ_∞ of the fully cured network are measured. It is found that:

• The glass transition temperature, T_g, versus conversion, x, curves follows the equation of Di Benedetto modified by Pascault and Williams
• There exists a linear relation between log σ/log σ₀ and T_g.

So, it is possible to predict both kinetic and dielectric behaviors of these epoxy-amine systems by the knowledge of T_g0, ΔC_p0, and σ₀, respectively, glass transition temperature, heat capacity, and conductivity of initial monomer mixture, T_g∞ and ΔC_p∞, and σ_∞, respectively, glass transition temperature and heat capacity and conductivity of fully cured network. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 2911–2921, 1998     

Polyesters containing ring units in the main chain. II. Effects of size and mobility of rings on molecular relaxations

Molecular relaxations of polyesters containing C₅–C₁₅ rings in the main chain have been studied by DSC, dielectric dispersion, and NMR. Results are discussed in relation to the size and mobility of the rings. The T_g or α-relaxation peak moves to higher temperature with an increase in the ring size from C₅ to C₁₂, but the effect is accompanied by an even–odd alternation with ring size. The β relaxations in dielectric dispersion reflect local-mode motion of ester groups and are affected by steric interactions with the rings. Motions of the ring methylenes of C₁₂ and C₁₅ ring units are detected below T_g by broad-line NMR.     

Vapour pressure of α-iodonaphthalene

Using three different techniques, the vapour pressure of α-iodonaphthalene was measured in the temperature range 322–422 K. The pressure equation log P(kPa) = 8.82 ± 0.29 ? (3719 ± 300) /T, was determined. The enthalpy of vaporization change, ΔH⁰₂₉₈ = 69.4 ± 4.0 kJ mole^?1, was determined as the average of the results obtained by second-and third-law treatment of the experimental data. Antoine's constants, A = 6.258, B = 2010 and C = 171, were also derived.     

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     

A computational study of surface tension determination of dense fluids as benzene,toluene, methanol,ammonia, ethylene,and carbon monoxide

In this study, the parameters of linear isotherm regularity, which called LIR equation state used to compute the surface tension of some dense fluids as benzene, toluene, methanol, ammonia, ethylene, and carbon monoxide. An expression has derived for radial distribution function (RDF) at constant temperature, g (σ), for a real fluid by the use of LIR. This expression, which is related to intermolecular interaction, can be used to describe the temperature–density dependency of RDF at constant temperature, g (σ, ρ, T). In addition, we derive an expression for surface tension of dense fluids (CO, C₆H₆, C₆H₅CH₃, CH₃OH, NH₃, and C₂H₄) using the LIR and g (σ, ρ, T). Unlike previous models, it has shown that, surface tension can obtain without employing ΔH and ΔS. Only P-V-T experimental data have been used to calculate the surface tension. Comparison of the calculated values of surface tension by LIR with the values obtained experimentally show this method is not precise. This problem has led us to try to obtain the expression for surface tension using the extended parameters A, B (A and B are the temperature-dependent parameters which noticeably are depended on attraction and repulsion). The obtained result shows that the accuracy of this method is very high and quite admissible.     

The equation of state and heat of fusion of poly(ether ether ketone)

The pressure-volume-temperature properties of poly(ether ether ketone) (PEEK) were studied experimentally at temperatures of 400°C and pressures to 200 MPa. Specific volume data were fitted successfully to the empirical Tait equation for T < T_g and T > T_m and to the theoretical Simha-Somcynsky equation of state for the melt. The pressure dependence of the glass-transition temperature is about 0.57–0.59°C/MPa and that of the melting point 0.483°C/MPa. The pressure dependence of the melting point, the specific volume of the melt at T_m, and the specific volume of the crystal at T_m determined from x-ray diffraction data at elevated temperatures were combined in the Clapeyron equation to calculate a heat of fusion of 161 ± 20 J/g for the PEEK crystal. This value is somewhat higher than the previously reported value of 130 J/g.     

Pressure and temperature effects on the vibrational frequencies of crystalline polymethylenes. II. Lattice anharmonicity and the equation of state

The lattice anharmonicity of crystalline polymethylene is interpreted from the observed pressure and temperature dependence of Raman active interchain lattice frequencies of the n-paraffins C₂₃H₄₈ and C₄₄H₉₀. The temperature dependence of the L_c′ interchain lattice frequency is separated into quasiharmonic and self-energy shifts. The former is due to the volume dependence of the force constant of the oscillator. The latter is due to the anharmonicity of the dynamic potential, and is obtained as a function of volume and phonon population. The setting angle of the carbon skeleton is predicted to be temperature-sensitive. While the potential surface of the crystal is asymmetric along the L_c′ normal coordinate, it is essentially symmetrical along the T_b′ coordinate. The well-known Mie–Gruneisen equation of state is generalized to include anharmonicities of oscillators through the temperature dependence of their vibrational frequencies.     

The characteristic dielectric behaviour in ferroelectric liquid crystals at a phase transition and the contribution of the soft mode

The characteristic dielectric behaviour of ferroelectric liquid crystals with a large spontaneous polarization has been studied as functions of the D.C. bias field, frequency, cell thickness and applied pressure. Under the condition in which the contribution of the Goldstone mode is suppressed, a sharp peak in the temperature dependence of the dielectric constant is clearly observed at the transition between S_A and S*_C phases T _{S C *s A}. The relaxation of the soft mode is observed both in the S_A and S*_C phases by eliminating the contribution of the Goldstone mode under a D.C. bias field. Another relaxation is also observed in the S*_C phase around several kHz in addition to that of the soft mode and the Goldstone mode. The pressure effect on the soft mode was also studied.     

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.     

Bulk and shear rheology of a symmetric three‐arm star polystyrene

The bulk and shear rheological properties of a symmetric three‐arm star polystyrene were measured using a self‐built pressurizable dilatometer and a commercial rheometer, respectively. The bulk properties investigated include the pressure–volume–temperature behavior, the pressure‐dependent glass transition temperature (T_g), and the viscoelastic bulk modulus and Poisson's ratio. Comparison with data for a linear polystyrene indicates that the star behaves similarly but with slightly higher T_gs at elevated pressures and slightly higher limiting bulk moduli in glass and rubbery states. The Poisson's ratio shows a minimum at short times similar to what is observed for the linear chain. The horizontal shift factors above T_g obtained from reducing the bulk and shear viscoelastic responses are found to have similar temperature dependence when plotted using T ? T_g scaling; in addition, the shift factors also exhibit a similar temperature dependence to linear polystyrene. The retardation spectra for the bulk and shear responses are compared and show that the long time molecular mechanisms available to the shear response are unavailable to the bulk. At short times, the two spectra have similar slopes, but the short‐time retardation spectrum for the shear response is significantly higher than that for the bulk, a finding that is, as yet, unexplained. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

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.